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SLAC-PUB-7129\
UCSD-96-05\
March 1996\
hep-ph/9603400
[**Hadro-production of Quarkonia in Fixed Target Experiments**]{}[^1]
[M. Beneke$^1$]{} and [I.Z. Rothstein$^2$]{}
*$^1$Stanford Linear Accelerator Center,*
*Stanford Univerity, Stanford, CA 94309, U.S.A.*
0.6truecm
*$^2$University of California, San Diego,*
*9500 Gilman Drive, La Jolla, CA 92093, U.S.A.*
**Abstract**
We analyze charmonium and bottomonium production at fixed target experiments. We find that inclusion of color octet production channels removes large discrepancies between experiment and the predictions of the color singlet model for the total production cross section. Furthermore, including octet contributions accounts for the observed direct to total $J/\psi$ production ratio. As found earlier for photo-production of quarkonia, a fit to fixed target data requires smaller color octet matrix elements than those extracted from high-$p_t$ production at the Tevatron. We argue that this difference can be explained by systematic differences in the velocity expansion for collider and fixed-target predictions. While the color octet mechanism thus appears to be an essential part of a satisfactory description of fixed target data, important discrepancies remain for the $\chi_{c1}/\chi_{c2}$ production ratio and $J/\psi$ ($\psi'$) polarization. These discrepancies, as well as, the differences between pion and proton induced collisions emphasize the need for including higher twist effects in addition to the color octet mechanism.
PACS numbers: 13.85.Ni, 13.88.+e, 14.40.Gx
Introduction
============
Quarkonium production has traditionally been calculated in the color singlet model (CSM) [@SCH94]. Although the model successfully describes the production rates for some quarkonium states, it has become clear that it fails to provide a theoretically and phenomenologically consistent picture of all production processes. In hadroproduction of charmonia at fixed target energies ($\sqrt{s} < 50\,$ GeV), the ratio of the number of $J/\psi$ produced directly to those arising from decays of higher charmonium states is under-predicted by at least a factor five [@VAE95]. The $\chi_{c1}$ to $\chi_{c2}$ production ratio is far too low, and the observation of essentially unpolarized $J/\psi$ and $\psi'$ can not be reproduced. At Tevatron collider energies, when fragmentation production dominates, the deficit of direct $J/\psi$ and $\psi'$ in the color singlet model is even larger. This deficit has been referred to as the ‘$\psi'$-anomaly’ [@BRA94; @ROY95].
These discrepancies suggest that the color singlet model is too restrictive and that other production mechanisms are necessitated. Indeed, the CSM requires that the quark-antiquark pair that binds into a quarkonium state be produced on the time scale $\tau\simeq 1/m_Q$ with the same color and angular momentum quantum numbers as the eventually formed quarkonium. Consequently, a hard gluon has to be emitted to produce a ${}^3 S_1$ state in the CSM and costs one power of $\alpha_s/\pi$. Since the time scale for quarkonium formation is of order $1/(m_Q v^2)$, where $v$ is the relative quark-antiquark velocity in the quarkonium bound state, this suppression can be overcome if one allows for the possibility that the quark-antiquark pair is in any angular momentum or color state when produced on time scales $\tau\simeq 1/m_Q$. Subsequent evolution into the physical quarkonium state is mediated by emission of soft gluons with momenta of order $m_Q v^2$. Since the quark-antiquark pair is small in size, the emission of these gluons can be analyzed within a multipole expansion. A rigorous formulation [@BOD95] of this picture can be given in terms of non-relativistic QCD (NRQCD). Accordingly, the production cross section for a quarkonium state $H$ in the process $$\label{proc}
A + B \longrightarrow H + X,$$
can be written as
$$\label{fact}
\sigma_H = \sum_{i,j}\int\limits_0^1 d x_1 d x_2\,
f_{i/A}(x_1) f_{j/B}(x_2)\,\hat{\sigma}(ij\rightarrow H)\,,$$
$$\label{factformula}
\hat{\sigma}(ij\rightarrow H) = \sum_n C^{ij}_{\bar{Q} Q[n]}
\langle {\cal O}^H_n\rangle\,.$$
Here the first sum extends over all partons in the colliding hadrons and $f_{i/A}$ etc. denote the corresponding distribution functions. The short-distance ($x\sim 1/m_Q \gg 1/(m_Q v)$) coefficients $C^{ij}_{\bar{Q} Q[n]}$ describe the production of a quark-antiquark pair in a state $n$ and have expansions in $\alpha_s(2 m_Q)$. The parameters[^2] $\langle {\cal O}^H_n\rangle$ describe the subsequent hadronization of the $Q\bar{Q}$ pair into a jet containing the quarkonium $H$ and light hadrons. These matrix elements can not be computed perturbatively, but their relative importance in powers of $v$ can be estimated from the selection rules for multipole transitions.
The color octet picture has led to the most plausible explanation of the ‘$\psi'$-anomaly’ and the direct $J/\psi$ production deficit. In this picture gluons fragment into quark-antiquark pairs in a color-octet ${}^3 S_1^{(8)}$ state which then hadronizes into a $\psi'$ (or $J/\psi$) [@BRA95; @CAC95; @CHO95]. Aside from this striking prediction, the color octet mechanism remains largely untested. Its verification now requires considering quarkonium production in other processes in order to demonstrate the process-independence (universality) of the production matrix elements $\langle {\cal O}_n^H\rangle$, which is an essential prediction of the factorization formula (\[factformula\]).
Direct $J/\psi$ and $\psi'$ production at large $p_t\gg 2 m_Q$ (where $m_Q$ denotes the heavy [*quark*]{} mass) is rather unique in that a single term, proportional to $\langle {\cal O}_8^H ({}^3 S_1)\rangle$, overwhelmingly dominates the sum (\[factformula\]). On the other hand, in quarkonium formation at moderate $p_t\sim 2 m_Q$ at colliders and in photo-production or fixed target experiments ($p_t\sim 1\,$GeV), the signatures of color octet production are less dramatic, because they are not as enhanced by powers of $\pi/\alpha_s$ or $p_t^2/m_Q^2$ over the singlet mechanisms. Furthermore, theoretical predictions are parameterized by more unknown octet matrix elements and are afflicted by larger uncertainties. In particular, there are large uncertainties due to the increased sensitivity to the heavy quark mass close to threshold. (The production of a quark-antiquark pair close to threshold is favored by the rise of parton densities at small $x$.) These facts complicate establishing color octet mechanisms precisely in those processes where experimental data is most abundant.
Cho and Leibovich [@CHO95II] studied direct quarkonium production at moderate $p_t$ at the Tevatron collider and were able to extract a value for a certain combination of unknown parameters $\langle {\cal O}_8^H({}^1 S_0)\rangle$ and $\langle {\cal O}_8^H({}^3 P_0)\rangle$ ($H=J/\psi,\psi',\Upsilon(1S),
\Upsilon(2S)$). A first test of universality comes from photo-production [@CAC96; @AMU96; @KO96], where a different combination of these two matrix elements becomes important near the elastic peak at $z\approx 1$, where $z=p\cdot k_\psi / p \cdot k_\gamma$, and $p$ is the proton momentum. A fit to photo-production data requires much smaller matrix elements than those found in [@CHO95II]. Taken at face value, this comparison would imply failure of the universality assumption underlying the non-relativistic QCD approach. However, the extraction from photo-production should be regarded with caution since the NRQCD formalism describes inclusive quarkonium production only after sufficient smearing in $z$ and is not applicable in the exclusive region close to $z=1$, where diffractive quarkonium production is important.
In this paper we investigate the universality of the color octet quarkonium production matrix elements in fixed target hadron collisions and re-evaluate the failures of the CSM in fixed target production [@VAE95] after inclusion of color octet mechanisms. Some of the issues involved have already been addressed by Tang and Vänttinen [@TAN95] and by Gupta and Sridhar [@GUP96], but a complete survey is still missing. We also differ from [@TAN95] in the treatment of polarized quarkonium production and the assessment of the importance of color octet contributions and from [@GUP96] in the color octet short-distance coefficients.
The paper is organized as follows: In Sect. 2 we compile the leading order color singlet and color octet contributions to the production rates for $\psi^\prime,~\chi_J,~J/\psi$ as well as bottomonium. In Sect. 3 we present our numerical results for proton and pion induced collisions. Sect. 4 is devoted to the treatment of polarized quarkonium production. As polarization remains one of the cleanest tests of octet quarkonium production at large $p_t$ [@WIS95; @BEN95], we clarify in detail the conflicting treatments of polarized production in [@BEN95] and [@CHO95II]. Sect. 5 is dedicated to a comparison of the extracted color-octet matrix elements from fixed target experiments with those from photo-production and the Tevatron. We argue that kinematical effects and small-$x$ effects can bias the extraction of NRQCD matrix elements so that a fit to Tevatron data at large $p_t$ requires larger matrix elements than the fit to fixed target and photo-production data. The final section summarizes our conclusions.
Quarkonium production cross sections at fixed target energies
=============================================================
Cross sections
--------------
We begin with the production cross section for $\psi'$ which does not receive contributions from radiative decays of higher charmonium states. The $2\to 2$ parton diagrams produce a quark-antiquark pair in a color-octet state or $P$-wave singlet state (not relevant to $\psi'$) and therefore contribute to $\psi'$ production at order $\alpha_s^2 v^7$. (For charmonium $v^2\approx
0.25 - 0.3$, for bottomonium $v^2\approx 0.08 - 0.1$.) The $2\to3$ parton processes contribute to the color singlet processes at order $\alpha_s^3 v^3$. Using the notation in (\[fact\]): $$\begin{aligned}
\label{psiprimecross}
\hat{\sigma}(gg\to\psi') &=& \frac{5\pi^3\alpha_s^2}{12 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\left[\langle {\cal O}_8^{\psi'} ({}^1 S_0)
\rangle+\frac{3}{m_c^2} \langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle
+\frac{4}{5 m_c^2} \langle {\cal O}_8^{\psi'} ({}^3 P_2)\rangle
\right]\nonumber\\[0.0cm]
&&\hspace*{-1.5cm}
+\,\frac{20\pi^2\alpha_s^3}{81 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\,\langle
{\cal O}_1^{\psi'} ({}^3 S_1)\rangle\,z^2\left[\frac{1-z^2+2 z
\ln z}{(1-z)^2}+\frac{1-z^2-2z \ln z}{(1+z)^3}\right]\\[0.2cm]
\hat{\sigma}(gq\to\psi') &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \psi') &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\psi'} ({}^3 S_1)
\rangle\end{aligned}$$
Here $z\equiv (2 m_c)^2/(s x_1 x_2)$, $\sqrt{s}$ is the center-of-mass energy and $\alpha_s$ is normalized at the scale $2 m_c$. Corrections to these cross sections are suppressed by either $\alpha_s/\pi$ or $v^2$. Note that the relativistic corrections to the color singlet cross section are substantial in specific kinematic regions $z\to 0,1$ [@JUN93]. For $\sqrt{s}>15\,$GeV these corrections affect the total cross section by less than $50\%$ and decrease as the energy is raised [@SCH94]. Furthermore, notice that we have expressed the short-distance coefficients in terms of the charm quark mass, $M_{\psi'}\approx 2 m_c$, rather than the true $\psi'$ mass. Although the difference is formally of higher order in $v^2$, this choice is conceptually favored since the short-distance coefficients depend only on the physics prior to quarkonium formation. All quarkonium specific properties which can affect the cross section, such as quarkonium mass differences, are hidden in the matrix elements.
The production of $P$-wave quarkonia differs from $S$-waves since color singlet and color octet processes enter at the same order in $v^2$ as well as $\alpha_s$ in general. An exception is $\chi_{c1}$, which can not be produced in $2\to 2$ parton reactions through gluon-gluon fusion in a color singlet state. Since at order $\alpha_s^2$, the $\chi_{c1}$ would be produced only in a $q\bar{q}$ collision, we also include the gluon fusion diagrams at order $\alpha_s^3$, which are enhanced by the gluon distribution. We have for $\chi_{c0}$,
$$\begin{aligned}
\label{chi0cross}
\hat{\sigma}(gg\to\chi_{c0}) &=& \frac{2\pi^3\alpha_s^2}{3 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c0}} ({}^3 P_0)
\rangle\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c0}) &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c0}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c0}} ({}^3 S_1)
\rangle\,,\end{aligned}$$
for $\chi_{c1}$,
$$\begin{aligned}
\label{chi1cross}
\hat{\sigma}(gg\to\chi_{c1}) &=& \frac{2\pi^2\alpha_s^3}{9 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c1}} ({}^3 P_1)
\rangle\nonumber\\
&&\hspace*{-1.5cm}
\times\Bigg[\frac{4 z^2\ln z \, (z^8+9 z^7+26 z^6+28 z^5+17 z^4+7 z^3-
40 z^2-4 z-4}{(1+z)^5 (1-z)^4}\nonumber\\
&&\hspace*{-1.5cm}
\,+\frac{z^9+39 z^8+145 z^7+251 z^6+119 z^5-153 z^4-17 z^3-147 z^2-8 z
+10}{3 (1-z)^3 (1+z)^4}\Bigg]
\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c1}) &=& \frac{8\pi^2\alpha_s^3}{81 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c1}} ({}^3 P_1)
\rangle\left[-z^2\ln z + \frac{4 z^3-9 z+5}{3}\right]\nonumber\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c1}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c1}} ({}^3 S_1)
\rangle\,,\end{aligned}$$
and for $\chi_{c2}$
$$\begin{aligned}
\label{chi2cross}
\hat{\sigma}(gg\to\chi_{c2}) &=& \frac{8\pi^3\alpha_s^2}{45 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c2}} ({}^3 P_2)
\rangle\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c2}) &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c2}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c2}} ({}^3 S_1)
\rangle\,.\end{aligned}$$
Note that in the NRQCD formalism the infrared sensitive contributions to the $q\bar{q}$-induced color-singlet process at order $\alpha_s^3$ are factorized into the color octet matrix elements $\langle {\cal O}_8^{\chi_{cJ}} ({}^3 S_1)\rangle$, so that the $q\bar{q}$ reactions at order $\alpha_s^3$ are truly suppressed by $\alpha_s$. The production of $P$-wave states through octet quark-antiquark pairs in a state other than ${}^3 S_1$ is higher order in $v^2$.
Taking into account indirect production of $J/\psi$ from decays of $\psi'$ and $\chi_{cJ}$ states, the $J/\psi$ cross section is given by
$$\label{jpsicross}
\sigma_{J/\psi} = \sigma(J/\psi)_{dir} + \sum_{J=0,1,2}
\mbox{Br}(\chi_{cJ}\to J/\psi X)\,\sigma_{\chi_{cJ}} +
\mbox{Br}(\psi'\to J/\psi X)\,\sigma_{\psi'}\,,$$
where ‘Br’ denotes the corresponding branching fraction and the direct $J/\psi$ production cross section $\sigma(J/\psi)_{dir}$ differs from $\sigma_{\psi'}$ (see (\[psiprimecross\])) only by the replacement of $\psi'$ matrix elements with $J/\psi$ matrix elements. Finally, we note that charmonium production through $B$ decays is comparatively negligible at fixed target energies.
The $2\to 2$ parton processes contribute only to quarkonium production at zero transverse momentum with respect to the beam axis. The transverse momentum distribution of $H$ in reaction (\[proc\]) is not calculable in the $p_t<\Lambda_{QCD}$ region, but the total cross section (which averages over all $p_t$) is predicted even if the underlying parton process is strongly peaked at zero $p_t$.
The transcription of the above formulae to bottomonium production is straightforward. Since more bottomonium states exist below the open bottom threshold than for the charmonium system, a larger chain of cascade decays in the bottomonium system must be included. In particular, there is indirect evidence from $\Upsilon(3 S)$ production both at the Tevatron [@PAP95] as well as in fixed target experiments (to be discussed below) that there exist yet unobserved $\chi_b(3P)$ states below threshold whose decay into lower bottomonium states should also be included. Our numerical results do not include indirect contributions from potential $D$-wave states below threshold.
All color singlet cross sections compiled in this section have been taken from the review [@SCH94]. We have checked that the color octet short-distance coefficients agree with those given in [@CHO95II], but disagree with those that enter the numerical analysis of fixed target data in [@GUP96].
Matrix elements
---------------
The number of independent matrix elements can be reduced by using the spin symmetry relations
$$\begin{aligned}
&&\langle {\cal O}^{\chi_{cJ}}_1 ({}^3 P_J) \rangle =
(2 J+1)\,\langle {\cal O}^{\chi_{c0}}_1 ({}^3 P_0) \rangle
\nonumber\\
&&\langle {\cal O}^{\psi}_8 ({}^3 P_J) \rangle =
(2 J+1)\,\langle {\cal O}^{\psi}_8 ({}^3 P_0) \rangle
\\
&& \langle {\cal O}^{\chi_{cJ}}_8 ({}^3 S_1) \rangle =
(2 J+1)\,\langle {\cal O}^{\chi_{c0}}_1 ({}^3 S_1) \rangle
\nonumber\end{aligned}$$
and are accurate up to corrections of order $v^2$ ($\psi=J/\psi,
\psi'$ – identical relations hold for bottomonium). This implies that at lowest order in $\alpha_s$, the matrix elements $\langle {\cal O}^H_8 ({}^1 S_0) \rangle$ and $\langle {\cal O}^H_8 ({}^3 P_0) \rangle$ enter fixed target production of $J/\psi$ and $\psi'$ only in the combination
$$\label{delta}
\Delta_8(H)\equiv \langle {\cal O}^H_8 ({}^1 S_0) \rangle +
\frac{7}{m_Q^2}\langle {\cal O}^H_8 ({}^3 P_0) \rangle\,.$$
Up to corrections in $v^2$, all relevant color singlet production matrix elements are related to radial quarkonium wave functions at the origin and their derivatives by
$$\label{wave}
\langle {\cal O}^H_1 ({}^3 S_1) \rangle =
\frac{9}{2\pi} |R(0)|^2
\qquad
\langle {\cal O}^H_1 ({}^3 P_0) \rangle =
\frac{9}{2\pi} |R'(0)|^2.$$
We are then left with three non-perturbative parameters for the direct production of each $S$-wave quarkonium and two parameters for $P$-states.
The values for these parameters, which we will use below, are summarized in tables \[tab1\] and \[tab2\]. Many of the octet matrix elements, especially for bottomonia, are not established and should be viewed as guesses. The numbers given in the tables are motivated as follows: All color singlet matrix elements are computed from the wavefunctions in the Buchmüller-Tye potential tabulated in [@EQ] and using (\[wave\]). Similar results within $\pm30\%$ could be obtained from leptonic and hadronic decays of quarkonia for some of the states listed in the tables. The matrix elements $\langle {\cal O}_8^H ({}^3 S_1)\rangle$ are taken from the fits to Tevatron data in [@CHO95II] with the exception of the $3S$ and $3P$ bottomonium states. In this case, we have chosen the numbers by (rather ad hoc) extrapolation from the $1S$, $2S$ and $1P$, $2P$ states.
$$\begin{array}{|c||c|c|c|c|c|}
\hline
\mbox{ME} & J/\psi & \psi' & \Upsilon(1S) & \Upsilon(2S) &
\Upsilon(3S) \\ \hline
\langle {\cal O}^H_1 ({}^3 S_1) \rangle &
1.16 & 0.76 & 9.28 & 4.63 & 3.54 \\
\langle {\cal O}^H_8 ({}^3 S_1) \rangle &
6.6\cdot 10^{-3} & 4.6\cdot 10^{-3} & 5.9\cdot 10^{-3} &
4.1\cdot 10^{-3} & 3.5\cdot 10^{-3} \\
\Delta_8(H)& \mbox{fitted} & \mbox{fitted} & 5.0\cdot 10^{-2} &
3.0\cdot 10^{-2} & 2.3\cdot 10^{-2} \\
\hline
\end{array}$$
$$\begin{array}{|c||c|c|c|c|}
\hline
\mbox{ME} & \chi_{c0} & \chi_{b0}(1P) & \chi_{b0}(2P) &
\chi_{b0}(3P) \\ \hline
\langle {\cal O}^H_1 ({}^3 P_0)\rangle/m_Q^2 &
4.4\cdot 10^{-2} & 8.5\cdot 10^{-2} & 9.9\cdot 10^{-2} &
0.11 \\
\langle {\cal O}^H_8 ({}^3 S_1) \rangle &
3.2\cdot 10^{-3} & 0.42 & 0.32 & 0.25 \\
\hline
\end{array}$$
The combination of matrix elements $\Delta_8(H)$ turns out to be the single most important parameter for direct production of $J/\psi$ and $\psi'$. For this reason, we leave it as a parameter to be fitted and later compared with constraints available from Tevatron data. For bottomonia we adopt a different strategy since $\Delta_8(H)$ is of no importance for the total (direct plus indirect) bottomonium cross section. We therefore fixed its value using the results of [@CHO95II] together with some assumption on the relative size of $\langle {\cal O}_8^H ({}^1 S_0)
\rangle$ and $\langle {\cal O}_8^H ({}^3 P_0)\rangle$ and an ad hoc extrapolation for the $3S$ state. Setting $\Delta_8(H)$ to zero for bottomonia would change the cross section by a negligible amount.
Results
=======
Figs. \[prifig\] to \[upsfig\] and table \[tab3\] summarize our results for the charmonium and bottomonium production cross sections. We use the CTEQ3 LO [@cteq] parameterization for the parton distributions of the protons and the GRV LO [@grv] parameterization for pions. The quark masses are fixed to be $m_c=1.5\,$GeV and $m_b=4.9\,$GeV, as was done in [@CHO95II]. The strong coupling is evaluated at the scale $\mu=2 m_Q$ ($Q=b,c$) and chosen to coincide with the value implied by the parameterization of the parton distributions (e.g., $\alpha_s(2 m_c)\approx 0.23$ for CTEQ3 LO). We comment on these parameter choices in the discussion below. The experimental data have been taken from the compilation in [@SCH94] with the addition of results from [@AKE93] and the $800\,$GeV proton beam at Fermilab [@SCH95; @ALE95]. All data have been rescaled to the nuclear dependence $A^{0.92}$ for proton-nucleon collisions and $A^{0.87}$ for pion-nucleon collisions. All cross sections are given for $x_F>0$ only (i.e. integrated over the forward direction in the cms frame where most of the data has been collected).
$\psi'$
-------
The total $\psi'$ production cross section in proton-nucleus collisions is shown in Fig. \[prifig\]. The color-singlet cross section is seen to be about a factor of two below the data and the fit, including color octet processes, is obtained with
$$\Delta_8(\psi') = 5.2\cdot 10^{-3}\,\mbox{GeV}^3\,.$$
The contribution from $\langle {\cal O}^{\psi'}_8({}^3 S_1)
\rangle$ is numerically irrelevant because gluon fusion dominates at all cms energies considered here. The relative magnitude of singlet and octet contributions is consistent with the naive scaling estimate $\pi/\alpha_s\cdot v^4
\approx 1$ (The color singlet cross section acquires an additional suppression, because it vanishes close to threshold when $4 m_c^2/(x_1 x_2 s)\to 1$).
It is important to mention that the color singlet prediction has been expressed in terms of $2 m_c=3\,$GeV and not the physical quarkonium mass. Choosing the quarkonium mass reduces the color singlet cross section by a factor of three compared to Fig. \[prifig\], leading to an apparent substantial $\psi'$ deficit[^3]. As explained in Sect. 2, choosing quark masses is preferred but leads to large normalization uncertainties due to the poorly known charm quark mass, which could only be partially eliminated if the color singlet wave function were extracted from $\psi'$ decays. If, as in open charm production, a small charm mass were preferred, the data could be reproduced even without a color octet contribution. Although this appears unlikely (see below), we conclude that the total $\psi'$ cross section alone does not provide convincing evidence for the color octet mechanism. If we neglect the color singlet contribution altogether, we obtain $\Delta_8(\psi') < 1.0\cdot 10^{-2}\,$GeV${}^3$. This bound is strongly dependent on the value of $m_c$. Varying $m_c$ between $1.3\,$GeV and $1.7\,$GeV changes the total cross section by roughly a factor of eight at $\sqrt{s}=30\,$GeV and even more at smaller $\sqrt{s}$. Compared to this normalization uncertainty, the variation with the choice of parton distribution and $\alpha_s(\mu)$ is negligible. This remark applies to all other charmonium cross sections considered in this section.
$J/\psi$
--------
The $J/\psi$ production cross section in proton-nucleon collisions is displayed in Fig. \[psifig\]. A reasonable fit is obtained for
$$\Delta_8(J/\psi)=3.0\cdot 10^{-2}\,\mbox{GeV}^3\,.$$
We see that the color octet mechanism substantially enhances the direct $J/\psi$ production cross section compared to the CSM, as shown by the dashed and dotted lines in Fig. \[psifig\]. The total cross section includes feed-down from $\chi_{cJ}$ states which is dominated by the color-singlet gluon fusion process. As expected from the cross section in Sect. 2, the largest indirect contribution originates from $\chi_{c2}$ states, because $\chi_{c1}$ production is suppressed by one power of $\alpha_s$ in the gluon fusion channel. The direct $J/\psi$ production fraction at $\sqrt{s}=23.7\,$GeV ($E=300\,$GeV) is $63\%$, in excellent agreement with the experimental value of $62\%$ [@ANT93]. Note that this agreement is not a trivial consequence of fitting the color octet matrix element $\Delta_8(J/\psi)$ to reproduce the observed total cross section since the indirect contribution is dominated by color singlet mechanisms and the singlet matrix elements are fixed in terms of the wavefunctions of [@EQ].
One could ask whether the large sensitivity to the charm quark mass could be exploited to raise the direct production fraction in the CSM, thus obviating the need for octet contributions altogether? As shown in Fig. \[ratfig\] this is not the case, since the charm mass dependence cancels in the direct-to-total production ratio. It should be mentioned, that expressing all cross sections in terms of the respective quarkonium masses increases $\sigma(J/\psi)_{dir}/\sigma_{J/\psi}$, because $M_{\chi_{cJ}}>M_{J/\psi}$. However, the total color singlet cross section then decreases further and falls short of the data by about a factor five. We therefore consider the the combination of total $J/\psi$ production cross section and direct production ratio as convincing evidence for an essential role of color octet mechanisms for direct $J/\psi$ production also at fixed target energies.
$$\begin{array}{|c||c|c|c||c|c|c|}
\hline
& pN\,\mbox{th.} & pN\,\mbox{CSM} & pN\,\mbox{exp.} &
\pi^- N\,\mbox{th.} & \pi^- N\,\mbox{CSM} & \pi^- N\,\mbox{exp.}\\
\hline
\sigma_{J/\psi} &
90\,\mbox{nb} & 33\,\mbox{nb} & 143\pm 21\,\mbox{nb} &
98\,\mbox{nb} & 38\,\mbox{nb} & 178\pm 21\,\mbox{nb}\\
\sigma(J/\psi)_{dir}/\sigma_{J/\psi} &
0.63 & 0.21 & 0.62\pm 0.04 &
0.64 & 0.24 & 0.56\pm 0.03 \\
\sigma_{\psi'}/\sigma(J/\psi)_{dir} &
0.25 & 0.67 & 0.21\pm 0.05 &
0.25 & 0.66 & 0.23\pm 0.05 \\
\chi\mbox{-fraction} &
0.27 & 0.69 & 0.31\pm 0.04 &
0.28 & 0.66 & 0.37\pm 0.03 \\
\chi_{c1}/\chi_{c2}\,\mbox{ratio} &
0.15 & 0.08 & - &
0.13 & 0.11 & 1.4\pm 0.4 \\
\hline
\end{array}$$
The comparison of theoretical predictions with the E705 experiment [@ANT93] is summarized in Tab. \[tab3\]. Including color octet production yields good agreement for direct $J/\Psi$ production, as well as the relative contributions from all $\chi_{cJ}$ states and $\psi'$. Note that the total cross section from [@ANT93] is rather large in comparison with other data (see Fig. \[psifig\]). In the CSM, the direct production cross section of $7\,$nb should be compared with the measured $89\,$nb, clearly demonstrating the presence of an additional numerically large production mechanism. Note also that our $\psi'$ cross section in the CSM is rather large in comparison with the direct $J/\Psi$ cross section in the CSM. A smaller value which compares more favorably with the data could be obtained if one expressed the cross section in terms of quarkonium masses [@VAE95]. From the point of view presented here, this agreement appears coincidental since the cross sections are dominated by octet production.
Perhaps the worst failure of the theory is the $\chi_{c1}$ to $\chi_{c2}$ ratio in the feed-down contribution that has been measured in the WA11 experiment at $E=185\,$GeV [@LEM82]. We see that the prediction is far too small even after inclusion of color octet contributions. The low rate of $\chi_1$ production is due to the fact, as already mentioned, that the gluon-gluon fusion channel is suppressed by $\alpha_s/\pi$ compared to $\chi_{c2}$ due to angular momentum constraints. Together with $J/\psi$ (and $\psi'$) polarization, discussed in Sect. 4, the failure to reproduce this ratio emphasizes the importance of yet other production mechanisms, presumably of higher twist, which are naively suppressed by $\Lambda_{QCD}/m_c$ [@VAE95].
Pion-induced collisions
-----------------------
The $\psi'$ and $J/\psi$ production cross section in pion-nucleon collisions are shown in Figs. \[pripifig\] and \[psipifig\]. The discussion for proton-induced collisions applies with little modification to the pion case. A breakdown of contributions to the $J/\psi$ cross section at $E=300\,$GeV is given in table \[tab3\]. The theoretical prediction is based on the values of $\Delta_8(H)$ extracted from the proton data. Including color octet contributions can add little insight into the question of why the pion-induced cross sections appear to be systematically larger than expected. This issue has been extensively discussed in [@SCH94]. The discrepancy may be an indication that, either the gluon distribution in the pion is not really understood (although using parameterizations different from GRV LO tends to yield rather lower theoretical predictions), or that a genuine difference in higher twist effects for the proton and the pion exists.
$\Upsilon(nS)$
--------------
If higher twist effects are important for fixed target charmonium production, their importance should decrease for bottomonium production and facilitate a test of color octet production. Unfortunately, data for bottomonium production at fixed target energies is sparse and does not allow us to complete this test.
Due to the increase of the quark mass, bottomonium production differs in several ways from charmonium production, from a theoretical standpoint. The relative quark-antiquark velocity squared decreases by a factor of three, thus, the color octet contributions to direct production of $\Upsilon(nS)$ are less important since they are suppressed by $v^4$ (at the same time $\alpha_s(2 m_Q)$ decreases much less). The situation is exactly the opposite for the production of $P$-wave bottomonia. In this case the color singlet and octet contributions scale equally in $v^2$. The increased quark mass, together with an increased relative importance of the octet matrix element $\langle {\cal O}_8^{\chi_{b0}}
({}^3 S_1)\rangle$ (extracted from Tevatron data in [@CHO95II]) as compared to the singlet wavefunction (compare $\chi_{c0}$ with $\chi_{b0}$ in Tab. \[tab2\]), leads to domination of quark-antiquark pair initiated processes. Consequently, the direct $\Upsilon(nS)$ production cross section is at least a factor ten below the indirect contributions from $\chi_b$-decays. This observation leads to the conclusion that the number of $\Upsilon(3S)$ observed by the E772 experiment [@ALD91] can only be explained if $\chi_{bJ}
(3P)$ states that have not yet been observed directly exist below the open bottom threshold. Such indirect evidence has also been obtained from bottomonium production at the Tevatron collider [@PAP95].
To obtain our numerical results shown in Fig. \[upsfig\], we assumed that these $\chi_{bJ}(3P)$ states decay into $\Upsilon(3S)$ with the same branching fractions as the corresponding $n=2$ states. The total cross sections are compared with the experimental value $195\pm 67\,$pb/nucleon obtained from [@ALE95] at $E=800\,$GeV for the sum of $\Upsilon(nS)$, $n=1,2,3$ and show very good agreement. The color-singlet processes alone would have led to a nine times smaller prediction at this energy. We should note, however, that integration of the $x_F$-distribution for $\Upsilon(1S)$ production given in [@ALD91] indicates a cross section about two to three times smaller than the central value quoted by [@ALE95]. The theoretical prediction for the relative production rates of $\Upsilon(1S):
\Upsilon(2S):\Upsilon(3S)$ is $1:0.42:0.30$ to be compared with the experimental ratio [@ALD91] $1:0.29:0.15$[^4]. This comparison should not be over interpreted since it depends largely on the rather uncertain octet matrix elements for $P$-wave bottomonia. Due to lack of more data we also hesitate to use this comparison for a new determination of these matrix elements.
$\psi'$ and $J/\psi$ Polarization
=================================
In this section, we deal with $\psi'$ and $J/\psi$ polarization at fixed target energies and at colliders at large transverse momentum. Before returning to fixed target production in Sect. 4.2, we digress on large-$p_t$ production. We recall that, at large $p_t^2\gg 4 m_Q^2$, $\psi'$ and direct $J/\psi$ production is dominated by gluon fragmentation into color octet quark-antiquark pairs and expected to yield transversely polarized quarkonia [@WIS95]. The reason for this is that a fragmenting gluon can be considered as on-shell and therefore transverse. Due to spin symmetry of NRQCD, the quarkonium inherits the transverse polarization up to corrections of order $4 m_c^2/p_T^2$ and $v^4$. Furthermore, it has been shown [@BEN95] that including radiative corrections to gluon fragmentation still leads to more than $90\%$ transversely polarized $\psi'$ (direct $J/\psi$). Thus, polarization provides one of the most significant tests for the color octet production mechanism at large transverse momenta. At moderate $p_t^2\sim 4 m_c^2$, non-fragmentation contributions proportional to $\langle {\cal O}_8^H ({}^1 S_0)\rangle$ and $\langle {\cal O}_8^H ({}^3 P_J)\rangle$ are sizeable [@CHO95II]. Understanding their polarization yield quantitatively is very important since most of the $p_t$-integrated data comes from the lower $p_t$-region. The calculation of the polarization yield has also been attempted in [@CHO95II]. However, the method used is at variance with [@BEN95] and leads to an incorrect result for $S$-wave quarkonia produced through intermediate quark-antiquark pairs in a color octet $P$-wave state. In the following subsection we expound on the method discussed in [@BEN95] and hope to clarify this difference.
Polarized production
--------------------
For arguments sake, let us consider the production of a $\psi'$ in a polarization state $\lambda$. This state can be reached through quark-antiquark pairs in various spin and orbital angular momentum states, and we are led to consider the intermediate quark-antiquark pair as a coherent superposition of these states. Because of parity and charge conjugation symmetry, intermediate states with different spin $S$ and angular momentum $L$ can not interfere[^5], so that the only non-trivial situation occurs for ${}^3 P_J$-states, i.e. $S=1$, $L=1$.
In [@CHO95II] it is assumed that intermediate states with different $J J_z$, where $J$ is total angular momentum do not interfere, so that the production cross section can be expressed as the sum over $J J_z$ of the amplitude squared for production of a color octet quark-antiquark pair in a ${}^3 P_{J J_z}$ state times the amplitude squared for its transition into the $\psi'$. The second factor can be inferred from spin symmetry to be a simple Clebsch-Gordon coefficient so that
$$\label{cho}
\sigma^{(\lambda)}_{\psi'} \sim \sum_{J J_z}
\sigma(\bar{c} c[{}^3 P^8_{J J_z}])\,|\langle J J_z|1
(J_z-\lambda);1\lambda\rangle|^2\,.$$
We will show that this equation is incompatible with spin symmetry which requires interference of intermediate states with different $J$.
A simple check can be obtained by applying (\[cho\]) to the calculation of the gluon fragmentation function into longitudinally polarized $\psi'$. Since the fragmentation functions into quark-antiquark pairs in a ${}^3 P^8_{J J_z}$ state follow from [@TRI95] by a change of color factor, the sum in (\[cho\]) can be computed. The result not only differs from the fragmentation function obtained in [@BEN95] but contains an infrared divergence which can not be absorbed into another NRQCD matrix element.
To see the failure of (\[cho\]) more clearly we return to the NRQCD factorization formalism. After Fierz rearrangement of color and spin indices as explained in [@BOD95], the cross section can be written as
$$\label{factor}
\sigma^{(\lambda)} \sim H_{ai;bj}\cdot S_{ai;bj}^{(\lambda)}\,.$$
In this equation $H_{ai;bj}$ is the hard scattering cross section, and $S_{ai;bj}$ is the soft (non-perturbative) part that describes the ‘hadronization’ of the color octet quark pair into a $\psi'$ plus light hadrons. Note that the statement of factorization entailed in this equation occurs only on the cross section and not on the amplitude level. The indices $ij$ and $ab$ refer to spin and angular momentum in a Cartesian basis $L_a S_i$ ($a,i=1,2,3=x,y,z$). Since spin-orbit coupling is suppressed by $v^2$ in the NRQCD Lagrangian, $L_z$ and $S_z$ are good quantum numbers. In the specific situation we are considering, the soft part is simply given by (the notation follows [@BOD95; @BEN95])
$$S_{ai;bj}^{(\lambda)}=
\langle 0|\chi^\dagger\sigma_i T^A\left(-\frac{i}{2}
\stackrel{\leftrightarrow}{D}_a\right)
\psi\,{a_{\psi'}^{(\lambda)}}^\dagger
a_{\psi'}^{(\lambda)}\,\psi^\dagger\sigma_j T^A\left(-\frac{i}{2}
\stackrel{\leftrightarrow}{D}_b\right)\chi|0\rangle\,,$$
where $a_{\psi'}^{(\lambda)}$ destroys a $\psi'$ in an out-state with polarization $\lambda$. To evaluate this matrix element at leading order in $v^2$, we may use spin symmetry. Spin symmetry tells us that the spin of the $\psi'$ is aligned with the spin of the $\bar{c} c$ pair, so $S_{ai;bj}^{(\lambda)}\propto
{\epsilon^i}^*(\lambda)\epsilon^j(\lambda)$. Now all vectors $S^{(\lambda)}_{ai;bj}$ can depend on have been utilized, and thus by rotational invariance, only the Kronecker symbol is left to tie up $a$ and $b$. The overall normalization is determined by taking appropriate contractions, and we obtain
$$\label{decomp}
S_{ai;bj}^{(\lambda)}=
\langle {\cal O}^{\psi^\prime}_8(^3\!P_0)\rangle\,
\delta_{ab}\,{\epsilon^i}^*(\lambda)\epsilon^j(\lambda)\,.$$
This decomposition tells us that to calculate the polarized production rate we should project the hard scattering amplitude onto states with definite $S_z=\lambda$ and $L_z$, square the amplitude, and then sum over $L_z$ ($\sum_{L_z}\epsilon_a(L_z)
\epsilon_b(L_z)=\delta_{ab}$ in the rest frame). In other words, the soft part is diagonal in the $L_z S_z$ basis.
It is straightforward to transform to the $J J_z$ basis. Since $J_z=L_z+S_z$, there is no interference between intermediate states with different $J_z$. To see this we write, in obvious notation,
$$\label{factornew}
\sigma^{(\lambda)} \sim \sum_{J J_z;J' J_z^\prime}
H_{J J_z;J' J_z^\prime}\cdot S_{J J_z;J' J_z^\prime}^{(\lambda)}\,,$$
and using (\[decomp\]) obtain,
$$S_{J J_z;J' J_z^\prime}^{(\lambda)} =
\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle
\sum_M \langle 1M;1\lambda|J J_z\rangle\langle J' J_z^\prime|1 M;1 \lambda
\rangle\,,$$
which is diagonal in $(J J_z)(J' J_z^\prime)$ only after summation over $\lambda$ (unpolarized production). In general, the off-diagonal matrix elements cause interference of the following $J J_z$ states: $00$ with $20$, $11$ with $21$ and $1(-1)$ with $2 (-1)$. While the diagonal elements agree with (\[cho\]), the off-diagonal ones are missed in (\[cho\]).
To assess the degree of transverse $\psi'$ (direct $J/\psi$) polarization at moderate $p_t$, the calculation of [@CHO95II] should be redone with the correct angular momentum projections.
Polarization in fixed target experiments
----------------------------------------
Polarization measurements have been performed for both $\psi$ [@AKE93] and $\psi^\prime$ [@HEI91] production in pion scattering fixed target experiments. Both experiments observe an essentially flat angular distribution in the decay $\psi\to \mu^+ \mu^-$ ($\psi= J/\psi,\psi'$),
$$\frac{d\sigma}{d\cos\theta }\propto 1+ \alpha \cos^2 \theta\,,$$
where the angle $\theta$ is defined as the angle between the three-momentum vector of the positively charged muon and the beam axis in the rest frame of the quarkonium. The observed values for $\alpha$ are $0.02\pm 0.14$ for $\psi'$, measured at $\sqrt{s}=21.8\,$GeV in the region $x_F>0.25$ and $0.028\pm 0.004$ for $J/\psi$ measured at $\sqrt{s}=15.3\,$GeV in the region $x_F>0$. In the CSM, the $J/\psi$’s are predicted to be significantly transversely polarized [@VAE95], in conflict with experiment.
The polarization yield of color octet processes can be calculated along the lines of the previous subsection. We first concentrate on $\psi'$ production and define $\xi$ as the fraction of longitudinally polarized $\psi'$. It is related to $\alpha$ by
$$\alpha=\frac{1-3\xi}{1+\xi}\,.$$
For the different intermediate quark-antiquark states we find the following ratios of longitudinal to transverse quarkonia:
$$\addtolength{\arraycolsep}{0.3cm}
\begin{array}{ccc}
{}^3 S_1^{(1)} & 1:3.35 & \xi=0.23\\
{}^1 S_0^{(8)} & 1:2 & \xi=1/3\\
{}^3 P_J^{(8)} & 1:6 & \xi=1/7\\
{}^3 S_1^{(8)} & 0:1 & \xi=0
\end{array}$$
where the number for the singlet process (first line) has been taken from [@VAE95][^6]. Let us add the following remarks:
\(i) The ${}^3 S_1^{(8)}$-subprocess yields pure transverse polarization. Its contribution to the total polarization is not large, because gluon-gluon fusion dominates the total rate.
\(ii) For the ${}^3 P_J^{(8)}$-subprocess $J$ is not specified, because interference between intermediate states with different $J$ could occur as discussed in the previous subsection. As it turns out, interference does in fact not occur at leading order in $\alpha_s$, because the only non-vanishing short-distance amplitudes in the $J J_z$ basis are $00$, $22$ and $2(-2)$, which do not interfere.
\(iii) The ${}^1 S_0^{(8)}$-subprocess yields unpolarized quarkonia. This follows from the fact that the NRQCD matrix element is
$$\label{above}
\langle 0|\chi^\dagger T^A{a_{\psi'}^{(\lambda)}}^\dagger
a_{\psi'}^{(\lambda)}\,\psi^\dagger T^A\chi|0\rangle =\frac{1}{3}\,
\langle {\cal O}_8^{\psi'}({}^1 S_0)\rangle\,,$$
independent of the helicity state $\lambda$. At this point, we differ from [@TAN95], who assume that this channel results in pure transverse polarization, because the gluon in the chromomagnetic dipole transition ${}^1 S_0^{(8)}\to {}^3 S_1^{(8)}+g$ is assumed to be transverse. However, one should keep in mind that the soft gluon is off-shell and interacts with other partons with unit probability prior to hadronization. The NRQCD formalism applies only to inclusive quarkonium production. Eq. (\[above\]) then follows from rotational invariance.
\(iv) Since the ${}^3 P_J^{(8)}$ and ${}^1 S_0^{(8)}$-subprocesses give different longitudinal polarization fractions, the $\psi'$ polarization depends on a combination of the matrix elements $\langle {\cal O}_8^{\psi'} ({}^1 S_0)\rangle$ and $\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle$ which is different from $\Delta_8(\psi')$.
To obtain the total polarization the various subprocesses have to be weighted by their partial cross sections. We define
$$\delta_8(H)=\frac{\langle {\cal O}_8^{H} ({}^1 S_0)\rangle}
{\Delta_8(H)}$$
and obtain
$$\begin{aligned}
\xi &=& 0.23\,\frac{\sigma_{\psi'}({}^3 S_1^{(1)})}{\sigma_{\psi'}} +
\left[\frac{1}{3}\delta_8(\psi')+\frac{1}{7} (1-\delta_8(\psi'))\right]
\frac{\sigma_{\psi'}({}^1 S_0^{(8)}+{}^3 P_J^{(8)})}{\sigma_{\psi'}}
\nonumber\\
&=& 0.16+0.11\,\delta_8(\psi')\,,\end{aligned}$$
where the last line holds at $\sqrt{s}=21.8\,$GeV (The energy dependence is mild and the above formula can be used with little error even at $\sqrt{s}=40\,$GeV). Since $0<\delta_8(H)<1$, we have $0.16<\xi<0.27$ and therefore
$$0.15 < \alpha < 0.44\,.$$
In quoting this range we do not attempt an estimate of $\delta_8(\psi')$. Note that taking the Tevatron and fixed target extractions of certain (and different) combinations of $\langle {\cal O}_8^{\psi'} ({}^1 S_0)\rangle$ and $\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle$ seriously (see Sect. 5.1), a large value of $\delta_8(\psi')$ and therefore low $\alpha$ would be favored. Within large errors, such a scenario could be considered consistent with the measurement quoted earlier. From a theoretical point of view, however, the numerical violation of velocity counting rules implied by this scenario would be rather disturbing.
In contrast, the more accurate measurement of polarization for $J/\psi$ leads to a clear discrepancy with theory. In this case, we have to incorporate the polarization inherited from decays of the higher charmonium states $\chi_{cJ}$ and $\psi'$. This task is simplified by observing that the contribution from $\chi_{c0}$ and $\chi_{c1}$ feed-down is (theoretically) small as is the octet contribution to the $\chi_{c2}$ production cross section. On the other hand, the gluon-gluon fusion process produces $\chi_{c2}$ states only in a helicity $\pm 2$ level, so that the $J/\psi$ in the subsequent radiative decay is completely transversely polarized. Weighting all subprocesses by their partial cross section and neglecting the small $\psi'$ feed-down, we arrive at
$$\xi = 0.10 + 0.11\,\delta_8(J/\psi)$$
at $\sqrt{s}=15.3\,$GeV, again with mild energy dependence. This translates into sizeable transverse polarization
$$0.31 < \alpha < 0.63\,.$$
The discrepancy with data could be ameliorated if the observed number of $\chi_{c1}$ from feed-down were used instead of the theoretical value. However, we do not know the polarization yield of whatever mechanism is responsible for copious $\chi_{c1}$ production.
Thus, color octet mechanisms do not help to solve the polarization problem and one has to invoke a significant higher-twist contribution as discussed in [@VAE95]. To our knowledge, no specific mechanism has yet been proposed that would yield predominantly longitudinally polarized $\psi'$ and $J/\psi$ in the low $x_F$ region which dominates the total production cross section. One might speculate that both the low $\chi_{c1}/\chi_{c2}$ ratio and the large transverse polarization follow from the assumption of transverse gluons in the gluon-gluon fusion process, as inherent to the leading-twist approximation. If gluons in the proton and pion have large intrinsic transverse momentum, as suggested by the $p_t$-spectrum in open charm production, one would be naturally led to higher-twist effects that obviate the helicity constraint on on-shell gluons.
Other processes
===============
Direct $J/\psi$ and $\psi'$ production is sensitive to the color octet matrix element $\Delta_8(H)$ defined in (\[delta\]). In this section we compare our extraction of $\Delta_8(H)$ with constraints from quarkonium production at the Tevatron and in photo-production at fixed target experiments and HERA.
Quarkonium production at large $p_t$
------------------------------------
An extensive analysis of charmonium production data at $p_t>5\,$GeV has been carried out by Cho and Leibovich [@CHO95; @CHO95II], who relaxed the fragmentation approximation employed earlier [@BRA95; @CAC95]. At the lower $p_t$ boundary, the theoretical prediction is dominated by the ${}^1 S_0^{(8)}$ and ${}^3 P_J^{(8)}$ subprocesses and the fit yields
$$\begin{aligned}
\label{tevme}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{3}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 6.6\cdot 10^{-2}\nonumber\\
\langle {\cal O}^{\psi'}_8 ({}^1 S_0) \rangle +
\frac{3}{m_c^2}\langle {\cal O}^{\psi'}_8 ({}^3 P_0) \rangle\,
= 1.8\cdot 10^{-2}\,,\end{aligned}$$
to be compared with the fixed target values[^7]
$$\begin{aligned}
\label{fixme}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 3.0\cdot 10^{-2}\nonumber\\
\langle {\cal O}^{\psi'}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{\psi'}_8 ({}^3 P_0) \rangle\,
= 0.5\cdot 10^{-2}\,.\end{aligned}$$
If we assume $\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle =
\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle/m_c^2$, the fixed target values are a factor seven (four) smaller than the Tevatron values for $J/\psi$ ($\psi'$). The discrepancy would be lower for the radical choice $\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle=0$.
While this comparison looks like a flagrant violation of the supposed process-independence of NRQCD production matrix elements, there are at least two possibilities that could lead to systematic differences:
\(i) The $2\to 2$ color octet parton processes are schematically of the form
$$\frac{\langle {\cal O}\rangle}{2 m_c}\,\frac{1}{M_f^2}\,
\delta(x_1 x_2 s-M_f^2)\,,$$
where $M_f$ denotes the final state invariant mass. To leading order in $v^2$, we have $M_f=2 m_c$. Note, however, that this is physically unrealistic. Since color must be emitted from the quark pair in the octet state and neutralized by final-state interactions, the final state is a quarkonium accompanied by light hadrons with invariant mass squared of order $M_f^2\approx (M_H+M_H v^2)^2$ since the soft gluon emission carries an energy of order $M_H v^2$, where $M_H$ is the quarkonium mass. The kinematic effect of this difference in invariant mass is very large since the gluon distribution rises steeply at small $x$ and reduces the cross section by at least a factor two. The ‘true’ matrix elements would therefore be larger than those extracted from fixed target experiments at leading order in NRQCD. Since the $\psi'$ is heavier than the $J/\psi$, the effect is more pronounced for $\psi'$, consistent with the larger disagreement with the Tevatron extraction for $\psi'$. Note that the effect is absent for large-$p_t$ production, since in this case, $x_1 x_2 s > 4 p_t^2 \gg M_f^2$. If we write $M_f=2 m_c+{\cal O}
(v^2)$, then the difference between fixed target and large-$p_t$ production stems from different behaviors of the velocity expansion in the two cases.
\(ii) It is known that small-$x$ effects increase the open bottom production cross section at the Tevatron as compared to collisions at lower $\sqrt{s}$. Since even at large $p_t$, the typical $x$ is smaller at the Tevatron than in fixed target experiments, this effect would enhance the Tevatron prediction more than the fixed target prediction. The ‘true’ matrix elements would therefore be smaller than those extracted from the Tevatron in [@CHO95II].
While a combination of both effects could well account for the apparently different NRQCD matrix elements, one must keep in mind that we have reason to suspect important higher twist effects for charmonium production at fixed target energies. Theoretical predictions for fixed target production are intrinsically less accurate than at large $p_t$, where higher-twist contributions due to the initial hadrons are expected to be suppressed by $\Lambda_{QCD}/p_t$ (if not $\Lambda_{QCD}^2/p_t^2$) rather than $\Lambda_{QCD}/m_c$.
Photo-production
----------------
A comparison of photo-production with fixed target production is more direct since the same combination of NRQCD matrix elements is probed and the kinematics is similar. All analyses [@CAC96; @AMU96; @KO96] find a substantial overestimate of the cross section if the octet matrix elements of (\[tevme\]) are used. The authors of [@AMU96] fit
$$\label{photome}
\langle {\cal O}^{J/\psi}_8 ({}^1 S_0) \rangle +
\frac{7}{m_c^2}\langle {\cal O}^{J/\psi}_8 ({}^3 P_0) \rangle\,
= 2.0\cdot 10^{-2}\,,$$
consistent with (\[fixme\]) within errors, which we have not specified. While this agreement is reassuring, it might also be partly accidental since the extraction of [@AMU96] is performed on the elastic peak, which is not described by NRQCD. Color octet mechanisms do not leave a clear signature in the total inelastic photo-production cross section. The authors of [@CAC96] argue that the color-octet contributions to the energy spectrum of $J/\psi$ are in conflict with the observed energy dependence in the endpoint region $z>0.7$, where $z=E_{J/\psi}/E_\gamma$ in the proton rest frame. This discrepancy would largely disappear if the smaller matrix element of (\[fixme\]) or (\[photome\]) were used rather than (\[tevme\]). Furthermore, since in a color octet process soft gluons with energy $M_H v^2$ must be emitted, but are kinematically not accounted for, the NRQCD-prediction for the energy distribution should be smeared over an interval of size $\delta z\sim v^2\sim 0.3$, making the steep rise of the energy distribution close to $z=1$ is not necessarily physical.
Conclusion
==========
We have reanalyzed charmonium production data from fixed target experiments, including color octet production mechanisms. Our conclusion is twofold: On one hand, the inclusion of color octet processes allows us to reproduce the overall normalization of the total production cross section with color octet matrix elements of the expected size (if not somewhat smaller) without having to invoke small values of the charm quark mass. This was found to be true for bottomonium as well as for charmonium. Comparing the theoretical predictions within this framework with the data implies the existence of additional bottomonium states below threshold which have not yet been seen directly.
On the other hand, the present picture of charmonium production at fixed target energies is far from perfect. The $\chi_{c1}/
\chi_{c2}$ production ratio remains almost an order of magnitude too low, and the transverse polarization fraction of the $J/\psi$ and $\psi'$ is too large. We thus confirm the expectation of [@VAE95] that higher twist effects must be substantial even after including the octet mechanism.
The uncertainties in the theoretical prediction at fixed target energies are substantial and preclude a straightforward test of universality of color octet matrix elements by comparison with quarkonium production at large transverse momentum. We have argued that small-$x$, as well as kinematic effects, could bias the extraction of these matrix elements in different directions at fixed target and collider energies. The large uncertainties involved, especially due to the charm quark mass, could hardly be eliminated by a laborious calculation of $\alpha_s$-corrections to the production processes considered here. To more firmly establish existence of the octet mechanism there are several experimental measurements which need to be performed. Data on polarization is presently only available for charmonium production in pion-induced collisions. A measurement of polarization at large transverse momentum or for bottomonium is of crucial importance, because higher twist effects should be suppressed. Furthermore, a measurement of direct and indirect production fractions in the bottom system would provide further confirmation of the color octet picture and constrain the color octet matrix elements for bottomonium.
[**Acknowledgments.**]{} We thank S.J. Brodsky, E. Quack and V. Sharma for discussions. IZR acknowledges support from the DOE grant DE-FG03-90ER40546 and the NSF grant PHY-8958081.
[99]{}
G.A. Schuler, ‘Quarkonium production and decays’, CERN report CERN-TH-7170-94 \[hep-ph/9403387\], and references therein
M. Vänttinen, P. Hoyer, S.J. Brodsky and W.-K. Tang, Phys. Rev. [**D51**]{}, 3332 (1995)
E. Braaten, M. Doncheski, S. Fleming and M. Mangano, Phys. Lett. [**B333**]{}, 548 (1994)
D.P. Roy and K. Sridhar, Phys. Lett. [**B339**]{}, 141 (1994)
G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. [**D51**]{}, 1125 (1995)
E. Braaten and S. Fleming, Phys. Rev. Lett. [**74**]{}, 3327 (1995)
M. Cacciari, M. Greco, M.L. Mangano and A. Petrelli, Phys. Lett. [**B356**]{}, 553 (1995)
P. Cho and A.K. Leibovich, Phys. Rev. [ **D53**]{}, 150 (1996)
P. Cho and A.K. Leibovich, CalTech report CALT-68-2026 \[hep-ph/9511315\]
M. Cacciari and M. Krämer, DESY report DESY 96-005 \[hep-ph/9601276\]
J. Amundson, S. Fleming and I. Maksymyk, University of Texas report UTTG-10-95 \[hep-ph/9601298\]
P. Ko, J. Lee and H.S. Song, Seoul University report \[hep-ph/9602223\]
W.-K. Tang and M. Vänttinen, SLAC report SLAC-PUB-95-6931 \[hep-ph/9506378\]; NORDITA report NORDITA-96/18 P \[hep-ph/9603266\]
S. Gupta and K. Sridhar, Tata Institute report TIFR/TH/96-04 \[hep-ph/9601349\]
P. Cho and M.B. Wise, Phys. Lett. [**B346**]{}, 129 (1995)
M. Beneke and I.Z. Rothstein, University of Michigan report UM-TH-95-23 \[hep-ph/9509375\], to appear in Phys. Lett. [**B**]{}
H. Jung, D. Krücker, C. Greub and D. Wyler, Z. Phys. [**C60**]{}, 721 (1993)
F. Abe [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 4358 (1995)
E. Eichten and C. Quigg, Phys. Rev. [**D52**]{}, 1726 (1995).
H.L. Lai [*et al.*]{}, Phys. Rev. [**D51**]{}, 4763 (1995)
M. Glück, E. Reya and A. Vogt, Z. Phys. [**C53**]{}, 651 (1992)
C. Akerlof [*et al.*]{}, Phys. Rev. [**D48**]{}, 5067 (1993)
M. H. Schub [*et al.*]{}, Phys. Rev. [**D52**]{}, 1307 (1995)
T. Alexopoulos [*et al.*]{}, Fermilab report FERMILAB-Pub-95/297-E
L. Antoniazzi [*et al.*]{}, Phys. Rev. Lett. [**70**]{}, 383 (1993)
Y. Lemoigne [*et al.*]{}, Phys. Lett. [**B113**]{}, 509 (1982)
D.M. Alde [*et al.*]{}, Phys. Rev. Lett. [**66**]{}, 2285 (1991)
P. Cho, M.B. Wise and S. Trivedi, Phys. Rev. [ **D51**]{}, 2039 (1995)
J.G. Heinrich [*et al.*]{}, Phys. Rev. [**D44**]{}, 1909 (1991)
[^1]: Research supported by the Department of Energy under contract DE-AC03-76SF00515.
[^2]: Their precise definition is given in Sect. VI of [@BOD95].
[^3]: This together with a smaller value for the color singlet radial wavefunction could at least partially explain the huge discrepancy between the CSM and the data that was reported in [@SCH95].
[^4]: These numbers were taken from the raw data with no concern regarding the differing efficiencies for the individual states.
[^5]: Technically, this means that NRQCD matrix elements with an odd number of derivatives or spin matrices vanish if the quarkonium is a $C$ or $P$ eigenstate.
[^6]: This number is $x_F$-dependent and we have approximated it by a constant at low $x_F$, where the bulk data is obtained from. The polarization fractions for the octet $2\to 2$ parton processes are $x_F$-independent.
[^7]: Since there is a strong correlation between the charm quark mass and the extracted NRQCD matrix elements, we emphasize that both (\[tevme\]) and (\[fixme\]) as well as (\[photome\]) below have been obtained with the same $m_c=1.5\,$GeV (or $m_c=1.48\,$ GeV, to be precise). On the other hand, the apparent agreement of predictions for fixed target experiments with data claimed in [@GUP96] is obtained from (\[photome\]) in conjunction with $m_c=1.7\,$GeV.
|
---
abstract: 'In this paper, we aim to understand whether current language and vision (LaVi) models truly grasp the interaction between the two modalities. To this end, we propose an extension of the MS-COCO dataset, FOIL-COCO, which associates images with both correct and ‘foil’ captions, that is, descriptions of the image that are highly similar to the original ones, but contain one single mistake (‘foil word’). We show that current LaVi models fall into the traps of this data and perform badly on three tasks: a) caption classification (correct vs. foil); b) foil word detection; c) foil word correction. Humans, in contrast, have near-perfect performance on those tasks. We demonstrate that merely utilising language cues is not enough to model FOIL-COCO and that it challenges the state-of-the-art by requiring a fine-grained understanding of the relation between text and image.'
author:
- |
Ravi Shekhar, Sandro Pezzelle, Yauhen Klimovich,\
**Aurélie Herbelot, Moin Nabi, Enver Sangineto, Raffaella Bernardi**\
University of Trento\
[{firstname.lastname}@unitn.it]{}
bibliography:
- 'raffa.bib'
- 'ravi.bib'
- 'moin.bib'
title: 'FOIL it! Find One mismatch between Image and Language caption'
---
Acknowledgments {#acknowledgments .unnumbered}
===============
We are greatful to the Erasmus Mundus European Master in Language and Communication Technologies (EM LCT) for the scholarship provided to the third author. Moreover, we gratefully acknowledge the support of NVIDIA Corporation with the donation of the GPUs used in our research.
|
---
abstract: 'We present correlations between 9 CO transitions ($J=4-3$ to $12-11$) and beam-matched far-infrared (far-IR) luminosities ($L_{\mathrm{FIR},\,b}$) among 167 local galaxies, using [*[Herschel]{}*]{} Spectral and Photometric Imaging ReceiverFourier Transform Spectrometer (SPIRE; FTS) spectroscopic data and Photoconductor Array Camera and Spectrometer (PACS) photometry data. We adopt entire-galaxy FIR luminosities ($L_{\mathrm{FIR},\,e}$) from the [*[IRAS]{}*]{} Revised Bright Galaxy Sample and correct to $L_{\mathrm{FIR},\,b}$ using PACS images to match the varying FTS beam sizes. All 9 correlations between $L''_{\mathrm{CO}}$ and $L_{\mathrm{FIR},\,b}$ are essentially linear and tight ($\sigma=0.2-0.3\,dex$ dispersion), even for the highest transition, $J=12-11$. This supports the notion that the star formation rate (SFR) is linearly correlated with the dense molecular gas ($n_{\mathrm{H}_2}\gtrsim10^{4-6}\,cm^{-3}$). We divide the entire sample into three subsamples and find that smaller sample sizes can induce large differences in the correlation slopes. We also derive an average CO spectral line energy distribution (SLED) for the entire sample and discuss the implied average molecular gas properties for these local galaxies. We further extend our sample to high-[*[z]{}*]{} galaxies with literature CO($J=5-4$) data from the literature as an example, including submillimeter galaxies (SMGs) and “normal” star-forming BzKs. BzKs have similar FIR/CO(5–4) ratios as that of local galaxies, and follow well the locally-determined correlation, whereas SMG ratios fall around or slightly above the local correlation with large uncertainties. Finally, by including Galactic CO($J=10-9$) data as well as very limited high-[*[z]{}*]{} CO($J=10-9$) data, we verify that the CO($J=10-9$)–FIR correlation successfully extends to Galactic young stellar objects, suggesting that linear correlations are valid over 15 orders of magnitude.'
author:
- 'Daizhong Liu , Yu Gao , Kate Isaak , Emanuele Daddi , Chentao Yang , Nanyao Lu , Paul van der Werf'
title: 'High-[*[J]{}*]{} CO Versus Far-Infrared Relations in Normal and Starburst Galaxies'
---
Introduction
============
Molecular gas is the raw material from which stars are formed, hence its presence and physical conditions correlate with the star formation rate (SFR). A well-known correlation between gas surface density and SFR surface density is the [*Kennicutt-Schmidt law*]{} (K-S law): $\Sigma_{SFR} = A\,\Sigma_{gas}^{\;N}$, with slope $N=1.4\pm0.15$ [@Kennicutt1998], where $\Sigma_{SFR}$ is based on $\mathrm{H}_{\alpha}$ luminosity or far-infrared (far-IR) luminosity ($L_{FIR}$, $40-400\mu{m}$) and $\Sigma_{gas}$ is the sum of the atomic gas (from $\mathrm{H}\mathrm{I}$ observations) and molecular gas (inferred from CO(1–0) line luminosity) surface density. This quantitative correlation serves as a fundamental input to most cosmological simulations [e.g. @Springel2003; @MacLow2004; @Krumholz2005; @Krumholz2007; @Narayanan2008b]. However, it brushes over and/or oversimplifies the relationship between the individual gas components and star formation (SF). First, observations have shown that the presence and state of the atomic gas component has little impact on SF [e.g. @Bigiel2008; @Schruba2013; @LiuLJ2012; @LiuLJ2015]. Second the correlation between the most widely used molecular gas tracer, CO(1–0), and $L_{FIR}$ is found to be more complex than just a single power law: [@Gao2004] showed that the CO(1–0)–FIR slope changes from 1.27 to 1.73 with different sample selections, e.g. including only normal star-forming galaxies (SFGs:$\;L_{\mathrm{IR}}<10^{11}\,L_{\odot}$), or adding (ultra-)luminous IR galaxies (LIRGs:$\;L_{\mathrm{IR}}=10^{11-12}\,L_{\odot}$, ULIRGs:$\;L_{\mathrm{IR}}>10^{13}\,L_{\odot}$), respectively. [@Greve2014] showed similar slope variations from 0.9 to 1.4 by analyzing a large number of data from the literature. It is thus questionable that a single power-law relation exists between CO(1–0) and FIR and whether it can be applied to all galaxies even if it exists.
CO(1–0) has a low critical density ($n_{\mathrm{H}_2,crit}\sim2\times10^{3}\,cm^{-3}$), and thus traces the total amount of molecular gas. It provides no physical insight into the higher density gas ($n_{\mathrm{H}_2}>10^{4}\,cm^{-3}$) that is known to be found at the formation sites of individual stars [@Kennicutt2012]. The sensitivity of the CO $(J_{u}{\to}J_{u}-1)$ transition to denser/warmer gas increases with an increasing upper level ($J_{u}$): for example, CO(10-9) has a critical density of $n_{\mathrm{H}_2,crit}\sim10^{6}\,cm^{-3}$ [@Carilli2013]. In contrast, the ground-[*[J]{}*]{} of high dipole-moment molecules such as HCN and HCO$^+$ already have $n_{\mathrm{H}_2,crit}\sim10^{6}\,cm^{-3}$, thus probing the dense gas in cold phase. Studies based on HCN [@Gao2004; @Wu2005; @Wu2010] and high-[*[J]{}*]{} CS [@Zhang2014] have found a unified linear correlation between dense gas and the SFR, describing the scenario in which all dense gas above a certain density threshold has a similar efficiency/timescale to collapse and form new stars, thus is linearly determining SFRs.
Until recently, published mid-to-high-[*[J]{}*]{} ($J_{u}\ge4$) CO data in local galaxies were still scarce due to the challenges of observing at frequencies $\gtrsim450\,\mathrm{GHz}$ from the ground. [@Bayet2009] performed large velocity gradient (LVG) modeling with ground-based $J_{u}\le7$ CO to extrapolate $J_{u}>7$ transitions for several local galaxies. They found that CO versus total IR luminosity ($L_{TIR}$, $8-1000{\mu}m$) correlations have decreasing slopes for increasing $J_{u}$, similar to what modelings and simulations predicted [@Narayanan2008b; @Juneau2009].
More recently, presented the [*[Herschel]{}*]{} Spectral and Photometric Imaging Receiver [SPIRE; @Griffin2010] Fourier Transform Spectrometer [FTS, @Naylor2010] high-[*[J]{}*]{} CO data from the HerCULES program [@vanderWerf2010; @Rosenberg2015]. They find that up to CO(7–6) the CO versus TIR or FIR relations are all roughly linear, while for higher-$J_{u}$ the correlations become sub-linear, similar to the results of [@Bayet2009]. In this work, we use a large SPIRE FTS data set to statistically determine the correlations between 9 $L'_{\mathrm{CO}}$ ($J=4-3\;\mathrm{to}\;12-11$) and beam-matched $L_{\mathrm{FIR},b}$, and to determine which transitions are the best tracers of star formation. We also verify the slope variations against $J_{u}$ and test the validity of dense gas versus SF relation at high-[*[z]{}*]{}. The sample, data, and method are described in Section \[Section2\]. The results and a discussion are given in Section \[Section3\]. We adopt $H_0=73\,km\,s^{-1},\;\Omega_{\Lambda}=0.73,\;\Omega_{M}=0.27$.
Sample and Data {#Section2}
===============
The sample was selected from all public FTS observations in the [*[Herschel]{}*]{} Science Archive (HSA) of local galaxies that are in the [*[IRAS]{}*]{} Revised Bright Galaxy Sample [@Sanders2003]. A further selection criterion requiring the availability of $70-160\,{\mu}m$ band imaging data taken with Photoconductor Array Camera and Spectrometer [PACS; @Poglitsch2010] for galaxies partially resolved by the FTS beam (see Section \[Section21\]) was imposed to enable beam-matching techniques to be used in the analysis. The final sample contains 167 local galaxies ($z<0.064$, $d_L<286\,\mathrm{Mpc}$), including 124 (U)LIRGs and 43 SFGs (see [@Rosenberg2015] and [@Lu2014] for details of the HerCULES and GOALS programs, respectively; details of the full data set are given in Liu et al. in preparation).
FTS CO Measurements {#Section21}
-------------------
FTS has two bolometer arrays: the SLW bolometer array ($446-989\,\mathrm{GHz}$) and the SSW bolometer array: ($959-1543\,\mathrm{GHz}$). Two observing modes are used: single-pointing for point-like sources (mostly with angular size less than FTS beam sizes, and $d_L>30\,Mpc$), and mapping for nearby extended sources ($d_L<30\,Mpc$). In single-pointing observations the central bolometer of each array is coaligned with the target, while off-axis bolometers point to the off-center sky. Mapping observations perform jiggling to scan extended targets, and each individual bolometer will produce one spectrum at its own pointing (R.A., decl.). All mapping data and $\sim30\%$ single-pointing data were reduced using SPIRE v12 calibration products and [*[Herschel]{}*]{} Interactive Processing Environment [HIPE v12.1.0, @Ott2010] pipelines, with the remainder reduced using SPIRE v10 calibration products+pipelines. We note that measured line fluxes for the two sets of the $\sim30\%$ single-pointing data show little difference ($\sigma\sim8\%$). For single-pointing data, we use central bolometers to extract CO lines. For mapping data, which are usually for spatially-extended galaxies, we select individual bolometers that are $>$half-beam-separated (Nyquist sampling) to extract CO lines.
Note that FTS covers a wide range of frequencies, and its beam size varies accordingly [^1]: $\sim43''$ at CO(4–3) to $\sim17''$ at CO(13–12), and is not a simple function of frequency. The beam size variation is well-calibrated for each central bolometer, but unavailable for off-axis bolometers. Thus we assign an additional 15% uncertainty for off-axis bolometers. To derive CO line flux, we use two different line profile fitting functions in HIPE: a Sinc and a Sinc-convolved-Gaussian (SCG). A SCG, where a Gaussian line profile is convolved with the Sinc instrument response, is appropriate for observations of sources with broad/resolved lines [e.g. @Zhao2013]: in such cases a Sinc fit would underestimate the line flux by $\sim40\%$. In the more common case where the CO line is not obviously resolved, the SCG-derived fluxes are systematically larger by $<20\%$. Given that a large fraction of single-pointing data have broad/resolved lines while mapping data do not, here we use SCG and Sinc for single-pointing and mapping data, respectively, although we note that using only SCG or Sinc for all data has no significant change to our correlation results. The CO flux uncertainties are derived from the rms in the baseline-subtracted spectra in the vicinity of each CO line rather than from formal line fitting uncertainties. Then the fluxes are converted to $L'_{\mathrm{CO}}$ according to [@Solomon1992].
Beam-matched FIR {#Section22}
----------------
As a starting point we took the entire-galaxy FIR luminosity ($L_{\mathrm{FIR},e}$) from [@Sanders2003]. Given the mismatch between the dramatically varying FTS CO beam sizes and the physical extent of nearby galaxies [e.g. @Galametz2013] and some merging/interacting (U)LIRGs [e.g. @Gao1999], we used a beam-scaling method based on the PACS imaging photometry [^2] to scale entire-galaxy $L_{\mathrm{FIR},e}$ down to the local region that matches the CO beam size area ($L_{\mathrm{FIR},b}$, for each CO line of each bolometer). PACS bands span the peak of the spectral energy distribution (SED) of typical FIR-luminous local galaxies, and thus provide a good proxy for FIR and are insensitive to dust temperatures ($T_{\mathrm{dust}}$) [^3]. By performing photometry with an aperture of CO beam size ($F_{\mathrm{PACS},b}$) and an aperture of entire-galaxy ($F_{\mathrm{PACS},e}$), respectively, we can determine the FIR luminosity that is appropriate to each CO beam size: $L_{\mathrm{FIR},b}={F_{\mathrm{PACS},b}}/{F_{\mathrm{PACS},e}}{\times}L_{\mathrm{FIR},e}$. In this way we calculate a value of $L_{\mathrm{FIR},b}$ for each CO line. This is essential for local galaxies that are partially resolved for FTS CO beam sizes .
Results and Discussion {#Section3}
======================
Correlations between $L'_{\mathrm{CO}}$ and Beam-matched $L_{\mathrm{FIR}}$ {#Section31}
---------------------------------------------------------------------------
Fig.\[Fig01\] shows the 9 CO–FIR correlations of local galaxies and spatially resolved regions of nearby galaxies. FTS can observe CO(4–3) to CO(13–12) except for galaxies at $z>0.032$ (mostly (U)LIRGs) whose CO(4–3) shift out of the FTS SLW waveband. Moreover, detections of CO(13–12) are sparse, and thus are not analyzed here.
To derive the slope and intercept, we fit all data points (excluding upper limits) with two linear fitting codes: an IDL least-squares fitting code based on MPFIT [@Markwardt2009], and an IDL Bayesian regression code LINMIX\_ERR [@Kelly2007]. The two methods give consistent results, thus we only show the latter results in Fig.\[Fig01\]. CO(4–3) has the largest number of detections, but the least dispersions are seen in the CO(6–5) and CO(7–6) panels (both with $\sigma=0.19\,dex$). This confirms the conclusion of [@Lu2014] that CO $J_{u}\sim6-7$ transitions are the best SFR tracers for (U)LIRGs. For the entire sample, we find no evidence of significant decrease in slopes with increasing $J_{u}$: $N\sim$1.0–1.1 at $J_{u}\sim4-6$, and 0.96–1.0 at $J_{u}\sim10-12$. Besides, the fittings are consistent with the average distribution of upper limits for non-detections in each transition.
[@Bayet2009] analyzed the IR beam corrections based on 850$\mu{m}$ images and found correction factors similar to ours for galaxies we had in common. They found that IR beam corrections have only minor effects on the slopes ($<5\%$) and thus did not apply them, while in contrast we found that slopes without IR beam corrections are smaller by 10–15%. This partially explains the discrepancy between our results and their decreasing slopes, i.e. $N\sim1$ at $J_{u}\sim3$, then $N\sim0.8$ at $J_{u}\sim7$, and finally $N=0.53\pm0.07$ at $J_{u}=12$.
In addition, small number statistics is another key factor behind the discrepancies between different studies. presented the high-[*[J]{}*]{} CO–FIR correlation with HerCULES FTS data [@vanderWerf2010] and data the from literature of high-$z$ submillimeter galaxies (SMGs). Their HerCULES sample contains 23 (U)LIRGs (excluding extended/merging galaxies). They evaluate IR beam correction with 870$\mu{m}$ or 350$\mu{m}$ maps, but the correction factors are small for their local sample (mostly (U)LIRGs). They found correlation slopes $N\sim1.0$ for $1\le{J_{u}}\le5$, which then decrease to $N=0.87\pm0.05$ at $J_{u}=7$ and then rapidly become $N=0.51\pm0.11$ at $J_{u}=12$.
[cccccccccccc]{} Entire & ${N}=$$1.06\pm0.03$ & $1.07\pm0.03$ & $1.10\pm0.03$ & $1.03\pm0.04$ & $1.02\pm0.03$ & $1.01\pm0.03$ & $0.96\pm0.04$ & $1.00\pm0.05$ & $0.99\pm0.09$\
& ${A}=$$1.49\pm0.24$ & $1.71\pm0.22$ & $1.79\pm0.24$ & $2.62\pm0.26$ & $2.82\pm0.27$ & $3.10\pm0.22$ & $3.67\pm0.25$ & $3.51\pm0.33$ & $3.83\pm0.67$\
& $\bar{A}=$$1.96\pm0.07$ & $2.27\pm0.07$ & $2.56\pm0.08$ & $2.86\pm0.07$ & $3.04\pm0.08$ & $3.20\pm0.09$ & $3.38\pm0.10$ & $3.56\pm0.11$ & $3.77\pm0.15$\
HerCULES & ${N}=$$0.75\pm0.62$ & $1.08\pm0.43$ & $0.98\pm0.29$ & $0.84\pm0.35$ & $0.81\pm0.24$ & $0.81\pm0.26$ & $0.73\pm0.25$ & $0.76\pm0.26$ & $0.65\pm0.31$\
& ${A}=$$4.46\pm0.57$ & $1.67\pm3.89$ & $2.78\pm2.61$ & $4.18\pm3.06$ & $4.68\pm2.04$ & $4.80\pm0.21$ & $5.56\pm0.02$ & $5.49\pm0.02$ & $6.58\pm0.00$\
& $\bar{A}=$$2.20\pm0.23$ & $2.40\pm0.21$ & $2.61\pm0.21$ & $2.85\pm0.21$ & $3.08\pm0.21$ & $3.28\pm0.22$ & $3.42\pm0.22$ & $3.65\pm0.23$ & $3.90\pm0.28$\
Pointing & ${N}=$$1.00\pm0.10$ & $1.10\pm0.09$ & $1.06\pm0.09$ & $1.05\pm0.08$ & $1.09\pm0.11$ & $1.02\pm0.11$ & $0.93\pm0.09$ & $1.05\pm0.12$ & $1.12\pm0.18$\
& ${A}=$$1.96\pm0.88$ & $1.43\pm0.79$ & $2.09\pm0.79$ & $2.51\pm0.65$ & $2.34\pm0.92$ & $3.02\pm0.84$ & $3.88\pm0.70$ & $3.12\pm0.91$ & $2.73\pm1.36$\
& $\bar{A}=$$1.93\pm0.13$ & $2.30\pm0.12$ & $2.62\pm0.11$ & $2.89\pm0.11$ & $3.02\pm0.13$ & $3.18\pm0.16$ & $3.34\pm0.16$ & $3.51\pm0.18$ & $3.65\pm0.24$\
Mapping & ${N}=$$1.17\pm0.10$ & $1.13\pm0.09$ & $1.11\pm0.10$ & $0.96\pm0.08$ & $0.94\pm0.08$ & $1.17\pm0.12$ & $1.12\pm0.18$ & $1.03\pm0.17$ & $0.66\pm0.42$\
& ${A}=$$0.64\pm0.71$ & $1.29\pm0.67$ & $1.78\pm0.74$ & $3.08\pm0.56$ & $3.44\pm0.56$ & $2.14\pm0.77$ & $2.68\pm1.14$ & $3.40\pm1.04$ & $5.99\pm0.00$\
& $\bar{A}=$$1.93\pm0.08$ & $2.22\pm0.10$ & $2.47\pm0.13$ & $2.82\pm0.12$ & $3.04\pm0.12$ & $3.19\pm0.12$ & $3.40\pm0.17$ & $3.58\pm0.18$ & $3.81\pm0.29$
For comparison, we divide our entire sample into three subsamples: a HerCULES subsample corresponding to the local sample (mostly (U)LIRGs, excluding extended/merging ones), all other single-pointing data (mixed SFGs+(U)LIRGs), and all mapping data (nearby resolved SFGs, dominating the faint-end). Table.\[Tab01\] lists the best-fit slope $N$, the intercept $A$, and the mean FIR/CO ($\bar{A}=\log\,(L_{\mathrm{FIR}}/L'_{\mathrm{CO}})$) for the entire sample and three subsamples. For the HerCULES subsample, similar to and [@Bayet2009], we find decreasing slopes, e.g. $N=0.65\pm0.31$ at CO(12–11). We overlay the significantly sub-linear CO(12-11)–FIR correlation of ($N=0.51\pm0.11$) in the CO(12–11) panel of Fig.\[Fig01\] for reference. Thus, with the largest high-[*[J]{}*]{} CO data set available to date, we conclude that within the luminosity range shown in Fig.\[Fig01\], $L'_{\mathrm{CO}}$ $7\le{J}_{u}\le9$ closely and linearly follow $L_{\mathrm{FIR}}$ and by extension the SFRs, while all other $4\le{J}_{u}\le12$ CO–FIR correlations are tight and not far from linear. These results are not in conflict with previous studies.
Average CO/FIR Spectral Line Energy Distribution (SLED) {#Section32}
-------------------------------------------------------
In Fig.\[Fig04\], we show a global FIR-normalized CO SLED, constructed from the products of the inversion of the best-fit FIR/CO normalization parameter for each transition ($\bar{A}$, which is equal to $L_{\mathrm{FIR}}/L'_{\mathrm{CO}}$, given in Table \[Tab01\]) $\times{J_u}^2$ (where ${J_u}^2$ is included to make the unit same as integrated line flux $Jy\:km\:s^{-1}$). Error bars are the dispersions of CO–FIR correlations (the $\sigma$ shown in Fig.\[Fig01\]).
This CO SLED represents the average CO excitation conditions in local galaxies, and reveals at least two components: a low-excitation component peaking around $J_{u}\sim3-4$, and a higher-excitation component peaking around $J_{u}\sim8$.
We used RADEX [@vanderTak2007] to construct two-component LVG models and performed least-$\chi^2$ fitting to search for the best-fits of gas kinetic temperature $T_{\mathrm{kin}}$ and density $n_{\mathrm{H}_2}$, deriving $T_{kin}\sim90_{-40}^{+100}\;K$, $\log\,n_{\mathrm{H}_2}\sim3.0_{-0.3}^{+0.3}\;cm^{-3}$ for the lower-excitation component, and $T_{kin}\gtrsim200\;K$, $\log\,n_{\mathrm{H}_2}\sim4.1_{-0.2}^{+0.2}\;cm^{-3}$ for the higher-excitation component.
In Fig.\[Fig04\], we also overlay the FIR-normalized CO SLEDs of a selection of galaxies that have at least six CO detections and span a wide $L_{\mathrm{FIR}}$ range. (U)LIRGs/starbursts have stronger high-[*[J]{}*]{} CO excitations and their normalized SLEDs are flatter at $J_{u}\gtrsim6$, whereas normal SFGs with weaker excitations show a bump at $J_{u}\sim4$ in their normalized SLEDs. Surprisingly, the largest difference between normalized SLEDs of (U)LIRGs and SFGs is seen not at a high-[*[J]{}*]{} (i.e. $J_{u}\sim10$) but at a low-$J$ (i.e. $J_{u}\sim4$), where the CO/FIR ratios vary within $\sim1.5\,dex$. Since the low-excitation component dominates the low-$J$ part of SLED, the large variation at a low-$J$ indicates that low-excitation gas is less correlated with FIR, while the high-excitation component is intrinsically better correlated with FIR [see also @Lu2014].
![Average CO/FIR SLED (i.e. CO SLED normalized by FIR) for the entire sample, overlaid with several FIR-normalized CO SLEDs of individual galaxies spanning a wide range of $L_{\mathrm{FIR}}$. The black open squares are the values of the average CO/FIR (i.e. $L'_{\mathrm{CO}}/L_{\mathrm{FIR}} \times J_{u}^2$). The black solid curve is the best-fit two-component LVG model. The other dashed curves are the best-fit one- or two-component LVG models for individual galaxies. The colors indicate their $\lg\,(L_{\mathrm{FIR}})$. The two embedded panels are the least-$\chi^2$ fitting results of low-excitation gas (left) and high-excitation gas (right) respectively. \[Fig04\]](Fig04){width="0.9\columnwidth"}
The CO(5–4) – FIR Correlation and Extension to High-[z]{} {#Section33}
---------------------------------------------------------
To extend the high-[*[J]{}*]{} CO–FIR correlation toward higher-[*[z]{}*]{}, we use the extensive compilation of high-[*[z]{}*]{} CO data in that contains 60 CO $J_{u}\ge4$ detections with literature $L_{\mathrm{IR}}$[^4]. They are the most extreme starbursts across all cosmic time, likely experiencing a short-term burst phase of SF resembling local ULIRGs, rather than the long-lasting mode in normal SFGs or the so-called ”main-sequence” (MS) galaxies at high-[*[z]{}*]{} . CO $J_{u}\ge4$ observations toward high-[*[z]{}*]{} MS galaxies are still very rare: [@Daddi2014] presented the first CO(5–4) detections in a sample of 4 $z\sim1.5$ BzK-color-selected MS galaxies (BzKs). Thus combining all high-[*[z]{}*]{} $J_{u}\ge4$ data, CO(5–4) has the second-most detections but covers the most diverse galaxy types. In Fig.\[One\_CO\_InBeam\_IR\_CO54\], we show the CO(5–4)–FIR correlation, including high-[*[z]{}*]{} galaxies as an example to illustrate the correlation between dense gas and SFR, as CO(5–4) traces a hundred times denser molecular gas than CO(1–0). We show the linear fits from Fig.\[Fig01\] only, as the inclusion of high-[*[z]{}*]{} SMGs and BzKs does not significantly change the results of the fit. The mean $\log\,(L_{\mathrm{FIR}}/L'_{\mathrm{CO}(5-4)})=2.27\pm0.07\;L_{\odot}\;(K\,km\,s^{-1}\,pc^{2})^{-1}$ is consistent with [@Daddi2014] considering a conversion factor of 1/1.3 from TIR to FIR. BzK galaxies fall on the local linear best-fit correlation (dashed line), whereas high-[*[z]{}*]{} SMGs are offset above by $\sim0.28\,dex$. Interestingly, using HCN(1-0) as a dense gas tracer, [@Gao2007] found higher FIR/HCN ratios in high-[*[z]{}*]{} galaxies (mostly QSOs/AGNs with only 5 HCN detections) than those of local ULIRGs/SFGs. Given the large uncertainties in $L_{\mathrm{FIR}}$ for a large fraction of SMGs [^5] compared to BzKs (from SED fitting), the amount of excess in FIR/CO(5–4) (i.e. dense gas SF efficiency) in SMGs should still be treated with caution. And this excess is not significant enough to break down a linear form of dense gas versus SF relation, as these starbursts are the most extreme systems at all redshifts and as such do not represent the dominant mode of SF observed in MS galaxies (e.g. BzKs). Thus, better FIR measurements, e.g. via SEDs, and additional high-[*[J]{}*]{} CO observations in normal MS galaxies are needed before making solid conclusions at high-[*[z]{}*]{}.
The CO(10–9) – FIR Correlation Extending to Galactic Young Stellar Objects (YSOs) {#Section34}
---------------------------------------------------------------------------------
In Fig.\[One\_CO\_InBeam\_IR\_SJ13\] we show the correlation between an even denser/warmer gas tracer CO(10–9) and $L_{\mathrm{FIR}}$, which has the largest number of observations of sources spanning high-[*[z]{}*]{} galaxies (from and [@ALMAPartnership2015]) to galactic YSOs/protostars (from [@SanJoseGarcia2013]). [@ALMAPartnership2015] presented spatially resolved ALMA CO(10–9) in a strongly lensed SMG SDP.81 at $z=3.042$, with demagnified $L_{\mathrm{TIR}}=5.1\times10^{12}\,L_{\odot}$, comparable to the BzKs. We show the global SDP.81 as well as its east and west components in Fig.\[One\_CO\_InBeam\_IR\_SJ13\]. The best fit for all sources is $N=1.04\pm0.01$ and $A=3.04\pm0.06$ [see also @SanJoseGarcia2013 who obtained the same $N=1.04$ with only six galaxies]; considering YSOs/protostars alone we get $N=1.26\pm0.09$, which is likely biased by the limited number of bright sources ($L_{bol}\approx10^5\,L_{\odot}$). A fixed-slope $N=1$ fit indicates that a mean $\log\,(L_{\mathrm{FIR}}/L'_{\mathrm{CO}(10-9)})=3.30\pm0.09\;L_{\odot}\,(K\,km\,s^{-1}pc^{2})^{-1}$ is valid for all sources within $\sigma\sim0.36\,dex$.
Conclusions {#Section35}
-----------
We use [*[Herschel]{}*]{} SPIRE FTS observations of 167 local galaxies, including mapping data, to determine $L'_{CO}$ $J=4-3$ to $12-11$, and derive corresponding beam-matched $L_{FIR,b}$ for each CO line using PACS photometry. We find that these CO–FIR correlations are all linear, and that the non-linear result reported in can be attributed to the comparatively small number of galaxies in their sample. The overall linearity suggests that a universal CO/FIR SLED exists among these galaxies, which further reveals two excitation states of gas. The high-excitation CO SLED peaks at $J\sim7$, where the transitions are the best tracers of SFR [e.g. @Lu2014].
At high-[*[z]{}*]{}, however, SMGs/QSOs have elevated FIR/CO(5–4) ratios relative to those seen in local galaxies and contemporary BzKs: this is because they are the most extreme types of galaxies and hence are not representative of the typical SFG population. A tight, linear correlation between CO(10–9)–FIR is shown to hold for the majority of local galaxies, resolved sub-kpc regions, Galactic YSOs, and also high-[*[z]{}*]{} galaxies. These results strongly support a fundamental linear relationship between dense gas and the SFR, and also provide the local benchmark for probing the gas and SF in more types of galaxies (e.g. MS galaxies) at high-[*[z]{}*]{}.
This work is supported by NSFC \#11173059, \#11390373, \#11420101002, and CAS \#XDB09000000. D. L. gratefully thanks T. Greve, Z. Zhang, P. Papadopoulos, S. Madden, and R. Wu for constructive discussions, and K. Okumura and B. Altieri for helpful instructions on PACS.
, A., [Vlahakis]{}, C., [Hunter]{}, T. R., et al. 2015, , 808, L4
Aniano, G., Draine, B. T., Gordon, K. D., Sandstrom, K., 2011, , 123, 1218
Bayet, E., Gerin, M., Phillips, T. G., Contursi, A., 2009, , 399, 264
Bigiel, F., Leroy, A., Walter, F., et al. 2008, , 136, 2846
Bothwell, M., Smail, I., Chapman, S., et al. 2013, , 429, 3047
Carilli, C. L., Walter, F., 2013, , 51, 105 (CW13)
Daddi, E., Elbaz, D., Walter, F., et al. 2010, , 714, L118
Daddi, E., Dannerbauer, H., Liu, D., et al. 2015, , 577, A46 Galametz, M., Kennicutt, R. C., Calzetti, D., et al. 2013, , 431, 1956
Gao, Y., Solomon, P. M., 1999, , 512, L99
Gao, Y., Solomon, P. M., 2004, , 606, 271
Gao, Y., Carilli, C. L., Solomon, P. M., Vanden Bout, P. A., 2007, , 660, L93
Greve, T. R., Bertoldi, F., Smail, I., et al. 2005, , 359, 1165
Greve, T. R., Leonidaki, I., Xilouris, E. M., et al., 2014, ApJ, 794, 142 (G14)
Griffin, M. J., Abergel, A., Abreu, A., et al. 2010, , 518, L3
Iono, D., Wilson, C. D., Yun, M. S., et al. 2009, , 695, 1537
Juneau, S., Narayanan, D. T., Moustakas, J., et al. 2009, , 707, 1217
Kennicutt, R. C., 1998, , 498, 541 Kennicutt, R. C. Jr., Evans, N. J., 2012, , 50, 531
Krumholz, M. R., McKee, C. F., 2005, , 630, 250
Krumholz, M. R., Thompson, T. A., 2007, , 669, 289
Kelly, B. C., 2007, , 665, 1489
Liu, L., Gao, Y., 2012, ScChG, 55, 347
Liu, L., Gao, Y., Greve, T. R., 2015, , 805, 31
Lu, N., Zhao, Y., Xu, C. K., et al. 2014, , 787, L23
, M. M., [Klessen]{}, R. S. 2004, Rev. Mod. Phys., 76, 125
Mao, R., Schulz, A., Henkel, C., et al. 2010, , 724, 1336
Markwardt, C. B., 2009, ASPC, 411, 251
Narayanan, D., Cox, T. J., Shirley, Y., Dav[é]{}, R., Hernquist, L., Walker, C. K., 2008, , 684, 996
Naylor, D. A., Baluteau, J. P., Barlow, M. J., et al. 2010, SPIE, 7731, E29
Ott, S. 2010, ASPC, 434, 139
Pineda, J. L., Langer, W. D., Goldsmith, P. F., 2014, , 570, 121
Poglitsch, A., Waelkens, C., Geis, N. et al. 2010, , 518, L2
, M., [Van der Werf]{}, P., [Aalto]{}, S., et al., 2015, , 801, 72
Roussel, H., 2013, , 125, 1126
Sanders, D. B., Mazzarella, J. M., Kim, D. C., Surace, J. A., Soifer, B. T., 2003, , 126, 1607
San Jos[é]{}-Garc[í]{}a, I., Mottram, J. C., Kristensen, L. E., et al. 2013, , 553, A125
Schruba, A., 2013, IAUS, 292, 311
, P. M., [Downes]{}, D., [Radford]{}, S. J. E., 1992, , 398, L29
Springel, V., Hernquist, L., 2003, , 339, 289
Van der Werf, P. P., Isaak, K. G., Meijerink, R., et al. 2010, , 518, L42
Van der Tak, F. S., Black, J. H., Schöier, F. L., et al. 2007, , 468, 627
, M. and [Genel]{}, S. and [Springel]{}, V., et al. 2014, , 444, 1518
Wu, J., Evans, N., Gao, Y., Solomon, P. M., Shirley, Y. L., Vanden Bout, P. A., 2005, , 635, L173
Wu, J., Evans, N., Shirley, Y. L., Knez, C., 2010, , 188, 313
Yang, C., Gao, Y., Omont, A., et al. 2013, , 771, L24
Zhao, Y., Lu, N., Xu, C. K., et al. 2013, , 765, L13
Zhang, Z., Gao, Y., Henkel, C., et al. 2014, , 784, L31 Papadopoulos, P. P., Van der Werf, P., Xilouris, E. M., Isaak, K. G., Gao, Y., Muehle, S., 2012, , 426, 2601
[^1]: Figure 5.18 of http://herschel.esac.esa.int/Docs/SPIRE/spire\_handbook.pdf
[^2]: All PACS data are updated to calibration 48 and post-processed with [*Scanamorphos v19.0*]{} [@Roussel2013].
[^3]: PACS bands: 70, 100, and 160${\mu}m$. $\lambda_{peak}\approx{290\,{\mu}m}/{T_{dust}}$, where $T_{\mathrm{dust}}\sim20-30\,K$ for typical local SFGs.
[^4]: The $L_{\mathrm{FIR}}$ for nine galaxies are derived from IR SEDs, a further 14 from FIR-radio correlation , and the remaining 37 scaled from single-band $850\mu{m}$ or $1200\mu{m}$ . Where required we converted $L_{\mathrm{TIR}}$ to $L_{\mathrm{FIR}}$ by a factor of $1/1.3$.
[^5]: For example, $850\mu{m}$-derived $L_{\mathrm{FIR}}$ (using conversion factor) are $\sim77-200\%$ of radio-derived $L_{\mathrm{FIR}}$ in [@Bothwell2013]. Lensing corrections also introduce uncertainties.
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---
abstract: 'We show that the maximum transmission distance of continuous-variable quantum key distribution in presence of a Gaussian noisy lossy channel can be arbitrarily increased using a linear noiseless amplifier. We explicitly consider a protocol using amplitude and phase modulated coherent states with reverse reconciliation. We find that a noiseless amplifier with amplitude gain $g$ can increase the maximum admissible losses by a factor $g^{-2}$.'
author:
- Rémi Blandino
- Anthony Leverrier
- Marco Barbieri
- Jean Etesse
- Philippe Grangier
- 'Rosa Tualle-Brouri'
title: 'Improving the maximum transmission distance of continuous-variable quantum key distribution using a noiseless amplifier'
---
\
Introduction
============
Cryptography is certainly one of the most advanced applications of quantum technologies. Within this field, the most studied primitive is quantum key distribution (QKD), which is the art of distributing a secret key to two distant parties, Alice and Bob, in an untrusted environment controlled by an adversary, Eve [@SBC08]. The security of QKD lies on the idea that an adversary trying to acquire some information about the secret key will necessarily introduce some noise in the quantum communication between Alice and Bob. A consequence of this idea is that if the quantum channel is too lossy or noisy, then it cannot be used to distill a secret key. This limits the maximum transmission distance between the legitimate parties. Developing QKD protocols resistant to losses and noise is therefore of great practical importance.
Among QKD protocols, those encoding information in the amplitude and phase of coherent states [@grosshans_continuous_2002] have the advantage of only requiring off-the-shelf telecom components, as well as being compatible with wavelength-division multiplexing [@qi2010feasibility], making an interesting solution for robust implementations [@cerf_quantum_2007; @weedbrook_gaussian_2012].
On the theoretical side, these continuous-variable (CV) protocols have been proven secure against arbitrary attacks provided that they are secure against collective attacks [@RC09]. This latter condition is in particular met for all CV protocols without postselection for which Gaussian attacks are known to be optimal within collective attacks [@GC06; @NGA06; @PBL08; @leverrier_simple_2010].
Protocols with postselection on the other hand [@SRL02; @LKL04], where Alice and Bob only use part of their data to extract a secret key, can increase the robustness of QKD to losses and noise but at the price of more involved security proofs. In particular, their security is only established against Gaussian attacks [@HL06; @HL07], or when an active symmetrization of the classical data is applied [@PhysRevA.85.022339].
In this paper, we consider the use of a noiseless linear amplifier (NLA) [@ralph_nondeterministic_2008; @ferreyrol_implementation_2010; @ferreyrol_experimental_2011; @barbieri_nondeterministic_2011; @zavatta_high-fidelity_2011; @xiang_heralded_2010] on the detection stage as a way to increase the robustness of CV QKD protocols against losses and noise. First, it should be noted that while amplifiers can effectively recover classical signals, they only offer limited advantages when working on quantum signals, as amplification is bound to preserve the original signal to noise ratio (SNR) [@caves_quantum_1982; @levenson_reduction_1993; @ferreyrol_experimental_2011]. This implies that ordinary linear amplifiers, as those realized by optical parametric processes [@loudon_quantum_2000], can only find limited applications in the context of QKD [@fossier_improvement_2009].
On the other hand, a *probabilistic* NLA can in principle amplify the amplitude of a coherent state while retaining the initial level of noise [@ralph_nondeterministic_2008]. Thus, when only considering its successful runs, the NLA can compensate the effect of losses and could therefore be useful for quantum communication [@ralph_quantum_2011], and to establish the nonlocal nature of quantum correlations thanks to a loophole-free Bell test [@PhysRevA.85.042116]. The availability of such a device has stimulated intense experimental activity over the past years, demonstrating the implementation of approximated versions [@ferreyrol_implementation_2010; @zavatta_high-fidelity_2011; @xiang_heralded_2010; @ferreyrol_experimental_2011; @barbieri_nondeterministic_2011], which have provided solid proof-of-principle.
The question arises if these more sophisticated devices can deliver a compensation of losses with a success rate such that it may represent a useful tool for quantum cryptography. Here we address this problem, by investigating the advantages and limitations of the most general NLA device, without making assumptions on the particular realization.
We find a regime in which the NLA leads to an improvement of the maximum transmission distance attainable on a noisy and lossy Gaussian channel. Because of the non-deterministic nature of the NLA, the security proofs considered here are similar to those concerning protocols with postselection, that is, they hold against Gaussian attacks, or collective attacks provided an additional symmetrization of the classical data is performed.
Description of the GG02 protocol
================================
We consider explicitly the case for the most common protocol for continuous-variable QKD, designed by Grosshans and Grangier (GG02) [@grosshans_continuous_2002]. In its prepare-and-measure (PM) version, Alice encodes information in the quadratures of coherent states which are then sent to Bob through the untrusted quantum channel. Alice chooses her preparation ${\vert\alpha{=}x_\mathrm{A}{+}ip_\mathrm{A}\rangle}$ from a Gaussian distribution for the two quadratures having zero mean and variance $V_\mathrm{A}$. Bob randomly decides whether to measure the $\hat{x}$ or the $\hat{p}$ quadrature, using homodyne detection. Alice and Bob finally extract a secret key from the correlated data by performing classical data manipulation and authenticated classical communication. This protocol offers a simple experimental implementation [@grosshans_quantum_2003; @LBG07; @fossier_field_2009; @jouguet2012field] and is secure against finite-size collective attacks [@LGG10] as well as arbitrary attacks in the asymptotic limit of arbitrary long keys [@RC09].
This protocol can be reformulated in an entanglement-based version (EB), in terms of entanglement distribution between Alice and Bob [@grosshans_virtual_2003]: the two parties initially share a two-mode squeezed vacuum state ${\vert\lambda\rangle}{=}\sqrt{1-\lambda^2}\sum_{n{=}0}^{\infty}\lambda^n{\vertn\rangle}{\vertn\rangle}$, with $\lambda{<}1$. Alice performs an measurement on her mode, which projects the other mode on a coherent state. The outcome of Alice’s measurement is random, but with a probability distribution depending on $\lambda$.
Although the EB version does not correspond to the actual implementation, it is fully equivalent to the PM version from a security point of view, and it provides a more powerful description for establishing security proofs against collective attacks through the covariance matrix $\gamma_\mathrm{AB}$ of the state shared by Alice and Bob before their respective measurements. In the case of a Gaussian channel with transmittance $T$, and input equivalent excess noise $\epsilon$ [@weedbrook_gaussian_2012]: $$\begin{aligned}
\gamma_\mathrm{AB}= \left(
\begin{array}{cc}
V(\lambda) \mathbb{I} &\sqrt{T(V(\lambda)^2-1)} \mathbb{Z} \\
\sqrt{T(V(\lambda)^2-1)} \mathbb{Z} & T(V(\lambda)+B+\epsilon) \mathbb{I}
\end{array} \right)
\label{CM}\end{aligned}$$ where $\mathbb{I}{=}\operatorname{diag}(1,1)$ and $\mathbb{Z}{=}\operatorname{diag}(1,{-}1)$, $V(\lambda){=}\frac{1{+}\lambda^2}{1{-}\lambda^2}$ is the variance of the thermal state $\operatorname{Tr}_\mathrm{A}{\vert\lambda\rangle}{\langle\lambda\vert}$ related to the modulation variance by $V_\mathrm{A}{=}V{-}1$, and $B{=}\frac{1-T}{T}$ is the input equivalent noise due to losses.
This matrix contains all the information needed to establish the secret key rate for collective attacks [@LBG07]: $$\begin{aligned}
\Delta I(\lambda,T,\epsilon,\beta){=}\beta I_\mathrm{AB}(\lambda,T,\epsilon)-\chi_\mathrm{BE}(\lambda,T,\epsilon),
\label{secretKey}\end{aligned}$$ where $I_\mathrm{AB}{=}\frac{1}{2}\log_{2}(\frac{V{+}B{+}\epsilon}{1{+}B{+}\epsilon})$ is the mutual information shared by Alice and Bob given by Shannon’s theory [@shannon_mathematical_1948], and $\chi_\mathrm{BE}$ is the Holevo bound for the mutual information shared by Eve and Bob (see Appendix \[appendixHolevo\]). The reconciliation efficiency $\beta{<}1$ accounts for the fact that in practical implementations of this protocol, Alice and Bob do not have sufficient resources to reach the Shannon limit. Steady progress has been made in recent years on the problem of error correction for CV QKD [@bloch_ldpc-based_2006; @LAB08; @jouguet2012high] and today procedures based on modern error correcting techniques achieve $\beta \approx95 \%$ for a large range of SNR [@JKL11].
Equivalent channel and squeezing {#equivalentChannel}
================================
Let us now consider the use of a NLA in the GG02 protocol. As usual, we will perform the security analysis of the EB version. Here, we restrict ourselves to the case of a Gaussian quantum channel, that is Eve is limited to performing Gaussian attacks. Since the secure key rate of the protocol depends only on the covariance matrix of Alice and Bob $\gamma_\mathrm{AB}$, it is sufficient to compute it in presence of the NLA.
In this modified version of the protocol, Alice and Bob implement GG02 as usual but Bob adds a NLA to his detection stage, before his homodyne detection, which is here assumed to be perfect. Then, only the events corresponding to a successful amplification will be used to extract a secret key. This scheme is therefore very similar to protocols with postselection.
Since the output of the NLA remains in the Gaussian regime, we can look for equivalent parameters of an EPR state sent through a Gaussian noisy channel. Their derivation is explained in detail in Appendix \[appendixEffectiveParameters\], where it is shown that the covariance matrix $\gamma_\mathrm{AB}(\lambda,T,\epsilon,g)$ of the amplified state is equal to the covariance matrix $\gamma_\mathrm{AB}(\zeta,\eta,\epsilon^g,g{=}1)$ of an equivalent system with an EPR parameter $\zeta$, sent through a channel of transmittance $\eta$ and excess noise $\epsilon^g$, without using the NLA. Those effective parameters are given by: $$\addtolength{\fboxsep}{4pt}
\begin{split}
\zeta&=\lambda \sqrt{\frac{\left(g^2-1\right) \left(\epsilon-2\right) T-2}{\left(g^2-1\right)
\epsilon T-2}}, \\
\eta&=\frac{g^2 T}{\left(g^2-1\right) T \left(\frac{1}{4} \left(g^2-1\right) \left(\epsilon-2\right) \epsilon T-\epsilon+1\right)+1} , \\
\epsilon^g&=\epsilon-\frac{1}{2} \left(g^2-1\right) \left(\epsilon-2\right) \epsilon T.
\label{parametresEffectifsMain}
\end{split}$$
![Equivalent channel and squeezing: a state ${\vert\lambda\rangle}$ sent through a Gaussian channel of transmittance $T$ and excess noise $\epsilon$, followed by a successful amplification, has the same Alice-Bob covariance matrix than a state ${\vert\zeta\rangle}$ sent through a Gaussian channel of transmittance $\eta$ and excess noise $\epsilon^g$, without the NLA.[]{data-label="schema"}](schemaEBPM4.pdf){width="\columnwidth"}
This identification easily provides the secret information $\Delta I ^g$ corresponding to the successful amplification, since Eq. \[secretKey\] can be used with the effective parameters: $$\begin{aligned}
\Delta I ^g (\lambda, T,\epsilon,\beta)= \Delta I(\zeta,\eta,\epsilon^g,\beta).
\label{deltaMaxNLA}
\end{aligned}$$
Those parameters can be interpreted as physical parameters of an equivalent system if they satisfy the physical meaning constraints $0{\leq}\zeta{<}1$, $0{\leq}\eta{\leq}1$, and $\epsilon^g{\geq}0$. Since $\lambda$ is a global factor in the expression of $\zeta$, the first condition is always satisfied if $\lambda$ is below a limit value: $$\begin{aligned}
0\leq\zeta < 1 \Rightarrow 0<\lambda < \left(\sqrt{\frac{\left(g^2{-}1\right) \left(\epsilon{-}2\right) T{-}2}{\left(g^2{-}1\right)
\epsilon T-2}}\right)^{{-}1}.\end{aligned}$$ As $\eta$ and $\epsilon^g$ do not depend on $\lambda$, the parameter $\zeta$ can be considered as independent of those two parameters, keeping in mind that this simply sets the value of $\lambda$.
The second and the third conditions are satisfied if the excess noise is smaller than 2, and if the gain is smaller than a maximum value given by Eq. \[gainMax\], and plotted on Fig. \[gMax\].
![$g_\mathrm{max}(T,\epsilon)$ against the losses in dB. $\epsilon{=}0.2$.[]{data-label="gMax"}](gMax.pdf){width="6.5cm"}
Increase of the maximum transmission distance
=============================================
The analysis of the equivalent state allows us to compare the secret key rate obtained with and without an ideal NLA. The comparison must be performed for a given channel with fixed losses $T$ and excess noise $\epsilon$, as those parameters cannot be controlled by Alice or Bob. However, since the relevant quantity is the maximum secret key rate achievable over this channel, Alice is allowed to optimize her modulation variance $V_\mathrm{A}$ (or equivalently, the parameter $\lambda$) in order to maximize the secret key rate.
The secret key rate without the NLA is given by $\Delta I(\lambda,T,\epsilon,\beta)$ (Eq. \[secretKey\]). The secret key rate with the NLA $\Delta I_\mathrm{NLA}$ is obtained by multiplying the secret key rate for successful amplifications $\Delta I^g$ by the probability of success $P_\mathrm{suc}$. If the NLA has a sufficient dynamics to neglect distortions, we can assume that $P_\mathrm{suc}$ is constant. This is a reasonable assumption if $\beta{<}1$, since in that case the optimal value of $V_\mathrm{A}$ is not infinite. The precise value of $P_\mathrm{suc}$ will depend on practical implementations, and is not important is our study, since it only acts as a scaling factor and does not change the fact that a negative secret key rate can become positive with a NLA. Therefore, $$\begin{aligned}
\Delta I_\mathrm{NLA}=P_\mathrm{suc}\Delta I(\zeta,\eta,\epsilon^g).\end{aligned}$$
In Appendix \[appendixProba\], we show that the probability of success for a NLA of gain $g$ is upper bounded by $1/g^2$. We can therefore use this bound, keeping in mind that the relevant conclusion which can be taken is only whether the secret key rate is positive or not. Both secret key rate with and without the NLA are computed using the formulae given in Appendix \[appendixHolevo\].
Since the expression of $\Delta I$ is relatively difficult to manipulate, we perform a series expansion at the first order in $T$, which corresponds to the strong losses regime (Appendix \[appendixDLHolevo\]). The approximate secret key rate is given by Eq. \[dlHolevo\]. Its expression gives us an intuition about two important behaviors: first, since $T$ appears inside the expansion and not only as a global factor, it explains why there can be a maximum transmission distance, or equivalently a value $T_\mathrm{lim}$ for which the secret key rate becomes null. Second, in this regime, the effect of the NLA is simply to replace the transmittance $T$ by $g^2 T$, the other physical parameters being the same. Hence, it is clear that the losses are reduced, which will increase the maximum distance of transmission.
Let us prove those statements more precisely. From Eq. \[dlHolevo\], we find an analytical value of $T_\mathrm{lim}$ when $g{=}1$ (*i.e* without the NLA), $$\begin{aligned}
&T_\mathrm{lim}=\frac{1}{\epsilon}2 \lambda ^{-\frac{4 \lambda ^4}{\epsilon \left(\lambda ^2-1\right)^2}}
e^{\frac{\lambda ^2 (2 \beta +\epsilon )-\epsilon }{\epsilon \left(\lambda
^2-1\right)}}.
\label{Tlim}\end{aligned}$$ This expression clearly tends to 0 when $\epsilon$ tends to 0, which shows that there is no maximum transmission distance without excess noise. Interestingly, there is a maximum transmission distance as soon as the excess noise $\epsilon$ is non zero, even if the reconciliation efficiency $\beta$ equals 1. When $\beta$ decreases or when $\epsilon$ increases, this maximum transmission distance decreases. There is no limitation of the distance of transmission only when $\epsilon{=}0$, and in that case Eq. \[dlHolevo\] takes a simple form: $$\begin{aligned}
\Delta I_\mathrm{NLA}&{\simeq}\frac{1}{g^2} g^2 T \lambda ^2 \frac{ \left(1{-}\lambda ^2\right) (\beta {-}2 \log \lambda
){+}2 \log \lambda}{\left(\lambda ^2{-}1\right)^2 \log (2)}{\simeq}\Delta I.
\label{tauxNoNoise}\end{aligned}$$ This shows that for strong losses without excess noise, the secret key rate using the NLA with the most optimistic probability of success is the same as the secret key rate without the NLA, and is always positive if $\lambda$ is optimized. $T_\mathrm{lim}$ can also be optimized (*i.e.* minimized) by optimizing $\lambda$. Interestingly, the optimal value $\lambda_\mathrm{opt}$ depends only on $\beta$, as shown by Eq. \[betaOpt\].
The same calculation with a NLA of gain $g$ shows that: $$\begin{aligned}
T_\mathrm{lim}^g=\frac{1}{g^2}T_\mathrm{lim}.\end{aligned}$$ Therefore, the losses for which the secret key rate is zero are increased by: $$\begin{aligned}
\Delta \mathfrak{L}=20 \log_{10} g \text{ dB}. \end{aligned}$$
Let us stress that this result does not depend on the probability of success of the NLA, which simply acts as a scaling factor for the secret key rate. Hence, even for a more realistic probability of success, the NLA increases the maximum distance of transmission in the same way.
Those results are compared with numerical results for the full expressions of $\Delta I$ and $\Delta I_\mathrm{NLA}$, on Fig. \[schemaEBPM3\] and \[contourPlot\]. For both figures, the secret key rate is computed without the NLA and with a NLA of gain $g{=}4$ (which is in the allowed region of Fig. \[gMax\]).
![Maximized secret key rate against losses in dB. The maximization is performed on $\lambda$ for the series expansion, and on $\zeta$ for the the numerical expression. The numerical curves are in excellent agreement with the analytical expansions. As explained in the main text, the secret key rate with the NLA is very optimistic due to the probability of success $1/g^2$, and hence its curve gives only information on its positivity. The other parameters are $\epsilon{=}0.05$, $\beta{=}0.95$ [@JKL11].[]{data-label="schemaEBPM3"}](tauxSecret.pdf){width="\columnwidth"}
![Maximal excess noise for which the secret key rate is positive, against losses. The curves do not depend on the probability of success chosen for the NLA. The maximization is performed on $\lambda$ for the series expansion, and on $\zeta$ for the numerical expression. For low losses, we see that the first order expansion is not enough, whereas it is in excellent agreement with the numerical curve for strong losses. The reconciliation efficiency is $\beta{=}0.95$ [@JKL11].[]{data-label="contourPlot"}](contourPlot.pdf){width="\columnwidth"}
Those figures clearly show that the secret key rate stays positive for losses increased by $\Delta\mathfrak{L}{=}12$ dB. Fig. \[contourPlot\] also shows that for given losses, the secret key rate stays positive for a higher value of excess noise. However, the gain in excess noise depends on the losses, and does not have a simple analytical expression.
Another important remark concerns the optimal gain. If the transmission can be intuitively increased by increasing the gain, this is not always the case for the secret key rate, as shown on Fig. \[keyRateVSgain\]. The first reason is the competition between the decreasing probability of success $1/g^2$ and the potential increase of the secret key rate for the successfully amplified states. The second reason is due to the dependance on the gain of the effective parameters (Eq. \[parametresEffectifsMain\]): the higher the gain, the higher $\eta$, but also the higher $\epsilon^g$. If the gain is too high, it is thus possible that the effective excess noise $\epsilon^g$ would be too important, for the transmittance $\eta$, to give a positive $\Delta I_\mathrm{NLA}$.
\
![Maximized secret key rate against the gain of the NLA, with a probability of success $1/g^2$. $\beta{=}1$, $\epsilon{=}0.1$, losses${=}$30 dB. With a gain $g{=}1$, the secret key rate is negative. The NLA can increase the secret key rate to positive values when the gain is increased up to a certain value, however if the gain is too important the secret key rate decreases and becomes negative again. The reason is that the effective excess noise becomes too important for the effective transmittance.[]{data-label="keyRateVSgain"}](tauxVSgain.pdf){width="8cm"}
Discussion and conclusion
=========================
In presence of excess noise, the secret key rate of the GG02 protocol against Gaussian collective attacks always becomes negative for a certain distance of transmission. We have shown that the noiseless linear amplifier can increase this distance by the equivalent of $20 \log_{10}g$ dB of losses. We have also shown that for given losses, the protocol is more robust against excess noise.
Our calculation of the secret key rate with the amplifier was based on an effective system for which the security proofs are well established. This approach could also find applications in other quantum communication protocols involving an EPR state sent through a quantum channel, followed by a noiseless amplifier. In particular, it could be applied to other CV QKD protocols, for instance protocols using squeezed states or with a heterodyne detection [@WLB04; @LSS05; @WLB06; @PhysRevLett.102.130501]
A further work would be to consider the experimentally demonstrated schemes of the NLA, which are only valid approximations of the ideal NLA up to a certain number of photons. If the state can be well approximated by this truncation, so that the Gaussian approximation still holds, the results presented in this paper are still valid. On the other hand, if the Gaussian approximation does not hold anymore, security proofs are more complicated to manipulate. This problem lies beyond the scope of the present work, and deserves further investigation.
We acknowledge support from the EU project ANR ERA-Net CHISTERA HIPERCOM. MB is supported by the Marie Curie contract PIEF-GA-2009-236345-PROMETEO.
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Derivation of the effective parameters {#appendixEffectiveParameters}
======================================
In this appendix, we detail the method used to obtain the effective parameters of Section \[equivalentChannel\]. Let us start by first computing the output of the NLA when the input state ${\mbox{\boldmath $\hat{\rho}$}}$ is a thermal state ${\mbox{\boldmath $\hat{\rho}$}}_\mathrm{th}(\lambda_\mathrm{ch}){=}(1{-}\lambda_\mathrm{ch}^2)\sum_{n{=}0}^{\infty}\lambda_\mathrm{ch}^{2n}{\vertn\rangle}{\langlen\vert}$ displaced by $\beta{=}\beta_\mathrm{x}{+}i\beta_\mathrm{y}$: $$\begin{aligned}
{\mbox{\boldmath $\hat{\rho}$}}{=}{\mbox{\boldmath $\hat{D}$}}(\beta){\mbox{\boldmath $\hat{\rho}$}}_\mathrm{th}(\lambda_\mathrm{ch}){\mbox{\boldmath $\hat{D}$}}({-}\beta) .\end{aligned}$$ This would be the state received by Bob if he knew the result of Alice’s heterodyne measurement. The state ${\mbox{\boldmath $\hat{\rho}$}}$ can be decomposed on an ensemble of coherent states using the $P$ function: $$\begin{aligned}
{\mbox{\boldmath $\hat{\rho}$}}=\int P(\alpha){\vert\alpha\rangle}{\langle\alpha\vert} \mathrm{d}\alpha.
\label{displThermState}
\end{aligned}$$ where $P(\alpha){=}\frac{e^{\vert\alpha \vert^2}}{\pi^2}\int e^{\vert u \vert^2} {\langle{-}u\vert}{\mbox{\boldmath $\hat{\rho}$}}{\vertu\rangle}e^{u^*\alpha{-}u \alpha^*} \mathrm{d}u$. Straightforward calculations show that ${\langle{-}u{-}\beta\vert}{\mbox{\boldmath $\hat{\rho}$}}_\mathrm{th}(\lambda_\mathrm{ch}){\vertu{-}\beta\rangle}=\\ (1{-}\lambda_\mathrm{ch}^2)e^{{-}\vert u \vert^2(1{+}\lambda_\mathrm{ch}^2){-}\vert \beta \vert ^2(1{-}\lambda_\mathrm{ch}^2){+}(u\beta^*{-}u^*\beta)(1{-}\lambda_\mathrm{ch}^2)}$, and therefore $P(\alpha_\mathrm{x}{+}i\alpha_\mathrm{y}){=}p(\alpha_\mathrm{x}) p(\alpha_\mathrm{y})$, with $$\begin{aligned}
p(\alpha_\mathrm{x})=\frac{1}{\sqrt{\pi}}\sqrt{\frac{1-\lambda_\mathrm{ch}^2}{\lambda_\mathrm{ch}^2}}e^{-\frac{1{-}\lambda_\mathrm{ch}^2}{\lambda_\mathrm{ch}^2}(\alpha_\mathrm{x}-\beta_\mathrm{x})^2}.
\label{pFunc}
\end{aligned}$$ In the absence of thermal noise ($\lambda_\mathrm{ch}{=}0$), the expression (\[pFunc\]) becomes proportional to a Dirac distribution $\delta(\alpha_\mathrm{x}{-}\beta_ \mathrm{x})$. The same statements hold for $p(\alpha_\mathrm{y})$.
The successful amplification can ideally be described by an operator ${\mbox{\boldmath $\hat{C}$}}{=}g^{\hat{n}}$, where $\hat{n}$ is the number operator in the Fock basis. The final state has to be normalized, but one has to be careful that the norm is not the success probability of the transformation, since ${\mbox{\boldmath $\hat{C}$}}$ is unbounded. The amplification of a coherent state ${\vert\alpha\rangle}$ leads to an amplified coherent state proportional to ${\vertg\alpha\rangle}$: $$\begin{aligned}
{\mbox{\boldmath $\hat{C}$}}{\vert\alpha\rangle}{=}e^{\frac{\vert \alpha \vert ^2}{2}(g^2{-}1)}{\vertg\alpha\rangle}.
\label{cohState}
\end{aligned}$$ Since ${\mbox{\boldmath $\hat{C}$}}$ is linear, the amplification of ${\mbox{\boldmath $\hat{\rho}$}}$ is simple to derive, using (\[pFunc\]) and (\[cohState\]) in the decomposition (\[displThermState\]): $$\begin{aligned}
{\mbox{\boldmath $\hat{\rho}$}}'&={\mbox{\boldmath $\hat{C}$}}{\mbox{\boldmath $\hat{\rho}$}}{\mbox{\boldmath $\hat{C}$}}\\
&=\int P(\alpha)e^{\vert \alpha \vert ^2(g^2-1)}{\vertg\alpha\rangle}{\langleg\alpha\vert}\mathrm{d}\alpha.\end{aligned}$$ By introducing the change of variable $u{=}g \alpha$, one gets $$\begin{aligned}
{\mbox{\boldmath $\hat{\rho}$}}'=\int P(u/g)e^{\frac{g^2-1}{g^2}\vert u^2 \vert}{\vertu\rangle}{\langleu\vert}\mathrm{d}u.\end{aligned}$$ As before, it is easy to see that $P(u/g){=}p(u_\mathrm{x}/g)p(u_\mathrm{y}/g)$. Since $\vert u^2 \vert{=}u_\mathrm{x}^2{+}u_\mathrm{y}^2$, we can consider only the term $p(u_\mathrm{x}/g)\exp\left(\frac{g^2-1}{g^2}u_\mathrm{x}^2\right)$, the results being similar for $u_\mathrm{y}$: $$\begin{aligned}
p(u_\mathrm{x}/g)e^{\frac{g^2{-}1}{g^2}u_\mathrm{x}^2}&{=}\frac{1}{\sqrt{\pi}}\sqrt{\frac{1{-}\lambda_\mathrm{ch}^2}{\lambda_\mathrm{ch}^2}}e^{{-}\frac{1{-}\lambda_\mathrm{ch}^2}{\lambda_\mathrm{ch}^2}(\frac{u_\mathrm{x}}{g}{-}\beta_\mathrm{x})^2+\frac{g^2{-}1}{g^2}u_\mathrm{x}^2}.\end{aligned}$$
The argument of the exponential can easily put in the form: $$\begin{aligned}
&{-}\frac{1{-}\lambda_\mathrm{ch}^2}{\lambda_\mathrm{ch}^2}\Big(\frac{u}{g}{-}\beta_\mathrm{x}\Big)^2{+}\frac{g^2{-}1}{g^2}u^2= \nonumber \\
&\underbrace{{-}\frac{1{-}g^2\lambda_\mathrm{ch}^2}{g^2\lambda_\mathrm{ch}^2}}_{\substack{\text{Thermal state $g\lambda_\mathrm{ch}$}}}\Big(u-\beta_\mathrm{x} \underbrace{g\frac{1{-}\lambda_\mathrm{ch}^2}{1{-}g^2\lambda_\mathrm{ch}^2}}_{\text{Effective gain}}\Big)^2{-}\underbrace{\beta_\mathrm{x}^2\frac{(1{-}g^2)(1{-}\lambda_\mathrm{ch}^2)}{1{-}g^2\lambda_\mathrm{ch}^2}}_{\substack{\text{Normalization term}}}.
\label{pEff}\end{aligned}$$ Thus, the expression (\[pEff\]) clearly corresponds to a thermal state ${\mbox{\boldmath $\hat{\rho}$}}_\mathrm{th}(g\lambda_\mathrm{ch})$ displaced by $g\frac{1-\lambda_\mathrm{ch}^2}{1-g^2\lambda_\mathrm{ch}^2}\beta$, up to a global unimportant normalization factor independent of the variable integrated $\alpha$ or $u$. We can conclude that: $$\begin{aligned}
{\mbox{\boldmath $\hat{\rho}$}}'\propto {\mbox{\boldmath $\hat{D}$}}(\tilde{g}\beta){\mbox{\boldmath $\hat{\rho}$}}_\mathrm{th}(g\lambda_\mathrm{ch}){\mbox{\boldmath $\hat{D}$}}(-\tilde{g}\beta).
\label{NLAdispTherm}\end{aligned}$$ where $\tilde{g}=g\frac{1-\lambda_\mathrm{ch}^2}{1-g^2\lambda_\mathrm{ch}^2}$. In order to keep a physical interpretation, we note that $g$ must be such that $g\lambda_\mathrm{ch}{<}1$.
Let us now find the values of $\beta$ and $\lambda_\mathrm{ch}$ corresponding to the Entanglement-Based protocol presented in the main text. When Alice obtains the results $\alpha_\mathrm{A}$ for her heterodyne measurement on one mode of the EPR state ${\vert\lambda\rangle}$, the second mode is projected on a coherent state with an amplitude proportional to $\lambda \alpha_\mathrm{A}$ [@grosshans_virtual_2003]. This state is then sent through the quantum channel of transmittance $T$, with transforms its amplitude to $\propto \sqrt{T}\lambda \alpha_\mathrm{A}$. The displacement $\beta$ can thus be taken as: $$\begin{aligned}
\beta=\sqrt{T}\lambda \alpha_\mathrm{A}.
\label{betaDis}\end{aligned}$$ The variance $\frac{1{+}\lambda_\mathrm{ch}^2}{1{-}\lambda_\mathrm{ch}^2}$ of the thermal state corresponds to Bob’s variance $1{+}T\epsilon$ when $V_\mathrm{A}{=}0$: $$\begin{aligned}
\frac{1+\lambda_\mathrm{ch}^2}{1-\lambda_\mathrm{ch}^2}=1+T\epsilon \Rightarrow \lambda_\mathrm{ch}^2=\frac{T\epsilon}{2+T\epsilon}.
\label{lambdaCH}\end{aligned}$$ Finally, the action of the NLA (Eq. \[NLAdispTherm\]) on a displaced thermal state given by Eq. \[betaDis\] and \[lambdaCH\] induces the transformations: $$\addtolength{\fboxsep}{4pt}
\begin{split}
\sqrt{T} \lambda \alpha_\mathrm{A} &\underset{\mathrm{NLA}}{\rightarrow} g\frac{1-\lambda_\mathrm{ch}^2}{1-g^2\lambda_\mathrm{ch}^2} \sqrt{T} \lambda \alpha_\mathrm{A}, \\
\frac{T\epsilon}{2+T\epsilon} &\underset{\mathrm{NLA}}{\rightarrow} g^2\frac{T\epsilon}{2+T\epsilon}.
\label{e29}
\end{split}$$ The next step is to consider the action of the NLA when Bob does not have any knowledge on Alice’s measurement outcome. In such a case, his state is a thermal state ${\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}{=}(1{-}\lambda^{\star2})\sum_{n{=}0}^{\infty}(\lambda^\star)^{2n}{\vertn\rangle}{\langlen\vert}$, whose variance is given by $\gamma_\mathrm{AB}$: $$\begin{aligned}
\frac{1{+}\lambda^{\star2}}{1{-}\lambda^{\star2}}=1{+}T V_\mathrm{A}{+}T \epsilon {\Rightarrow} \lambda^{\star2}{=}\frac{T \left(\lambda ^2 \left(2{-}\epsilon\right){+}\epsilon\right)}{2{-}\lambda ^2 \left(2{+}T\left(\epsilon{-}2\right)\right){+}T \epsilon} .
\label{e31}
\end{aligned}$$ Since the NLA always transforms a thermal state of parameter $\lambda^{\star}$ into another thermal state of parameter $g\lambda^{\star}$, Eq. \[e31\] shows that the NLA performs the transformation: $$\begin{aligned}
\frac{T \left(\lambda ^2 \left(2{-}\epsilon\right){+}\epsilon\right)}{2{-}\lambda ^2 \left(2{+}T\left(\epsilon{-}2\right)\right){+}T \epsilon} \underset{\mathrm{NLA}}{\rightarrow} g^2 \frac{T \left(\lambda ^2 \left(2{-}\epsilon\right){+}\epsilon\right)}{2{-}\lambda ^2 \left(2{+}T\left(\epsilon{-}2\right)\right){+}T \epsilon}.
\label{e32}\end{aligned}$$
We have now all the required equations to find the expression of the effective parameters $\zeta$, $\eta$ and $\epsilon^g$. Using Eq. \[e29\] and \[e32\], those parameters must satisfy: $$\begin{aligned}
\begin{split}
\sqrt{\eta} \zeta &{=} g\frac{1{-}\lambda_\mathrm{ch}^2}{1{-}g^2\lambda_\mathrm{ch}^2} \sqrt{T} \lambda, \\
\frac{\eta\epsilon^g}{2{+}\eta\epsilon^g} &{=} g^2\frac{T\epsilon}{2{+}T\epsilon},\\
\frac{\eta \left(\zeta ^2 \left(2{-}\epsilon^g\right){+}\epsilon^g\right)}{2{-}\zeta ^2 \left(2+\eta\left(\epsilon^g{-}2\right)\right){+}\eta \epsilon^g} &{=} g^2 \frac{T \left(\lambda ^2 \left(2{-}\epsilon\right){+}\epsilon\right)}{2{-}\lambda ^2 \left(2{+}T\left(\epsilon{-}2\right)\right){+}T \epsilon}.
\end{split}\end{aligned}$$ This system can be solved, leading to: $$\addtolength{\fboxsep}{4pt}
\begin{split}
\zeta&=\lambda \sqrt{\frac{\left(g^2-1\right) \left(\epsilon-2\right) T-2}{\left(g^2-1\right)
\epsilon T-2}} ,\\
\eta&=\frac{g^2 T}{\left(g^2-1\right) T \left(\frac{1}{4} \left(g^2-1\right) \left(\epsilon-2\right) \epsilon T-\epsilon+1\right)+1} , \\
\epsilon^g&=\epsilon-\frac{1}{2} \left(g^2-1\right) \left(\epsilon-2\right) \epsilon T.
\label{parametresEffectifs}
\end{split}$$
Finally, the expression of the maximum gain $g_\mathrm{max}(T,\epsilon)$ for which those parameters take physical values is given by:
$$\begin{aligned}
&g_\mathrm{max}(T,\epsilon)=\\
&\sqrt{\frac{\epsilon \left(T \left(\epsilon{-}4\right){+}2\right){+}4 \sqrt{\frac{T \left(\epsilon-2\right){+}2}{\epsilon}}{-}2 \sqrt{\epsilon \left(T \left(\epsilon{-}2\right){+}2\right)}{+}4
T{-}4}{T \left(\epsilon{-}2\right){}^2}}.
\label{gainMax}
\end{aligned}$$
\
Let us stress some important comments about those effective parameters, which confirm the validity of their expression. First, they naturally reduce to the real physical parameters without the NLA, for $g{=}1$: $$\begin{aligned}
g=1 \Rightarrow \left\{
\begin{array}{l}
\zeta=\lambda\\
\eta=T\\
\epsilon^g= \epsilon
\end{array} \right.\end{aligned}$$ Then, when there is no excess noise ($\epsilon{=}0$), they match previous results [@ralph_nondeterministic_2008]: $$\begin{aligned}
\epsilon=0 \Rightarrow \left\{
\begin{array}{l}
\zeta=\lambda \sqrt{1+(g^2-1)T} \\
\eta=\frac{g^2T}{1+(g^2-1)T}\\
\epsilon^g= 0
\end{array} \right.\end{aligned}$$
Expressions used to compute the Holevo bound $\chi_{BE}$ {#appendixHolevo}
========================================================
The Holevo bound $\chi_\mathrm{BE}$ is given by [@LBG07]: $\chi_\mathrm{BE}=G\left[ \frac{\mu_1-1}{2} \right]+G\left[ \frac{\mu_2-1}{2} \right]-G\left[ \frac{\mu_3-1}{2} \right]-G\left[ \frac{\mu_4-1}{2} \right]
$ where $$\begin{aligned}
G[x]&=(x+1)\log_2 [x+1]-x\log_2[x] \text{ si x $\neq$0, et } G[0]=0 \\
\mu_{1,2}^2&=\frac{1}{2}\left(A \pm \sqrt{A^2-4E}\right) \qquad \mu_{3,4}^2=\frac{1}{2}\left(C \pm \sqrt{C^2-4D}\right) \\
A&=V^2(1-2T)+2 T +T^2(V+\chi_\mathrm{line})^2 \qquad E=T^2(V \chi_\mathrm{line}+1)^2 \\
C&=\frac{V \sqrt{E}+T(V+\chi_\mathrm{line})}{T(V+\chi_\mathrm{line})} \qquad D=\frac{\sqrt{E}V}{T(V+\chi_\mathrm{line})}\\\end{aligned}$$ $V{=}V_\mathrm{A}{+}1$ is the variance of Alice’s thermal state (see text for details), and $\chi_\mathrm{line}=\frac{1-T}{T}+\epsilon$ is the total equivalent input noise. Bob’s homodyne detection is assumed to be perfect.
First order expansion in T {#appendixDLHolevo}
==========================
The first order expansion in T of the secret key rate given in Appendix \[appendixHolevo\] using the NLA is: $$\begin{aligned}
&\Delta I_\mathrm{NLA} {\simeq} \nonumber\\
&P_\mathrm{suc} g^2 T\frac{ (-2 \beta \lambda^2 (-1 {+} \lambda^2) - \epsilon (-1 {+} \lambda^2)^2 (1 {+}
\log[2]) {+} (-1 {+} \lambda^2) (\epsilon (-1 {+} \lambda^2)(\log[ \epsilon] {+} \log[g^2 T]) {+}
4 \lambda^2 \log[\lambda]) {+}
2 \lambda^2 \log[\lambda^2]) }{(2 (-1 {+} \lambda^2)^2 \log[2]}.
\label{dlHolevo}\end{aligned}$$
The equation that must satisfy the optimal value $\lambda_\mathrm{opt}$ to maximize the transmission distance (Eq. \[Tlim\]), and maximize the secret key rate (Eq. \[dlHolevo\]), is given by : $$\begin{aligned}
\frac{\lambda_\mathrm{opt} ^2 \left(\lambda_\mathrm{opt} ^2-4 \log (\lambda_\mathrm{opt} )-1\right)}{1-\lambda_\mathrm{opt} ^2}=\beta.
\label{betaOpt}\end{aligned}$$
Fig. \[betaVSLambdaOpt\] shows $\lambda_\mathrm{opt}$ as a function of $\beta$:
![$\lambda_\mathrm{opt}$ against $\beta$. $\beta{=}0.95$ gives $\lambda_\mathrm{opt}{\simeq}0.806$.[]{data-label="betaVSLambdaOpt"}](betaVSLambdaOpt.pdf){width="8cm"}
Success probability {#appendixProba}
===================
The success probability of the NLA can depend on many experimental factors. Here, we are interested in deriving an upper bound based on very general principles, when the success probability can be considered as being a constant value. In this way, we can obtain an optimistic estimate of its performance, but certainly we will not overlook interesting regimes. In both EB and PM versions of the GG02 protocol, Bob’s state prior to any classical communication with Alice is the thermal state ${\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(\lambda^{\star}){=}(1{-}\lambda^{\star2})\sum_{n{=}0}^{\infty}(\lambda^\star)^{2n}{\vertn\rangle}{\langlen\vert}$.
Consider now that the NLA produces an amplified state ${\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(g\lambda^{\star})$ with a success probability $P_\mathrm{suc}$. When the amplification fails, the protocol is aborted, and the state is simply replaced by the vacuum ${\vert0\rangle}{\langle0\vert}$. Without post-selection, the NLA can therefore be represented as a trace preserving operation $\mathcal{T}$ described by: $$\begin{aligned}
\mathcal{T}( {\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(\lambda^{\star}))=P_\mathrm{suc} {\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(g\lambda^{\star})+(1-P_\mathrm{suc}){\vert0\rangle}{\langle0\vert}.
\label{NLAT}
\end{aligned}$$ Naturally, $\mathcal{T}$ applied on the vacuum also gives the vacuum, regardless of the value of $P_\mathrm{suc}$. Since any trace preserving quantum operation cannot decrease the fidelity $\mathcal{F}$ between two quantum states [@nielsen_quantum_2000], $\mathcal{T}$ must verify: $$\begin{aligned}
\mathcal{F} \Big({\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(\lambda^{\star}), {\vert0\rangle}{\langle0\vert}\Big)&\leq \mathcal{F} \Big(\mathcal{T}({\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(g\lambda^{\star}), {\vert0\rangle}{\langle0\vert}\Big).
\label{bornFid0}\end{aligned}$$ which gives us an upper bound on $P_\mathrm{suc}$. Indeed, inserting the expression for Bob’s transformed state (Eq. \[NLAT\]) in the constraints on fidelities (Eq. \[bornFid0\]), we find that $P_\mathrm{suc}$ must verify: $$\begin{aligned}
{\langle0\vert} {\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(\lambda^{\star}) {\vert0\rangle}&\leq {\langle0\vert}\Big(P_\mathrm{suc}{\mbox{\boldmath $\hat{\rho}$}}_\mathrm{B}(g\lambda^{\star}){+}(1{-}P_\mathrm{suc}){\mbox{$|0\rangle\langle 0|$}}\Big){\vert0\rangle} .
\label{bornFid}\end{aligned}$$ which is satisfied if: $$\begin{aligned}
P_\mathrm{suc}\leq\frac{1}{g^2}.\end{aligned}$$
|
[**BIDUAL AS A WEAK NONSTANDARD HULL**[^1]]{}
Siu-Ah Ng[^2]
Abstract
Every normed linear space $X$ extends naturally to the bidual $X^{\prime\prime}.\,$ Less well-known among functional analysts is that $X$ also has a natural extension to a nonstandard version ${\,^*\!}X$ using methods from the Nonstandard Analysis. In a sense, any extension of $X$ with respect to some formal properties can always be identified from ${\,^*\!}X.$ The aim of this article is to relate $X^{\prime\prime}$ and ${\,^*\!}X$ and exploit the link between them. The main tool comes from a modification of W.A.J. Luxemburg’s nonstandard hull construction ([@A]).
In §1 we give a very brief summary of the methodology and terminologies from nonstandard analysis. A generalization of the nonstandard hull construction is given. In §2 a representation of the bidual of a standard normed linear space $X$ as the weak nonstandard hull of $X$ is constructed from ${\,^*\!}X.$ In §3 applications of this representation to some sequence spaces are given. In §4 we apply our results to C\*-algebras. Based on the weak nonstandard hull representation of the bidual, we produce a simple nonstandard proof of the Sherman-Takeda Theorem that the bidual of a C\*-algebra forms a von Neumann algebra. In particular, this shows that the weak nonstandard hull of a C\*-algebra is always a von Neumann algebra. Moreover a natural representation is provided for the Arens product(s) on the bidual.
Preliminaries and the general nonstandard hull construction
===========================================================
Background from nonstandard analysis is summarized as follows.
The nonstandard extension of a standard mathematical object $\,X\,$ is denoted by $\,{\,^*\!}X.\,$ (Note the various usages of the star symbol in this article.) The extensions are done simultaneously for all ordinary mathematical objects under consideration and with the preservation of all set theoretical properties among the extensions expressible in the first order logic in the language consisting of the membership symbol. This is referred to as the *Transfer Principle*. In particular we have extensions such as ${\,^*\!}{{\mathbb{N}}},\,$ ${\,^*{\mathbb R}}\,$ and ${\,^*\!}{{\mathbb C}}$ which behave with respect to each other in the same formal manner as ${{\mathbb{N}}},$ ${{\mathbb{R}}}$ and ${{\mathbb C}}.$ Since we regard $X\subset {\,^*\!}X,$ any element $a$ in $X$ is also written as ${\,^*\!}a,$ depending on the emphasis. An element from some $\,{\,^*\!}X\,$ is referred to as an *internal* set. Since $X\in\mathcal{P} (X),$ the power set, each $\,{\,^*\!}X\,$ itself is internal; but there are internal sets not of this form. Non-internal sets are called *external*. We identify a property with the set it defines, so we may speak of ${\,^*\!}P$ when $P$ is a standard mathematical property.
Elements in the set ${\,^*\!}{{\mathbb{N}}}$ are called *hyperfinite*; a set counted internally by a hyperfinite number is also called hyperfinite (this is the same as ${\,^*\!}$finite); given $r, s\in{\,^*\!}{{\mathbb{R}}},\,$ if ${\left\vertr-s\right\vert}<1/n$ for all $n\in{{\mathbb{N}}},$ we write $r\approx s$ (*infinitely close*); $r$ is called *infinitesimal* when $r\approx 0;$ a finite element $r$ of ${\,^*\!}{{\mathbb{R}}}$ (written ${\left\vertr\right\vert}<\infty$) is one with ${\left\vertr\right\vert}<n$ for some $n\in{{\mathbb{N}}}$; such $r$ is $\approx s$ for a unique $s\in{{\mathbb{R}}}$ (called the *standard part*; in symbol: $s={\,^\circ}r$). We use similar notions for elements in $\,{\,^*\!}{{\mathbb C}}.\,$
For some uncountable cardinal $\kappa$ sufficiently large for our purpose, we assume throughout that the so-called *$\kappa$-Saturation Principle* is satisfied in our universe of nonstandard objects (which is possible under a weaker form of the Axiom of Choice), namely:
> If *$\,\mathcal{F}$ is a family of no more than $\kappa$ internal sets such that $\displaystyle{\,\bigcap{\mathcal F}_0\neq\emptyset}$ for any finite subfamily $\mathcal{F}_0$ of $\,\mathcal{F}\,$ (i.e. $\mathcal{F}$ satisfies the finite intersection property),* then $\displaystyle{\,\bigcap{\mathcal F}\neq\emptyset}.$
We refer the readers to [@A] for details of the construction of nonstandard universe and the methodology of nonstandard analysis.
Alternatively, material in this article can be formulated in the less intuitive and more complicated language of ultraproducts, namely one regards ${\,^*\!}X$ as some ultrapower $\prod_{\text{U}}X$ and internal subsets of ${\,^*\!}X$ as the some ultraproduct $\prod_{\text{U}}X_i$ for some $X_i\subset X$ and a strong enough but fixed ultrafilter $\text{U}.$
From a standard normed linear space, the nonstandard hull construction, due to Luxemburg ([@A]), produces a Banach space extension in the standard sense. Here we describe a generalization of this method.
Let $X$ be an internal linear space over ${\,^*\!}{{\mathbb C}}.$ Let $\rm W$ be a (possibly external) set of internal seminorms on $X.\,$ So for each $p\in {\rm W},\,$ $\,p:X\to{\,^*\!}[0,\infty)\,$ and $$\forall x,y\in X\,\forall \alpha\in{\,^*\!}{{\mathbb C}}\; \big[p(x+y)\leq p(x)+p(y)\,\land\, p(\alpha x)={\left\vert\alpha\right\vert}p(x)\big].$$
Write $\text{Fin}(X)$ for $\{x\in X\,\vert\, \sup_{p\in {\rm W}} {\,^\circ}p(x)<\infty \},$ the finite part of $X$ w.r.t. $\rm W.$ On $\text{Fin}( X)$ an equivalence relation $\approx_{\rm{w}}$ is defined: $x_1\approx_{\rm{w}} x_2$ iff $\,\forall p\in{\rm W}\, [p(x_1)\approx p(x_2)].$ Let $\mathfrak{I}:=\{x\in \text{Fin}(X)\,\vert\,x\approx_{\rm{w}} 0\}.$ Noticing that $\text{Fin}(X)$ is a linear space in the standard sense and $\mathfrak{I}$ is a subspace in the standard sense, i.e. closed under addition and multiplication by $\alpha\in{{\mathbb C}}\,$ (in fact even by finite $\alpha\in{\,^*\!}{{\mathbb C}}$), we can form the quotient space $\text{Fin}(X)/\mathfrak{I},\,$ and denote it by $\widehat{X}^{\rm{w}}.\,$ Elements $x+\mathfrak{I}$ of $\widehat{X}^{\rm{w}}\,$ are denoted by $\widehat{x},\,$ where $x\in \text{Fin}(X).\,$ For $\widehat{x},\widehat{y}\in \widehat{X}^{\rm{w}}$ and $\alpha\in {{\mathbb C}},\; \widehat{x}+\alpha \widehat{y}\,$ is defined as $\widehat{x+\beta y},$ for any $\beta\approx\alpha.$ Moreover, $${\left\Vert\widehat{x}\right\Vert}_{\rm{w}}:=\sup\big\{\,{\,^\circ}p(x)\;\vert\; p\in{\rm W}\,\big\}.$$ It is straightforward to check that all these are well-defined.
${\left\Vert\cdot \right\Vert}_{\rm{w}}$ forms a norm on $\widehat{X}^{\rm{w}}$ under which $\widehat{X}^{\rm{w}}$ is a standard Banach space.
It is easy to see that $\widehat{X}^{\rm{w}}$ is a standard linear space normed by ${\left\Vert\cdot \right\Vert}_{\rm{w}}.$
Completeness follows from the $\kappa$-saturation with $\kappa$ chosen to be $\geq (\omega_1+{\left\vert{\rm W}\right\vert})^+\,$ as follows.
Let $\{ \widehat{x}_n\,\vert\,n\in{{\mathbb{N}}}\}\,$ be a Cauchy sequence in $\widehat{X}^{\rm{w}}.$ For $m\in{{\mathbb{N}}}$ let $k_m\in{{\mathbb{N}}}\,$ so that $$\forall n\in{{\mathbb{N}}}\, \big[ n>k_m \Rightarrow {\left\Vert\widehat{x}_n-\widehat{x}_{k_m}\right\Vert}_{\rm{w}}<\frac{1}{2m}\,\big].$$ By $\omega_1$-saturation, we can extend $\{ x_n\,\vert\,n\in{{\mathbb{N}}}\}\,$ to some internal hyperfinite sequence $\{ x_n\,\vert\,n<N\}\,$ in $X,\,$ for some $N\in{\,^*\!}{{\mathbb{N}}}.\,$ Let $\mathcal{F}=\{\mathcal{F}_{p,m}\,\vert\, p\in{\rm W},\, m\in {{\mathbb{N}}}\,\},\,$ where $\mathcal{F}_{p,m}:= \{ x_n\,\vert\, n<N\;\land\; p({x_n-x_{k_m}})\leq 1/2m\,\}.$ Clearly, $\mathcal{F}$ is a family of internal sets having the finite intersection property. Therefore, by $\kappa$-saturation, we can find some $x\in\bigcap\mathcal{F}.\,$ From the definitions, we have $$\forall n, m\in{{\mathbb{N}}}\;\big[ n>k_m\Rightarrow \sup_{p\in {\rm W}} {\,^\circ}p(x-x_n)\leq\frac{1}{m}\,\big]$$ i.e. $\;x\in \text{Fin}(X)\,$ and $\displaystyle{\lim_{n\to\infty} {\left\Vert\widehat{x}-\widehat{x}_n \right\Vert}_{\rm{w}} =0. }$
Hence $\widehat{X}^{\rm{w}}$ is a Banach space under ${\left\Vert\cdot \right\Vert}_{\rm{w}}.$
When ${\rm W}=\{\,{\left\Vert\cdot \right\Vert} \},$ i.e. an internal norm ${\left\Vert\cdot\right\Vert}$ on $X,$ this construction coincides with Luxemburg’s nonstandard hull.
The weak nonstandard hull and the bidual
========================================
From now on $X$ always stands for a standard normed linear space over ${{\mathbb C}}.\,$ The dual space is denoted by $X^\prime$ and hence the bidual (second dual) by $X^{\prime\prime}.$ Each bounded linear form $\phi\in X^\prime$ defines an internal seminorm on ${\,^*\!}X:$ $$p_\phi (x) \,:=\,{\left\vert{\,^*\!}\phi (x)\right\vert},\quad\text{where} \; x\in{\,^*\!}X.$$
We further restrict ourselves to the case ${\rm W}=\{\, p_\phi\,\vert\, \phi\in X^\prime,\,{\left\Vert\phi\right\Vert}\leq 1\,\}\,$ and write $\widehat{X},\, \approx$ and $\,{\left\Vert\cdot\right\Vert}\,$ instead of $\widehat{{\,^*\!}X}^{\rm{w}},\,\approx_{\rm w}\,$ and $\,{\left\Vert\cdot\right\Vert}_{\rm{w}}.\,$
We call $\widehat{X}$ *the weak nonstandard hull* of $X.$
Note that $\text{Fin}({\,^*\!}X)$ includes $\{x\in{\,^*\!}X\,\vert\, {\left\Vertx\right\Vert}<\infty \}.\,$ It is generally a proper subset. Moreover, on $\text{Fin}({\,^*\!}X),$ $x_1\approx x_2$ iff $\,\forall \phi\in X^\prime\,\big[ {\,^*\!}\phi (x_1)\approx {\,^*\!}\phi (x_2)\big].$
Also $\mathfrak{I}=\{x\in \text{Fin}({\,^*\!}X)\,\vert\,{\,^*\!}\phi (x)\approx 0,\,\phi\in X^\prime\}.$
We identify $X\subset \widehat{X}\,$ as a subspace via the isomorphic embedding $x\mapsto \widehat{x}.\,$
The following result identifies the bidual with the weak nonstandard hull of $X.$
\[bidual\] The Banach space $\widehat{X}$ is isometrically isomorphic to $X^{\prime\prime}.$
We first define $\pi : \widehat{X}\to X^{\prime\prime}.\,$ For $\widehat{x}\in \widehat{X}$ and $\phi\in X^\prime,$ we let $\pi (\widehat{x})\big(\phi\big):= {\,^\circ}\big({\,^*\!}\phi(x)\big).$ So $\pi (\widehat{x})$ is a well-defined bounded linear form on $X^\prime.$
Clearly $\pi$ is injective and a homomorphism. We now show that it is surjective and an isometry.
Under enough saturation, let $A$ be a hyperfinite set such that $\,X^\prime \subset A\subset {\,^*\!}\big( X^\prime \big).$ Here the inclusion $X^\prime\subset {\,^*\!}\big(X^\prime\big)$ is given by identifying each $\phi\in X^\prime$ with ${\,^*\!}\phi.$ Let $x^{\prime\prime}\in X^{\prime\prime}$ and $0<\epsilon\approx 0.$ By transferring Helley’s Theorem ([@M] 1.9.12), there is $a\in {\,^*\!}X$ such that $${\left\Verta\right\Vert}\leq {\left\Vert{\,^*\!}x^{\prime\prime}\right\Vert} +\epsilon\quad\text{and}\quad \forall\,\phi\in A\,\big[\phi(a) ={\,^*\!}x^{\prime\prime} (\phi)\big].$$ In particular, $a\in \text{Fin}({\,^*\!}X)$ and $\forall\,\phi\in X^\prime \,\big[{\,^*\!}\phi(a) \approx \big(x^{\prime\prime}\big) (\phi)\big].$
Consequently, $\pi (\widehat{a}) = x^{\prime\prime}$ and ${\left\Vert\widehat{a}\right\Vert}= {\left\Vertx^{\prime\prime}\right\Vert}.$
Therefore $\pi$ is surjective and an isometry.
We remark that only $\kappa$-saturation for an uncountable $\kappa$ greater than the algebraic dimension of $X^\prime$ is needed in the above proof.
From now on, $\widehat{X}$ is identified with $X^{\prime\prime}.$
\[wc\] $X$ is a reflexive Banach space iff $X=\widehat{X}\,$ iff the closed unit ball of $X$ is weakly compact.
By the above theorem and the canonical embedding of $X$ into $\widehat{X},\,$ reflexivity is equivalent to $X=\widehat{X},\,$ which by definition is equivalent to every point $a$ in the unit ball of ${\,^*\!}X$ is infinitely close to a standard point $c\in X\,$ in the weak topology, i.e. ${\,^*\!}\phi(a)\approx \phi (c)\,$ for all $\phi\in X^\prime.\,$ So by Robinson’s characterization of compactness ([@A]), it is equivalent to the weak compactness of the close unit ball of $X.$
A similar application leads to a proof of Goldstine’s Theorem: The closed unit ball of $X$ is weak\* dense in the closed unit ball of $X^{\prime\prime}.$
The following gives an alternative way of computing the norm in $\widehat{X}$ from the norm in ${\,^*\!}X.$
\[norm\] Let $\widehat{a}\in \widehat{X}.$ Then ${\displaystyle{\left\Vert}\right\Vert}{\widehat{a}} = \inf\big\{ {\,^\circ}{\left\Vertx\right\Vert}\,\big\vert\, x\in{\rm{Fin}}({\,^*\!}X),\, x\approx a\,\big\}.$
In other words, ${\displaystyle{\left\Vert}\right\Vert}{\widehat{a}} = \inf_{\varepsilon\in\mathfrak{I}} {\,^\circ}{\left\Verta+\varepsilon\right\Vert}.$
For convenience, we write ${\left\Vert\widehat{a}\right\Vert}_{\rm{v}} = \inf_{\varepsilon\in\mathfrak{I}} {\,^\circ}{\left\Verta+\varepsilon\right\Vert}.$
We let $B_X$ and $\bar{B}_{X^\prime}$ denote the open ball $\{x\in X\,\vert\, {\left\Vertx\right\Vert}<1\}$ and the closed ball $\{\phi\in {X^\prime}\,\vert\, {\left\Vert\phi\right\Vert}\leq 1\}$ respectively.
Let $\varepsilon\in\mathfrak{I},$ then $${\left\Vert\widehat{a}\right\Vert}\,=\,\sup_{\phi\in \bar{B}_{X^\prime}}{\,^\circ}{\left\vert{\,^*\!}\phi(a)\right\vert}\,=\,\sup_{\phi\in \bar{B}_{X^\prime}}{\,^\circ}{\left\vert{\,^*\!}\phi(a+\varepsilon )\right\vert}\,\leq\,{\,^\circ}{\left\Verta+\varepsilon\right\Vert},$$ hence ${\left\Vert\widehat{a}\right\Vert}\leq {\left\Vert\widehat{a}\right\Vert}_{\rm{v}}.$
To show ${\left\Vert\widehat{a}\right\Vert}_{\rm{v}}\leq {\left\Vert\widehat{a}\right\Vert}$ we first assume without loss of generality that ${\left\Vert\widehat{a}\right\Vert} = 1,$ and will show that ${\left\Vert\widehat{a}\right\Vert}_{\rm{v}} \leq 1.$
*Claim*: Let $A\subset \{\phi\in X^\prime\,\vert\,{\left\Vert\phi\right\Vert}=1\,\}$ be finite and $n,m\in{{\mathbb{N}}}.$ Then there exists some $c\in (1+\frac{1}{m}) B_X$ such that $\forall \phi\in A\, \forall n\in N\;{\,^\circ}{\left\vert{\,^*\!}\phi(a)-\phi(c)\right\vert}\leq \frac{1}{n}.$
We denote ${\,^\circ}({\,^*\!}\phi(a))$ by $r_\phi.$ Suppose the claim fails. Then for some finite $A\subset \{\phi\in X^\prime\,\vert\,{\left\Vert\phi\right\Vert}=1\,\}$ and $n,m\in{{\mathbb{N}}},$ we have $$\forall x\in X\, \Big[\bigwedge_{\phi\in A}\Big({\left\vert\phi(x)-r_\phi\right\vert}\leq\frac{1}{n}\Big)\,\Rightarrow\,x\notin \big(1+\frac{1}{m}\big) B_X\,\Big].$$ i.e. the set $\big\{ x\in X\,\vert \bigwedge_{\phi\in A}{\left\vert\phi(x)-r_\phi\right\vert}\leq\frac{1}{n}\big\}\,$ is disjoint from $(1+\frac{1}{m}) B_X.$ Moreover, both sets are convex and the latter is open. So by the Hahn-Banach Separation Theorem ([@R] 3.4), for some $\theta\in X^\prime$ and $\ell\geq 0,$ $$\begin{aligned}
\big(1+\frac{1}{m}\big) B_X\,&\subset\, \{x\in X\,\vert\, {\rm Re}(\theta(x))<\ell\}\\
\big\{ x\in X\,\vert \bigwedge_{\phi\in A}{\left\vert\phi(x)-r_\phi\right\vert}\leq\frac{1}{n}\big\}\,&\subset\,\{x\in X\,\vert\, {\rm Re}(\theta(x))\geq\ell\}.\end{aligned}$$ That is, for any $x,y\in X,\,$ whenever ${\left\Vertx\right\Vert}<1$ and $\bigwedge_{\phi\in A}{\left\vert\phi(y)-r_\phi\right\vert}\leq\frac{1}{n},$ then $$\label{e1}
{\rm Re}(\theta (x))\,<\,\frac{\ell}{1+1/m}\,<\,\ell\,\leq\, {\rm Re}(\theta (y)).$$
By scaling, we can assume that ${\left\Vert\theta\right\Vert}=1.$
By saturation, for some $x\in{\,^*\!}B_X,$ ${\left\Vert\theta\right\Vert}\approx {\left\vert{\,^*\!}\theta(x)\right\vert}.$ Replace such $x$ by $x e^{-i{\rm Arg}(\theta (x))},$ we can assume that ${\left\Vert\theta\right\Vert}\approx {\,^*\!}\theta(x)\in{\,^*\!}{{\mathbb{R}}}.$ Then by transferring (\[e1\]), $$1\,=\,{\left\Vert\theta\right\Vert}\approx {\,^*\!}\theta(x)\, < \,\frac{\ell}{1+1/m}\,<\,\ell\,\leq\, {\rm Re}(\theta (a))\,\leq\,{\left\vert\theta(a)\right\vert}$$ which gives ${\left\Vert\widehat{a}\right\Vert} > 1,$ a contradiction. Hence the Claim is proved.
From the Claim, by saturation and the transfer, we have $$\exists c\in{\,^*\!}X\, \big[{\,^\circ}{\left\Vertc\right\Vert}\leq 1\, \land \,\forall \phi\in X^\prime\,\big( {\,^*\!}\phi(a)\approx {\,^*\!}\phi(c)\big)\big],$$ i.e. $c\approx a$ and ${\,^\circ}{\left\Vertc\right\Vert}\leq 1.$ Therefore we have ${\left\Vert\widehat{a}\right\Vert}_{\rm{v}}\leq 1$ as desired.
\[ca\] Let $\widehat{a}\in \widehat{X}.$ Then ${\displaystyle{\left\Vert}\right\Vert}{\widehat{a}} \approx {\left\Verta+\varepsilon\right\Vert}$ for some $\varepsilon\in\mathfrak{I}.$
By the above theorem, for each $n\in {{\mathbb{N}}}$ there is $\varepsilon_n\in\mathfrak{I}$ so that $$\Big\vert{\left\Vert\widehat{a}\right\Vert}\,-\,{\left\Verta+\varepsilon_n\right\Vert}\Big\vert\,\leq\,\frac{1}{n}.$$ Extend $\{\varepsilon_n\}$ to an internal sequence, let $\varepsilon =\varepsilon_n$ for any $n\in{\,^*\!}{{\mathbb{N}}}\setminus{{\mathbb{N}}}\,$ and notice for each $n\in{\,^*\!}{{\mathbb{N}}}$ that ${\displaystyle{\left\Vert}\right\Vert}{\varepsilon_n}\leq 2{\left\Verta\right\Vert}+\frac{1}{n} <\infty.$
The spaces $c_0,\, \ell_1,\, \ell_\infty\,$ and $ba({{\mathbb{N}}})$
====================================================================
In this section, we draw from results in the previous section some interesting conclusions about certain Banach spaces of complex sequences. We let $c_0,\, \ell_1,\, \ell_\infty\,$ respectively denote the Banach space of complex sequences which converge to $0,$ which are summable and which are uniformly bounded. $c_0$ and $\ell_\infty$ are both given the supremum norm and each $a\in \ell_1$ is given the norm $\sum_{n\in{{\mathbb{N}}}}{\left\verta_n\right\vert}.\,$ We let $ba({{\mathbb{N}}})\,$ to denote the Banach space of finitely additive complex measures defined for all subsets of ${{\mathbb{N}}},\,$ with the total variation norm. (See [@D].)
It is well-known that $c_0^\prime = \ell_1,\,$ $\ell_1^\prime =\ell_\infty\,$ and $\ell_\infty^\prime = ba({{\mathbb{N}}}).$
Let $X=c_0.$ Then $${\rm Fin}({\,^*\!}X)\,=\,{\rm Fin}({\,^*\!}c_0)\,=\,\big\{a\in{\,^*\!}c_0\,\vert\, \forall b\in \ell_1\,{\left\vert\sum_{n\in{\,^*\!}{{\mathbb{N}}}} a_n{\,^*\!}b_n\right\vert}<\infty\, \big\}$$ and, in it, $a\approx c$ iff ${\displaystyle\forall} b\in \ell_1\,\sum_{n\in{\,^*\!}{{\mathbb{N}}}} a_n{\,^*\!}b_n\approx \sum_{n\in{\,^*\!}{{\mathbb{N}}}} c_n{\,^*\!}b_n.\,$
Let $\pi : \widehat{X}\to X^{\prime\prime}$ be given by Theorem \[bidual\], i.e. $\pi : \widehat{c_0}\to \ell_\infty.\,$
Let $\widehat{a}\in\widehat{c_0}\,$ and let $c\in\ell_\infty\,$ denote $\pi(\widehat{a}).\,$ Then from the definitions and transfer, we have for all $b\in\ell_1\,$ that $$\sum_{n\in{\,^*\!}{{\mathbb{N}}}} a_n{\,^*\!}b_n ={\,^*\!}b (a)\approx \pi (\widehat{a}) (b)= c(b)=\sum_{n\in{{\mathbb{N}}}}c_n b_n = \sum_{n\in{\,^*\!}{{\mathbb{N}}}}{\,^*\!}c_n {\,^*\!}b_n.$$ By taking $b={\rm 1}_{\{ n\}},\,n\in {{\mathbb{N}}},\,$ we have $a_n\approx c_n\,$ for all $n\in{{\mathbb{N}}}.$
Therefore, $\pi(\widehat{a}) = ({\,^\circ}a_n)_{n\in {{\mathbb{N}}}}.\,$ In particular, for $a\in {\rm Fin}({\,^*\!}c_0),$ $$\forall b\in\ell_1\,\sum_{n\in{\,^*\!}N}a_n{\,^*\!}b_n\approx 0\quad\text{iff}\quad \forall n\in{{\mathbb{N}}}\; a_n\approx 0.$$
Hence, given any $a\in{\,^*\!}c_0\,$ with $a_n\approx 0$ for all $n\in {{\mathbb{N}}},\,$ if ${\displaystyle\sum}_{n\in{\,^*\!}{{\mathbb{N}}}}a_n{\,^*\!}b_n\not\approx 0\,$ for some $b\in\ell_1,\,$ then ${\displaystyle\sum}_{n\in{\,^*\!}{{\mathbb{N}}}}a_n{\,^*\!}b_n\,$ is infinite for some other $b\in\ell_1.\,$
Moreover, as a consequence of Theorem \[norm\], given any $a\in {\rm Fin}({\,^*\!}c_0),\,$ there exists $c\in {\rm Fin}({\,^*\!}c_0),\,$ such that $$\forall b\in\ell_1\,\sum_{n\in{\,^*\!}{{\mathbb{N}}}}a_n{\,^*\!}b_n\approx\sum_{n\in{\,^*\!}{{\mathbb{N}}}}c_n{\,^*\!}b_n\quad\text{and}\quad {\,^\circ}\sup_{n\in{{\mathbb{N}}}}{\left\vertc_n\right\vert}=\sup_{b\in\ell_1,\,{\left\Vertb\right\Vert}=1}{\,^\circ}\sum_{n\in{\,^*\!}{{\mathbb{N}}}}a_n{\,^*\!}b_n.$$ $\Box$
A similar application of Theorem \[bidual\] and Theorem \[norm\] gives the following concrete representation of measures in $ba({{\mathbb{N}}}).\,$
Let $X=\ell_1.$ Then $${\rm Fin}({\,^*\!}X)\,=\,{\rm Fin}({\,^*\!}\ell_1)\,=\,\big\{a\in{\,^*\!}\ell_1\,\vert\, \forall b\in \ell_\infty\,{\left\vert\sum_{n\in{\,^*\!}{{\mathbb{N}}}} a_n{\,^*\!}b_n\right\vert}<\infty\, \big\}$$ and here $a\approx c$ iff ${\displaystyle\forall} b\in \ell_\infty\,\sum_{n\in{\,^*\!}{{\mathbb{N}}}} a_n{\,^*\!}b_n\approx \sum_{n\in{\,^*\!}{{\mathbb{N}}}} c_n{\,^*\!}b_n.\,$
Let $\pi : \widehat{X}\to X^{\prime\prime}$ be given by Theorem \[bidual\], i.e. $\pi : \widehat{\ell_1}\to ba({{\mathbb{N}}}).\,$
Therefore for each $\mu\in ba({{\mathbb{N}}}),\,$ there is $a\in{\rm Fin}({\,^*\!}\ell_1)\,$ such that $\pi(\widehat{a})=\mu.\,$
Let $S\subset {{\mathbb{N}}},\,$ so ${\rm 1}_S\in\ell_\infty\,$ and $$\mu (S) = \int_S d\mu = \mu ({\rm 1}_S)=\pi(\widehat{a})({\rm 1}_S)\approx \sum_{n\in{\,^*\!}{{\mathbb{N}}}} a_n {\,^*\!}{\rm 1}_S(n).$$
That is, by the above and Theorem \[norm\], for any $\mu\in ba({{\mathbb{N}}}),\,$ there is $a\in{\,^*\!}\ell_1\,$ $$\forall S\in {{\mathbb{N}}}\; \mu (S) ={\,^\circ}\sum_{n\in{\,^*\!}S} a_n\quad\text{and}\quad {\left\Vert\mu\right\Vert} ={\,^\circ}\sum_{n\in{\,^*\!}{{\mathbb{N}}}}{\left\verta_n\right\vert}.$$ $\Box$
The bidual of a C\*-algebra
===========================
In this section we will show that the bidual of a C\*-algebra forms a von Neumann algebra.
Following the tradition, the involution of an element $x$ in an C\*-algebra is denoted by $x^*,\,$ not to be confused with the nonstandard extension ${\,^*\!}x.$
Throughout this section $X$ is taken to be a standard C\*-algebra. Recall that $X^{\prime\prime}$ is identified with $\widehat{X},$ hence has elements of the form $\widehat{a},\,$ where $ a\in {\,^*\!}X.$
Let $H$ be the Hilbert space corresponding to the universal representation of $X$ ([@B] III.5.2.1). Let $\mathcal{B}(H)$ denote the C\*-algebra of bounded linear operators on $H$ and identify $X$ as a C\*-subalgebra of $\mathcal{B}(H)$ under this representation.
For $\xi,\rho\in H,$ we let $\omega_{\xi,\rho}$ denote the linear form in $\big(\mathcal{B}(H)\big)^\prime$ that takes $x\in \mathcal{B}(H)$ to $\langle x(\xi ),\rho\rangle.$ We also regard $\omega_{\xi,\rho}\in X^\prime\,$ when dealing only with its restriction on $X.\,$ The definition extends internally to $\omega_{\xi,\rho}$ for $\xi,\rho\in {\,^*\!}H.$ It is clear that ${\left\Vert\omega_{\xi,\rho}\right\Vert}\leq {\left\Vert\xi\right\Vert}\,{\left\Vert\rho\right\Vert}.$ In particular, $\,\omega_{\xi,\rho}\in X^\prime\,$ for $\,\xi,\rho\in H.\,$
\[pl\] Each element in $X^\prime\,$ is a linear combination of the $\omega_{\xi,\xi}\,$ where $\,\xi\in H.$
By the universal representation ([@B] III.5.2.1), positive linear forms from $X^\prime$ are precisely the $\omega_{\xi,\xi}$ for some $\xi\in H.$ On the other hand, each element in $X^\prime$ is represented as a canonical linear combinations of some positive ones ([@B] II.6.3.4).
Given $\phi\in X^\prime,\,$ we regard $\phi\in \mathcal{B}(H)^\prime\,$ via the above canonical representation. Also, from Proposition \[pl\], the following is immediate.
\[lpe\] Let $a,b\in{\rm Fin}({\,^*\!}X).\,$ Then $a\approx b\,$ iff ${\,^*\!}\omega_{\xi,\rho}(a)\,=\,{\,^*\!}\omega_{\xi,\rho}(b)\,$ for all $\xi,\rho\in H.\,$
Since $\,\omega_{\xi,\rho}\in X^\prime\,$ for $\,\xi,\rho\in H,\,$ one direction is trivial.
For the other one, assume that ${\,^*\!}\omega_{\xi,\rho}(a)\,=\,{\,^*\!}\omega_{\xi,\rho}(b)\,$ for all $\xi,\rho\in H.\,$ Then by Proposition \[pl\], $\,\forall \phi\in X^\prime\, {\,^*\!}\phi (a)\,=\, {\,^*\!}\phi ( b),\,$ i.e. $\,a\,\approx\, b.$
For $a\in{\rm Fin}({\,^*\!}X)$ and $\xi,\rho\in H,\,$ $$\label{omega}
\overline{{\,^*\!}\omega_{\rho,\xi}(a)}\,=\,\overline{\langle a ({\,^*\!}\rho),{\,^*\!}\xi\rangle}\,=\,\langle {\,^*\!}\xi,a ({\,^*\!}\rho)\rangle\,=\,\langle a^* ({\,^*\!}\xi),{\,^*\!}\rho\rangle\,=\,\,{\,^*\!}\omega_{\xi,\rho}(a^*).$$ Therefore we have:
Let $a,b\in{\rm Fin}({\,^*\!}X).\,$ Then $a\approx b\,$ iff $a^*\approx b^*.$$\Box$
For fixed $a\in{\rm Fin}({\,^*\!}X)$ and $\rho\in H,$ by (\[omega\]), the mapping $H\ni \xi \mapsto \overline{{\,^*\!}\omega_{\rho,\xi}(a)}$ is a bounded linear form on $\,H.\,$ Hence, by the Riesz-Frèchet Theorem, there is a unique $\eta\in H$ such that $\forall \xi\in H\; \langle \xi, \eta \rangle = {\,^\circ}\overline{{\,^*\!}\omega_{\rho,\xi}(a)}.$
Moreover, for $\xi,\rho\in H,$ if $b\in{\rm Fin}({\,^*\!}X)$ and $a\approx b,$ then $\overline{{\,^*\!}\omega_{\rho,\xi}(a)}\,\approx\,\overline{{\,^*\!}\omega_{\rho,\xi}(b)},\,$ since $\omega_{\rho,\xi }(x)\in X^\prime.$ We thus define $$\pi: X^{\prime\prime} \to \mathcal{B}(H)$$ by letting $\pi (\widehat{a})$ be the bounded operator on $H$ that takes $\rho\in H$ to this unique $\eta\in H.\,$ That is, for $\rho\in H,\,$ $\pi(\widehat{a})(\rho )\,$ is the unique element in $H$ such that $$\forall\xi\in H\;\big[ \langle \xi, \pi(\widehat{a})(\rho )\rangle \,=\,{\,^\circ}\langle {\,^*\!}\xi, a({\,^*\!}\rho)\rangle\big].$$
In case the preimage or the image of $\pi$ is in $X,$ we have the following.
\[ppg\] Let $a\in{\rm Fin}({\,^*\!}X).\,$
1. If $a\in X,$ then $\pi (a) =a.$
2. If $\pi (\widehat{a})\in X\,$ then ${\displaystyle\widehat}{a}\,=\,\widehat{{\,^*\!}(\pi (\widehat{a}))}.\,$ In particular $a\approx c\,$ for some $c\in X.$
(i): If $a\in X,\,$ then $\,\forall\,\xi,\rho\in H,\,$ we have $$\langle\xi,\pi(\widehat{a})(\rho) \rangle\,\approx\,\langle{\,^*\!}\xi,a({\,^*\!}\rho) \rangle\,=\,\langle\xi,a(\rho) \rangle.$$
(ii): Let $\,b=\pi (\widehat{a})\in X.\,$ For $\,\xi,\rho\in H,\,$ we have $$\overline{{\,^*\!}\omega_{\rho,\xi}(a)}\,\approx\,\langle\xi,\pi(\widehat{a})(\rho) \rangle\,=\,\langle\xi,b(\rho) \rangle\,\approx\,\langle{\,^*\!}\xi, {\,^*\!}b({\,^*\!}\rho) \rangle\,=\,\overline{{\,^*\!}\omega_{\rho,\xi}({\,^*\!}b)}.$$ So we have the required $\,a\approx {\,^*\!}b$ as a consequence of Lemma \[lpe\]. Note $b\in X.$
The following shows that $\pi$ is an isometry.
\[le\] Let $a\in{\rm Fin}({\,^*\!}X).$ Then ${\left\Vert\widehat{a}\right\Vert}_{X^{\prime\prime}}\,=\,{\left\Vert\pi(\widehat{a})\right\Vert}_{\mathcal{B}(H)}.$
First note that $$\begin{aligned}
\label{le1}
{\left\Vert\pi(\widehat{a})\right\Vert}_{\mathcal{B}(H)}\,&=\,\sup\big\{\langle\xi,\pi(\widehat{a})(\rho) \rangle\,\big\vert\, \xi,\rho\in H\,\land\, {\left\Vert\xi\right\Vert}={\left\Vert\rho\right\Vert}=1\,\big\}\\
\nonumber &=\,\sup\big\{{\,^\circ}\langle{\,^*\!}\xi,a ({\,^*\!}\rho) \rangle\,\big\vert\, \xi,\rho\in H\,\land\, {\left\Vert\xi\right\Vert}={\left\Vert\rho\right\Vert}=1\,\big\}\\
\nonumber &=\,\sup\big\{{\,^\circ}{\left\vert{\,^*\!}\omega_{\rho,\xi}(a)\right\vert}\,\big\vert\, \xi,\rho\in H\,\land\, {\left\Vert\xi\right\Vert}={\left\Vert\rho\right\Vert}=1\,\big\}\,\leq \,{\left\Vert\widehat{a}\right\Vert}_{X^{\prime\prime}}.\end{aligned}$$
We define for an internal subspace $K\subset {\,^*\!}H$ the following $$\Theta (K)\,=\, \sup\big\{{\left\vert{\,^*\!}\omega_{\xi,\rho}(a)\right\vert}\,\big\vert\, \xi,\rho\in K\,\land\, {\left\Vert\xi\right\Vert}={\left\Vert\rho\right\Vert}=1\,\big\}.$$
Let $\,r\,=\,{\left\Vert\pi(\widehat{a})\right\Vert}_{\mathcal{B}(H)}.$ Then by (\[le1\]) and the transfer, the family ${\displaystyle\{}\mathcal{F}_{H_0, n}\,\},\,$ where $$\mathcal{F}_{H_0, n}\,:=\,\big\{K\,\vert\, K\; \text{an internal subspace so that}\; H_0\subset K\subset {\,^*\!}H\;\land\;{\left\vert\Theta (K)-r\right\vert}\leq\frac{1}{n}\,\big\},\,$$ with indices ranging over all finite dimensional subspace $H_0\subset H\,$ and $n\in{{\mathbb{N}}},$ has the finite intersection property. Therefore, as a consequence of enough saturation, we can fix some (hyperfinite dimensional) internal subspace $K\subset {\,^*\!}H\,$ such that $\,H\subset K\,$ and $\Theta (K)\,\approx\,r.$
Let $\varrho_K$ denote the projection of ${\,^*\!}H$ onto this $K.$ Then for all $\xi,\rho\in H\,$ we have ${\,^*\!}\omega_{\xi,\rho}(a)\,=\,{\,^*\!}\omega_{\xi,\rho}(a\,\varrho_K ).\,$ (If $\varrho_K\in {\,^*\!}X,$ we have $\,a\,\approx\, a\,\varrho_K\,$ by Lemma \[lpe\].)
Now notice that $$\begin{aligned}
{\left\Vert\widehat{a}\right\Vert}_{X^{\prime\prime}}\,&=\,\sup\big\{{\,^\circ}{\left\vert{\,^*\!}\phi (a)\right\vert}\,\big\vert\, \phi\in X^\prime,\; {\left\Vert\phi\right\Vert}=1 \big\}\\
&=\,\sup\big\{{\,^\circ}{\left\vert{\,^*\!}\phi (a\,\varrho_K )\right\vert}\,\big\vert\, \phi\in X^\prime,\; {\left\Vert\phi\right\Vert}=1 \big\}\,\lessapprox\,{\left\Verta\,\varrho_K\right\Vert}_{{\,^*\!}\mathcal{B}(H)},\end{aligned}$$ where in the second equality, the $\phi$ is regarded as an element in $\mathcal{B}(H)^\prime\,$ via the canonical linear combination of positive linear functionals.
Observing that ${\left\Verta\,\varrho_K\right\Vert}_{{\,^*\!}\mathcal{B}(H)}\,=\,\Theta (K),\,$ we therefore have $$\label{le2}
{\left\Vert\widehat{a}\right\Vert}_{X^{\prime\prime}}\,\lessapprox\,{\left\Verta\,\varrho_K\right\Vert}_{{\,^*\!}\mathcal{B}(H)}\,=\,\Theta (K)\,\approx\, r\,=\,{\left\Vert\pi(\widehat{a})\right\Vert}_{\mathcal{B}(H)},$$
Now the conclusion follows from (\[le1\]) and (\[le2\]).
\[lpf\] The embedding $\pi:X^{\prime\prime}\to\mathcal{B}(H)\,$ is a Banach space isometric isomorphism.
For $a,b\in{\rm Fin}({\,^*\!}X)\,$ and $\lambda\in{{\mathbb C}},\,$ if $\,\xi,\rho\in H,\,$ $$\begin{aligned}
\langle\xi,\pi(\widehat{a}+\lambda\widehat{b})(\rho)\rangle\,&\approx\,{\,^*\!}\omega_{\xi,\rho}((a+\lambda b)^*)\,=\,{\,^*\!}\omega_{\xi,\rho}(a^*+\bar{\lambda} b^*)\\
& \,=\,{\,^*\!}\omega_{\xi,\rho}(a^*)+\bar{\lambda} {\,^*\!}\omega_{\xi,\rho}(b^*))\\
& \,=\,\langle\xi,\pi(\widehat{a})(\rho)\rangle+\bar{\lambda}\langle\xi,\pi(\widehat{b})(\rho)\rangle
\,=\,\langle\xi,\,\pi(\widehat{a})(\rho)+\lambda\pi(\widehat{b})(\rho)\rangle,\end{aligned}$$ hence $\pi$ is a linear operator.
It now follows from Lemma \[le\] that $\pi$ is an isometric isomorphism.
In the remainder of this section, we will show that $\pi$ is in fact an isometric C\*-isomorphism.
Given $\, \widehat{a}, \widehat{b} \in X^{\prime\prime},\,$ the mapping that takes $\,\omega_{\xi,\rho}\,$ (restricted on $X$), where $\,\xi,\rho\in H,\,$ to $\,\langle \xi,\, \pi(\widehat{a})\big(\pi (\widehat{b}) (\rho )\big)\rangle$ extends to a bounded linear form in $\,X^\prime\,$ via Proposition \[pl\]. So this bounded linear form is the unique $\widehat{c^* }$ for some $\, c \in{\rm Fin}({\,^*\!}X).\,$ Then for $\,\xi,\rho\in H,\,$ $$\langle \xi,\, \pi(\widehat{a})\big(\pi (\widehat{b}) (\rho )\big)\rangle\,=\,\widehat{c^*} (\omega_{\xi,\rho})\,\approx\,{\,^*\!}\omega_{\xi,\rho}(c^*)\,=\, \langle {\,^*\!}\xi,\, c({\,^*\!}\rho)\rangle\,\approx\, \langle \xi,\, \pi(\widehat{c})(\rho)\rangle,$$ where the second equality follows from (\[omega\]).
In other words, we have $\pi(\widehat{c})\,=\,\pi(\widehat{a})\pi(\widehat{b}).\,$ Hence the image of $\pi$ is closed under the product in $\mathcal{B}(H).\,$
Also for $\,\xi,\rho\in H,\,$ $$\begin{aligned}
\langle \xi,\, (\pi(\widehat{a}))^* (\rho)\rangle\,&=\,\langle \pi(\widehat{a})(\xi),\,\rho\rangle\,=\,\overline{\langle \rho,\,\pi(\widehat{a})(\xi)\rangle}\,\approx\, \overline{\langle {\,^*\!}\rho,\,a({\,^*\!}\xi)\rangle}\\
&\approx\, \langle a({\,^*\!}\xi),\,{\,^*\!}\rho \rangle\,=\,\langle {\,^*\!}\rho,\, a^* ({\,^*\!}\xi) \rangle.\end{aligned}$$ i.e. $\,(\pi (\widehat{z})^*\,=\,\pi (\widehat{a^*})\,$ and the image of $\pi$ is closed under the involution in $\mathcal{B}(H).\,$
These together with Lemma \[lpf\] prove that:
\[tcs\] $\pi$ is an isometric C\*-isomorphism embedding $\,X^{\prime\prime}\,$ into $\mathcal{B}(H).\,$ $\Box$
Recall from [@B] that the weak operator topology is the weakest topology on $\mathcal{B}(H)$ that makes all $\omega_{\xi,\rho},\,\xi,\rho\in H,\,$ continuous. It is therefore generated by open sets of the form $\,\{x\in \mathcal{B}(H)\,\vert\, \langle \xi,\,x(\rho) \rangle<\epsilon\,\},\;\xi,\rho\in H,\, \epsilon>0.\,$ A von Neumann algebra is a C\*-algebra which is closed in the weak operator topology in some representation as a C\*-subalgebra of the algebra of bounded operators on some Hilbert space.
[(Sherman-Takeda Theorem)]{} The von Neumann algebra generated by a C\*-algebra $X\,$ is isometrically C\*-isomorphic to $X^{\prime\prime}.$
By Proposition \[ppg\] (i) and Theorem \[tcs\], $\pi$ extends $X$ isometrically and isomorphically to a C\*-subalgebra of $\mathcal{B}(H).\,$ Moreover, the image is the weak operator topological closure of $X:$
Let $\widehat{a}\in X^{\prime\prime}\,$ and consider $\pi(\widehat{a}).\,$ Suppose $\,\xi_i,\rho_i\in H,\, \epsilon_i>0,\; i=1,\dots, n\in{{\mathbb{N}}},$ $$\bigwedge_{i=1}^n\,{\left\vert\langle \xi_i,\,\pi(\widehat{a})(\rho_i) \rangle\right\vert}<\epsilon_i.$$ Then for some $\epsilon^{\prime}_i<\epsilon_i,$ we have $$\bigwedge_{i=1}^n\,{\left\vert\langle {\,^*\!}\xi_i,\,a({\,^*\!}\rho_i) \rangle\right\vert}\leq\epsilon^{\prime}_i,\quad\text{consequently}\quad \exists x\in{\,^*\!}X\bigwedge_{i=1}^n\,{\left\vert\langle {\,^*\!}\xi_i,\,x({\,^*\!}\rho_i) \rangle\right\vert}\leq\epsilon^{\prime}_i.$$ By transfer, we then have $$\exists x\in X\bigwedge_{i=1}^n\,{\left\vert\langle \xi_i,\,x(\rho_i) \rangle\right\vert}\leq\epsilon^{\prime}_i.$$ So $\pi(\widehat{a})$ is in the weak operator topological closure of $X.\,$
Hence the image of $\pi$ is the von Neumann algebra generated by $X$ in $\mathcal{B}(H).$
1. The above shows that the weak nonstandard hull of a C\*-algebra is always a von Neumann algebra.
2. The predual of the von Neumann algebra generated by $X$ is simply $X^\prime.$ (See also Sakai’s Theorem ([@B] III.2.4.2).)
3. The bicommutant of $X$ in $\mathcal{B}(H)$ is just $X^{\prime\prime}.$ $\Box$
By results in [@G], the product on $X^{\prime\prime}$ given by $\widehat{a}\,\widehat{b} := \pi^{-1}\big(\pi(\widehat{a})\pi (\widehat{b})\big)$ is the same as the left and the right Arens product. But in the current setting there is a better representation of the Arens product deriving from the convergent nets in the weak operator topology.
\[arens\] Given $a,b\in{\rm Fin}({\,^*\!}X),\,$ there is $a_0\in{\rm Fin}({\,^*\!}X),\,a_0\approx a,\,$ such that $\pi(\widehat{a})\pi(\widehat{b})\,=\,\pi(\widehat{a_0 b}).\,$
Similarly, there is $b_0\in{\rm Fin}({\,^*\!}X),\,b_0\approx b,\,$ such that $\pi(\widehat{a})\pi(\widehat{b})\,=\,\pi(\widehat{ab_0}).\,$
Moreover, $a_0$ (resp. $b_0\,$) can be chosen from the internal extension of a net converging to $\pi(\widehat{a})$ (resp. $\pi(\widehat{b})\,$) in the weak operator topology.
Let $a,b\in{\rm Fin}({\,^*\!}X),\,$ write $c=a^*.\,$ Let $c_i\in X,\,i\in I,$ a net, and $c_i\to \pi(\widehat{c})\,$ in the weak operator topology. Then for any finite list of $\xi,\,\rho\in H,\,n\in {{\mathbb{N}}},\,$ we have $${\left\vert\langle c_i({\,^*\!}\xi),\,{\,^*\!}\big(\pi (\widehat{b})(\rho )\big)\rangle\,-\,\langle c({\,^*\!}\xi),\,{\,^*\!}\big(\pi (\widehat{b})(\rho )\big)\rangle\right\vert}\leq\,n^{-1}\quad\text{for all large}\; i\in I.$$ Note for $i\in I\,$ that $$\langle c_i({\,^*\!}\xi),\,{\,^*\!}\big(\pi (\widehat{b})(\rho )\big)\rangle\,=\,\langle c_i(\xi),\,\big(\pi (\widehat{b})(\rho )\big)\rangle\,\approx\,
\langle {\,^*\!}(c_i(\xi)),\,b({\,^*\!}\rho )\big)\rangle$$ Therefore the family ${\displaystyle\{}\mathcal{F}_{H_0, n, i}\,\}\,$ has the finite intersection property, where the indices range over all finite dimensional subspace $H_0\subset H,\,$ $n\in {{\mathbb{N}}}\,$ and $i\in I,\,$ and $\mathcal{F}_{H_0, n, i}$ is the internal set of the $(K,k),$ where $K$ is an internal subspace of ${\,^*\!}H$ and $K$ includes $H_0$ as a subspace, $i\leq k\in {\,^*\!}I,\,$ for all $\, \xi,\rho\in K\,$ with ${\left\Vert\xi\right\Vert}={\left\Vert\rho\right\Vert}=1,$ the following is satisfied: $${\left\vert\langle c_k({\,^*\!}\xi),\,b({\,^*\!}\rho )\big)\rangle\,-\,\langle c({\,^*\!}\xi),\,{\,^*\!}\big(\pi (\widehat{b})(\rho )\big)\rangle\right\vert}\leq\,n^{-1}.$$ By enough saturation, we let $(K,k)\,$ be an element in the intersection of the family ${\displaystyle\{}\mathcal{F}_{H_0, n, i}\,\}.\,$ Then for $\xi,\,\rho\in H\,$ (hence ${\,^*\!}\xi,\,{\,^*\!}\rho\in K\,$), we have $$\langle c_k({\,^*\!}\xi),\,b({\,^*\!}\rho )\big)\rangle\,\approx\, \langle c({\,^*\!}\xi),\,{\,^*\!}\big(\pi (\widehat{b})(\rho )\big)\rangle\,=\,
\langle {\,^*\!}\xi,\,a\big({\,^*\!}\big(\pi (\widehat{b})(\rho )\big)\big)\rangle\,\approx\, \langle \xi,\,\pi(a)\big(\pi(\widehat{b})(\rho)\big)\rangle.$$
On the other hand, $$\langle c_k({\,^*\!}\xi),\,b({\,^*\!}\rho )\big)\rangle\,=\,\langle {\,^*\!}\xi,\, c_k^* \big(b({\,^*\!}\rho )\big)\rangle\,=\, \langle \xi,\, \pi (c_k b)({\,^*\!}\rho )\rangle.$$ Since $c_k\approx c\,= a^*,\,$ if we let $a_0=c_k^*,\,$ then $a_0\approx a\,$ and the above shows that $\pi(\widehat{a})\pi(\widehat{b})\,=\,\pi(\widehat{a_0 b}).\,$ Note that $a_0\in{\rm Fin}({\,^*\!}X).$
The second statement of the theorem follows a dual argument, i.e apply the above to $b^* a^*.$
In general, for $a,b\in{\rm Fin}({\,^*\!}X)\,$ the product $\widehat{a}\widehat{b}$ in $X^{\prime\prime}$ is not the same as $\widehat{ab}.\,$ For example, take $H=\ell^2({{\mathbb{N}}})$ and $X=\mathcal{B}(H).$ Fix $N\in{\,^*\!}{{\mathbb{N}}}\setminus{{\mathbb{N}}}.\,$ Let $a\in{\,^*\!}X$ be the operation that interchanges the first and the $N$th coordinates in each $\xi\in H.\,$ Then $a^2=1\,$ so $\widehat{a^2}=1\,$ but $\widehat{a}\,$ is the operator that replaces the first coordinate in each $\xi\in H$ by $0.$$\Box$
[10]{}
S. Albeverio, J.-E. Fenstad, R. H[ø]{}egh-Krohn and T. Lindstr[ø]{}m, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986.
B. Blackader, Operator Algebras, theory of $C^*$-algebras and von Neumann algebras. Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2006.
N. Dunford and J.T. Schwartz, Linear operators. Part I. General theory. Reprint of the 1958 original. John Wiley & Sons, Inc., New York, 1988.
G. Godefroy and B. Iochum, Arens-regularity of Banach algebras and the geometry of Banach spaces, J. Funct. Anal. 80 (1988), no.1, 47–59.
R.E. Megginson, An Introduction to Banach Space Theory. Graduate Texts in Mathematics, 183. Springer-Verlag, New York, 1998.
W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.
M. Takesaki, Theory of Operator Algebra I. Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, V. Springer-Verlag, Berlin, 2002.
[^1]: [*Mathematics Subject Classification*]{} Primary: 46L05. Secondary: 03H05 26E35 46S20.\
Key words: nonstandard hull, bidual, C\*-algebra, von Neumann algebra, Sherman-Takeda Theorem.
[^2]: Address: School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209 South Africa\
website: `http://www.maths.unp.ac.za/ siuahn/default.htm` email: ngs@ukzn.ac.za
|
---
abstract: 'Group-buying websites represented by Groupon.com are very popular in electronic commerce and online shopping nowadays. They have multiple slots to provide deals with significant discounts to their visitors every day. The current user traffic allocation mostly relies on human decisions. We study the problem of automatically allocating the user traffic of a group-buying website to different deals to maximize the total revenue and refer to it as the Group-buying Allocation Problem (). The key challenge of is how to handle the tipping point (lower bound) and the purchase limit ( upper bound) of each deal. We formulate as a knapsack-like problem with variable-sized items and majorization constraints. Our main results for can be summarized as follows. (1) We first show that for a special case of , in which the lower bound equals the upper bound for each deal, there is a simple dynamic programming-based algorithm that can find an optimal allocation in pseudo-polynomial time. (2) The general case of is much more difficult than the special case. To solve the problem, we first discover several structural properties of the optimal allocation, and then design a two-layer dynamic programming-based algorithm leveraging those properties. This algorithm can find an optimal allocation in pseudo-polynomial time. (3) We convert the two-layer dynamic programming based algorithm to a fully polynomial time approximation scheme (FPTAS), using the technique developed in [@ibarra1975fast], combined with some careful modifications of the dynamic programs. Besides these results, we further investigate some natural generalizations of , and propose effective algorithms.'
author:
- 'Weihao Kong Jian Li Tao Qin Tie-Yan Liu'
bibliography:
- 'groupon.bib'
title: Optimal Groupon Allocations
---
|
---
abstract: |
We collect data at well sampled frequencies from the radio to the $\gamma$–ray range for the following three complete–samples of blazars: the Slew Survey and the 1 Jy samples of BL Lacs and the 2 Jy sample of Flat Spectrum Radio–Loud Quasars (FSRQs). The fraction of objects detected in $\gamma$–rays (E $\gs 100$ MeV) is $\sim$ 17 %, 26 % and 40 % in the three samples respectively. Except for the Slew Survey sample, $\gamma$–ray detected sources do not differ either from other sources in each sample, nor from all the $\gamma$-ray detected sources, in terms of the distributions of redshift, radio and X–ray luminosities and of the broad band spectral indices (radio to optical and radio to X–ray).
We compute average spectral energy distributions (SEDs) from radio to $\gamma$–rays for each complete sample and for groups of blazars binned according to radio luminosity, irrespective of the original classification as BL Lac or FSRQ.
The resulting SEDs show a remarkable continuity in that: i) the first peak occurs in different frequency ranges for different samples/ luminosity classes, with most luminous sources peaking at lower frequencies; ii) the peak frequency of the $\gamma$–ray component correlates with the peak frequency of the lower energy one; iii) the luminosity ratio between the high and low frequency components increases with bolometric luminosity.
The continuity of properties among different classes of sources and the systematic trends of the SEDs as a function of luminosity favor a unified view of the blazar phenomenon: a single parameter, related to luminosity, seems to govern the physical properties and radiation mechanisms in the relativistic jets present in BL Lac objects as well as in FSRQ. The general implications of this unified scheme are discussed while a detailed theoretical analysis, based on fitting continuum models to the individual spectra of most $\gamma$-ray blazars, is presented in a separate paper (Ghisellini et al. 1998).
author:
- |
G. Fossati$^1$, L. Maraschi$^2$, A. Celotti$^{1,3}$, A. Comastri$^4$ and G. Ghisellini$^2$\
$^1$ S.I.S.S.A., via Beirut 4, I–34014, Trieste, Italy\
$^2$ Osservatorio Astronomico di Brera, via Brera 28, I-20121, Milano, Italy\
$^3$ Institute of Astronomy, Madingley Road, Cambridge CB3 0HA\
$^4$ Osservatorio Astronomico di Bologna, via Zamboni 33, I-40126, Bologna, Italy\
date: 'Received \*\*\*; in original form \*\*\*'
title: A Unifying View of the Spectral Energy Distributions of Blazars
---
\#1[[**\#1**]{}]{}
quasars: general – BL Lacertae objects: general – X–rays: galaxies – X–rays: general – radiative mechanisms: non–thermal – surveys
Introduction
============
The discovery of BL Lac objects and the paradoxes associated with their violent variability led to a major step forward in the theory of Active Galactic Nuclei (AGN), that is to the concept of relativistic jets. Flat spectrum, radio–loud quasars (Angel & Stockman 1980) share basically all of the properties of BL Lac objects related to the presence of a strong non–thermal broad band continuum, except for the absence of broad emission lines. Hence the common designation of blazars proposed by Ed Spiegel in 1978.
It was initially supposed that BL Lacs represented the most extreme version of FSRQs, i.e. those with the most highly boosted continuum. Instead, it has been recognized later (e.g. Ghisellini, Madau & Persic 1987; Padovani 1992a; Ghisellini et al. 1993) that the [*amount of relativistic beaming*]{} and the [*intrinsic*]{} power in the lines are lower in BL Lacs than in FSRQs, implying some intrinsic difference between the two classes. Differences are also found in the extended radio emission and jet structure (e.g. Padovani 1992a; Gabuzda et al. 1992). Nevertheless the continuity of several observational properties including the luminosity functions (Maraschi & Rovetti 1994), the radio to X–ray SEDs (Sambruna et al. 1996) and the luminosity of the lines (Scarpa & Falomo 1997) suggests that blazars can still be considered as a single family, where the physical processes are essentially similar allowing for some scaling factor(s). The identification of these scaling factors would represent a substantial progress in the understanding of the blazar phenomenon.
A special class of BL Lacs was found from identification of X–ray sources. X–ray selected BL Lacs (XBL) differ from the classical radio–selected BL Lacs (RBL) in a lesser degree of “activity” (including polarization), in the radio to optical emission and in the relative intensity of their X–ray and radio emission. This led to the suggestion that the X–ray radiation was less beamed than the radio one and that XBLs were observed at larger inclination to the jet axis (e.g. Urry & Padovani 1995 for a review).
Giommi & Padovani (1994) quantified the differences in SEDs between XBLs and RBLs, and Padovani & Giommi (1995) introduced the distinction between ’High–energy peak BL Lacs’ (HBL) and ’Low–energy peak BL Lacs’ (LBL), for objects which emit most of their synchrotron power at high (UV–soft–X) or low (far–IR, near–IR) frequencies respectively. Quantitatively a distinction can be done on the basis of the ratio between radio and X-ray fluxes (see also §3.2.2). We will use the broad band spectral index $\alpha_{\rm
RX}$[^1] and call HBL and LBL objects having $\alpha_{\rm RX} \ls 0.75$, $\gs
0.75$, respectively. Giommi and Padovani also proposed that HBL represent a small fraction of the BL Lac population and are numerous in X–ray surveys only due to selection effects. An alternative hypothesis (Fossati et al. 1997) relates the spectral properties to the source luminosity in such a way that low luminosity objects (with high space density) are HBLs while high luminosity objects (with low space density) are LBLs.
We will include here X–ray selected BL Lacs together with “classical” BL Lacs in the blazar family, again assuming that the basic physical processes by which the continuum is produced are common to the whole family.
The detection by EGRET, on board the Compton Gamma–Ray Observatory (CGRO), of many blazars at $\gamma$–ray energies (E $\gs 30$ MeV) revealed that a substantial fraction and in some cases the bulk of their power is emitted in this very high energy band. The $\gamma$–ray emission is therefore of fundamental importance in the SED of blazars.
From the theoretical point of view the radio to UV continuum is universally attributed to synchrotron emission from a relativistic jet, while a flat inverse Compton component due to upscattering of the low energy photons is expected to emerge at high energies as originally discussed in Jones, O’Dell & Stein (1974). The latter process is therefore a plausible candidate to explain the $\gamma$–ray emission. The soft photons to be upscattered could be either the synchrotron photons themselves (synchrotron self–Compton process, SSC, e.g. Maraschi, Ghisellini & Celotti 1992; Bloom & Marscher 1993) or photons produced by the disk and/or scattered /reprocessed in the broad line region (Blandford 1993; Dermer & Schlickeiser 1993; Sikora, Begelman & Rees 1994; Ghisellini & Madau 1996). Understanding whether/how the $\gamma$–ray properties differ among subclasses is essential to assess the role of different mechanisms and to verify whether the idea of blazars as a unitary class can be maintained.
Here we study the systematics of the SEDs of blazars using data from the radio to the $\gamma$-ray band. We confirm and extend previous results of Maraschi et al. (1995) and Sambruna et al. (1996) by: i) [*extending the SED to the $\gamma$-ray range*]{}; ii) using a much larger complete sample of FSRQ; iii) using the richer and brighter sample of X–ray selected BL Lacs recently derived from the Slew Survey. We use the available $\gamma$–ray data for each sample but also indirect information derived from the $\gamma$–ray detected (not complete) sample discussed by Comastri et al. (1997). Since we find that the continuity hypothesis among blazars holds we also consider a merged “global” sample subdivided in luminosity bins irrespective of the original classification of the objects.
The structure of the paper is as follows. In section 2 we describe how the data for the SEDs were collected and treated for each sample. The $\gamma$–ray properties of different samples are also discussed. In section 3 we build average SEDs for the three sub–samples and for the global sample subdivided according to luminosity and present our results. These are discussed in section 4, while our conclusions are drawn in section 5.
The data
========
The samples
-----------
We decided to consider the following three samples of blazars: the Slew Survey Sample and the 1 Jy sample of BL Lac objects and the FSRQ sample derived from the 2 Jy sample of Wall & Peacock (1985), motivated by the need of completeness, sufficient number of objects and observational coverage at other frequencies, as detailed below.
### BL Lacs, X–ray selected: the Slew survey sample
The [*Einstein*]{} Slew survey (Elvis 1992) was derived from data taken with the IPC while the telescope scanned the sky in between different pointings. It has limited sensitivity (flux limit of $\simeq 5 \times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ in the IPC band \[0.3 $-$ 3.5 keV\]), but covers a large fraction of the sky ($\sim$ 36000 deg$^2$). In a restricted region of the sky Perlman (1996a) selected a sample of 48 BL Lacs (40 HBL, 8 LBL) which can be regarded as being practically complete. This is the largest available X–ray selected sample of BL Lacs. The redshift is known for 41 out of the 48 objects, and 8/48 have been detected at $\gamma$–ray energies, (6 with EGRET, E$\gs$ 100 MeV, 1 with Whipple, E$\gs$ 0.3 TeV, 1652+398, and 1 with both instruments, 1101+384).
### BL Lacs, radio selected: the 1 Jy sample
This is the largest complete radio sample of BL Lacs compiled so far. The complete 1 Jy BL Lac sample was derived from the catalog of extragalactic sources with $F_{\rm 5GHz} \ge 1$ Jy (Kühr 1981) with additional requirements on radio flatness ($\alpha_{\rm R} \le 0.5$), optical brightness ($m_V \le 20$) and the weakness of optical emission lines (EW$_{\lambda} \le 5$ Å, evaluated in the source rest frame) (Stickel 1991). This yielded 34 (2 HBL, 32 LBL) sources matching the criteria, 26 with a redshift determination and 4 with a lower limit on it (Stickel, Meisenheimer & Kühr 1994). Out of these 34 objects, 9 have been detected at $\gamma$–ray energies (8 with EGRET, plus 1 with Whipple, 1652+398).
### Flat Spectrum Quasars: Wall & Peacock sample
For FSRQs we considered the sample drawn by Padovani & Urry (1992) from the “2 Jy sample” (Wall & Peacock 1985), a complete flux–limited catalogue selected at 2.7 GHz, covering 9.81 sr, and including 233 sources with $F_{\rm 2.7GHz} > 2$ Jy, and $\alpha_{\rm R} \le 0.5$. It consists of 50 sources with almost complete polarization data, of which 20 are detected in $\gamma$–rays (all with EGRET).
### The Total Blazar sample
Combining the three samples yields a total of 126 blazars (six of them are present in both the radio and X–ray selected samples of BL Lacs), of which 33 detected in $\gamma$–rays. We will refer to them as the [*total blazar sample*]{}.
Multi-frequency Data
--------------------
In view of building average SEDs minimizing the bias introduced by incompleteness, we decided to focus on a few well covered frequencies, at which fluxes are available for most objects. In a separate paper (Ghisellini et al. 1998) we consider a sub–sample of $\gamma$–ray loud blazars with extensive coverage in frequency with the scope of carrying out detailed model fitting for each source.
We chose the following seven well sampled frequencies, that are sufficient to give the basic information on the SED shape from the radio to the X–ray band: radio at 5 GHz, millimeter at 230 GHz, far infrared (IRAS data) at 60 and 25 $\mu$m, near infrared (K band) at 2.2 $\mu$m, optical (V band) at 5500 Å, and soft X–rays at 1 keV. Data were collected from a careful search in the literature and extensive usage of the NASA Extragalactic Database (NED)[^2]. In the radio and optical bands the coverage is complete for all the objects in the three samples, while unfortunately for mm, far and near IR and X–ray fluxes data for some sources are lacking (see Table 1). The worse case is the far IR (25 $\mu$m) band where only 28/126 objects have measured fluxes.
For each source, at each frequency [*from radio to optical*]{} we assigned the average of all the fluxes found in literature. Given the large variability these averages were performed logarithmically (magnitudes).
In principle a suitable alternative to the averaging would be to consider in each band the maximum detected flux (see for instance Dondi & Ghisellini 1995). On one hand this choice could be particularly meaningful in view of the fact that in the $\gamma$–ray band, due to the limited sensitivity of detectors, we are biased towards measuring the brightest states. On the other hand also this option is biased since the value of the maximum flux is strongly dependent on the observational coverage and for most of the objects we only have a few (sometimes a single) observations. Moreover the strength of this bias is ’band–dependent’ and can thus significantly affect the determination of the broad band spectral shape. As both choices present advantages and disadvantages, and since our goal is a statistical analysis, we consider them equally good. The ’averages’ option has been preferred because it is likely to be more robust with respect to the definition of radio–optical SED properties.
In Table 1 a summary of the collected broad band data is reported, with the computed average flux values for each object.
### X–ray data
The knowledge of the X–ray properties is of special relevance because in this band both the synchrotron and inverse Compton processes can contribute to the emission. Since the first mechanism is expected to produce a steep continuum in this band while the second one should give rise to a flat component ($\alpha \le 1$, rising in a $\nu
F_{\nu}$ plot) the shape of the X–ray spectrum can give a fundamental hint for disentangling the two components and inferring the respective peak frequencies.
We privileged the large and homogeneous [*ROSAT*]{} data base. In fact, a large fraction of the 126 sources (90/126) has been observed with the [*ROSAT*]{} PSPC allowing to uniformly derive X–ray fluxes and in many cases, that is for 73 targets of pointed observations, spectral shapes in the 0.1–2.4 keV range (Brunner et al. 1994; Lamer et al. 1996; Perlman et al. 1996b; Urry et al. 1996; Comastri et al. 1995, 1997; Sambruna 1997). X–ray spectral indices were derived from the same observation and, when available, we adopted the $\alpha_{\rm X}$ resulting from fits with neutral hydrogen column density N$_{\rm H}$ allowed to vary. Some of these 90 objects (17) have been only detected in the [*ROSAT*]{} All Sky Survey (RASS) and fluxes are published by Brinkmann, Siebert & Boller (1994) and Brinkmann et al. (1995). Monochromatic fluxes (at 1 keV) for these sources have been derived from the 0.1 – 2.4 keV integrated flux adopting the average spectral index of the sample to which they belong (see Table 4) and the value of the Galactic column in the source direction (Elvis et al 1989; Dickey & Lockman 1990; Lockman & Savage 1995; Murphy et al 1996). When more than one observation was available we give the average flux.
Of the remaining 36 sources, 24 belong to the Slew survey sample and for them we used directly the [*Einstein*]{} IPC flux from Perlman et al. (1996a). The fluxes at 2 keV listed by Perlman et al. (1996a) were converted to 1 keV using the average [*ROSAT*]{} spectral index of the Slew survey sample ($\langle \alpha_{\rm X}\rangle$ =1.40), derived from the 24 sources with a [*ROSAT*]{} measured value.
For other 3 sources, without ROSAT data, we used an [*Einstein*]{} IPC flux, bringing the total number of sources with measured X–ray flux to 117/126.
### $\gamma$–ray data
Within the three samples only a fraction of blazars were detected in $\gamma$–rays, namely 9/34 in the 1 Jy sample, 8/48 in the Slew sample, 20/50 in the FSRQ sample. Four of these sources (0235+164, 0735+178, 0851+202 and 1652+398) are present in both the BL Lac object samples, giving a net number of $\gamma$–ray detections of 33 out of 126 blazars. All but one of them have been observed by EGRET in the 30 MeV – 30 GeV band. For 28/32 a $\gamma$–ray spectral index has been determined. One source, 1652+398 (Mkn 501), has only been detected at very high energies, beyond 0.3 TeV by ground based Cherenkov telescopes (Whipple and HEGRA, Weekes et al. 1996; Bradbury et al. 1997), while EGRET yielded only an upper limit. It is worth noticing that the detected fraction is significantly different between quasars and BL Lacs, being respectively $40 \pm 10.6$ % and $17.1 \pm 5.1$ % for XBLs and RBLs together. However for RBLs only the fraction detected in $\gamma$–rays is $26.5 \pm 9.9$ %, consistent with that of quasars while XBLs only yield $16.7 \pm 6.4$ %.
------------------ ------------- ---------------- ------------------ -----------------------
(1) (2) (3) (4) (5)
IAU name z F$_{\rm 5GHz}$ F$_{\rm 100MeV}$ $\alpha_\gamma$
(Jy) (nJy)
[ 0130$-$171 ]{} [ 1.022 ]{} [ 1.00 ]{} [ 0.122 ]{} [ ... ]{}
[ 0202+149 ]{} [ 1.202 ]{} [ 2.40 ]{} [ 0.383 ]{} [ 1.5 $\pm$ 0.1 ]{}
[ 0234+285 ]{} [ 1.213 ]{} [ 2.36 ]{} [ 0.296 ]{} [ 1.7 $\pm$ 0.3 ]{}
[ 0446+112 ]{} [ 1.207 ]{} [ 1.22 ]{} [ 0.470 ]{} [ 0.8 $\pm$ 0.3 ]{}
[ 0454$-$234 ]{} [ 1.009 ]{} [ 2.2 ]{} [ 0.143 ]{} [ ... ]{}
[ 0458$-$020 ]{} [ 2.286 ]{} [ 2.04 ]{} [ 0.364 ]{} [ ... ]{}
[ 0506$-$612 ]{} [ 1.093 ]{} [ 2.1 ]{} [ 0.062 ]{} [ ... ]{}
[ 0521$-$365 ]{} [ 0.055 ]{} [ 9.7 ]{} [ 0.139 ]{} [ 1.16 $\pm$ 0.36 ]{}
[ 0804+499 ]{} [ 1.433 ]{} [ 2.05 ]{} [ 0.322 ]{} [ 1.72 $\pm$ 0.38 ]{}
[ 0805$-$077 ]{} [ 1.837 ]{} [ 1.04 ]{} [ 0.404 ]{} [ 1.4 $\pm$ 0.6 ]{}
[ 0827+243 ]{} [ 0.939 ]{} [ 0.67 ]{} [ 0.226 ]{} [ 1.21 $\pm$ 0.47 ]{}
[ 0829+046 ]{} [ 0.18 ]{} [ 1.65 ]{} [ 0.132 ]{} [ ... ]{}
[ 0917+449 ]{} [ 2.18 ]{} [ 1.03 ]{} [ 0.075 ]{} [ 0.98 $\pm$ 0.25 ]{}
[ 1156+295 ]{} [ 0.729 ]{} [ 1.65 ]{} [ 1.727 ]{} [ 1.21 $\pm$ 0.52 ]{}
[ 1222+216 ]{} [ 0.435 ]{} [ 0.81 ]{} [ 0.278 ]{} [ 1.50 $\pm$ 0.21 ]{}
[ 1229$-$021 ]{} [ 1.045 ]{} [ 1.1 ]{} [ 0.250 ]{} [ 1.92 $\pm$ 0.44 ]{}
[ 1313$-$333 ]{} [ 1.210 ]{} [ 1.47 ]{} [ 0.098 ]{} [ 0.8 $\pm$ 0.3 ]{}
[ 1317+520 ]{} [ 1.060 ]{} [ 0.66 ]{} [ 0.079 ]{} [ ... ]{}
[ 1331+170 ]{} [ 2.084 ]{} [ 0.713 ]{} [ 0.091 ]{} [ ... ]{}
[ 1406$-$076 ]{} [ 1.494 ]{} [ 1.08 ]{} [ 1.013 ]{} [ 1.03 $\pm$ 0.12 ]{}
[ 1604+159 ]{} [ 0.357 ]{} [ 0.50 ]{} [ 0.260 ]{} [ 0.99 $\pm$ 0.50 ]{}
[ 1606+106 ]{} [ 1.227 ]{} [ 1.78 ]{} [ 0.312 ]{} [ 1.20 $\pm$ 0.30 ]{}
[ 1622$-$297 ]{} [ 0.815 ]{} [ 1.92 ]{} [ 2.416 ]{} [ 1.2 $\pm$ 0.1 ]{}
[ 1622$-$253 ]{} [ 0.786 ]{} [ 2.2 ]{} [ 0.336 ]{} [ 1.3 $\pm$ 0.2 ]{}
[ 1730$-$130 ]{} [ 0.902 ]{} [ 6.9 ]{} [ 0.258 ]{} [ 1.39 $\pm$ 0.27 ]{}
[ 1739+522 ]{} [ 1.375 ]{} [ 1.98 ]{} [ 0.236 ]{} [ 1.23 $\pm$ 0.38 ]{}
[ 1933$-$400 ]{} [ 0.966 ]{} [ 1.48 ]{} [ 0.158 ]{} [ 1.4 $\pm$ 0.2 ]{}
[ 2032+107 ]{} [ 0.601 ]{} [ 0.77 ]{} [ 0.192 ]{} [ 1.5 $\pm$ 0.3 ]{}
[ 2344+514 ]{} [ 0.044 ]{} [ 0.215 ]{} [ 0.8$^{a}$ ]{} [ ... ]{}
[ 2356+196 ]{} [ 1.066 ]{} [ 0.70 ]{} [ 0.311 ]{} [ ... ]{}
------------------ ------------- ---------------- ------------------ -----------------------
: Basic data for the 30 $\gamma$–ray detected sources not included in our samples : (1): IAU name; (2): redshift; (3): radio flux at 5 GHz; (4): $\gamma$–ray flux at 100 MeV; (5): EGRET spectral index
[**References of Table 2:**]{}
References for the data here reported are listed in the Notes to Table 5.
$^{(a)}$ source detected only by Whipple. The given value is the integrated flux measured at E $>$ 300 GeV, in units of $10^{-11}$ photons cm$^{-2}$ s$^{-1}$.
Many other blazars ($\sim$ 30) have been detected by EGRET, but do not fall in our samples. One can consider the group of $\gamma$–ray detected objects as a sample in its own right, though not a complete one at present, since a significant fraction of the sky has been surveyed though not uniformly. This larger sample comprises 66 sources (Fichtel et al. 1994; von Montigny et al. 1995; Thompson et al. 1995, 1997; Mattox et al. 1997 and references therein), of which 60 have a measured redshift, and 48 an estimate of the spectral index. To this set we can add Mkn 501 (already included in both our BL Lac samples), and 2344+514, detected only by the Whipple telescope (Fegan 1996). We will use this additional information to discuss whether the $\gamma$–ray properties of our samples can be representative of the whole $\gamma$–ray loud population and if so, to increase the statistics (see section 3.1). We therefore collected basic data also on all of the 30 (29 EGRET plus 2344+514) $\gamma$–ray detected AGN/blazars not included in the complete samples. They are reported in Table 2.
Given the large amount of observations and analysis of the same data by different authors, for the selection of the flux and spectral index we used the following criteria: i) spectral index and flux referring to the same observation, ii) data corresponding to a single viewing period, iii) if data were analyzed by different authors, the results of the most recent analysis are preferred.
$\gamma$–ray data are usually given in units of photons cm$^{-2}$ s$^{-1}$ above an energy threshold (e.g. for EGRET E $\gs$ 100 MeV). We converted them to monochromatic fluxes at 100 MeV integrating a power law in photons with the measured or assumed (the average) spectral index.
-- --------------- ------ --------- --------- -------
band Slew 1 Jy FSRQ refs.
[(1)]{} (2) (3) (4) (5)
radio 0.20 $-$0.27 $-$0.30 1
mm 0.32 0.32 0.48 2
IRAS 0.60 0.80 1.00 3
IR–opt 0.67 1.21 1.52 4
X–rays 1.40 1.25 0.83 5
$\gamma$–rays 0.98 1.26 1.21 5
-- --------------- ------ --------- --------- -------
: spectral indices used for K–correction of monochromatic fluxes: (1) spectral band; (2) Slew; (3) 1Jy; (4) FSRQ; (5) references.
[**References to Table 3:**]{}
\(1) Stickel et al. 1994; (2) Gear et al. 1994; (3) derived from IRAS data; (4) Bersanelli et al. 1992; (5) this work, see Table 5.
### Luminosities and K–correction
All fluxes were K-corrected and luminosities were computed with the following choices:
- we considered lower limits on redshift (4 sources) as detections, while we assigned the average redshift of the sample to the few sources without any estimate (4 in the 1 Jy sample, for which $\langle z \rangle = 0.492$, and 6 in the Slew survey sample, $\langle z \rangle = 0.194$);
- luminosity distances were calculated adopting H$_0$=50 km s$^{-1}$ Mpc$^{-1}$ and q$_0$=0;
- fluxes were K–corrected according to the following prescriptions. For radio–to–optical data we used average spectral indices derived from the literature (see Table 3). For X–ray and $\gamma$–ray data we used measured power law spectral indices, when available, or the average index derived for the sources of the same sample (see Table 3).
Results
=======
Distributions of properties
---------------------------
Since the fraction of objects detected in $\gamma$-rays in the three samples is rather small, it is important to ask whether the detected sources are representative of each sample as a whole or are distinguished in other properties from the rest of the objects in it. Moreover we want to verify whether the $\gamma$-ray detected sources in general differ from those belonging to the complete samples.
We therefore computed the distributions of various quantities, i.e. redshift, luminosities and broad band spectral indices, for objects belonging to each sample. These are shown in Figs. 1–5 for the three samples and the total blazar one, evidentiating those sources detected in the $\gamma$–ray band as grey shaded areas in the histograms. The redshift, radio (at 5GHz) and X–ray (at 1keV) luminosities (expressed as $\nu$L$_\nu$, erg/sec) are shown in Figs. 1–3, while the distributions of the broad band spectral indices $\alpha_{\rm RO}$ and $\alpha_{\rm RX}$ are plotted in Figs. 4 and 5, respectively.
The redshift distributions (Fig. 1) of the three complete samples show the known tendency towards the detection of FSRQs at higher $z$, being the latter ones more powerful radio sources, as shown in Fig. 2.
In the same figure and in Fig. 3, the tendency for XBL to have similar X–ray but lower radio luminosities compared to RBL is also apparent. Correspondingly, it appears from Figs. 4,5 that $\alpha_{\rm RO}$ and $\alpha_{\rm RX}$ increase from XBL to RBL while FSRQ have $\alpha_{\rm RO}$ slightly larger and $\alpha_{\rm RX}$ similar to RBLs. Later on (section 3.2) we will show that there is a relationship between these spectral indices and the peak frequency of the synchrotron component.
We note that for all of the distributions there is continuity in properties not only between the two BL Lac samples, but also between BL Lacs and FSRQs.
It is clear from Figs. 1-5 that for the RBL and FSRQ samples the $\gamma$–ray detected sources do not differ from non–detected ones in any of the considered quantities while for the Slew sample there is a tendency for $\gamma$–ray loud sources to have larger L$_{\rm 5GHz}$, $\alpha_{\rm RX}$ and $\alpha_{\rm RO}$. This indicates that either the radio luminosity or radio–loudness are important in determining the $\gamma$–ray detection. On the contrary, and in some sense surprisingly, the X–ray luminosity does not seem to play an important role with respect to the $\gamma$–ray emission, although the X-ray band is the closest in energy to the $\gamma$–rays . We will come back to this issue later on (§3.5). We checked the possible difference of means and variances of the distributions with the t–student’s test and only for the $\alpha_{\rm RX}$ of the Slew sample the significance is higher than 95 per cent.
Except for the case of the Slew survey, we conclude that the $\gamma$–ray detected sources are representative of the samples as a whole, being indistinguishable from the others in terms of radio to X-ray broad band properties and power.
We also checked that the $\gamma$-ray detected sources belonging to our samples are homogeneous with respect to all of the $\gamma$-ray blazars detected so far. In Fig. 6 we compare the redshift distributions, the radio luminosities and fluxes and the $\gamma$–ray spectral index. The grey shaded areas represent the $\gamma$–ray sources belonging to the complete samples considered here. We conclude that there is no significant difference.
Nevertheless, we are aware that the limited sensitivity of the EGRET instrument implies that at a given radio flux, only the $\gamma$–ray loudest sources are detected. Therefore the non detected ones are probably on average weaker in $\gamma$-rays. Impey (1996) quantified this effect by taking into account the correlation between radio and $\gamma$–ray luminosities (see §3.3.1), and other observables. Assuming a Gaussian distribution of the $\gamma$–ray to radio flux ratio he estimated the width of the distribution and the “true” ratio referring to the whole population, which could be a factor 10 lower than the observed one. There could be a real spread in the intrinsic properties of the blazar population, the $\gamma$-ray detected blazars being intrinsically louder than the rest of the population. Alternatively this may be due to variability, i.e. a source is detected only when it undergoes a flare. The observed ratio would thus refer to flaring states, while the average level of each source would be lower.
Since variability is a distinctive property of blazars and has been observed to occur also in $\gamma$–rays, often with extremely large amplitude (greater than a factor 10) (3C 279, Maraschi et al 1994; PKS 0528+134, Mukherjee et al. 1996; 1622$-$297, Mattox et al. 1997) the latter alternative is likely although the problem remains an open one. We conclude that the average $\gamma$–ray luminosities computed here are necessarily overestimated. However we chose not to correct for this effect given the uncertainties. In particular the “bias factor” for different classes of blazars could be different if their $\gamma$–ray variability properties (amplitude and duty cycle) are (Ulrich, Maraschi & Urry 1997).
Synchrotron peak frequency
--------------------------
As previously noted the SEDs clearly show a broad peak between radio and UV–X–rays. In order to determine the position of the peak of the synchrotron component in individual objects with an objective procedure, we fitted the data points for each source (in a $\nu$ vs. $\nu$L$_\nu$ diagram) with a third degree polynomial, which yields a complex SED profile, with an upturn allowing for X–ray data points not to lay on the direct extrapolation from the lower energy spectrum. In many cases there is evidence that the X–ray component, even in the soft [*ROSAT*]{} PSPC band, is due to the inverse Compton process (e.g. Sambruna 1997; Comastri et al. 1997). Thus to impose that the X–ray point smoothly connects to the lower energy data, as would happen in a parabolic fit, could be misleading for a determination of the synchrotron peak frequency. We used a simple parabola when the cubic fit was not able to find a maximum, which typically happens when the peak occurs at energies higher than X–rays. In fact when the peak moves to high enough frequencies (typically beyond the IR band), the X–ray flux is completely dominated by the synchrotron emission, and the results given by the cubic and parabolic fits are fully consistent. In 8 cases neither the cubic nor the parabolic fit were able to determine a peak frequency/luminosity mainly due to the paucity of data points.
### Synchrotron Peak Frequency vs. Luminosity
The peak frequencies derived with the above procedure (defined as the frequencies of the maximum in the fitted polynomial function) are plotted in Fig. 7a,b,c versus the radio and $\gamma$–ray luminosities and vs the corresponding peak luminosities, as determined from the fits. Let us stress once again the continuity between the different samples. Considering the samples together strong correlations are present between these quantities, in the sense of $\nu_{\rm peak,sync}$ decreasing with increasing luminosity. The results of Kendall’s $\tau$ statistical test (Table 4) show that the correlations are highly significant.
Since on one hand in flux limited samples spurious correlations can be introduced by the luminosity/redshift relation and on the other hand the correlations might be due to evolutionary effects genuinely related to redshift, we checked its role in two ways. We estimated the possible correlation of the relevant quantities with redshift directly, and performed partial correlation tests between two quantities subtracting out the common dependence on $z$ (Padovani 1992b)(see results in Table 4). In addition, in order to have an independent check on the redshift bias, we also considered the significance of the correlations restricting them to objects with $z < 0.5$ (see Table 4).
The correlation between $\nu_{\rm peak,sync}$ and L$_{\rm 5 GHz}$ still holds after subtraction of the very strong dependence on redshift. The same is true for the relation between $\nu_{\rm peak,sync}$ and the $\gamma$-ray luminosity, although the significance is much smaller, due to the smaller number of sources. On the other hand the correlation between $\nu_{\rm peak,sync}$ and L$_{\rm peak,sync}$ is strongly weakened when subtracting the redshift effect.
Considering only the $z < 0.5$ interval the significance of the first two correlations persists and does not change when the redshift dependence is subtracted. These values can then be considered as irreducible, being the signature of a [*true dependence*]{} of $\nu_{\rm peak,sync}$ on luminosity. This result can also be read as a check of the reliability of the partial correlation procedure. On the contrary the correlation $\nu_{\rm peak,sync}$ vs. L$_{\rm peak,sync}$ disappears at low redshifts, due to the narrow interval of values spanned by L$_{\rm peak,sync}$, that varies less with the change of peak frequency than both radio and $\gamma$–ray luminosity do.
[@ccccccccccccccc]{} & & && &&\
\
&[x$_1$]{} & [x$_2$]{} && [x$_1$/x$_2$]{} && [x$_1$/z]{} & [x$_2$/z]{} & [x$_1$/x$_2$$-Z$]{} && [x$_1$/x$_2$]{} && [x$_1$/z]{} & [x$_2$/z]{} & [x$_1$/x$_2$$-Z$]{}\
&(1)&(2)&& (3)&&(4)&(5)&(6)&& (7)&&(8)&(9)&(10)\
&&&&&&&&&&&&&&\
&$\nu_{\rm peak, sync}$ & L$_{\rm 5 GHz}$ && 1.2e-16 && \[1.1e-9\] & \[1.9e-31\] & 1.1e-11 && 1.3e-8 && \[ — \] & \[4.0e-4\] & 9.2e-9\
&&&&&&&&&&&&&&\
&$\nu_{\rm peak, sync}$ & L$_{\rm peak, sync}$ && 4.9e-8 && \[1.1e-9\] & \[9.5e-27\] & 6.5e-2 && — && \[ — \] & \[1.2e-7\] & —-\
&&&&&&&&&&&&&&\
&$\nu_{\rm peak, sync}$ & $\alpha_{\rm RO}$ && 1.1e-23 && \[1.1e-9\] & \[7.7e-11\] & 4.7e-19 && 1.8e-7 && \[ — \] & \[ — \] & 2.9e-7\
&&&&&&&&&&&&&&\
&$\nu_{\rm peak, sync}$ & $\alpha_{\rm RX}$ && 4.2e-17 && \[1.1e-9\] & \[6.0e-5\] & 1.9e-14 && 4.1e-15 && \[ — \] & \[ — \] & 8.4e-15\
&&&&&&&&&&&&&&\
\
&&&&&&&&&\
& & && &&\
\
&$\nu_{\rm peak, sync}$ & L$_\gamma$ && 2.1e-3 && \[3.4e-2\] & \[1.5e-10\] & 1.8e-2 && 2.4e-2 && \[ — \] & \[2.4e-3\] & 6.4e-2\
&&&&&&&&&&&&&&\
&L$_{\rm 5GHz}$ & L$_\gamma$ && 6.0e-11 && \[5.5e-10\] & \[1.5e-10\] & 4.2e-5 && 3.9e-3 && \[5.2e-2\] & \[2.4e-3\] & 2.4e-2\
&&&&&&&&&&&&&&\
&$\nu_{\rm peak, sync}$ & L$_\gamma$/L$_{\rm peak,sync}$ && 2.7e-5 && \[3.4e-2\] & \[1.5e-5\] & 2.2e-4 && 1.4e-3 && \[ — \] & \[7.3e-1\] & 4.1e-3\
&&&&&&&&&&&&&&\
&$\nu_{\rm peak, sync}$ & L$_\gamma$/L$_{\rm 5500\AA}$ && 2.5e-6 && \[3.4e-2\] & \[4.2e-5\] & 1.7e-5 && 2.4e-3 && \[ — \] & \[ — \] & 6.8e-3\
&&&&&&&&&&&&&&\
&&&&&&&&&\
& & && &&\
\
&L$_{\rm 5GHz}$ & L$_\gamma$ && 4.8e-15 && \[4.8e-15\] & \[1.8e-14\] & 2.4e-6 && 1.8e-3 && \[1.1e-2\] & \[1.4e-4\] & 4.1e-2\
&&&&&&&&&&&&&&\
$^{(a)}$ we considered only sources with at least a lower limit on redshift, and for which it has been possible to determine the “synchrotron" peak frequency by means of the polynomial fit.
### Synchrotron Peak Frequency vs. Broad Band Spectral indices
The relations between the synchrotron peak frequency and each of the two point spectral indices $\alpha_{\rm RO}$ and $\alpha_{\rm RX}$ are shown in Fig. 8. Also these quantities are strongly correlated (see Table 4). In fact recent papers (e.g. Maraschi et al. 1995; Comastri et al. 1995; Comastri et al. 1997), suggested that the position of the synchrotron peak could be devised from the values of broad band spectral indices.
We see that the knowledge of any of the two spectral indices is enough to guess the position of the peak of the synchrotron component, except for some ranges, namely $\nu_{\rm
peak,sync} > 10^{16-17}$ Hz for both $\alpha_{\rm RO}$ and $\alpha_{\rm RX}$, and $\nu_{\rm peak,sync} < 10^{14}$ Hz for $\alpha_{\rm RX}$. These “failures” can be explained bearing in mind the typical shape of the blazar SEDs (see inset in Fig. 8): when the spectrum peaks at low frequencies, X–rays are typically dominated by the inverse Compton, flat spectrum, component whose luminosity level is strongly correlated with the radio one (Fossati et al. 1997), and then the X–ray/radio ratio (i.e. $\alpha_{\rm RX}$) tends to a fixed value. Conversely the Compton component begins to dominate the ([*ROSAT*]{}) X-ray band when $\alpha_{\rm RX}
\sim 0.75$, corresponding to $\nu_{\rm peak,sync} \lg 3 \times 10^{14}$ Hz. It is interesting to note that the adopted dividing threshold between LBL and HBL has been set to this same value from purely practical purposes, while in the light of the result above it assumes a more “physical" meaning. LBL sources have Compton–dominated soft–X–ray emission, while in HBL this is pure synchrotron.
At the other end of the spectrum a problem arises when $\nu_{\rm peak,sync}$ moves at energies higher than that used to compute the broad band spectral index. The reason is that the ratio between, for instance, optical and radio luminosity is no longer sensitive to the peak moving further towards higher frequencies, because both the radio and optical bands lay on the same (rising) branch of the synchrotron “bump”.
For comparison we draw in Fig. 8 the loci of $\alpha_{\rm RO}$–$\nu_{\rm peak,sync}$ and $\alpha_{\rm RX}$–$\nu_{\rm peak,sync}$ obtained from a set of SEDs of the kind reported in the inset, and that we are going to discuss in more detail in section 3.5. The parameterization describes the observed features very well.
### Synchrotron Peak Frequency vs. $\gamma$–ray dominance
In Fig. 9 $\nu_{\rm peak,sync}$ is plotted against the $\gamma$–ray dominance parameter, defined as the ratio between the $\gamma$–ray and the synchrotron peak luminosities. A strong correlation (see Table 4) is present over four orders of magnitude in $\nu_{\rm peak,sync}$, in the sense of a decrease in the $\gamma$–ray dominance with an increase of the synchrotron peak frequency. In the same figure we plotted also the ratio between the $\gamma$–ray and optical luminosities, to check if the latter could eventually be a good indicator of the $\gamma$–ray dominance, with the advantage of being an observed quantity. In fact there is little difference, at most a factor 3 for a quantity spanning more than three decades.
Average SEDS
------------
Having discussed extensively the possible biases introduced by the limited number of $\gamma$–ray detected sources in the complete samples we construct here the average SEDs for each sample. We will come back later to the bias problem (Section 4).
The averaging procedure has been performed on the logarithms of the luminosities at each frequency. Apart from the problems in the $\gamma$-ray range discussed above the incompleteness of the data coverage at some frequencies could also introduce a bias in the average values. For instance in the Slew survey sample only 10/48 objects have a flux measured at 230 GHz, and they are the more luminous sources at 5 GHz. Averaging independently L$_{\rm 230GHz}$ (for 10 objects) and L$_{\rm 5GHz}$ (for 48 objects) we would obtain a ratio between the two luminosities higher than that derived considering only the subsample of 10 sources, and presumably higher than the actual one, too.
To reduce this kind of effect we first normalized the monochromatic luminosities to the radio luminosity for each source, we computed average ratios $\langle \log(L_{\nu^*}/L_{\rm 5GHz}) \rangle|_{sub}$, considering only the subsample of sources with a measured flux at $\nu^*$, and used that ratio to compute the average monochromatic luminosity at $\nu*$ for all sources in the sample as $\langle \log(L_{\nu^*}) \rangle|_{all} =
\langle \log(L_{\rm 5GHz}) \rangle|_{all} +
\langle \log(L_{\nu^*}/L_{\rm 5GHz}) \rangle|_{sub}$. In this way we basically averaged the spectral shape between $\nu^*$ and 5 GHz for the measured objects and assigned that spectral shape to the sample.
The X–ray and $\gamma$–ray spectral indices have been averaged with a simple mean, without weighting.
The average broad band spectra for each of the three samples are shown in Fig. 10. The 6 sources common to the radio and the X–ray selected BL Lac samples are considered in both of them. Average luminosities entering Fig. 10 are reported in Table 5 together with the number of sources contributing at each frequency.
It is apparent from Fig. 10 that the three samples refer to objects with different average integrated luminosities and that the peak frequency of the power emitted between the radio and the X-ray band moves from the X-ray to the far infrared band going from the XBL to the FSRQ samples as anticipated from the analysis of single objects in the previous section 3.2. Correspondingly the $\gamma$–ray luminosities increase and the $\gamma$–ray spectra steepen suggesting that also the peak frequency of the high energy emission moves to lower frequencies. The overall similarity and regularity of the SEDs of the different samples as well as the continuity in the properties of the individual objects discussed in section 3.2 suggest a basic similarity of all blazars irrespective of their original classification and different appearance in a specific spectral band.
[@clccccccccc ]{} &&&&\
\
& Band & Slew & 1 Jy & W&P && $<$42 & 42$-$43 & 43$-$44 & 44$-$45 & $>$45\
&&&&&&&&&&\
& 5 GHz & 41.71 & 43.69 & 44.81 && 41.24 & 42.47 & 43.71 & 44.54 & 45.39\
& & 48 & 34 & 50 && 38 & 10 & 17 & 44 & 17\
&&&&&&&&&&\
& 230 GHz & 43.11 & 45.01 & 46.11 && 42.64 & 43.77 & 45.13 & 45.83 & 46.63\
& & 10 & 34 & 50 && 5 & 7 & 15 & 44 & 17\
&&&&&&&&&&\
& 60 $\mu$m & 44.17 & 45.94 & 46.84 && 43.73 & 44.65 & 46.09 & 46.65 & 47.61\
& & 12 & 19 & 13 && 6 & 5 & 8 & 16 & 2\
&&&&&&&&&&\
& 25 $\mu$m & 44.25 & 46.07 & 46.93 && 43.74 & 44.95 & 46.08 & 46.79 & 47.69\
& & 10 & 15 & 8 && 4 & 6 & 7 & 9 & 2\
&&&&&&&&&&\
& K–band & 44.64 & 45.86 & 46.49 && 44.42 & 45.04 & 45.96 & 46.27 & 47.21\
& & 23 & 31 & 28 && 13 & 10 & 15 & 32 & 6\
&&&&&&&&&&\
& V–band & 44.91 & 45.68 & 46.58 && 44.61 & 45.01 & 45.82 & 46.27 & 47.21\
& & 48 & 34 & 50 && 38 & 10 & 17 & 44 & 17\
&&&&&&&&&&\
& 1 keV & 44.94 & 44.72 & 45.98 && 44.81 & 44.11 & 44.92 & 45.66 & 46.50\
& & 48 & 32 & 43 && 38 & 10 & 15 & 42 & 12\
&&&&&&&&&&\
& 100 MeV & 44.45 & 46.50 & 47.93 && 44.24 & 44.79 & 46.67 & 47.71 & 48.68\
& & 7 & 8 & 20 && 3 & 5 & 9 & 33 & 12\
&&&&&&&&&&\
& $\alpha_{\rm X}$ & 1.40 $\pm$ 0.07 & 1.25 $\pm$ 0.09 & 0.83 $\pm$ 0.08 && 1.37 $\pm$ 0.09 & 1.55 $\pm$ 0.15 & 1.16$\pm$ 0.14& 1.11 $\pm$ 0.08 & 0.57 $\pm$ 0.13\
& & 24 & 31 & 24 && 16 & 8 & 14 & 26 & 9\
&&&&&&&&&&\
& $\alpha_\gamma$& 0.98 $\pm$ 0.32 & 1.26 $\pm$ 0.26 & 1.21 $\pm$ 0.09 && 0.64 $\pm$ 0.07 & 0.73 $\pm$ 0.47 & 1.37 $\pm$ 0.31 & 1.06 $\pm$ 0.13 & 1.30 $\pm$ 0.08\
& & 6 & 6 & 18 && 2 & 4 & 7 & 25 & 11\
&&&&&&&&&&\
(5 and 230 GHz): Becker, White & Edwards 1991; Bloom et al. 1994; Gear et al. 1986; Gear 1993a; Gear et al. 1994; Kühr et al. 1981; Kühr & Schmidt 1990; Perlman et al. 1996a; Reuter et al. 1997; Steppe et al. 1988, 1992, 1993; Stevens et al. 1994; Stickel et al. 1991; Stickel et al. 1993; Stickel, Meisenheimer & Kühr 1994;Tornikovski et al. 1993, 1996; Terasranta et al. 1992.
(IR–optical data): Allen, Ward & Hyland 1982; Ballard et al. 1990; Bersanelli et al. 1992; Bloom et al. 1994; Brindle et al. 1986; Brown et al. 1989; Elvis et al. 1994; Falomo et al. 1988; Falomo et al. 1993a; Falomo et al. 1993b; Falomo, Scarpa & Bersanelli 1994; Gear et al. 1986; Gear 1993b; Glass 1979, 1981; Holmes et al. 1984; Impey & Brand 1981; Impey & Brand 1982; Impey et al. 1982; Impey et al. 1984; Impey & Neugebauer 1988; Impey & Tapia 1988, 1990; Jannuzi, Smith & Elston 1993, 1994; Landau et al. 1986; Lepine, Braz & Epchtein 1985; Lichtfield et al. 1994; Lorenzetti et al. 1990; Mead et al. 1990; O’Dell et al. 1978; Pian et al. 1994; Sitko & Sitko 1991; Smith et al. 1987; Stevens et al. 1994; Wright, Ables & Allen 1983.
(X–rays): Brinkmann et al. 1994; Brinkmann et al. 1995; Brunner et al. 1994; Comastri et al 1995; Comastri et al 1997; Lamer, Brunner & Staubert 1996; Maraschi et al. 1995; Perlman et al. 1996a; Perlman et al. 1996b; Sambruna 1997; Urry et al. 1996.
($\gamma$–rays): Bertsch et al. 1993; Catanese et al. 1997; Chiang et al. 1995; Dingus et al. 1996; Fichtel et al. 1994; Hartman et al. 1993; Lin et al. 1995; Lin et al. 1996; Madejski et al. 1996; Mattox et al. 1997; Mukherjee et al. 1995, 1996; Nolan et al. 1996; Quinn et al. 1996; Radecke et al. 1995; Shrader et al. 1996; Sreekumar et al. 1996; Thompson et al. 1993; Thompson et al. 1995; Thompson et al. 1996; Vestrand, Stacy & Sreekumar 1995; von Montigny et al. 1995.
We therefore considered the merged total sample with the scope of finding the key parameter(s) governing the whole blazar phenomenology. Since luminosity appears to have an important role in that it correlates with the main spectral parameters we decided to bin the total blazar sample according to luminosity, irrespective of the original classification. We used the 5 GHz radio luminosity which is available for all objects. It may be desirable to use the total integrated luminosity which in all cases is close to the $\gamma$–ray one. However the latter is only available for a few objects. We note that a correlation between $\gamma$–ray and radio luminosity has been claimed by many authors using different techniques (Dondi & Ghisellini, 1995; Mattox et al. 1997). It is however still being debated whether it is true or it arises from selection effects, connected with the common redshift dependence of luminosities, and with the exclusion of upper limits, which could favour the appearance of a spurious correlation. It is worth mentioning that Mücke et al. (1997) using a technique designed to take into account both these effects did not find any significant correlation between radio and $\gamma$-ray data for a sample of 38 extragalactic EGRET sources.
We also checked this correlation on both the 31 EGRET detected sources included in our samples and the larger “comparison sample" of 62 EGRET sources. In Table 4 we report the significance of the correlation, together with its value after subtracting the common redshift dependence through a partial correlation test, and its significance for samples restricted to $z<0.5$. In all cases the radio and $\gamma$–ray luminosities correlate significantly.
In Figs. 11 $\alpha_{\rm X}$ and $\alpha_\gamma$ for individual sources are plotted against the radio power, both showing a good correlation with it. Comastri et al. (1997) discussed the interesting consequences of the apparent anti–correlation between X–ray and $\gamma$-ray spectral indices, without relating it to any “absolute” parameter, such as luminosity. Here again we see that these other spectral properties have a dependence on radio luminosity.
Since in some luminosity bins the number of $\gamma$-ray detected sources is small, we used an indirect procedure to associate $\gamma$–ray fluxes and spectra to our average SEDs, taking advantage of the whole body of information regarding the $\gamma$–ray properties of blazars. Namely for each luminosity bin $\langle$L$_{\gamma}\rangle$ and $\langle\alpha_\gamma\rangle$ were computed from blazars in the general EGRET–detected sample falling into the same L$_{\rm 5GHz}$ bin. The basic assumption is the uniformity of the spectral properties, as discussed in section 3.1. The resulting SEDs are shown in Fig. 12 and average luminosities, X–ray and $\gamma$–ray spectral indices, and number of sources are reported in Table 5. The most interesting result is that the trends pointed out for the three separate sub–classes of blazars (Fig. 10) hold for the total blazar sample, irrespective of the original classification of sources, when the radio luminosity is adopted as the key parameter characterizing each object.
Discussion
==========
In Fig. 12 we superimposed to the averaged data a set of (dashed) lines, whose main goal is to guide the eye. The radio–X–ray SED is approximated with a power law starting in the radio domain continuously connecting at $\nu \simeq 5 \times 10^{11}$ Hz with a parabolic branch. This latter describes the peak of the SED and its steepening beyond it. In soft X–rays a rising power law, representing the onset of the hard inverse Compton component is summed to this curved “synchrotron" component. The normalization of this second X–ray component is kept fixed relative to the radio one. Based on our findings (see Fig. 7a), we then assume that the peak frequency of the synchrotron spectral component is (inversely) related to radio luminosity. The simplest hypothesis of a straight unique relationship between $\nu_{\rm peak,sync}$ and L$_{\rm 5GHz}$ does not give a good result when compared with the average SEDs. We then allow for a different SED–shape/luminosity dependence for high and low luminosity objects, a distinction that turns out to roughly correspond also to that between objects with and without prominent emission lines. We adopted a “two–branch" relationship between $\nu_{\rm peak,sync}$ and L$_{\rm 5GHz}$ in the form of two power laws $\nu_{\rm peak,sync}
\propto$ L$_{\rm 5GHz}^{-\eta}$, with $\eta = 0.6$ or $\eta = 1.8$ for log(L$_{\rm 5GHz}$) higher or smaller than 42.5, respectively. The shape of the analytic SEDs is parabolic with a smooth connection to a fixed power law in the radio and the loci of the maxima as defined above. A full description of the parameterization can be found in Fossati et al. (1997) where a similar scheme was proposed to account for the source number densities of BL Lacs with different spectral properties (LBL and HBL).
The analytic representation of the second spectral component (X–ray to $\gamma$–rays) is a parabola of the same width as the synchrotron one, and has been obtained assuming that: (a) the ratio of the frequencies of the high and low energy peaks is constant ($\nu_{\rm peak, Compt}/\nu_{\rm peak, sync}~\simeq 5 \times 10^8$), (b) the high energy ($\gamma$–ray) peak and radio luminosities have a fixed ratio, $\nu_\gamma$L$_{\rm peak,gamma} / \nu_{\rm 5GHz}$L$_{\rm 5GHz}
\simeq 3 \times 10^3$. Given the extreme simplicity of the latter assumptions, it is remarkable that the phenomenological analytic model describes the run of the average SEDs reasonably well. The worst case refers to the second luminosity bin: the analytic model predicts a $\gamma$-ray luminosity larger than the computed bin average by a factor of 10 (but predicts the correct spectral shape). We note that only 5 $\gamma$–ray detected objects fall in this bin.
The results derived from the above analysis (see in particular Figs. 10–12) can then be summarized as follows:
1. [*two peaks*]{} are present in all the SEDs. The first one (synchrotron) is anticorrelated with the source luminosity (see Figs. 7 and Table 4), moving from $\sim 10^{16}-10^{17}$ Hz for less luminous sources to $\sim 10^{13}-10^{14}$ Hz for the most luminous ones.
2. the [*X–ray spectrum*]{} becomes harder while the $\gamma$[*–ray spectrum*]{} softens with increasing luminosity, indicating that the second (Compton) peak of the SEDs also moves to lower frequencies from $\sim
10^{24}-10^{25}$ Hz for less luminous sources to $\sim 10^{21}-10^{22}$ Hz for the most luminous ones;
3. therefore [*the frequencies of the two peaks are correlated*]{}: the smaller the $\nu_{\rm peak,sync}$ the smaller the peak frequency of the high energy component; a comparison with the analytic curves shows that the data are consistent with a constant ratio between the two peak frequencies;
4. increasing L$_{\rm 5GHz}$ increases the $\gamma$[*–ray dominance*]{}, the ratio of the power emitted in the inverse Compton and synchrotron components, estimated with the ratio of their respective peak luminosities (see also Fig. 9).
The fact that the trends present when comparing the different samples (Fig. 10), persist when the total blazar sample is considered and binned according to radio luminosity only, suggests that we deal with a [*continuous spectral sequence*]{} within the blazar family, rather than with separate spectral classes. In particular the “continuity” clearly applies also to the HBL – LBL subgroups: HBL have the lowest luminosities and the highest peak frequencies.
An interesting result apparent from the average SEDs is the variety and complexity of behaviour shown in the X–ray band. As expected the crossing between the synchrotron and inverse Compton components can occur below or above the X-ray band affecting the relation between the X–ray luminosity and that in other bands. A source can be brighter than another at 1 keV being dimmer in the rest of the radio–$\gamma$–ray spectrum except probably in the TeV range. This effect narrows the range of values spanned by L$_{\rm 1keV}$ and explains why $\gamma$–ray detected sources do not select a particular range in the X–ray luminosity distributions (see Fig. 3) while this happens for L$_{\rm 5GHz}$.
Using this simple scheme of SED parameterization we can compute the luminosities in the EGRET (30 MeV – 3 GeV) and Whipple (0.3 – 10 TeV) bands. These are plotted in Fig. 13 (bottom panel) together with their ratio with the radio and X–ray luminosities (top and middle panel respectively).
It is easy to recognize that for a given radio flux sources with $\nu_{\rm peak,sync}$ around $10^{14}$ have the largest relative flux in the EGRET band, because the peak of the Compton component falls right there (Fig. 13, top panel). For higher values of $\nu_{\rm peak,sync}$ the $\gamma$–ray peak moves to higher energies too and the contribution in the EGRET band is reduced. For sufficiently high $\nu_{\rm peak,sync}$ the $\gamma$–ray peak reaches the TeV band where it becomes detectable.
Qualitatively the same general behaviour is present also in the ratios between EGRET/Whipple fluxes and the X–ray one (Fig. 13, middle panel). There are however a couple of significant differences: firstly the EGRET/X–ray ratio profile, while still peaking around $10^{14}$ Hz, is sharper than in the EGRET/radio case, secondly the TeV relative flux distribution is broader and skewed towards lower values of the synchrotron peak frequency. Thus, for a given X–ray flux (as would be the case in a flux limited X–ray selected sample) only those sources falling in the restricted interval $\nu_{\rm peak,sync} \sim 10^{13}-10^{15}$ Hz would have a flux in the EGRET band high enough to be detectable. On the other hand, for the TeV band it turns out that the chance of being observable is not confined to very extreme HBLs, with X-ray synchrotron peaks, but also intermediate BL Lacs could reach a comparable TeV flux.
Since $\nu_{\rm peak,sync}$ is directly related to both $\alpha_{\rm RO}$ and $\alpha_{\rm RX}$ we can understand now the tendency (section 3.1) of $\gamma$–ray detected sources in the Slew survey to have larger values of $\alpha_{\rm RO}$ and $\alpha_{\rm RX}$. Moreover, due to the fact that in the Slew sample LBLs are only a small fraction, the discussion above explains also the lower EGRET detection rate with respect to other blazar samples.
The proposed scenario relates the shape of the continuum to the total source power. It follows that predictions of this unifying scheme on both the detectability of blazars at $\gamma$–ray energies (in view of more sensitive $\gamma$–ray detectors, GLAST, improved Cherenkov telescopes, etc.), and their contribution to the $\gamma$–ray diffuse background depend on the combined effects of the SED shape, the luminosity functions and possibly evolution (Fossati et al., in preparation; see also Stecker, de Jager & Salamon 1996). In particular, an interesting and testable prediction of the scheme is the absence of high luminosity sources with synchrotron peaks in the X-ray range and strong associated TeV emission.
Interpretation
--------------
The extreme “regularity” of the SEDs of blazars and in particular the trends discussed above must derive from the common underlying physical processes. The common scenario envisaged is that of a relativistic jet pointing close to the line of sight. Assuming the simple case of a single (homogeneous) zone model the shape of the SED depends on the spectrum of the high energy electrons radiating via synchrotron and inverse Compton, the magnetic field and the nature of seed photons for the inverse Compton process. The latter could be the synchrotron photons themselves (synchrotron self Compton, SSC) or photons outside the jet (“external Compton" EC). In the following we discuss qualitatively the implications of the suggested trends for the two scenarios.
Let us assume a constant bulk Lorentz factors in all blazars. Should the (homogeneous) SSC model be valid for all sources, it is easy to see that the (approximately) constant ratio between the high and low peak frequencies yields an (approximately) constant value for the energy of the particles radiating at the peaks (e.g. Ghisellini, Maraschi, Dondi 1996). If the energy of the radiating particles is similar in all sources the different peak frequencies should result from a systematic variation in magnetic field strength, HBLs having the highest, FSRQs the lowest random field intensity.
Should instead the soft photons upscattered to the $\gamma$–ray range be produced outside the jet at a “typical” frequency (the same for all sources) the condition of a constant ratio between the peak frequencies implies a constant value of the magnetic field (Ghisellini, Maraschi, Dondi 1996). As a consequence the energy of the particles radiating at the peaks should vary along the spectral sequence being lower in FSRQs and higher in HBLs.
It could also be that there is a smooth transition between the SSC and EC mechanisms depending on the physical conditions outside the jet. In all cases however the role of the luminosity, which phenomenologically appears dominant, does not find an immediate physical justification, although one could find plausible arguments to link it to the parameters mentioned above and in particular to the conditions surrounding the jet.
In a separate paper (Ghisellini et al. 1998), we perform model fits to the spectral energy distributions of 51 individual objects, deriving the (model dependent) physical parameters for each source. These computations indeed suggest the idea that the blazar sequence follows from a transition from the SSC to the EC scenario, RBLs being the intermediate objects. The computed radiation energy densities, which determine the amount of radiative cooling, increase with increasing source luminosity and may be responsible for the lower energy of the particles radiating at the peaks in higher luminosity sources.
The likely possibility that the external photon field involved in the EC process is (or is related to) the radiation reprocessed as broad emission lines, seems to be at least qualitatively in agreement with the observational evidence concerning the emission line luminosity in the suggested blazar sequence.
Conclusions
===========
The main conclusion of this work is that despite the differences in the continuum shapes of different sub–classes of blazars, a unitary scheme is possible, whereby Blazar continua can be described by a family of analytic curves with the source luminosity as the fundamental parameter. The “scheme” (admittedly empirical) determines both the frequency and luminosity of the peaks in the synchrotron and inverse Compton power distributions (and therefore also the $\gamma$–ray luminosity in the EGRET range) starting from the radio luminosity only. The main suggested trend is that with increasing luminosity both the synchrotron peak and the inverse Compton peak move to lower frequencies and that the latter becomes energetically more dominant. The scheme is testable, for instance it predicts that sources emitting strongly in the TeV band have relatively low intrinsic luminosity.
The “spectral sequence” finds a plausible interpretation in the framework of relativistic jet models radiating via the synchrotron and inverse Compton processes if the physical parameters (magnetic field and/or critical energy of the radiating electrons) vary with luminosity or if photons outside the jet become increasingly important as seed photons for the inverse Compton process in sources of larger luminosity. The latter alternative is supported at least qualitatively by the increasing dominance of emission lines in higher luminosity objects.
The proposed scenario, in which the intrinsic jet power regulates, in a continuous sequence, the observational properties from the weaker HBL, through LBL, to the most powerful FSRQs, also fits in very nicely with the unification of FR I and FR II type radio galaxies as proposed by Bicknell (1995). After a long debate the prevailing view is that FR I and FR II radio galaxies both contain relativistic jets which can be decelerated giving rise to the FR I morphology depending on the kinetic power in the jet and the pressure of the ambient medium.
The whole radio–loud AGN population could be unified in a two parameter space one being the intrinsic jet power, the other the viewing angle. An interesting point for future discussion is whether a third parameter associated with the luminosity of an accretion disk is necessary or is already implicitly and uniquely linked to the jet power.
Acknowledgments {#acknowledgments .unnumbered}
===============
The Italian MURST (GF, Annalisa Celotti) and the Institute of Astronomy PPARC Theory Grant (Annalisa Celotti) are acknowledged for financial support. Andrea Comastri acknowledges financial support from the Italian Space Agency under contract ARS-96-70 This research has made use of NASA’s Astrophysics Data System Abstract Service and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration.
References {#references .unnumbered}
==========
Adam G., 1985, A&AS, 1985, 61, 225
Allen D.A., Ward M.J., Hyland A.R., 1982, MNRAS, 199, 969
Angel J.R.P., Stockman H.S., 1980, ARA&A, 8, 321
Ballard K.R., Mead A.R.G., Brand P.W.J.L., Hough J.H., 1990, MNRAS, 243, 640
Becker R.L., White R.L., Edwards A.L., 1991, ApJS, 75, 1
Bersanelli, M., Bouchet P., Falomo R., Tanzi E.G., 1992, AJ, 104, 28
Bertsch D.L. et al., 1993, ApJ, 405, L21
Bicknell G.V., 1995, ApJS, 101, 29
Blandford R.D., 1993, in Friedlander M., Gehrels N., Macomb D.J., eds, Proc. CGRO AIP 280. New York, p. 533
Bloom S.D., Marscher A.P., 1993, in Friedlander M., Gehrels N., Macomb D.J., eds, Proc. CGRO AIP 280. New York, p. 578
Bloom S.D., Marscher A.P., Gear W.K., Terasranta H., Valtaoja E., Aller H.D., Aller M.F. 1994, AJ, 108, 398
Bradbury S.M. et al., 1997, A&A, 320, L5
Brindle C., Hough J.H., Bailey J.A., Axon D.J., Hyland A.R., 1986, MNRAS, 221, 739
Brinkmann W., Siebert J., Boller T., 1994, A&A, 281, 355
Brinkmann W. et al., 1995, A&AS, 109, 147
Brown L.M.J. et al., 1989, ApJ, 340, 129
Brunner H., Lamer G., Worrall D.M., Staubert R., 1994, A&A, 287, 436
Catanese M. et al., 1997, ApJ, 480, 562
Chiang J, Fichtel C.E., von Montigny C., Nolan P.L., Petrosian V.,1995, ApJ, 452, 156
Comastri A., Molendi S., Ghisellini G., 1995, MNRAS, 277, 297
Comastri A., Fossati G., Ghisellini G., Molendi S., 1997, ApJ, 480, 534
Dermer C.D., Schlickeiser R., 1993, ApJ, 416, 458
Dickey J. M., Lockman F. J., 1990, ARA&A, 28, 215
Dingus B.L. et al., 1996, ApJ, 467, 589
Dondi L., Ghisellini, G. 1995, MNRAS, 273, 583
Elvis M., Lockman F. J., Wilkes B. J. 1989, AJ, 97, 777
Elvis M., Plummer D., Schachter J., Fabbiano G., 1992, ApJS, 80, 257
Elvis M. et al., 1994, ApJS, 95, 1
Falomo R., Bersanelli M., Bouchet P., Tanzi E.G., 1993a, AJ, 106, 11
Falomo R., Treves A., Chiappetti L., Maraschi L., Pian E., Tanzi E.G., 1993b, ApJ, 402, 532L
Falomo R., Scarpa R., Bersanelli M., 1994, ApJS, 93, 125
Fegan D.J., 1996, private communication
Fichtel C.E. et al., 1994, ApJS, 94, 551
Fossati G., Celotti A., Ghisellini G., Maraschi L., 1997, MNRAS, 289, 136
Gabuzda D.C., Cawthorne T.V., Roberts D.H., Wardle J.F.C., 1992, ApJ, 338, 40
Gear W.K., 1993a, MNRAS, 264, L21
Gear W.K., 1993b, MNRAS, 264, 919
Gear W.K. et al., 1985, ApJ, 291, 511
Gear W.K. et al., 1986, ApJ, 304, 295
Gear W.K. et al., 1994, MNRAS, 267, 167
Ghisellini G., Madau P., Persic M., 1987, MNRAS, 224, 257
Ghisellini G., Madau P. 1996, MNRAS, 280, 67
Ghisellini G., Maraschi L., Dondi L., 1996, A&AS, 120, 503
Ghisellini G., Padovani P., Celotti A., Maraschi L., 1993, ApJ, 407, 65
Ghisellini G., Celotti A., Fossati G., Maraschi L., Comastri A., 1998, MNRAS, submitted
Giommi P., Padovani P., 1994, MNRAS, 268, L51
Glass I.S., 1979, MNRAS, 186, L29
Glass I.S., 1981, MNRAS, 194, 795
Hartman R.C. et al., 1993, ApJ, 407, L41
Holmes P.A., Brand P.W.J.L., Impey C.D., Williams P.M., 1984, MNRAS, 210, 961
Impey C.D. 1996, AJ, 112, 2667
Impey C.D., Brand P.W.J.L., 1981, Nature, 292, 814
Impey C.D., Brand P.W.J.L., 1982, MNRAS, 201, 849
Impey C.D., Brand P.W.J.L., Wolstencroft R.D., Williams P.M., 1982, MNRAS, 200, 19
Impey C.D., Brand P.W.J.L., Wolstencroft R.D., Williams P.M., 1984, MNRAS, 209, 245
Impey C.D., Neugebauer G., 1988, AJ, 95, 307
Impey C.D., Tapia S., 1988, ApJ, 333, 666
Impey C.D., Tapia S., 1990, ApJ, 354, 124
Jannuzi B.T., Smith P.S., Elston R., 1993, ApJS, 85, 265
Jannuzi B.T., Smith P.S., Elston R., 1994, ApJ, 428, 130
Jones T.W., O’Dell S.L., Stein W.A., 1974, ApJ, 188, 353
Kühr H., Witzel A., Pauliny–Toth I.K., Nauber U., 1981, A&A, 45, 367
Lamer G., Brunner H., Staubert R., 1996, A&A, 311, 384
Landau R. et al., 1986, ApJ, 308, 78
Lepine J.R.D., Braz M.A., Epchtein N., 1985, A&A, 149, 351
Lin Y.C. et al., 1995, ApJ, 442, 96
Lichtfield S.J., Robson E.I., Stevens J.A., 1994, MNRAS, 270, 341
Lockman F. J., Savage B. D. 1995, ApJS, 97,1
Lorenzetti D., Massaro E., Perola G.C., Spinoglio L., 1990, A&A, 235, 35
Madejski G, et al., 1996, ApJ, 459, 156
Maraschi L., Rovetti F., 1994, ApJ, 436, 79
Maraschi L., Fossati G., Tagliaferri G., Treves A., 1995, ApJ, 443, 578
Maraschi L., Ghisellini G., Celotti A., 1992, ApJ, 397, L5
Maraschi L., Ghisellini G., Tanzi E.G., Treves A., 1986, ApJ, 310, 325
Mattox J.R. et al., 1997, ApJ, 476, 692
Mattox J.R., Schachter J., Molnar L., Hartman R.C., Patnaik A.R., 1997, ApJ, 481, 95
Mead A.R.G., Ballard K.R., Brand P.W.J.L., Hough J.H., Brindle C., Bailey J.A., 1990, A&AS, 83, 183
Morris S.L., Stocke J.T., Gioia I.M., Schild R.E., Wolter A., Maccacaro T., Della Ceca R., 1991, ApJ, 380, 49
Mücke A. et al., 1997, A&A, 320, 33
Mukherjee R. et al., 1995, ApJ, 445, 189
Mukherjee R. et al., 1996, ApJ, 470, 831
Murphy E.M., Lockman F.J., Laor A., & Elvis M. 1996, ApJS, 105, 369
Nolan P.L. et al., 1993, ApJ, 414, 82
Nolan P.L. et al., 1996, ApJ, 459, 100
O’Dell S.L., Puschell J.J., Stein W.A., Warner J.W., 1978, ApJS, 38, 267
Padovani P., 1992a, MNRAS, 257, 404
Padovani P., 1992b, A&A, 256, 399
Padovani P., Giommi P., 1995, ApJ, 444, 567
Padovani P., Urry C.M., 1992, ApJ, 387, 449
Pian E., Falomo R., Scarpa R., Treves A., 1994, ApJ, 432, 547
Perlman E.S., Stocke J.T., Schachter J.F., Elvis M., Ellingson E., Urry C.M., Potter M., Impey C.D., Kolchinsky P., 1996a, ApJS, 104, 251 (SLEW)
Perlman E.S., Stocke J.T., Wang Q.D., Morris S.L., 1996b, ApJ, 456, 451 (EMSS)
Quinn J. et al., 1996, ApJ, 456, L83
Radecke H.–D. et al., 1995, ApJ, 438, 659
Reuter H.-P. et al., 1997, A&AS, 122, 271
Riecke G.H., Lebofski M.J., 1985, ApJ 288, 618
Sambruna R.M., 1997, ApJ, 487, 536
Sambruna R.M., Maraschi L., Urry C.M., 1996, ApJ, 463, 444
Scarpa , Falomo R., 1997, A&A, 325, 109
Shrader C.R., Hartman R.C., Webb J.R., 1996, A&AS, 120, 599
Sikora M., Begelman M.C., Rees M.J., 1994, ApJ, 421, 153
Sitko M.L., Schmidt G.D., Stein W.A., 1985, ApJS, 59, 323
Sitko M.L., Sitko A.K., 1991, PASP, 103, 160
Smith P., Balonek T.J., Elston R., Heckert P.A., 1987, ApJS, 64, 459
Sreekumar P. et al., 1996, ApJ, 464, 628
Stecker F.W., de Jager O.C., Salamon M.H., 1996, ApJ, 473, L75
Steppe H. et al., 1988, A&AS, 75, 317
Steppe H. et al., 1992, A&AS, 96, 441
Steppe H. et al., 1993, A&AS, 102, 611
Stevens J.A. et al 1994, ApJ, 437, 91
Stickel M. et al., 1991, ApJ, 374, 431
Stickel M., Meisenheimer, K., Kühr H. 1994, A&AS, 105, 211
Terasranta H. et al., 1992, A&AS, 94, 121
Thompson D.J. et al., 1993, ApJ, 415, L13
Thompson D.J. et al., 1995, ApJS, 101, 259
Thompson D.J. et al., 1996, ApJS, 107, 227
Tornikovski M. et al., 1993, AJ, 105, 1680
Tornikovski M. et al., 1996, A&AS, 116, 157
Ulrich M.–H., Maraschi L., Urry C.M., 1997, ARA&A, 35
Urry C.M., Padovani P., 1995, PASP, 107, 803
Urry C.M., Sambruna R.M., Worrall D.M., Kollgaard R.I., Feigelson E.D., Perlman E.S., Stocke J.T., 1996, ApJ, 463, 424
Vestrand, W.T., Stacy J.G., Sreekumar P., 1995, ApJ, 454, L93
von Montigny C. et al., 1995, ApJ, 440, 525
Wall J.V., Peacock J.A., 1985, MNRAS, 216, 173
Weekes T.C. et al., 1996, A&AS, 120, 603
Wolter A., Caccianiga A., Della Ceca R., Maccacaro T., 1994, ApJ, 433, 29
Worrall D.M., Wilkes B.J., 1990, ApJ, 360, 396
Wright A., Ables J.G., Allen D.A., 1983, MNRAS, 205, 793
[^1]: Hereinafter we define spectral indices as F$_\nu \propto \nu^{-\alpha}$. In broad band indices radio, optical and X–ray fluxes are taken at 5 GHz, 5500 Å and 1 keV, respectively.
[^2]: The optical magnitudes have been de–reddened using values of A$_{\rm
V}$ derived from the A$_{\rm B}$ reported in the NED database according to the law A$_{\rm V} =$ A$_{\rm B}/1.324$ (Riecke & Lebofski 1985)
|
---
abstract: 'In this paper we study smooth complex projective varieties $X$ containing a Grassmannian of lines $\G(1,r)$ which appears as the zero locus of a section of a rank two nef vector bundle $E$. Among other things we prove that the bundle $E$ cannot be ample.'
address:
- 'Departamento de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, 28933-Móstoles, Madrid, Spain'
- 'Dipartimento di Matematica, Università di Trento, Via Sommarive 14, I-38050, Povo (Trento), Italy '
- 'Departamento de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, 28933-Móstoles, Madrid, Spain'
author:
- Roberto Muñoz
- Gianluca Occhetta
- 'Luis E. Solá Conde'
title: 'An extension of Fujita’s non extendability theorem for Grassmannians'
---
[^1]
Introduction {#sec:intro}
============
A natural problem in algebraic geometry is to study to which extent the geometry of a smooth irreducible variety $X$ is determined by the geometry of its smooth subvarieties $Y \subset X$, under certain positivity conditions on the embedding $Y\subset X$. A typical result of this kind would characterize $X$ (possibly saying that it cannot exist) by containing a particular subvariety $Y \subset X$.
The classical setting in which the problem arose was the classification of smooth projective embedded varieties $X \subset \P^N$ in terms of their smooth linear sections $Y =X \cap \P^{N-k}$, for example the classification of low degree embedded varieties. Later on, it evolved in different settings. For instance one may impose positivity conditions on the normal bundle $N_{Y/X}$.
If it is (generically) globally generated then there exists a family of deformations of $Y$ sweeping out $X$. That is the case, for example, of the varieties swept out by linear subspaces of small codimension, see for instance [@S] and [@NO]. In a recent paper (cf. [@MS]) the first and third author have dealt with embedded varieties $X\subset\P^N$ swept out by codimension two Grassmannians, that may be regarded as a projectively-embedded counterpart of this paper. The case of quadrics has been also studied, see [@Fu] and [@BIquadrics].
If $N_{Y/X}$ is an ample line bundle, then it is well known that, up to a birational transformation, $Y$ can be considered as an ample divisor on $X$. See [@HL] for a foundational reference on ample subvarieties. With no assumption on the codimension, the hypothesis on $N_{Y/X}$ to be ample joint to some topological assumptions constitute the setup of [@BdFL]. In that paper it is shown how some structural maps (RC-fibrations, nef-value morphism, Mori contractions) of $Y$ extend to $X$.
In the context of complex geometry, Lefschetz Theorem shows us how the topology of $Y$ is reflected on the topology of a variety $X\subset\P^N$ containing $Y$ as a linear section. Moreover an extension of this result, due to Sommese (cf. [@sommesefund] and [@sommese]), allows to work under weaker assumptions on the embedding $Y\subset X$. In this way, Lefschetz-Sommese Theorem provides an important tool for the type of problems we are considering here.
In this paper we will consider varieties $Y$ appearing as the zero locus of a regular section of an ample vector bundle $E$ on $X$. An interesting survey on this matter has been recently written by Beltrametti and Ionescu, see [@BI]. It deals mostly with the divisor case, but it also provides references for higher codimension. Among different results of this kind, let us recall a theorem by T. Fujita (cf. [@F1 Thm. 5.2]). It states that, apart of the obvious cases, Grassmannians cannot appear as ample divisors on a smooth variety. Our goal here is to show how this result can be extended to codimension two:
\[thm:fujita\] For $r \geq 4$ the Grassmannian of lines in $\P^r$ cannot appear as the zero locus of a section of an ample vector bundle of rank two over a smooth complex projective variety.
Let us observe that the Grassmannian of lines in $\P^r$, say $\G(1,r)$, is embedded naturally in $\G(1,r+1)$ as the zero locus of the universal quotient bundle $\cQ$ which is not ample but globally generated. It is then natural to look for a broader positivity assumption on $E$ in which this situation is included. Taking in account Lefschetz-Sommese Theorem, it makes sense to consider the notion of $k$-ampleness introduced by Sommese in [@sommese Def. 1.3] (see also Definition \[def:k-ample\] below). The main result of this paper, from which Theorem \[thm:fujita\] is a straightforward corollary, is the following:
\[thm:main\] Let $X$ be a smooth complex projective variety of dimension $2r$ and $G\subset X$ a subvariety isomorphic to the Grassmannian of lines in $\P^r$, $r\geq 4$. We further assume that $G$ equals the zero set of a section of a $(2r-4)$-ample vector bundle $E$ on $X$ of rank two. Then $X$ is isomorphic to the Grassmannian of lines in $\P^{r+1}$ and $E$ is the universal quotient bundle of this Grassmannian.
Our proof relies on proving that the normal bundle of $G$ in $X$ must be uniform, and on the classification of uniform vector bundles of low rank on Grassmannians.
The structure of the paper is the following. In Section \[sec:preliminars\] we recall some generalities on Grassmannians, positive vector bundles and vanishing results that we will use along the paper. In particular we find a lower bound on the degree of $E$ in terms of the index of $X$. Moreover one may show that this index is at most $\dim X-2$, a fact that is crucial in our argumentation; this is the purpose of Section \[sec:ruleout\]. Section \[sec:uniform\] deals with the classification of uniform vector bundles on Grassmannians, and in Section \[sec:E\] we determine the possible values of the restriction $E|_G$. In Section \[sec:proof\] we present the proof of Theorem \[thm:main\], and finally in Section \[sec:applications\] we use the results in [@BdFL] in order to derive from Theorem \[thm:fujita\] a non-extendability result for Grassmannian fibrations.
[**Acknowledgements:**]{} We would like to thank Tommaso de Fernex for his useful comments regarding Grassmannian fibrations.
Conventions and definitions. {#ssec:conventions}
----------------------------
Along this paper $X$ will denote a smooth complex projective variety of dimension $2r$ and $G\subset X$ a subvariety isomorphic to the Grassmannian of lines in $\P^r$, $\G(1,r)$, with $r\geq 4$. We further assume that $G$ equals the zero set of a section of a vector bundle $E$ on $X$ of rank two which is $(2r-4)$-ample in the sense of Sommese, see Definition \[def:k-ample\]. Denoting by $\cO(1)$ the ample generator of $\Pic(G)\cong\Z$, the determinant of $E|_G$ is isomorphic to $\cO(c)$, for some $c\in \Z$. We call $c$ the [*degree of*]{} $E$. The notation $\cO(1)$ will be also used to denote the ample generator of a variety of Picard number one and the tautological line bundle on a projective bundle. Subscripts will be used if necessary.
Finally, on a Fano variety of Picard number one, a rational curve of degree one with respect to $\cO(1)$ will be called a [*line*]{}. By definition, the family of lines in $X$ is [*unsplit*]{}, i.e. the subscheme of $\Chow(X)$ parametrizing them is proper. That amounts to say that a line is not algebraically equivalent to a reducible cycle.
We will write $\mathbb{Q}^n$ (or just $\mathbb{Q}$ when its dimension is not relevant) for a $n$-dimensional smooth quadric.
Preliminaries {#sec:preliminars}
=============
Generalities on Grassmannians {#ssec:grass}
-----------------------------
Let us recall some well-known facts on Grassmannians. We follow the conventions of [@A]. As said before, the Grassmannian of lines in $\P^r$ is denoted by $\G(1,r)$. We will denote by $\cQ$ the rank two [*universal quotient bundle*]{} and by $\mathcal{S}^\vee$ the rank $r-1$ [*universal subbundle*]{}, related in the universal exact sequence: $$0 \to \mathcal{S}^\vee \to \cO^{\oplus r+1} \to \cQ \to 0.$$
The projectivization of $\cQ$ provides the universal family of lines in $\P^r$: $$\xymatrix{&\P(\cQ)\ar[ld]_{p_1}\ar[rd]^{p_2}&\\ \G(1,r)&&\P^r.}$$ From right to left, this diagram may be thought of as the universal family of $\P^{r-1}$’s in $\G(1,r)$. These $\P^{r-1}$’s have degree one with respect to the Plücker polarization and their normal bundles in $\G(1,r)$ are isomorphic to $T_{\P^{r-1}}(-1)$. Finally we recall that the Chow ring of $\G(1,r)$ is generated by a well determined type of cycles, called [*Schubert cycles*]{}. The generators in dimension two are given by: the cycle parameterizing lines in a $\P^3\subset\P^r$ passing by a point, and the cycle parameterizing lines in a $\P^2\subset\P^r$ (we denote it by $\G(1,2)$). They are called $a$ and $b$–planes, respectively.
\[rmk:chern2\] [In particular, the second Chern class of a vector bundle $E$ on $\G(1,r)$ is given by two integers, corresponding to the second Chern classes of the restrictions of $E$ to the planes described above.]{}
Positivity, topology and vanishing results {#ssec:topo}
------------------------------------------
The hypotheses on $X$ in \[ssec:conventions\] impose severe restrictions on its topology. In order to describe them explicitly let us recall the definition of $k$-ampleness in the sense of Sommese, see [@sommese Def. 1.3]:
\[def:k-ample\] [Let $E$ be a semiample vector bundle over a projective variety, i.e. $\cO_{\P(E)}(m)$ is free for $m$ big enough. The vector bundle $E$ is said [*$k$-ample*]{} if every fiber of the morphism $\phi:\P(E)
\to \P(H^0(\P(E),\cO_{\P(E)}(m)))$ has dimension less than or equal to $k$.]{}
In particular any $k$-ample vector bundle is nef and is ample if it is $0$-ample. Sommese’s extension of Lefschetz Hyperplane Section Theorem [@lazarsfeld II, Thm. 7.1.1] admits an extension to $k$-ample vector bundles, see [@sommese Prop. 1.16] quoted in [@lazarsfeld II, Rmk. 7.1.9], which applies to our case giving the following relations between the topologies of $X$ and $G$.
\[lemma:lefschetz\] Let $X$, $G$ and $E$ be as in \[ssec:conventions\]. The restriction map $r:\Pic(X)\to\Pic(G)$ is an isomorphism.
Denote by $r_i:H^{i}(X,\Z) \to H^{i}(G,\Z)$ the corresponding restriction morphisms. By [@sommese Prop. 1.16] we get that $r_1$ is an isomorphism and $r_2$ is injective with torsion free cokernel. Furthermore we may compare the exponential sequences of $X$ and $G$ to get the following diagram: $$\xymatrix{ H^1(X,\Z)\ar[r] \ar[d]_{r_1} & H^1(X,\cO) \ar[r] \ar[d]_{r_{1,1}} &
\Pic(X) \ar[r] \ar[d]_{r}& H^2(X,\Z) \ar[d]_{r_2}\ar[r]& H^2(X,\cO)
\ar[d]_{r_{2,0}}
\\
H^1(G,\Z)\ar[r] & 0 \ar[r] & \Pic(G) \ar[r]^{\cong} & \Z \ar[r]& 0}$$ Since $r_1$ is an isomorphism and is compatible with the Hodge decomposition then $r_{1,1}$ is an isomorphism. Since $r_2$ is injective and with torsion free cokernel then it is an isomorphism and moreover $r_{2,0}$ is an isomorphism. This implies that $r$ is an isomorphism.
We will denote by $\cO_X(1)$ the ample generator of $\Pic(X)$, whose restriction to $G$ is the Plücker line bundle. The degree of the canonical sheaf of $X$ equals $\deg(K_G)-\deg(E)$, that is $K_X=\cO(-r-1-c)$ and $X$ is Fano. Hence Kobayashi-Ochiai Theorem [@KO] provides the bound $$\label{eq:c<=r} c\leq r.$$
Along this paper we will make use several times of the following variant of a vanishing theorem due to Griffiths [@lazarsfeld II, Variant 7.3.2]:
\[thm:vanishing\] Let $M$ be a smooth complex projective variety of dimension $n$, $L$ an ample line bundle on $M$ and $F$ a nef vector bundle of rank $k$ on $X$, then: $$H^i(M,\omega_M\otimes S^mF\otimes\det F\otimes L) = 0 \mbox{ for all }i>0,
m\geq 0.$$
Applied to our setting, the previous theorem provides the following vanishing.
\[lem:vanishing\] Under the assumptions in \[ssec:conventions\] and for every positive integer $l$, it follows that $$H^i(X,S^m E(l-r-1))=0, \mbox{ for all } i\geq 1, m\geq 0.$$
Being $G$ the subscheme of zeroes of a section of the rank two vector bundle $E$, the ideal sheaf of $G$ in $X$ has the following locally free presentation:
$$\label{eq:presenta}
0\rightarrow\det(E^\vee)\cong\cO(-c)\longrightarrow E^\vee\cong
E(-c)\longrightarrow\cI_{G/X}\rightarrow 0.$$
Combining it with Lemma \[lem:vanishing\] we immediately obtain:
\[lem:h1(I)\] With the assumptions of \[ssec:conventions\], the restriction maps $$H^0(X,\cO(k))\to H^0(G,\cO(k))$$ are surjective for all $k>0$.
In fact, it is enough to check that $H^1(X,\cI_{G/X}(k))=0$. Taking cohomology on sequence (\[eq:presenta\]), it suffices to show that $H^1(X,E(k-c))=H^2(X,\cO(k-c))=0$. By Lemma \[lem:vanishing\], the first vanishing holds whenever $k-c+r+1\geq 1$. Since $c\leq r$, see (\[eq:c<=r\]), that inequality is fulfilled for every positive $k$. For the second vanishing note that since $\Pic(X) \cong \Z$, Kodaira vanishing implies that line bundles on $X$ have no intermediate cohomology.
Let us take a projective space of maximal dimension contained in $G$, say $\P^{r-1} \cong M \subset G$, and denote by $E_M$ the restriction of $E$ to $M$. Later on we will need to apply Theorem \[thm:vanishing\] to $E_M$:
\[lem:vanishing2\] With the same assumptions as in \[ssec:conventions\] and for every positive integer $l$ it follows that: $$H^i(M,S^m E_M(l+c-r))=0, \mbox{ for } i\geq 1, m\geq 0.$$
High index Fano varieties containing codimension two Grassmannians {#sec:ruleout}
==================================================================
With the same assumptions as in \[ssec:conventions\], we will rule the cases $c=r,\; r-1$ and $r-2$ out, which correspond to projective spaces, quadrics and Del Pezzo varieties, respectively. In order to do that, it suffices to show that $h^0(X,\cO(1))<r(r+1)/2=h^0(G,\cO(1))$ contradicting Lemma \[lem:h1(I)\].
In the case $c=r$ we get $h^0(X,\cO(1))=2r+1$ and hence it is smaller than $r(r+1)/2$ whenever $r\geq 4$.
If $c=r-1$, $h^0(X,\cO(1))$ equals $2r+2$, which is smaller than $r(r+1)/2$ if $r\geq 5$. The case $r=4$ would correspond to a smooth quadric $\Q^8\subset\P^9$ containing a Grassmannian $G\cong \G(1,4)$, embedded in $\P^9$ via the Plücker map. But quadrics containing $\G(1,4)$ are given by $4\times 4$ pfaffians, hence singular.
In order to rule out the case of Del Pezzo varieties, we will make use of Fujita’s classification (cf. [@F2 8.11, p. 72], see also [@K V,1.12]). Being $\dim X=2r\geq 8$, the only possible values of $h^0(X,\cO(1))$ are $2r,2r+1,2r+2$ and $2r+3$, which are smaller that $r(r+1)/2$ except in the following cases:
- $X$ is a smooth cubic hypersurface in $\P^9$ containing a Plücker embedded Grassmannian $G\cong \G(1,4)$. Recall the notation on Grassmannians established in \[ssec:conventions\] and note that the normal bundle of $G$ in $\P^9$ is $\cO(1)\otimes\wedge^2\mathcal{S}\cong\cO(2)\otimes\mathcal{S}^\vee$ where $\mathcal{S}^\vee$ denotes the universal subbundle, see for instance [@manivel Prop. 4.5.1]. In particular, denoting by $E_G$ the restriction of $E$ to $G$ we get the following exact sequence: $$0 \to E_G \to
\mathcal{S}^\vee(2) \to \cO(3) \to 0.$$ Tensoring by $\cO(-3)$ we get $H^1(G,
E_G(-3))\ne 0$. Now use Serre duality to get $h^1(G,E_G(-3))=h^5(G,E_G^\vee(3)
\otimes \omega_G)=h^5(G, E_G(1) \otimes \omega_G)$. Since $E_G(1)$ is ample we get a contradiction with Le Potier Vanishing Theorem [@lazarsfeld Thm. 7.3.5].
- $X$ is a smooth complete intersection of two quadrics $\mathbb{Q}_1$ and $\mathbb{Q}_2$ in $\P^{10}$ containing a Plücker embedded Grassmannian $G\cong
\G(1,4)$. We may argue as before: observe on one hand that as a consequence of Theorem \[thm:vanishing\] we get that $h^1(G,E_G(-2))=0$. But on the other hand taking cohomology on the following exact sequences we get the contradiction $h^1(G,E_G(-2)) \ne0$: $$\begin{array}{c}\vspace{0.2cm}\xymatrix{
0\ar[r] & E_G \ar[r] & N_{G/\P^{10}} \ar[r]& \cO(2)^{\oplus 2} \ar[r] &0
,\\
}\\\xymatrix{0\ar[r] & S^\vee(2) \ar[r] & N_{G/\P^{10}} \ar[r]& \cO(1)
\ar[r] &0
.}\end{array}$$
As a corollary of what we have proved and recalling that $E$ is nef we get:
\[lem:0<c<r-2\] Under the assumptions of \[ssec:conventions\] we get that $0 \leq c <r-2$.
Uniform vector bundles on Grassmannianns {#sec:uniform}
========================================
Uniform vector bundles of low rank on Grassmannians have been classified by Guyot, cf. [@G]. For the sake of completeness we present here a proof for rank two vector bundles $E$ on $\G(1,r)$, using minimal sections of $E$ over its lines. Although we need only the case $r \geq 4$ we include a proof working for any $r \geq 2$.
Let us recall that a rank $k$ vector bundle $E$ on $\G(1,r)$ is [*uniform of type*]{} $(a_1, \dots, a_k)$ ($a_1 \leq \dots\leq
a_k)$ if for any line $\ell \subset \G(1,r)$ the restriction of $E$ to $\ell$ splits as $\cO(a_1) \oplus \dots \oplus \cO(a_k)$. The result is the following:
\[prop:class\] Every uniform rank two vector bundle $E$ on $G:=\G(1,r)$ of type $(0,1)$ is isomorphic either to $\cO\oplus\cO(1)$ or to the universal bundle $\cQ$.
Note that for $r=2$ the result is due to Van de Ven (cf. [@VV], [@OSS Thm. 2.2.2]), and we may assume that $r\geq 3$.
First we show that there exists a family of linear subspaces of $G$ of maximal dimension verifying that $E|_G\cong\cO\oplus\cO(1)$. In fact, if $r\geq 4$, the restriction of $E$ to a $\P^{r-1}$ is isomorphic to $\cO\oplus\cO(1)$ by the classification of uniform vector bundles on projective spaces (cf. [@EHS], [@OSS Thm. 3.2.3]). For the case $r=3$ recall that the Grassmannian $\G(1,3)$ contains two families of $\P^2$’s that we call $a$ and $b$-planes, see Section \[ssec:grass\]. Let us prove that the restriction of $E$ could not be isomorphic to $T_{\P^2}(-1)$ for both families. If this occurs then $c_2(E)$ equals the union of two planes, one of each family, see Remark \[rmk:chern2\]. Assume by contradiction that this is the case. Consider two $a$-planes $a_1$ and $a_2$ and denote by $P$ their intersection and by $r$ the corresponding line in $\P^3$. For every plane $M$ containing $r$ (determining a $b$-plane $b_M$ containing $P$) we get two lines $r_1(M)=b_M\cap a_1$ and $r_2(M)=b_M\cap a_2$. For each line $r_i$ ($i=1,2$) we get a lifting into $\P(E)$ determined by the unique surjective map $E|_{r_i(M)}\to\cO$. Denote by $R_1(M)$ and $R_2(M)$ the intersections of these liftings with the fiber over $P$. By hypothesis $E|_{a_i}\cong T(-1)$, hence the maps sending $M\to R_1(M)$ and $M\to R_2(M)$ are isomorphisms from the set of planes containing $r$ to the fiber over $P$. In particular there exists ${M_0}$ such that $R_1(M_0)=R_2(M_0)$. Now we consider the $b$-plane $b_{M_0}$. It contains two lines whose distinguished liftings meet at one point. Then the restriction of $E$ to $b_{M_0}$ cannot be $T(-1)$.
Recall that the family of $\P^{r-1}$’s of the previous paragraph is parameterized by a projective space $\mathcal{M} \cong \P^r$. Each element of this family admits a lifting to $\P(E)$ given by the unique surjective morphism $E|_{\P^{r-1}}\to\cO$ and we have the following diagram: $$\xymatrix{&\P(\cQ)\ar[ld]_{p_2}\ar[rd]^{p_1}\ar[r]^g&\P(E)\ar[d]^{\pi}\\
\mathcal{M} \cong \P^r& &\G(1,r)}$$ where $\cQ$ stands for the universal quotient bundle on $\G(1,r)$.
Now consider the restriction of $E$ to any $\G(1,2) \subset G$. The restriction $E|_{\G(1,2)}$ is either decomposable or isomorphic to $T(-1)$ by Van de Ven’s result. We claim that in the former case $E$ is decomposable. In fact take a point $x\in G$ and two $\P^{r-1}$’s, say $M_1$ and $M_2$, passing by $x$. We may find a $\G(1,2)$ meeting $M_1$ and $M_2$ in two lines. The (unique) lifting of this two lines to $\P(E)$ as curves of degree $0$ with respect to $\cO(1)$ meet in one point, since the two lines lie in $\G(1,2)$ and $E|_{\G(1,2)}=\cO\oplus \cO(1)$. In particular $g(\P(\cQ))$ meets the fiber $\pi^{-1}(x)$ in one point, hence $\pi:\P(E)\to G$ has a section and so $E$ splits as a sum of line bundles.
From now on we assume that $E|_{\G(1,2)}\cong T_{\P^2}(-1)$ for any $\G(1,2)\subset G$. Arguing as in the previous paragraph, we may prove that in this case the map $g$ is surjective. Moreover there cannot be two liftings of $\P^{r-1}$’s passing by the same point of $\P(E)$. In fact, if this occurs, we push it down to $G$ and we find a $\G(1,2)$ meeting the two $\P^{r-1}$’s in two lines. But the (unique) lifting of this two lines to $\P(E)$ as curves of degree $0$ with respect to $\cO(1)$ do not meet, since they lie in $\G(1,2)$ and $E|_{\G(1,2)}\cong T_{\P^2}(-1)$, a contradiction.
Summing up, the morphism $g:\P(\cQ)\to\P(E)$ is bijective, and the proof is finished.
Determining $E$ {#sec:E}
===============
In this section we will prove the following:
\[prop:E\] Under the assumptions of \[ssec:conventions\] the vector bundle $E$ verifies that the restriction of $E$ to $G$ is isomorphic to $\cQ$, where $\cQ$ stands for the universal quotient bundle.
We begin by studying the restriction $E_M$ of $E$ to a projective space of dimension $r-1$, $\mathbb{P}^{r-1}\cong M \subset G$. As a consequence of the upper bound on $c$ of \[lem:0<c<r-2\] and of the numerical characterization of rank two Fano bundles onto projective spaces, see for example [@APW], we get:
\[lemma:splitting\] Under the conditions above $E_M$ splits either as $E_M =\cO(1)^{\oplus 2}$ or as $E_M=\cO(1)\oplus \cO$.
Take the projective bundle $\pi: \P(E_M) \to M$. Since $-K_{\mathbb{P}(E_M)}=\cO(2) \otimes \pi^*\cO(r+1-c)$ then $\mathbb{P}(E_M)$ is a Fano variety, i.e. $E_M$ is a rank two Fano bundle. Hence we can use the classification of rank two Fano bundles, see [@APW Main Thm.] and [@SW Thm. (2.1)], to get that either $E_M$ splits as a sum of line bundles or $r=4$, $c=2$ and $E_M=\mathcal{N}(1)$, being $\mathcal{N}$ a null correlation bundle. This last possibility is excluded by the bound $c<r-2$ of \[lem:0<c<r-2\] so that $E_M$ splits as as a sum of line bundles.
Moreover the Bend and Break lemma leads to the following vanishing: $$\label{eq:vanishing}
H^0(M, E_M(-2))=0.$$ In fact, consider the exact sequence: $$0\rightarrow T_{M}(-3)\cong N_{M/G}(-2)\longrightarrow
N_{M/X}(-2)\longrightarrow E_M(-2)\rightarrow 0.$$ By Lemma \[lem:vanishing2\] we get $H^1(M,E_M(-2))=0.$ Taking cohomology in the Euler sequence tensored with $\cO(-2)$ we get that $H^1(M,T_{M}(-3))=0$ and therefore $H^1(M,N_{M/X}(-2))=0$. In particular the subscheme $\mathcal{M}_{\mathbb{Q}}$ of the Hilbert scheme parametrizing deformations of $M$ in $X$ containing a fixed smooth quadric $\mathbb{Q}\subset M$ is smooth at the point $[M]$ and its dimension equals $H^0(M,N_{M/X}(-2))$.
But $\mathcal{M}_{\mathbb{Q}}$ must be zero dimensional, otherwise given two general points $p,q\in \mathbb{Q}$, for every deformation $M_t$ of $M$ we could consider the line $\ell_t\subset
M_t$ joining $p$ and $q$. Then a Bend and Break argument (cf. [@De 3.2]) provides a reducible cycle $C$ algebraically equivalent to $\ell_t$, contradicting the fact that $\ell_t$ has degree $1$ with respect to $\cO(1)$. This implies that $H^0(M,N_{M/X}(-2))=0$, and so $H^0(M,E_M(-2))=0$, too.
Since $E_M$ is nef then, by the splitting of $E_M$ and (\[eq:vanishing\]), we get that $E_M=\cO(a_1)\oplus \cO(a_2)$ with $0 \leq a_1 \leq a_2 \leq 1$ and $a_1+a_2=c$. Hence $c\leq 2$, being $E_M=\cO(1)^{\oplus 2}$ if $c=2$ and $E_M=\cO(1)\oplus
\cO$ if $c=1$. If $c=0$ then denote by $E_G$ the restriction of $E$ to $G$. The rank two vector bundle is uniform with respect to the family of lines and in fact it is trivial, see [@AW (1.2)]. This contradicts the fact that the Picard number of $X$ is one, see [@MS Lemma 3.6].
Now we can complete the proof of Proposition \[prop:E\].
First we prove that the case $c=2$ cannot occur. Recall that $E_G$ stands for the restriction of $E$ to $G$. As $E_M=\cO(1)^{\oplus 2}$ we get that for any line $\ell \subset G$ the restriction of $E_G$ to $\ell $ is $\cO_{\ell}(1)^{\oplus 2}$. This implies uniformity of $E_G$ with respect to the family of lines and moreover $E_G=\cO(1)^{\oplus 2}$, [@AW (1.2)]. Consider the exact sequence of (\[eq:presenta\]) $$\label{eq:presntc=2}0 \to\cO(-2) \to E(-2) \to \mathcal{I}_{G/X} \to 0$$ and tensor it by $E(-1)$ to get: $$0 \to\ E(-3) \to E\otimes E(-3) \to E \otimes \mathcal{I}_{G/X}(-1) \to 0.$$ By the usual decomposition $E \otimes E \cong S^2 E \oplus \wedge^2 E$ and the vanishing of Lemma \[lem:vanishing\] we get $h^1(X,E \otimes
\mathcal{I}_{G/X}(-1))=0$. Now consider the exact sequence $$0 \to E \otimes \mathcal{I}_{G/X}(-1) \to
E(-1) \to E_G(-1)=
\cO^{\oplus 2} \to 0$$ to get that $h^0(X,E(-1))\geq 2$ and that $E(-1)$ is generically globally generated. Hence, see [@MS Lemma 3.5], $E(-1)=\cO^{\oplus 2}$. Tensoring the exact sequence (\[eq:presntc=2\]) by $\cO(1)$ we observe that $\mathcal{I}_{G/X}(1)$ is globally generated. This implies, see [@BSbook Cor. 1.7.5], that there exists a smooth element in the linear system $|\cO(1)|$ containing $G$, which contradicts [@F1 Thm. 5.2].
If $c=1$ we have shown in Proposition \[prop:class\] that $E_G$ is either as stated or splits as $E_G=\cO\oplus \cO(1)$. If $E_G$ splits, exactly as in the proof of the case $c=2$, we get $H^0(X,E(-1))\ne 0$. But this is a contradiction: in fact the exact sequence of (\[eq:presenta\]) $$0 \to\cO(-1)
\to E(-1) \to \mathcal{I}_{G/X} \to 0$$ gives $H^0(X,E(-1))=0$. This concludes that $E_G=\cQ$.
Proof of the main Theorem {#sec:proof}
=========================
Let us give the proof of Theorem \[thm:main\].
Let us recall that as a consequence of what we proved in the Section \[sec:E\] we can suppose that $c=1$ and that $E_G=\cQ$. Consider the projective bundle $\pi:\P(E)\to X$, which is a Fano variety. Recall that $E$ is nef by hypothesis and not ample as $c=1$. Hence we get that for $m$ big enough the linear system $|\cO(m)|$ defines an extremal ray contraction $\varphi$ leading to the following diagram: $$\xymatrix{\P(E)\ar[d]_{\pi}\ar[r]^{\varphi}&Z, \\X}$$ where $Z$ is normal. For $G \subset X$ we get that $E_G=\cQ$ so that, taking care of the Mori cone of $\P(E_G)$, the following diagram appears: $$\label{eq:diagramm}\xymatrix{ & \P^r \ar[dr]^{f}\\ \P(\cQ)
\ar@{^{(}->}[r]\ar[ur]^{\varphi_1}\ar[d]_{\pi_1} & \P(E)
\ar[d]_{\pi}\ar[r]^{\varphi} & Z,\\ G \ar@{^{(}->}[r]& X}$$ being $\pi_1$ and $\varphi_1$ the corresponding contractions of $\P(\cQ)$ and $f$ finite onto its image, which implies that $\dim Z \geq r$.
Now we claim that the general fiber $F$ of $\varphi$ is isomorphic to $\P^{r}$. In fact, $F$ is irreducible and smooth by Bertini’s Theorem and, if it is not a single point, adjunction formula tells us that $$-K_{F}=-K_{\P(E)}|_F=\pi^*\cO(1)^{\otimes(r+1)}.$$ But $\pi|_{F}$ is finite, hence $\pi^*\cO(1)|_{F}$ is ample and the above formula implies that, if not a point, $F$ is a Fano manifold of index greater than or equal to $r+1$. Recall that $\dim F \leq r+1$, hence either $F$ is a point or $F \cong \P^r$ and $\pi^*\cO(1)|_F=\cO(1)$ or $F$ is a smooth quadric of dimension $r+1$. In order to exclude the first and the last possibility let us introduce some notation. Since $N_{G/X}=E_G=\cQ$, which is globally generated, then there exists a $(r+1)$-dimensional irreducible variety $\mathcal{G}$ parameterizing deformations of $[G]$ which in fact contains the point corresponding to $G$, say $[G] \in\mathcal{G}$, as a smooth point. The family $\mathcal{G}$ dominates $X$. By rigidity of Grassmannians, the general point $[G']\in \mathcal{G}$ is isomorphic to $\mathbb{G}(1,r)$. Moreover $E_{G'}$ is nef and its Chern polynomial is that of $E_{G}$. In particular it is uniform so that $E_{G'}=\cQ$, see Proposition \[prop:class\]. Thus, for the general point $y \in \P(E)$ there exists $[G_y] \in \mathcal{G}$ such that $y
\in \P(E_{G_y})$ and provides a diagram as the one of (\[eq:diagramm\]). Therefore $$\label{eq:linearfiber} \varphi_1^{-1}f^{-1}(\varphi(y))
\supset \P^{r-1} \subset F$$ and this inclusion excludes the possibility of $F$ to be a point or a smooth quadric, being $r\geq 4$. Summing up we have shown that $F\cong \P^r$ and $\pi^*\cO(1)|_F=\cO(1)$ so that $\pi(F)\cong\P^r \subset X$. Moreover, since the fibers of $\varphi$ dominates $X$ via $\pi$, then the normal bundle $N_{\pi(F)/X}$ is generically globally generated.
We claim that $N_{\pi(F)/X}=T_{\P^r}(-1)$. Consider the Euler sequence $$0 \to \cO \to \pi^*(E^\vee) \otimes \cO(1) \to T_{\P(E)/X} \to 0$$ and restrict it to $F$ to get that $$T_{\P(E)/X} \otimes \cO_F=\cO(-1).$$ Then, since $\pi$ is an isomorphism, identifying isomorphic objects, we get the following diagram: $$\xymatrix{ & & 0 \ar[d] & 0 \ar[d] &
\\
& & T_{\P(E)/X} \otimes \cO_F \ar[r]^{\cong} \ar[d] & \cO(-1) \ar[d] &
\\
0\ar[r] & T_F \ar[r] \ar[d]_{\cong} & T_{\P(E)} \otimes \cO_F \ar[r] \ar[d] &
N_{F/\P(E)}\cong\cO^{\oplus r+1} \ar[r] \ar[d] & 0
\\
0\ar[r] & T_{\pi(F)} \ar[r] & T_X \otimes \cO_{\pi(F)} \ar[r]\ar[d] &
N_{\pi(F)/X} \ar[r]\ar[d] & 0.
\\ & & 0 & 0 &
}$$ The last vertical sequence is that of Euler and $N_{\pi(F)/X}=T_{\P^r}(-1)$ as claimed.
Now we claim that $\varphi$ is equidimensional. Let us suppose the existence of a fiber $F_0$ such that $\dim(\pi(F_0))>r$. Recall that for the general point $x
\in X$ there passes the image by $\pi$ of a general fiber $F$ of $\varphi$ and moreover $\pi(F)\cong \P^{r}$, $N_{\pi(F)/X}=T_{\P^{r}}(-1)$ and $$\label{eq:selfintersection} c_r(N_{\pi(F)/X})=1.$$ Hence there exists a component $\mathcal{M}$ of the Hilbert scheme of $\P^r$’s in $X$ containing $[\pi(F)]$ as a smooth point and sweeping out $X$. Through the general point $x \in \pi(F_0)$ there exists $[M] \in \mathcal{M}$ such that $x \in M \cong \P^r \subset X$. Since $\dim(M \cap \pi(F_0)) \geq 1$ and $E|_M=\cO\oplus \cO(1)$ then the intersection $M \cap \pi(F_0)$ admits a unique section into $\P(E)$ contracted by $\varphi$. It follows that $F_0$ intersects the only section $M_0$ over $M$ contracted by $\varphi$ so that it contains it, i.e. $M_0 \subset F_0$. Now consider a general $y \in \P(E)$. Recall that $F_y=\varphi^{-1}(\varphi(y)) \cong \P^r$ and $[\pi(F_y)] \in
\mathcal{M}$. Moreover, since $E|_{\pi(F_y)}=\cO \oplus \cO(1)$ then $F_y$ is the unique section of $E|_{\pi(F_y)}$ contracted by $\varphi$. But now observe that as a consequence of the selfintersection formula and (\[eq:selfintersection\]) any element in $\mathcal{M}$ is meeting $M$ and therefore $\pi(F_y) \cap M\ne \emptyset$ which in particular gives $\pi(F_y)
\cap \pi(F_0) \ne \emptyset$. But this leads to the contradiction $F_y \subset
F_0$ .
From the fact that $\varphi$ is equidimensional it follows that $\varphi: \P(E)
\to Z$ is a $\P^r$-bundle, that is all fibers are linear and $\varphi$ is providing the structure of projective bundle, see [@F3 2.12] quoted in [@BSbook Prop. 3.2.1]. In particular $Z$ is smooth.
Recall that $\varphi$ is defined by the system $|\cO(m)|$. We claim that we may assume $m=1$. In fact, take $x \in G\subset X$ and the fiber of $\pi$ over it, that is $\ell_x \cong \P^1=\pi^{-1}(x)$. Consider $y \in \ell_x$ and the fiber $F_y$ of $\varphi$ through $y$. Now observe that $[\pi(F_y)] \in \mathcal{M}$ and that $F_y$ corresponds to the only section of $E|_{\pi(F_y)}^\vee$. Then $F_y \cap \ell_x=\{y\}$ so that $\varphi|_{\ell_x}$ is a one-to-one map from $\P^1$ onto its image in $Z$. Hence the restriction of $f$ to $\varphi_1(\ell_x)$ is an isomorphism, for every $x\in G$. Since $G$ parametrizes all the lines of $\P^r$, it follows that $f$ itself is an isomorphism. Therefore we may consider $\P^r$ as an effective divisor in the smooth variety $Z$. Since $\Pic(Z)=\Z$, then $\P^r\subset Z$ is ample and Kobayashi-Ochiai Theorem tells us that $Z\cong\P^{r+1}$ and $\cO_Z(\P^r)\cong\cO_{\P^{r+1}}(1)$. In particular, fibers of $\pi$ map onto lines of $Z\cong\P^{r+1}$.
The next step in the proof is to observe that through any two points $x, y\in
X$ there cannot pass two elements of $\mathcal{M}$. In fact, by the self intersection formula and (\[eq:selfintersection\]) it holds that two possible different elements $M_1, M_2$ of $\mathcal{M}$ through $x$ and $y$ must meet in a positive dimensional subvariety $P=M_1 \cap M_2$. But $E_{M_i}=\cO \oplus
\cO(1)$ for $i=1,2$ so that, exactly as in the proof of the equidimensionality of $\varphi$, the corresponding unique sections $\P^r \cong F_i \subset \P(E)$ such that $\pi(F_i)=M_i$ are going to the same point by $\varphi$, contradicting the fact that $\varphi: \P(E) \to Z$ is a $\P^r$-bundle.
Recall that $\ell_x:=\varphi(\pi^{-1}(x))$ is a line in $Z\cong\P^{r+1}$ for all $x\in X$. This provides a map $g: X \to \mathbb{G}(1,r+1)$ sending $x$ to $\ell_x$. Since $X$ and $\mathbb{G}(1,r+1)$ are smooth of Picard number one then we conclude the proof of the theorem by showing that $g$ is surjective and generically injective. It is then enough to prove that for the general $x \in X$ there is no $y \in X$ different from $x$ such that $\ell_x= \ell_y$. Suppose on the contrary the existence of such $y \in X\setminus\{x\}$. For any point $z
\in \ell_x$ we get that $\varphi^{-1}(z)=\P^{r}$ is meeting the lines $\pi^{-1}(x)$ and $\pi^{-1}(y)$. This implies that $\pi(\varphi^{-1}(z))$ is the only element $\P^r =M \in \mathcal{M}$ through $x$ and $y$. This provides a one dimensional family of sections of $E_M^\vee$, which is a contradiction.
\[rmk:picardhypothesis\] [Let us remark that, as has been seen in the course of the proof, the hypothesis on the $(2r-4)$-ampleness of $E$ can be substituted by the hypothesis on the restriction map $r:\Pic(X)\to\Pic(G)$ to be an isomorphism. Note that $G$ appears as the zero set of a $(2r-2)$-ample vector bundle on, for instance, the product $X=G\times\P^2$, but $\Pic(X)\neq\Z$. A similar situation appears by considering the desingularization of a cone over $G$ with vertex a line. We do not know yet of any example in which $E$ is $(2r-3)$-ample and the restriction $r$ is not an isomorphism.]{}
Low values of $r$
=================
The case $r=3$ can be seen as a particular case of the general problem of quadrics appearing as the zero locus of sections of positive rank two vector bundles. This is well understood in the case in which $E$ is ample [@LM] (in fact in any codimension). Here we can prove the following:
\[prop:quadrics\] Let $X$ be as smooth complex projective variety of dimension $n \geq 6$. Suppose the existence of a rank two nef vector bundle $E$ on $X$ and a section of $E$ vanishing on a smooth quadric $\mathbb{Q} \subset X$. If the restriction map $r:\Pic(X) \to \Pic(\mathbb{Q})$ is an isomorphism then $(X,E)$ is either
- $(\P^n, \cO(2) \oplus \cO(1))$, or
- $(\mathbb{Q}, \cO(1) \oplus \cO(1))$, or
- $(\mathbb{G}(1,4), \cQ)$.
Denote as usual by $\cO(1)$ the ample generator of $\Pic(X)$ and by $c$ the degree of the determinant of $E$. Recall that since $E$ is nef and has a section vanishing on $E$ then $c>0$. Now use adjunction formula to get that $K_X=\cO(-(n-2)-c)$. This implies that either $c=3$ and $X=\P^n$ or $c=2$ and $X=\mathbb{Q}$ or $c=1$. Hence we can suppose that $c=1$ which means that $X$ is a Del Pezzo Variety. Now we apply [@MS Prop. 4.5] to get that $X$ is $\mathbb{G}(1,4)$. If $X \cong \mathbb{G}(1,4)$ then $E$ either splits as a sum of line bundles or $E \cong \cQ$, see Proposition \[prop:class\]. But in case $E=\cO \oplus \cO(1)$ there are no sections vanishing on a codimension two variety and the result follows.
The case $r=2$ can be seen as a particular case of the general problem of linear spaces appearing as the zero locus of sections of positive rank two vector bundles. See [@LM] for the case in which $E$ is ample. Here we can prove the following:
\[prop:linear\] Let $X$ be a smooth complex projective variety of dimension $n \geq 4$. Suppose the existence of a rank two nef vector bundle $E$ on $X$ and a section of $E$ vanishing on a linear space $\P^{n-2} \subset X$. If the restriction map $r:\Pic(X) \to \Pic(G)$ is an isomorphism then $(X,E)$ is either
- $(\P^n, \cO(1) \oplus \cO(1))$,
- $(\mathbb{G}(1,3), \cQ)$.
With the same notation as before we get by adjunction that $K_X=\cO(-(n-1)-c)$. Then either $c=2$ and $X \cong \P^n$ or $c=1$ and $X$ is a smooth quadric so that $\dim(X) \leq 4$, in fact equal by hypothesis. Since $E$ is uniform then either $E=\cO \oplus \cO(1)$ and no section vanishes in a codimension two subvariety or $E=\cQ$ and we conclude.
\[rmk:changeofhypothesis\] [For Propositions \[prop:quadrics\] and \[prop:linear\] let us remark that if we impose on $E$ to be $(n-4)$-ample then we get that the restriction morphism $r:\Pic(X) \to
\Pic(G)$ is an isomorphism, exactly as in Lemma \[lemma:lefschetz\].]{}
Fibrations in Grassmannians of lines {#sec:applications}
====================================
Inspired by [@BdFL Def. 5.1] we can give the following definition.
\[def:Grassmannianfibration\] [A surjective morphism $\pi:Y
\to Z$ between a smooth projective variety $Y$ and a normal projective variety $Z$ is called a ${\mathbb G}(1,r)$-[*fibration*]{} if $\pi$ is an elementary Mori contraction and there is a line bundle $L$ on $Y$ such that the general fiber $G$ of $\pi$ is isomorphic to ${\mathbb
G}(1,r)$ and $L|_G$ is the Plücker line bundle.]{}
If $\dim(Z) \leq 2$ it suffices to check this hypothesis on a fiber:
\[lem:codimensiontwofibrations\] Let $\pi:Y \to Z$ be a morphism between a smooth projective variety $Y$ and a normal projective variety $Z$ such that $\dim(Z) \leq 2$. If there exists $z$ a smooth point of $Z$ and $L \in \Pic(Y)$ such that $G:=\pi^{-1}(z)$ is isomorphic to $\mathbb{G}(1,r)$ and $L|_{G}$ is the Plücker line bundle then $\pi:Y \to Z$ is a $\mathbb{G}(1,r)$-fibration. Moreover $Z$ is smooth and all smooth fibers of $\pi$ are isomorphic to $\mathbb{G}(1,r)$.
Since the normal bundle $N_{G/Y}$ is trivial then, in particular is generically globally generated and its determinant is also trivial. Up to replacing $L$ with $L\otimes\pi^*A$, with a suitable ample line bundle $A$ on $Z$ we may assume that $L$ is ample and we can apply [@MS Lemma 2.5] to get that $\pi$ is the contraction of an extremal ray. Moreover, by [@AW Cor. 1.4], $\pi$ is equidimensional and $Z$ smooth. By rigidity of Grassmannians, see for instance [@HM], any smooth fiber is isomorphic to $\mathbb{G}(1,r)$ and the lemma follows.
\[prop:fibrations\] For $r \geq 4$ a $\mathbb{G}(1,r)$-fibration $Y$ cannot appear either as an ample divisor or as the zero locus of a section of a rank two ample vector bundle $E$ over a smoooh projective variety $X$.
Suppose on the contrary that $Y \subset X$ appears as the zero locus of a section of $E$. Then, by Lefschetz-Sommesse Theorem, the restriction map from $\Pic(X)$ to $\Pic(Y)$ is an isomorphism. Hence we may use [@BdFL Thm 4.1] to get a diagram: $$\xymatrix{Y\ar[d]_{\pi}\ar@{^{(}->}[r] & X \ar[d]_{\phi} \\ Z \ar[r]^{\delta} &
S,}$$ where $\phi$ is an elementary Mori contraction on $X$ and $\delta$ is a finite morphism.
Consider a general point $s \in S$ and denote by $F_s=\phi^{-1}(s)$ the fiber of $\phi$ over $s$, which is connected. Since $\delta$ is finite then $\delta^{-1}(s)=\{z_1,\dots,z_d\}$. Denote by $G_i=\pi^{-1}(z_i)$, $1 \leq i
\leq d$, so that $F_s \cap Y=G_1 \cup \dots \cup G_d$, where $G_i \cap
G_j=\emptyset$ for $i \ne j$. Recall that $Y$ is defined as the zero locus of a section of an ample vector bundle and, since $E|_{F_s}$ is ample then $F_s \cap
Y$ is also the zero locus of a section of an ample vector bundle. Then, by Lefschetz-Sommesse Theorem, $F_s \cap Y$ is connected so that $d=1$, that is $F_s \cap Y=G_1 \cong \mathbb{G}(1,r)$. But $\mathbb{G}(1,r)$ cannot appear either as an ample divisor on $F_s$ by [@F1 Thm. 5.2] or as the zero locus of a section of $E|_{F_s}$ by Theorem \[thm:fujita\]. This concludes the result.
\[rem:fibrations\] [Using [@BdFL Thm. 3.6], a similar statement holds under different hypotheses. We could have assumed that there exists an unsplit covering family $V$ of rational curves in $X$ verifying the following: it restricts to a family $V_Y$ covering $Y$ and the general equivalence class in $Y$ with respect to $V_Y$ is isomorphic to $\G(1,r)$.]{}
[9999]{}
Ancona, V., Peternell, T. and Wiśniewski. J. [*Fano bundles and splitting theorems on projective spaces and quadrics*]{}, Pacific J. Math. [**163**]{}, no. 1, 17-41 (1994).
Andreatta, M. and Wiśniewski, J. [*On manifolds whose tangent bundle contains an ample locally free subsheaf*]{}, Invent. Math. [**146**]{}, 209-217 (2001).
Arrondo, E. [*Subvarieties of Grassmannians*]{}, Lecture Note Series Dipartimento di Matematica Univ. Trento, 10 (1996).
Beltrametti, M.C., de Fernex, T. and Lanteri, A. [*Ample subvarieties and rationally connected fibrations*]{}, Math. Ann. [**341**]{}, 897-926 (2008).
Beltrametti, M.C., and Ionescu, P. [*On manifolds swept out by high dimensional quadrics*]{}. Math. Z. [**260**]{}, 229-234 (2008).
Beltrametti, M.C., and Ionescu, P. [*A view on extending morphisms from ample divisors*]{}. To appear in Contemporary Mathematics.
Beltrametti, M. C. and Sommese, A. J. [*The Adjunction Theory of Complex Projective Varieties*]{}, De Gruyter Expositions in Mathematics 16, De Gruyter, Berlin-New York, 1995.
Beltrametti, M. C., Sommese, A. J. and Wiśniewski, J. [*Results on varieties with many lines and their applications to adjunction theory*]{}, Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Mathematics 1507, Springer, Berlin, 1992, pp. 16–38.
Debarre, O. [*Higher-Dimensional Algebraic Geometry*]{}, Springer-Verlag New York, 2001.
Elencwajg, G., Hirschowitz, A. and Schneider, M. [*Les fibres uniformes de rang au plus [$n$]{} sur [${\bf P}\sb{n}({\bf C})$]{} sont ceux qu’on croit*]{}. Vector bundles and differential equations (Proc. Conf., Nice, 1979), Progr. Math., [**7**]{}, 37-63 (1980).
Fu, B. [*Inductive characterizations of hyperquadrics*]{}. Math. Ann. [**340**]{}, no. 1, 185-194 (2008).
Fujita, T. [*Vector bundles on ample divisors*]{}, J. Math. Soc. Japan [**33**]{}, no. 3, 405-414 (1981).
Fujita, T. [*On Polarized manifolds whose adjoint bundles are not semipositive*]{}. in [*Algebraic Geometry, Sendai 1985.*]{} Adv. Stud. Pure Math. [**10**]{}, 167-178 (1987).
Fujita, T. [*Classification Theories of Polarized Varieties*]{}. London Mathematical Society Lecture Note Series, no. 155. Cambridge University Press, Cambridge, 1990.
Guyot, M. [*Caractérisation par l’uniformité des fibrés universels sur la Grassmannienne*]{}. Math. Ann. [**270**]{}, 47-62 (1985).
Hartshorne, R. [*Ample subvarieties of algebraic varieties*]{}. Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York 1970.
Hwang, J-M. and Mok, N. [*Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation*]{}, Invent. Math. [**131**]{}, 393-418 (1998).
Kobayashi, S., Ochiai, T. [*Characterization of complex projective spaces and hyperquadrics*]{}, J. Math. Kyoto Univ. [**13**]{}, 31-47 (1973).
Kollár J. [*Rational curves on algebraic varieties*]{}, Berlin, Springer-Verlag, 1996.
Lazarsfeld, R. [*Positivity in Algebraic Geometry II.*]{} Springer-Verlag, Berlin-Heidelberg, 2004.
Lanteri, A. and Maeda, H. [*Ample vector bundles with section vanishing on projective spaces or quadrics*]{}, Int. J. Math.[**6**]{}, no. 4, 587-600 (1995).
Manivel, L. [*Gaussian maps and plethysm. Algebraic geometry (Catania, 1993/Barcelona, 1994),*]{} Lecture Notes in Pure and Appl. Math., 200, Dekker, New York, 1998, 91-117.
Muñoz, R. and Solá-Conde, L. E. [*Varieties swept out by Grassmannians of lines*]{} to appear in Contemporary Mathematics.
Novelli, C., Occhetta, G. [*Projective manifolds containing a large linear subspace with nef normal bundle*]{}, preprint 2008 arxiv: 0712.3406v2.
Sato, E. [*Projective manifolds swept out by large dimensional linear spaces*]{}, Tohoku Math. J. [**49**]{}, 299-321 (1997).
Okonek, C., Schneider, M. and Spindler, H. [ *Vector Bundles on Complex Projective Spaces*]{}. Progress in Mathematics 3, Birkhäuser, Boston, 1980.
Sommese, A. J. [*On manifolds that cannot be ample divisors*]{}, Math. Ann. [**221**]{}, no. 1, 55–72 (1976).
Sommese, A. J. [*Submanifolds of Abelian varieties*]{}, Math. Ann. [**233**]{}, no. 4 229-256 (1978).
Szurek, M. and Wiśniewski. J. [*Fano bundles over $P^3$ and $Q_3$*]{}, Pacific J. Math. [**141**]{}, no. 1 197-208 (1990).
Van de Ven, A. [*On uniform vector bundles*]{}. Math. Ann. [**195**]{}, no. 4, 245-248 (1972).
[^1]: Partially supported by the Spanish government project MTM2006-04785 and by MIUR (PRIN project: Proprietà geometriche delle varietà reali e complesse)
|
---
author:
- 'Eshan D. Mitra'
- 'William S. Hlavacek [^1]'
bibliography:
- 'references.bib'
title: Bayesian Uncertainty Quantification for Systems Biology Models Parameterized Using Qualitative Data
---
Abstract {#abstract .unnumbered}
========
**Motivation:** Recent work has demonstrated the feasibility of using non-numerical, qualitative data to parameterize mathematical models. However, uncertainty quantification (UQ) of such parameterized models has remained challenging because of a lack of a statistical interpretation of the objective functions used in optimization.\
**Results:** We formulated likelihood functions suitable for performing Bayesian UQ using qualitative data or a combination of qualitative and quantitative data. To demonstrate the resulting UQ capabilities, we analyzed a published model for IgE receptor signaling using synthetic qualitative and quantitative datasets. Remarkably, estimates of parameter values derived from the qualitative data were nearly as consistent with the assumed ground-truth parameter values as estimates derived from the lower throughput quantitative data. These results provide further motivation for leveraging qualitative data in biological modeling.\
**Availability:** The likelihood functions presented here are implemented in a new release of PyBioNetFit, an open-source application for analyzing SBML- and BNGL-formatted models, available online at [www.github.com/lanl/PyBNF](www.github.com/lanl/PyBNF).\
Introduction
============
Mathematical models of the dynamics of cellular networks, such as those defined using BioNetGen Language (BNGL) [@Faeder2009] or Systems Biology Markup Language (SBML) [@Hucka2003], require parameterization for consistency with experimental data. Conventional approaches use quantitative data such as time courses and dose-response curves to parameterize models. We and others have demonstrated that it is also possible to use non-numerical, qualitative data in automated model parameterization [@Oguz2013; @Pargett2013; @Pargett2014; @Mitra2018a]. Our demonstration [@Mitra2018a] used qualitative data in combination with quantitative data.
In the method of [@Mitra2018a], the available qualitative data are used to formulate inequality constraints on outputs of a model. Parameterization is performed by minimizing a sum of static penalty functions [@Smith1997] derived from the inequalities. Given a list of $n$ inequalities of the form $g_i<0$ for $i=1,...,n$, where the $g_i$ are functions of model outputs, the objective function is defined as $$\sum_{i=1}^n C_i\cdot\max(0, g_i)
\label{eq:static}$$
Static penalty functions have long been used in the field of constrained optimization [@Smith1997]. Each violated inequality contributes to the objective function a quantity equal to a distance from constraint satisfaction (e.g., the absolute difference between the left-hand side and right-hand side of the inequality), multiplied by a problem-specific constant weight $C_i$. The objective function of Equation \[eq:static\] becomes smaller as inequalities move closer to satisfaction, thus guiding an optimization algorithm toward a solution satisfying more of the inequalities. In the study of [@Mitra2018a], the approach proved effective in obtaining a reasonable point estimate for the parameters of a 153-parameter model of yeast cell cycle control developed by Tyson and coworkers [@Chen2000; @Chen2004; @Csikasz-Nagy2006; @Oguz2013; @Kraikivski2015], which had previously been parameterized by hand tuning.
The static penalty function approach has limitations. Most notably, the approach requires choosing problem-specific weights $C_i$ for the objective function. Although heuristics exist to make reasonable choices for the weights [@Mitra2018a], there is no rigorous method to do so. A related challenge in using qualitative data is performing uncertainty quantification (UQ).
Bayesian UQ (described in many studies, such as [@Kozer2013] and [@Klinke2009]) is a valuable approach that generates the multivariate posterior probability distribution of model parameters given data. This distribution can be used for several types of analyses. 1) The marginal distribution of each parameter can be examined to find the most likely value of that parameter and a credible interval. 2) Marginal distributions of pairs of parameters can be examined to determine which parameters are correlated. 3) Prediction uncertainty can be quantified by running simulations using parameter sets drawn from the distribution. Unfortunately, meaningful Bayesian UQ cannot be performed for models parameterized using qualitative data and penalty function-based optimization, because the penalty functions are heuristics. They are not grounded in statistical modeling.
Here, we present likelihood functions that can be used in parameterization and UQ problems incorporating both qualitative and quantitative data. We first present a likelihood function that can be used with binary categorical data, and then a more general form to use with ordinal data comprising three or more categories. We implemented the option to use these likelihood functions in fitting and in Bayesian UQ in our software PyBioNetFit [@Mitra2019a]. We built on existing PyBioNetFit support for qualitative data, which previously allowed only the static penalty function approach. In the first section of Results, we derive the new likelihood functions, which have similarities to both the chi squared likelihood function commonly used in curve fitting with quantitative data, and the logistic function commonly used to model classification error in machine learning. In the second section, we describe how we have added support for the new likelihood functions in PyBioNetFit and provide a guide to using them in optimization and UQ. In the third section, we provide an example application of the new software features. This example shows that qualitative datasets are potentially valuable resources for biological modeling.
Methods
=======
Likelihood functions presented in Results were implemented as options in PyBioNetFit v1.1.0, available online at <https://github.com/lanl/pybnf>. PyBioNetFit supersedes the earlier BioNetFit [@Thomas2016; @Hlavacek2018].
To illustrate use of the new functionality, we configured and solved an example UQ problem (described in Section \[sec:application\]) using PyBioNetFit v1.1.0. Configuration, model, and synthetic data files used for this example are available online (<https://github.com/RuleWorld/RuleHub/tree/2019Aug27/Contributed/Mitra2019Likelihood>). The model that we used has been published in BNGL format [@Faeder2009] in earlier work [@Harmon2017]. We took the published parameterization to be the ground truth. We adapted the simulation commands included in the BNGL file to produce degranulation outputs for specific conditions, as appropriate for our synthetic datasets described below.
We considered 11 instances of the problem using different qualitative and quantitative datasets. To generate synthetic quantitative data, we simulated the model with the assumed ground-truth parameterization, and added Gaussian noise to the desired degranulation outputs. To generate synthetic two-category qualitative data, we performed the same procedure, but recorded only whether the noise-corrupted primary degranulation response was greater or less than the noise-corrupted secondary degranulation response. To generate synthetic three-category qualitative data, we followed the same procedure, but recorded that the primary and secondary responses were approximately equal if the difference between the two responses was less than a designated threshold, which was set at $4.2\times 10^4$ arbitrary units.
We performed MCMC sampling using PyBioNetFit’s parallel tempering algorithm. For each dataset considered, we performed four independent runs and combined all samples obtained. Each run consisted of four Markov chains for each of nine temperatures, for a total of 36 chains, with samples saved from the four chains at temperature 1, run for a total of 50,000 steps including an unsampled 10,000-step burn-in period. Each run was performed using all 36 cores of a single Intel Broadwell E5-2695 v4 cluster node. Complete configuration settings are provided in the PyBioNetFit configuration file online.
Results
=======
Mathematical derivation
-----------------------
### Notation {#sec:problem}
By way of introduction to our newly proposed likelihood function for qualitative data, we begin by reviewing Bayesian UQ and its associated likelihood function with a more conventional quantitative dataset.
We are given an experimental dataset $\mathbf{y}=\{y_1,...,y_n\}$ and a model $f$. There is no restriction on what type of numerical measurement each $y_i$ represents; for example, it could represent a single data point of a time course, a sample mean of several independent and identically distributed measurements, or an arbitrary function of multiple measured quantities. Within a Bayesian framework, the $y_i$ are taken to be samples from the random variables $\{Y_1,...,Y_n\}$. The model $f$ takes as input a parameter vector ${\boldsymbol{\theta} }$ to predict the expected value of each data point $Y_i$, that is, $f_i({\boldsymbol{\theta} }) = E(Y_i)$. ${\boldsymbol{\theta} }$ is the realization of the random variable $\mathbf{\Theta}$. $f$ is assumed to be deterministic (e.g., an ODE model). Stochastic models would require additional treatment that is beyond the intended scope of this study.
In Bayesian UQ, parameter uncertainty is quantified by the posterior probability distribution $P({\boldsymbol{\theta} }|\mathbf{y})$, the probability of a particular parameter set given the data. Markov chain Monte Carlo (MCMC) algorithms can be used to sample the posterior distribution using the fact that, by Bayes’ law, $P({\boldsymbol{\theta} }|\mathbf{y}) \propto P(\mathbf{y}|{\boldsymbol{\theta} })P({\boldsymbol{\theta} })$. The change in the value of $P(\mathbf{y}|{\boldsymbol{\theta} })P({\boldsymbol{\theta} })$ is used to determine whether a proposed move by the MCMC algorithm is accepted. $P({\boldsymbol{\theta} })$ is a user-specified distribution representing prior knowledge about the parameters. Therefore, an important prerequisite for performing Bayesian UQ is an expression for the *likelihood*, $P(\mathbf{y}|{\boldsymbol{\theta} })$.
### Chi squared likelihood function
When performing conventional Bayesian UQ using only quantitative data, a common choice of likelihood function (e.g., see [@Kozer2013] and [@Harmon2017]) is the chi squared function.
$$-\log P(\mathbf{y}|{\boldsymbol{\theta} }) \propto \chi^2({\boldsymbol{\theta} }) = \sum_{i=1}^n \frac{(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2}
\label{eq:chisq}$$
Here $\sigma_i$ is the standard deviation of the measurement $y_i$. If $y_i$ represents the sample mean of several independent trials, it is common to estimate $\sigma_i$ as the standard error of the mean.
This likelihood function has a strong theoretical motivation. The underlying assumption is that each $Y_i$ has an independent Gaussian distribution with mean $f_i({\boldsymbol{\theta} })$ and standard deviation $\sigma_i$. Then the probability of a single data point $y_i$ given ${\boldsymbol{\theta} }$ is
$$P(y_i|{\boldsymbol{\theta} }) = \frac{1}{\sqrt{2\pi}\sigma_i} \exp(\frac{-(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2})$$
Given that the $Y_i$ are independent, the probability of the complete dataset $\mathbf{y}$ given ${\boldsymbol{\theta} }$ is given by the product
$$P(\mathbf{y}|{\boldsymbol{\theta} }) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}\sigma_i} \exp(\frac{-(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2})
\label{eq:product}$$
When performing MCMC sampling, we typically only need a value *proportional to* $P(\mathbf{y}|{\boldsymbol{\theta} })$ to calculate the ratio $P(\mathbf{y}|{\boldsymbol{\theta} }_1) / P(\mathbf{y}|{\boldsymbol{\theta} }_2)$ for two parameter sets ${\boldsymbol{\theta} }_1$ and ${\boldsymbol{\theta} }_2$. This ratio is used to determine, for example, the probability of transitioning from ${\boldsymbol{\theta} }_1$ to ${\boldsymbol{\theta} }_2$ in the Metropolis-Hastings algorithm. We therefore can ignore proportionality constants in Equation \[eq:product\] that are independent of ${\boldsymbol{\theta} }$.
$$P(\mathbf{y}|{\boldsymbol{\theta} }) \propto \prod_{i=1}^n \exp(\frac{-(y_i-f_i({\boldsymbol{\theta} }))^2}{2\sigma_i^2})
\label{eq:propproduct}$$
Taking the negative logarithm of Equation \[eq:propproduct\] results in the conventional chi squared function (Equation \[eq:chisq\]). Therefore, under the assumptions stated in this section, the chi squared function represents the kernel of the negative log likelihood and can be rigorously used in Bayesian UQ algorithms.
### Likelihood function for qualitative data
We now consider the situation in which the experimental data are qualitative. By qualitative data, we specifically mean observations that can be expressed as inequality constraints to be enforced on outputs of a model.
Our problem statement is nearly identical to that presented in Section \[sec:problem\], except we are no longer given the dataset $\mathbf{y}$. Instead, for each $Y_i$, we are given a constant $c_i$, and told whether $y_i < c_i$ or $y_i > c_i$ was observed. $y_i$ is the sample generated from $Y_i$ and is never observed. $y_i < c_i$ (or $y_i > c_i$) is the observation, which has two possible outcomes. We explicitly write down the procedure to generate these qualitative observations from the $Y_i$, which we refer to as our *sampling model*:
To generate observation $i$, sample $y_i$ from $Y_i$ and report whether $y_i < c_i$ or $y_i > c_i$.
Without loss of generality, we assume all given observations have the form $y_i < c_i$. If some quantity $A$ yielded an observation $a > k$, we could set $Y_i=-A$ and $c_i=-k$. This form also supports the case of an inequality $A<B$ between two measured quantities, as we could set $Y_i=A-B$ and $c_i=0$.
To perform Bayesian analysis, we require an expression for the probability of observing $y_i < c_i$ for all $i$ (rather than observing $y_i > c_i$ for some $i$), given a parameter set ${\boldsymbol{\theta} }$. As shorthand, we will write this as $P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} })$, where $\mathbf{y}$ is a vector of the $y_i$ and $\mathbf{c}$ is a vector of the $c_i$.
Following the example of the chi squared likelihood function, we assume each $Y_i$ has a Gaussian distribution with a known standard deviation $\sigma_i$. The mean of the distribution is, as before, taken to be given by the model prediction $f_i({\boldsymbol{\theta} })$. With this distribution, the probability of observing $y_i < c_i$ is, by definition, given by the Gaussian cumulative distribution function (CDF). We will write the CDF of a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$ evaluated at a point $x$ as $\textrm{cdf}(\mu,\sigma,x)$. The conditional probability of interest is as follows:
$$P(y_i<c_i|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i)
\label{eq:qualsingle}$$
We note that for ease of implementation, $\textrm{cdf}(\mu,\sigma,x)$ can be written in terms of the error function $\textrm{erf}(x)$, which is implemented in many standard libraries, including the Python and C++ standard libraries.
$$\textrm{cdf}(\mu,\sigma,x) = \mu + \frac{1 + \textrm{erf}(\frac{x}{\sigma \sqrt{2}})}{2}$$
As shown in Figure \[fig:logistic\], Equation \[eq:qualsingle\] is intuitively reasonable. If the true mean value of $Y_i$ is much smaller than $c_i$ (relative to the scale of $\sigma_i$), we are very likely to observe $y_i<c_i$, whereas if the mean of $Y_i$ is much larger than $c_i$, we are very unlikely to observe $y_i<c_i$. If the true mean of $Y_i$ is close to $c_i$, we are uncertain whether the observation will be $y_i<c_i$ or $y_i>c_i$ in the face of measurement noise. We note that this function has a similar appearance to the logistic function, which is commonly used to model binary categorization in machine learning.
Assuming independence of the $Y_i$, the probability of the entire dataset is given by the product.
$$P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \prod_{i=1}^n \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i)
\label{eq:qualproduct}$$
Finally, we take the negative logarithm to obtain
$$-\log P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \sum_{i=1}^n -\log \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i)
\label{eq:qualobj}$$
This function can be used for Bayesian UQ when considering qualitative data in an equivalent way to how the chi squared likelihood function is used when considering quantitative data.
![The proposed form for $P(y_i<c_i|{\boldsymbol{\theta} })$ (Equation \[eq:qualsingle\]). []{data-label="fig:logistic"}](Fig1.eps)
### Likelihood function for qualitative data with model discrepancy {#sec:twocat}
The likelihood function in Equation \[eq:qualobj\] has a remaining limitation when it comes to real-world experimental data. To illustrate this concern, we point to the model developed by Tyson and co-workers of yeast cell cycle control [@Chen2000; @Chen2004; @Csikasz-Nagy2006; @Oguz2013; @Kraikivski2015]. Several versions of this model have been parameterized using qualitative data (viability status of yeast mutants) by hand-tuning [@Chen2000; @Chen2004; @Csikasz-Nagy2006; @Kraikivski2015] and with optimization algorithms [@Oguz2013; @Mitra2018a]. In all of these parameterization studies, most but not all of the qualitative observations were satisfied by the reported best-fit parameterization. A few of the observations, however, were different from the model predictions. Due to such anomalous observations, a likelihood model as we have described could give the dataset a very low likelihood given the model and parameters, even though there intuitively is good agreement between the parameterized model and dataset.
How can we reconcile anomalous observations? An explanation given by Tyson and co-workers is that a model has a limited amount of detail, which is unable to capture every qualitative observation in the data [@Chen2004]. This explanation suggests using a statistical approach known as model discrepancy or model inadequacy [@Kennedy2001]. The principle of model discrepancy is that when calculating the likelihood of a dataset, one should take into account the difference between the model and reality. Although many statistical studies ignore model discrepancy, it has been shown to be important for performing effective statistical inference for certain problems [@Brynjarsdottir2014]. Given that qualitative data may be generated by high-throughput screening that could easily step outside the scope of a particular model, we believe model discrepancy is an especially important consideration for our applications.
Existing treatments of model discrepancy often describe discrepancy with its own probability distribution, such as a Gaussian distribution that is autocorrelated in time [@Brynjarsdottir2014]. Such an approach, which uses an assumption that model discrepancy is correlated for similar observations, is hard to apply to our problem formulation in which the $Y_i$ are taken to be independent (possibly coming from different model outputs). Thus, we take a more generic approach of expressing model discrepancy as a constant probability $\epsilon_i$ for each qualitative observation. $\epsilon_i$ relates to the probability that a given observation is outside the scope of the model. We say that when an observation is made, there is a probability $\epsilon_i$ that $y_i < c_i$ is reported regardless of the expected value of $Y_i$ given by the model. Likewise, there is also a probability $\epsilon_i$ that $y_i > c_i$ reported regardless of $Y_i$. These statements can be formalized as part of our sampling model:
To generate observation $i$, make a weighted random choice of one of the following possibilities:
- With probability $1-2\epsilon_i$, sample $y_i$ from $Y_i$ and report whether $y_i<c_i$ or $y_i>c_i$
- With probability $\epsilon_i$, report $y_i<c_i$
- With probability $\epsilon_i$, report $y_i>c_i$
With this modification, we have the probability distribution
$$P(y_i<c_i|{\boldsymbol{\theta} }) = \epsilon_i + (1-2\epsilon_i) \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i)$$
and the likelihood function
$$-\log P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \sum_{i=1}^n -\log (\epsilon_i + (1-2\epsilon_i) \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i))
\label{eq:qualobjfinal}$$
Equation \[eq:qualobjfinal\] gives our recommended form for a likelihood function incorporating qualitative data with two possible categorical outcomes ($y_i<c_i$ or $y_i > c_i$). We will refer to this function as the *two-category likelihood function*.
Note that although we introduced $\epsilon_i$ for dealing with model structure problems, it could also represent a shortcoming of our postulated Gaussian error model. For example, if an experimental instrument had some probability of reporting a false positive or negative, regardless of whether the mean of $Y_i$ is close to the threshold $c_i$, this non-Gaussian error could be accounted for by increasing the value of $\epsilon_i$.
### Likelihood function for ordinal data with more than two categories {#sec:threecat}
We next derive a likelihood function for ordinal categorical data with more than two categories. For simplicity, we suppose an observation has three possible outcomes: $y_i<c_{i,1}$, $c_{i,1}<y_i<c_{i,2}$, and $y_i>c_{i,2}$, for constants $c_{i,1}$ and $c_{i,2}$. An example would be if we were making an ordinary qualitative observation ($y_i<c_i$ or $y_i>c_i$), but another possible outcome of the experiment is $y_i=c_i$ to within the experimental error. Then the cutoffs $c_{i,1}$ and $c_{i,2}$ could be chosen on either side of $c_i$ such that the outcome $c_{i,1}<y_i<c_{i,2}$ corresponds to $y_i$ within measurement error.
From the definition of the Gaussian CDF we have
$$\label{eq:yltc1}
P(y_i<c_{i,1}|{\boldsymbol{\theta} }) = 1-\textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma, c_{i,1})$$
$$P(c_{i,1}<y_i<c_{i,2}|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,1}) - \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2})
\label{eq:pmiddle}$$
$$\label{eq:ygtc2}
P(y_i>c_{i,2}|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2})$$
A simplification is possible under the assumption that $c_{i,1}$ and $c_{i,2}$ are far enough separated that for any $E(Y_i)$, at most two of the three categories have non-negligible probability. That is, if $E(Y_i)$ is close enough to $c_{i,2}$ that observing $y_i>c_{i,2}$ is a probable outcome, $E(Y_i)$ is also high enough above $c_{i,1}$ that observing $y_i<c_{i,1}$ has a probability close to zero. Thus, we assume that for all ${\boldsymbol{\theta} }$, either $\textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,1}) = 1$ or $\textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2}) = 0$. This assumption is reasonable because if it were false, it would mean the experiment cannot reliably distinguish between the three categories, and so the data would be better analyzed as two-category data. With this assumption, Equation \[eq:pmiddle\] can be rewritten as
$$P(c_{i,1}<y_i<c_{i,2}|{\boldsymbol{\theta} }) = \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,1}) * (1 - \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i, c_{i,2}))
\label{eq:pmiddle2}$$
Note that Equation \[eq:pmiddle2\] is equivalent to Equation \[eq:qualproduct\] for two independent constraints $c_{i,1}<y_i$ and $y_i<c_{i,2}$ arising from two-category observations. This makes for a convenient implementation: rather than explicitly considering the two-sided observation $c_{i,1}<y_i<c_{i,2}$, we can rewrite the observation as two independent one-sided observations $c_{i,1}<y_i$ and $y_i<c_{i,2}$ described by Equation \[eq:qualproduct\].
A modification to the two-category case is necessary when model discrepancy is included as in Equation \[eq:qualobjfinal\]. Here, care must be taken to ensure that in the sampling model the probability of all possible outcomes sums to 1. For example, a reasonable sampling model for a three-category observation would be the following:
To generate observation $i$, make a weighted random choice of one of the following possibilities:
- With probability $1-3\epsilon_i$, sample $y_i$ from $Y_i$ and report whether $y_i<c_{i,1}$ or $c_{i,1}<y_i<c_{i,2}$ or $c_{i,2}<y_i$
- With probability $\epsilon_i$, report $y_i<c_{i,1}$
- With probability $\epsilon_i$, report $c_{i,1}<y_i<c_{i,2}$
- With probability $\epsilon_i$, report $c_{i,2}<y_i$
Recall that in Equation \[eq:qualobjfinal\], in the case of model discrepancy, the observation is equally likely to be $y_i>c_i$ or $y_i<c_i$ (each of these events is assumed to have probability $\epsilon_i$). In contrast, using the above sampling model, it is half as likely to report $y_i<c_{i,1}$ (probability $\epsilon_i$) as to report $y_i>c_{i,1}$ (probability $2\epsilon_i$).
We generalize Equation \[eq:qualobjfinal\] to account for the case of three-category observations by allowing for two separate parameters. We define the positive discrepancy rate $\epsilon_i^+$ as the probability that a constraint in the data is satisfied regardless of $Y_i$, and the negative discrepancy rate $\epsilon_i^-$ as the probability a constraint is violated regardless of $Y_i$. For example, with the above sampling model, for the observation $c_{i,2}<y_i$, we would use $\epsilon_i^+ = \epsilon_i$ and $\epsilon_i^- = 2\epsilon_i$
Our modified likelihood function is
$$-\log P(\mathbf{y}<\mathbf{c}|{\boldsymbol{\theta} }) = \sum_{i=1}^n -\log (\epsilon_i^+ + (1-\epsilon_i^+-\epsilon_i^-) \textrm{cdf}(f_i({\boldsymbol{\theta} }),\sigma_i,c_i))
\label{eq:qualobjfinalplus}$$
We will refer to this function as the *many-category likelihood function*.
The same formulation can be extended to allow for an arbitrary number of ordinal categories. For example, with four categories defined by the thresholds $c_{i,1}$, $c_{i,2}$, and $c_{i,3}$, we could write expressions analogous to Equations \[eq:yltc1\]-\[eq:ygtc2\] for $P(y_i<c_{i,1}|{\boldsymbol{\theta} })$, $P(c_{i,1}<y_i<c_{i,2}|{\boldsymbol{\theta} })$, $P(c_{i,2}<y_i<c_{i,3}|{\boldsymbol{\theta} })$, and $P(y_i>c_{i,3}|{\boldsymbol{\theta} })$.
We illustrate the use of Equation \[eq:qualobjfinalplus\] with a concrete example. Suppose we have a quantity of interest with the corresponding random variable $A$, and we make a qualitative observation with three possible outcomes: $a<100$, $a\approx100$, or $a>100$. Suppose also that based on the sensitivity of the assay, we know that any value of $a$ in the range 85–115 would be reported as “$a\approx100$.” Given this knowledge of the assay sensitivity, we take the standard deviation of $A$ to be 5, that is, we can only confidently report $a<100$ if $a$ is 3 standard deviations below the threshold of 100. We choose the sampling model shown in Figure \[fig:ex3\]A, giving a base probability of 0.03 to each possible outcome due to model discrepancy. Note that this sampling model follows the requirement that the probabilities of all possible outcomes sum to 1. We then formulate the constraint(s) as shown in Figure \[fig:ex3\]B, depending on whether the actual observation is $a<100$, $a\approx100$, or $a>100$. The resulting probabilities are shown in Figure \[fig:ex3\]C as a function of the expected value of $A$ predicted by the model.
When using the many-category likelihood function, it is important to consider the underlying sampling model, and choose $\epsilon_i^+$ and $\epsilon_i^-$ in a way such that the probabilities in the sampling model sum to 1. An example of how to correctly choose $\epsilon_i^+$ and $\epsilon_i^-$ given a sampling model is presented in Section \[sec:application\].
![Example constraints and probabilities arising from a qualitative observation with three possible categorical outcomes. (A) The sampling model associated with the observation. (B) Inequalities and $\epsilon^{+}$ and $\epsilon^{-}$ values associated with each possible observation outcome (C) Plots and equations giving the probability of each possible observation outcome as a function of the expected value of model output $A$. []{data-label="fig:ex3"}](Fig2.eps)
### Combined likelihood function
If independent quantitative and qualitative data are available, it is straightforward to combine the chi squared likelihood function for quantitative data with one of the newly presented likelihood functions for qualitative data. One would simply sum Equations \[eq:chisq\] and \[eq:qualobjfinal\] (or \[eq:qualobjfinalplus\]) to obtain the kernel of the negative log likelihood for the combined dataset.
The relative weighting of the two datasets is determined by the standard deviations for the quantitative data points and the values of $\sigma_i$ and $\epsilon_i$ for the qualitative observations.
Software implementation
-----------------------
We implemented the likelihood functions described in the previous section in PyBioNetFit v1.1.0. PyBioNetFit supports both the two-category (Equation \[eq:qualobjfinal\]) and many-category (Equation \[eq:qualobjfinalplus\]) likelihood functions for qualitative data, and supports combining these functions with the chi squared likelihood function for quantitative data.
The new options were added via an extension of the Biological Property Specification Language (BPSL) supported by PyBioNetFit. As previously described [@Mitra2019a], a BPSL statement consists of an inequality, followed by an enforcement condition, followed by a weight. For example, in the statement $$\texttt{A<4 at time=1 weight 2}$$ the inequality is `A<4` (referring to some modeled quantity $A$), the enforcement condition is `time=1` (referring to time 1 in a time course), and the weight is declared by `weight 2`. This weight declaration refers to $C_i$ in the previously described static penalty function (Equation \[eq:static\]). Using this formulation, the term added to an objective function for this constraint would be $2 \cdot \max(0,A(1)-4)$, where $A(1)$ is model output $A$ evaluated at time = 1.
In PyBioNetFit v1.1.0, we added an alternative to the weight clause to specify parameters of the new likelihood functions. As described in Section \[sec:twocat\], for each inequality in the data, the two-category likelihood function has two user-configurable parameters: the probability $\epsilon_i$ of measuring $y_i<c_i$ regardless of the distribution of $Y_i$, and the standard deviation $\sigma_i$ of the quantity $Y_i$. The value of $1-2\epsilon_i$ (i.e., the probability that the distribution of $Y_i$ is relevant to the experimental result) is supplied to PyBioNetFit with the `confidence` keyword. $\sigma_i$ is supplied to PyBioNetFit with the `tolerance` keyword. Therefore, an example BPSL statement using the two-category likelihood function is
$$\texttt{A<4 at time=1 confidence 0.98 tolerance 0.5}$$
This statement would result in using the likelihood function of Equation \[eq:qualobjfinal\] with $\epsilon_i=0.01$, $\sigma_i=0.5$, $c_i=4$, and $Y_i=A(1)$. The resulting term added to the likelihood function is $-\textrm{log}(0.01+0.98\cdot\textrm{cdf}(A(1),0.5,4))$.
PyBioNetFit also supports the use of the many-category likelihood function (Equation \[eq:qualobjfinalplus\]) through the specification of separate positive and negative discrepancy rates. In this case, the `confidence` keyword is replaced with the keywords `pmin` to specify $\epsilon_i^-$ (i.e., the minimum value of $P(y_i<c_i|{\boldsymbol{\theta} })$) and `pmax` to specify $1-\epsilon_i^+$ (i.e., the maximum value of $P(y_i<c_i|{\boldsymbol{\theta} })$). For example, the BPSL statement
$$\texttt{A<4 at time=1 pmin 0.01 pmax 0.98 tolerance 0.5}$$
would use Equation \[eq:qualobjfinalplus\] with $\epsilon_i^+=0.02$, $\epsilon_i^-=0.01$, $\sigma_i=0.5$, $c_i=4$, and $Y_i=A(1)$. The resulting term added to the likelihood function is $-\textrm{log}(0.01+0.97\cdot\textrm{cdf}(A(1),0.5,4))$.
When writing these statements in BPSL, care must be taken to ensure that results are statistically valid. First, note that the `tolerance` specifies the standard deviation of the final random variable $Y_i$ used to sample $y_i$ in Equation \[eq:qualobjfinal\]. For example in the above statement, it refers to the standard deviation of $A(1)$. In the statement `A>B at time=5 confidence 0.98 tolerance 0.5`, `tolerance` refers to the standard deviation of $A(5)-B(5)$, i.e., the sum of the standard deviations of $A(5)$ and $B(5)$. In the statement `A>4 always confidence 0.98 tolerance 0.5`, `tolerance` refers to the standard deviation of $\min(A(t))$, rather than the value of $A$ at any particular time.
Second, it is important to keep in mind the underlying sampling model to correctly set `confidence` or `pmin` and `pmax`. For example, in the sampling model of Fig \[fig:ex3\]A, there are three possible constraints each with probability 0.03 to be satisfied due to model discrepancy and probability 0.06 to be violated due to model discrepancy. Therefore, the correct setting is `pmin 0.03 pmax 0.94`.
Third, when using PyBioNetFit’s enforcement keywords `always`, `once`, and `between`, it is important to be sure the possible categories in the sampling model are mutually exclusive and cover all possible outcomes. For example, if one of two possible categorical outcomes is `A>4 always`, the other must be `A<4 once` (not `A<4 always`). Likewise, if one category is `A>4 between time=5,time=10`, its negation is `A<4 once between time=5,time=10`. We note that the `once between` enforcement condition used here is a new feature of BPSL introduced in PyBioNetFit v1.1.1.
The sampling model is never explicitly input into PyBioNetFit, as equations \[eq:qualobjfinal\] and \[eq:qualobjfinalplus\] are defined regardless of whether the sampling model is well-defined. It is the user’s responsibility to choose a well-defined sampling model and specify constraints accordingly to obtain meaningful results.
![Configuration of the example problem in BPSL. As described in the text, we considered the problem assuming either (A) two possible observation categories or (B) three possible categories. The left column shows an example BPSL statement for each possible category. In these BPSL statements, `p1` refers to the primary degranulation and `p3_`$<t>$ refers to the secondary degranulation after a delay of $t$ minutes. Note that in the three-category case, the middle category requires two separate BPSL statements. The right column shows simulated trajectories of the primary (left) and secondary (right) degranulation responses that are consistent with the BPSL statement. For the three-category case, `degrHigh` and `degrLow` are functions defined in the BNGL model file for use in the BPSL statements.[]{data-label="fig:setup"}](Fig3.eps)
Example application {#sec:application}
-------------------
To demonstrate the use of qualitative likelihood functions in PyBioNetFit, we performed Bayesian UQ on a synthetic example problem based on the study of [@Harmon2017]. The model of [@Harmon2017] describes the degranulation of mast cells in response to two consecutive stimuli with multivalent antigen. In the original study, it was found that depending on the time delay between the two stimuli, the secondary response could be either stronger or weaker than the primary response. The original data consisted of quantitative degranulation measurements for six different time delays.
In our synthetic problem, we suppose that the experimental data took a different form. Rather than quantitative measurements, we assume that it is only possible to measure whether the secondary degranulation is higher or lower than the primary degranulation. These measurements can be seen as case-control comparisons between several conditions of interest (secondary degranulation at various time delays) and a control (primary degranulation). We assume that these measurements can be made at a larger number of time delays than were used in the original study (i.e., we have a less precise but higher throughput instrument than in the actual study).
We generated synthetic data of this form using the published parameter values of the model as ground truth. For each time delay in the data, we ran a simulation, and added Gaussian noise to the primary and secondary degranulation outputs before recording whether the primary or secondary was higher. We generated datasets ranging from 4 to 64 time delays. The resulting datasets were implemented in BPSL as illustrated in Figure \[fig:setup\]A. Note that we set the `confidence` to 0.98, allowing for a 0.02 chance of model discrepancy (although there is no true model discrepancy in this synthetic problem). We set the `tolerance` to $1.4 \times 10^4$, which is the standard deviation of the difference between the primary and secondary degranulation values (i.e., twice the standard deviation of the added noise for each individual degranulation value).
We configured PyBioNetFit jobs to perform Bayesian UQ by parallel tempering for each dataset. The results are shown in Figure \[fig:bayes\]A-D and Figures S1–S5. Not surprisingly, as the number of qualitative observations increases, we obtain a narrower distribution of parameter values, and these narrower distributions include the ground truth parameter values. This result demonstrates that with a sufficient amount of qualitative data, it is possible to find nontrivial credible intervals for parameter values.
To demonstrate the use of the many-category likelihood function (Equation \[eq:qualobjfinalplus\]), we repeated the analysis using three-category synthetic data. Our three categories allow the secondary degranulation to be measured as smaller, larger, or within error of the primary degranulation. The three-category dataset was declared in BPSL as illustrated in Figure \[fig:setup\]B. Compared to the two-category synthetic data, modifications were required as described in Section \[sec:threecat\]. The assumed sampling model used for the constraints in Figure \[fig:setup\]B is the following, where $Y_i$ represents the primary degranulation minus the secondary degranulation:
To generate observation $i$, make a weighted random choice of one of the following possibilities:
- With probability 0.97, sample $y_i$ from $Y_i$ and report whether $y_i<-4.2\times 10^4$ or $-4.2\times 10^4<y_i<4.2\times 10^4$ or $4.2\times 10^4<y_i$
- With probability 0.01, report $y_i<-4.2\times 10^4$
- With probability 0.01, report $-4.2\times 10^4<y_i<4.2\times 10^4$
- With probability 0.01, report $4.2\times 10^4<y_i$
We have chosen a threshold of $4.2\times 10^4$ for the difference between primary and secondary degranulation that qualifies as “within error.” This value is three times the standard deviation of $Y_i$, giving the separation of categories required in Section \[sec:threecat\] (i.e., any sampled $y_i$ is consistent with at most two possible categories). This condition allows us to define the middle category ($-4.2\times 10^4<y_i<4.2\times 10^4$) using two independent BPSL statements. The choice of threshold is reflected in the BPSL by the use of the model outputs referred to as `degrHigh` and `degrLow`. Based on the sampling model, each category has a minimum probability of 0.01 due to model discrepancy, and a maximum probability of 0.98 (because the other two categories each have a minimum of 0.01). Therefore, we set `pmin` to 0.01 and `pmax` to 0.98 instead of using the `confidence` keyword. Finally, the `tolerance` is set to $1.4\times 10^4$, the same as for the two-category dataset.
The results of parallel tempering using this dataset are illustrated in Figure \[fig:bayes\]E and Figures S6–S10. As expected, compared to the results with two-category dataset of the same size, some parameters are bounded more tightly around their ground truth values.
For comparison, we also performed the analysis using synthetic quantitative data generated at the same time delays as in the original study (Figure \[fig:bayes\]F and Figure S11). The quantitative dataset produced distributions even tighter than those of the three-category qualitative data. It is notable how close we can get to the results with quantitative data by using purely qualitative data.
Discussion
==========
Here we have presented a new statistical framework for using qualitative data in conjunction with Bayesian UQ for biological models. In these models, unidentifiable parameters are common, but Bayesian analysis can determine which parameters and correlations are identifiable, and to what extent the model has predictive value despite unidentifiable parameters.
We see this framework as a more statistically rigorous improvement upon our previously described static penalty function approach [@Mitra2018a] (Equation \[eq:static\]). Our new framework can be used for statistical analysis, whereas the previous formulation was simply a heuristic for finding a single reasonable parameter set. Our new likelihood function has applications beyond Bayesian UQ. It can also, like the static penalty function, be used with optimization algorithms to find a point estimate of the best parameters. In such a problem, the global minimum (assuming it can be found by an optimization algorithm) is the maximum likelihood estimate, i.e., the maximum of the posterior distribution. The new likelihood function may also be used for UQ by profile likelihood analysis [@Kreutz2013].
The static penalty function may remain more efficient at point estimation. The cdf-based likelihood function has the limitation that when far from constraint satisfaction, its gradient is near zero, and so it cannot effectively guide the optimization algorithm toward constraint satisfaction. In contrast, the static penalty function provides useful information for optimization at any distance from constraint satisfaction. One potential workflow could be to use the static penalty function for initial optimization, followed by the likelihood function for refinement and evaluation of the best fit.
We note that under our new framework, each constraint now has two adjustable settings: $\epsilon_i$ and $\sigma_i$. This may appear worse than the single weight parameter $C_i$ in the static penalty formulation, but the advantage is that both of these parameters have a statistical interpretation. $\epsilon_i$ represents the probability of model discrepancy resulting in a qualitative observation that occurs regardless of the model and its predicted mean. $\sigma_i$ represents the standard deviation of the quantity considered in the constraint. This value might seem challenging to estimate, given we may not even be able to quantitatively measure the quantity of interest. However, much of the same intuition holds as when dealing with Gaussian-distributed quantitative data. In particular, if there is a difference of $2\sigma_i$ between a threshold and the mean, we can be reasonably confident (probability 97.7%) that an observation would yield the correct result (greater or less than the threshold). With a difference of $3\sigma_i$, we can be extremely confident (probability 99.87%). To choose $\sigma_i$, a reasonable thought process would be to ask, “How large of a difference would there have to be for the experiment to be sure to detect the difference?”, and set $\sigma_i$ equal to one third of that difference. Both parameters can be seen as optional. If we don’t expect a scenario in which a constraint is impossible to reconcile with our model, we can set $\epsilon_i=0$, ignoring this aspect of the likelihood function. Likewise, if we have no way to estimate the standard deviation of the measured quantity, we could set $\sigma_i=0$ and use $\epsilon_i$ to set a fixed probability of satisfying the constraint. Thus, the two adjustable constants should be seen as an opportunity to provide all available information about a qualitative observation of interest, rather than as a burden for manual adjustment.
We expect that our new formulation of a likelihood function derived from qualitative data will be useful in future modeling studies and will help facilitate the wider adoption of qualitative data as a data source for model parameterization.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge computational resources provided by the Institutional Computing program at Los Alamos National Laboratory, which is operated by Triad National Security, LLC for the NNSA of DOE under contract 9233218CNA000001. We thank Steven Sanche for useful discussions.
Funding {#funding .unnumbered}
=======
This work has been supported by NIH/NIGMS grant R01GM111510.
[^1]: Corresponding author. wish@lanl.gov
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abstract: 'The spatially averaged density of states, $\langle$$N(0)$$\rangle$, of an unconventional $d$-wave superconductor is magnetic field dependent, proportional to $H^{1/2}$, owing to the Doppler shift of quasiparticle excitations in a background of vortex supercurrents. [@vol88; @vol93] This phenomenon, called the Volovik effect, has been predicted to exist for a sign changing $s\pm$ state, [@ban10] although it is absent in a single band $s$-wave superconductor. Consequently, we expect there to be Doppler contributions to the NMR spin-lattice relaxation rate, $1/T_1 \propto$$\langle$$N(0)^2$$\rangle$, for an $s\pm$ state which will depend on magnetic field. We have measured the $^{75}$As $1/T_1$ in a high-quality, single crystal of Ba$_{0.67}$K$_{0.33}$Fe$_{2}$As$_{2}$ over a wide range of field up to 28 T. Our spatially resolved measurements show that indeed there are Doppler contributions to $1/T_1$ which increase closer to the vortex core, with a spatial average proportional to $H^2$, inconsistent with recent theory. [@ban11]'
author:
- 'Sangwon Oh$^{1}$, A. M. Mounce$^{1}$, W. P. Halperin$^{1}$, C. L. Zhang$^{2}$, Pengcheng Dai$^{2}$, A. P. Reyes$^{3}$, P. L. Kuhns$^{3}$'
date: Version
title: 'Magnetic field dependence of spin-lattice relaxation in the s$\pm$ state of Ba$_{0.67}$K$_{0.33}$Fe$_{2}$As$_{2}$'
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The spin-lattice relaxation rate, $1/T_1$, in the superconducting state can provide valuable information about gap structure and about the effects of vortices on the quasiparticle density of states. There have been a number of reports on the temperature dependence of $1/T_1$ at low magnetic fields for various pnictide superconductors, [@kam08; @nak10; @nin10; @fuk09; @yas09; @li11; @gra08] which have been interpreted in terms of order parameter structure. However, impurities of unknown origin and concentration can play an important role in the analysis. For this reason, conclusions about the superconducting state based on temperature dependences can be ambiguous.
An alternative approach to explore unconventional character of the order parameter is the magnetic field dependence of the density of states, which can be specific to a particular order parameter symmetry, easily probed through specific heat or thermal conductivity measurements. [@yip92] The unconventional structures of the $s\pm$ and $d$-wave states each lead to characteristic magnetic field dependences of the spatially averaged density of states, attributable to the Doppler shift of quasiparticle excitations, known as the Volovik effect. [@vol88; @vol93] For the $d$-wave case $c$-axis line nodes in the gap give rise to a field dependence, $\propto H^{1/2}$ and a non-linear Meissner effect. [@yip92] In contrast, according to Bang, [@ban10] in the case of $s\pm$ symmetry for a multiband superconductor, the spatially averaged density of states at the Fermi energy is proportional to the magnetic field. Since $1/T_1$ is proportional to the square of the local density of states, one might think that the predicted Volovik effect should be $\propto H^2$. However, the Volovik effect pertains to the [*spatial average*]{} over the vortex unit cell which decreases in area inversely proportional to the magnetic field. According to the theory, there is a region of normal state excitations surrounding the vortex core of radius $\xi(\Delta_2/\Delta_1)$, where $\xi$ is the core radius $(\sim 30\,\, \AA)$, equal to the coherence length, and $\Delta_2/\Delta_1$ is the ratio of large to small gaps, leading to the prediction, $1/T_1T \propto H$ at low magnetic fields, $H \leq H_{c2} (\Delta_1/\Delta_2)^2 \sim 3$ T; otherwise it should be constant. [@ban11] In this Letter, we report $^{75}$As NMR measurements in single crystals of Ba$_{0.67}$K$_{0.33}$Fe$_{2}$As$_{2}$ covering a wide range of magnetic fields. Our results show that indeed there is a Doppler contribution to the spatially averaged spin-lattice relaxation but that in the low temperature limit, $1/T_1T \propto H^2$ over the whole range of magnetic field, inconsistent with prediction. [@ban11]
We performed our $^{75}$As NMR measurements at Northwestern University and the National High Magnetic Field Laboratory, from 4 K to room temperature with external magnetic field from 6.4 to 28 T. The fields were parallel to the $c$-axis of the single crystals, Ba$_{0.67}$K$_{0.33}$Fe$_{2}$As$_{2}$ (BaK122) that had a zero-field $T_c = 38$ K and were grown at the University of Tennessee by the self-flux method. [@zha11] To increase signal intensity in the superconducting state, the crystals were cleaved to dimensions of 3$\times$3$\times$0.1 mm$^{3}$ and total mass of 17 mg. Typically, spin echo sequences ($\pi/2$ - $\pi$) were used to obtain the spectrum, Knight shift, and $1/T_1$ for the central transition (-1/2 $\Leftrightarrow$ 1/2) with a $\pi$-pulse $\approx$ 7 $\mu$sec. The spin-lattice relaxation was measured with the full recovery method (28 to 300 K) and progressive saturation techniques, [@mit01a] (4 to 26 K) the latter being more accurate for very long relaxation times at low temperatures. The average rate was measured with the $\pi$-pulse centered on the spectrum. Frequency-resolved spin-lattice relaxation was also measured by dividing the spectrum into many small frequency windows and the relaxation was determined separately in each window. Knight shift measurements were performed with a frequency sweep method.
![$^{75}$As NMR spectra of Ba$_{0.67}$K$_{0.33}$Fe$_{2}$As$_{2}$ measured by a frequency sweep technique in 13 T with $H|| c$-axis of the crystals. Below $T_c = 32$ K (blue trace) the spectra shift noticeably to lower frequency with decreasing temperature. The linewidths of the spectra decrease in the superconducting state as reported in some other compounds, [@che07; @oh11] where this was attributed to reduction in the local field distribution from impurities. []{data-label="fig1"}](fig1.eps){width="40.00000%"}
Early experiments on optimally, hole-doped, single crystals of Ba$_{0.6}$K$_{0.4}$Fe$_{2}$As$_{2}$ [@muk09] grown with tin flux did not detect any signal below 20 K due to linewidth broadening from paramagnetic impurities on the As sites at a level of $\approx 1 \%$. However, there have been substantial improvements in lowering the impurity concentration using the self-flux method. [@li11; @zha11] The frequency-swept $^{75}$As NMR spectra of our crystals in 13 T with $H||c$-axis, are shown in Fig. \[fig1\]. The $T_c$ in $H=13$ T is 32 K, and a shift of the spectra can be easily seen. This decrease of the Knight shift indicates spin-singlet pairing in the superconducting state. On cooling, the linewidth slowly broadens from 60 kHz at $T=300$ K to 70 kHz at $T_c$. Below $T_c$ the linewidth increases up to 110 kHz near $20$ K, and then it decreases to 80 kHz at 4 K, and is independent of magnetic field from 6.4 to 16.5 T, to within 10%. The weak dependence of the linewidth on magnetic field and temperature in the normal state indicates that few magnetic impurities are present, comparable to the cleanest cuprate crystals such as Bi$_2$SrCa$_2$Cu$_2$O$_{8+\delta}$ (Bi2212). [@che07] This point is also consistent with the similar results we find from our comparison of the zero field extrapolations of $1/T_1$ with those of clean Bi2212 crystals which we discuss later. The Knight shift, $K = K_s + K_{orb}$, was determined from the first moment of the NMR spectrum where $K_s$ and $K_{orb}$ are the spin and orbital parts of the shift, respectively. The orbital part is temperature and field independent, consequently the temperature dependence of the shift in Fig. \[fig2\] can be associated with $K_s$, decreasing below $T_c$ on cooling. The solid curve in the figure is the temperature dependence of $K_s$ that we describe with a phenomenological model for the density of states, Eq. \[eq1\], based on the parameters obtained from $1/T_1$ measurements.
![The total Knight shift K(T) is shown for $H=13$ T where the temperature dependence is associated with the spin part, $K_s$, that decreases below $T_c$, consistent with spin-singlet superconductivity. The black arrow indicates $T_c = 32$ K. The data can be fit phenomenologically assuming that the low temperature spin shift is proportional to an average density of states, Eq. \[eq1\], represented by the solid curve, with an orbital shift, $K_{orb}=0.21\%$.[]{data-label="fig2"}](fig2.eps){width="40.00000%"}
The behavior of the spin-lattice relaxation in the superconducting state is the main focus of our present work where we measure the temperature and magnetic field dependence for $H= 6.4, 10.8, 14, 16.5, 27$ and $28$ T, parallel to the $c$-axis of the crystals. The rates were measured with the spectrometer frequency set at the peaks of the spectra. A coherence peak below $T_c$ was not observed, and the suppression of $T_c$ by the magnetic field was minimal, from $T=32$ to 30 K when the external field was increased from 6.4 to 27 T. In low magnetic fields, [*i.e.*]{} 6.4 and 10.8 T, the temperature dependence of $1/T_1$ could be approximately described as $T^3$ at intermediate temperature, as has often been reported elsewhere. [@gra08; @fuk09] But in higher fields, 14, 16.5, 27 T, below $T=10$ K, we find $1/T_1 \propto T$ , indicating a constant average density of states at zero energy , $\langle$$N(0)$$\rangle$. Recently Li ${{\it et\ al.}}$ [@li11] observed an exponential temperature dependence of the rate in a magnetic field of $H=7.5$ T consistent with the presence of a full gap. A comparison of our data with that of Li ${{\it et\ al.}}$ shows that they are identical except at our lowest temperature, $T=4$ K, where our higher value of $1/T_1$ might be understood as the effect of residual impurities in our crystal obscuring exponential behavior. Increasing the magnetic field we find that the spin-lattice relaxation at 4 K increases systematically indicating the existence of a field dependent density of states at the Fermi surface. This observation is a characteristic signature of a Volovik effect.
![The spin lattice relaxation rate, $1/T_1$, of Ba$_{0.67}$K$_{0.33}$Fe$_{2}$As$_{2}$ in magnetic fields, $H=6.4, 10.8, 14, 16.5$, and 27 T. The rates were measured at the peak position in the spectrum. The data is consistent with our two-gap model (solid curves) provided the magnetic field dependence at low temperature is $\propto H^2$. The black arrow indicates $T_c$.[]{data-label="fig3"}](fig3.eps){width="40.00000%"}
We use a phenomenological model to fit $1/T_1$ in various magnetic fields. We express the thermal and spatial average over the density of states at the Fermi surface as, $$\langle N(0) \rangle = a(H) + b_0 e^{-\Delta_1/k_BT} + c_0 e^{-\Delta_2/k_BT}
\label{eq1}$$ where $a(H) = a_0 + a_1H + . . $ and $a_0$ represents possible contributions from non-magnetic impurities. The two gaps, $\Delta_1$ and $\Delta_2$, appear in exponential terms with relative weights, $b_0$ and $c_0$, as might be expected for the low temperature limit. Since $1/T_1T \propto \langle N(0)^2 \rangle$, our model for $1/T_1T$ becomes, $$1/T_1T \propto [ a(H) + b_0 e^{-\Delta_1/k_BT} + c_0 e^{-\Delta_2/k_BT} ]^2.
\label{eq2}$$ At low temperatures the two exponential terms are of little importance and the rate is determined by $a(H)$. Our numerical analysis provides fits for all of the parameters of the model. Below $H =16.5$ T, we take them to be magnetic field independent. However, at this and higher magnetic fields we find that the relative weight of the exponential term from the smaller gap, $b_0$, must be reduced compared to the larger gap weight, $c_0$, in order to fairly represent the data. As stated previously, these gap parameters are not important in the low temperature limit where we seek to describe the field dependence of the relaxation rate and so we do not ascribe specific importance to this additional field dependence other than it allows us to represent the high temperature behavior in each field. Nonetheless, we point out that our results for the temperature dependence at low magnetic field are identical to those from Li [*et al.*]{} [@li11] for clean crystals, except for the lowest temperature point at 4 K.
![(a)-(e) $1/^{75}T_1T$ with the best fit curve in each magnetic field. The unit of the $a(H), b_0,$ and $c_0$ is $(s^{-1}K^{-1})$, and the gaps are in meV. (f) $1/^{75}T_1T$ in BaK122 at 4 K, is shown for comparison with the Zeeman contributions to $1/^{17}T_1T$ in Y123 and Bi2212 at 5 K. Assuming that the electronic $g$-factor is the same for BaK122 as for the cuprates, we argue that the Zeeman contribution to the field dependence of the average rate we have measured in BaK122 is significantly smaller than from Doppler contributions. In the case of the cuprates, the Zeeman contributions to the spin-lattice relaxation rate were isolated using frequency resolved measurements performed at the saddle point of the local field distribution (peak of the spectrum) where Doppler contributions cancel based on symmetry of the supercurrents from near-neighbor vortices. []{data-label="fig4"}](fig4.eps){width="50.00000%"}
Our analysis in each field is shown in Fig. \[fig4\](a)-(e), where $\Delta_1$ and $\Delta_2$ are 2.1 $\pm$ 0.2 meV and 12.1 $\pm$ 1.4 meV respectively. The sizes of the gaps correspond well to the sizes of the 3D superconducting gap function from ARPES measurements, 2.07 meV and 12.3 meV. [@xu11] The ratio of the coefficients, $b_0$ and $c_0$, decreases at $H=16.5$ and 27 T, indicating a possible suppression of the smaller superconducting gap, $\Delta_1$, by the external magnetic field. The low temperature magnetic field dependence of $1/T_1T$ is given by a(H) shown in Fig. \[fig4\](f). The $H^2$ behavior might be associated with Doppler shifted quasiparticles, although the field dependence is different from that predicted by theory. [@ban11] It should be noted that the electronic Zeeman interaction also contributes to the quasiparticle energy giving a $H^2$ dependence to $1/T_1T$. [@mit01b] In Fig. \[fig4\](f) we show the field dependence of $1/T_1T$ for $^{17}$O NMR from YBa$_2$Cu$_3$O$_{7+\delta}$(Y123) aligned powders [@mit01b] and Bi2212 crystals [@mou11] which has been attributed to this Zeeman term. From comparison with these compounds, allowing for the 27% larger gyromagnetic ratio of arsenic compared to oxygen, it is reasonable to conclude that the significantly larger field dependence of $1/T_1T$ for BaK122, cannot be attributed to the Zeeman term. It is notable that the three materials have a similar value of the spin-lattice relaxation in the limit of zero field. Although, to some extent, this might be fortuitous, it nonetheless suggests that our BaK122 crystal does not have significantly more impurity scattering than these high quality cuprate materials. Since the NMR linewidth at $T=40$ K is independent of magnetic field for $H \leq16.5$ T to within 10%, we do not associate the field dependence of the rate with magnetic impurities. However, this possibility can be investigated further by measurement of the frequency-resolved spin-lattice rate which we describe next.
For unconventional superconductors $1/T_1$ can depend on the position of the probe nucleus relative to the vortex core. [@vol88; @tak99; @mit01b; @mou11] The increase in the supercurrent momentum, ${\bf p}_s$, approaching the core leads to a corresponding increase in the Doppler shift of the energy of quasiparticle excitations, ${\bf v}_F \cdot {\bf p}_s$, where ${\bf v}_F$ is Fermi velocity. The vortex core, having the highest local magnetic field, corresponds to the largest frequency in the NMR spectrum. We have looked for evidence of this spatial dependence of $1/T_1$ through frequency-resolved, [*i.e.*]{} spatially resolved, measurements performed across the spectrum, as shown in Fig. \[fig5\].
In the normal state (40 K) we find a flat $1/T_1$ distribution throughout the spectrum as expected in the absence of Doppler terms or magnetic impurity contributions to the rate. In the superconducting state, there is an increase of $1/T_1$ with frequency, developing markedly at $T=26$ K with more than an order of magnitude variation across the spectrum.
![ Spin-lattice relaxation rate across spectrum in the normal state (a) and superconducting states(b),(c),(d) in 16.5 T, with $H||c$-axis. In the normal state, at 40 K, there is no significant frequency dependence in $1/T_1$. However, the rate becomes dependent on frequency as the sample is cooled deep into the superconducting state. []{data-label="fig5"}](fig5.eps){width="50.00000%"}
We note that the linewidth, $\sim 80$ kHz at 4 K in $H=16.5$ T is somewhat broader than our calculation from Ginzburg-Landau theory using Brandt’s algorithm [@bra97] for a perfect vortex lattice, $\sim 23$ kHz. However, even in a somewhat disordered vortex structure, the high field portion of the spectrum can be associated with nuclei in the vortex core. This is the case for the distribution in $1/T_1$ observed in Y123, which was attributed to the Doppler shift [@mit01b] of quasiparticle energy from vortex supercurrents. Our frequency-resolved measurements of $1/T_1$ in BaK122, Fig. 5, show the existence of a spatially inhomogeneous distribution which onsets with superconductivity. We ascribe this to the vortex state for which the most likely explanation is a Volovik effect. Another explanation was suggested some years ago to explain observations in superconducting vanadium compounds. [@sil66; @sil67] There it was argued that spin-diffusion from relaxation sources in the vortex core might produce a spatially inhomogeneous distribution of $1/T_1$. Later measurements and theoretical work by Genack and Redfield [@gen73; @gen75] showed that this suggestion was incorrect, and that spin diffusion is quenched on very short time scales owing to depletion of the dipole energy reservoir, an effect even further suppressed with increasing field. We measured the spin lattice relaxation rates in higher fields, 24 T and 28 T, as shown in Fig. \[fig6\]. An inhomogeneous spin-lattice relaxation rate distribution was found similar to that of H = 16.5 T, Fig. \[fig5\], and rules out spin diffusion as a possible mechanism. [@sil66; @sil67]
![ Spin-lattice relaxation rate across spectrum at 4 K in 24 T (a) and 28 T (b) with $H||c$-axis. The inhomogeneous frequency dependence of the rate is observed in both magnetic fields similar to H = 16.5 T, Fig.\[fig5\] (d). Additionally, the increase in the high frequency part in the spectra can be understood as an asymmetry from the vortex lattice.[]{data-label="fig6"}](fig6.eps){width="50.00000%"}
With reports from experiments in cuprates a decade ago [@cur00; @mit01b] this mechanism was studied theoretically by Wortis, [@wor98] who came to the same conclusion. A more detailed discussion has been provided by Mounce [*et al*]{}. [@mou11b] We point out that in the recent theory [@ban10; @ban11] of the Volovik effect in $s\pm$ superconductors the combined effects of the Zeeman interaction and vortex supercurrents have not been taken into account. Their importance was indicated in the work of Mitrović [*et al.*]{} on YBa$_2$Cu$_3$O$_7$ [@mit01b] and might be an important component missing from the theory. We conclude that our observations are most likely a consequence of vortex supercurrents but for which there is not yet a satisfactory theoretical explanation.
In summary, we have studied the $^{75}$As Knight shift and spin-lattice relaxation rate in slightly underdoped Ba$_{0.67}$K$_{0.33}$Fe$_{2}$As$_{2}$ crystals in the superconducting mixed state. We found that $1/T_1T$ approaches a constant at low temperatures in high magnetic field and is proportional to the square of the field. Although this is inconsistent with a theory for the Volovik effect, [@ban10; @ban11] our results can be accounted for by a phenomenological model which is based on $s\pm$ symmetry with two isotropic gaps, and non-magnetic impurities. The distribution of $1/T_1$ across the spectrum resembles that observed in a vortex solid of an unconventional superconductor associated with spatially resolved Doppler contributions to the quasiparticle excitation spectrum.
We thank Y. Bang, G.E. Volovik, P.J. Hirschfeld, and J.A. Sauls for helpful discussions. Research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Awards DE-FG02-05ER46248 (Northwestern University) and No. DE-FG02-05ER46202 (the University of Tennessee). Work at high magnetic field was performed at the National High Magnetic Field Laboratory supported by the National Science Foundation and the State of Florida.\
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abstract: 'We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.'
address:
- 'Departamento de Geometría y Topología, c/ Tarfia s/n, Universidad de Sevilla, Sevilla 41012, Spain'
- 'Departamento de Didáctica de las Matemáticas, Facultad de Ciencias de la Educación, c/ Pirotecnia s/n, Universidad de Sevilla, Sevilla 41013, Spain'
- 'LAMAV, Université de Valenciennes, 59313, Valenciennes Cedex 9, France Departement Wiskunde, KU Leuven, Celestijnenlaan 200 B, 3001, Leuven, Belgium'
author:
- Alfonso Carriazo
- 'Verónica Martín-Molina'
- Luc Vrancken
title: 'A classification of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces'
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[^1]
Introduction
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Although under a different name, $(\kappa,\mu)$-spaces were introduced by D. E. Blair, T. Koufogiorgos and B. J. Papantoniou in [@blair95] (for technical details, we refer to the Preliminaries section). Actually, these manifolds have proven to be really useful, because they provide non-trivial examples for some important classes of contact metric manifolds (for instance, the unit tangent sphere bundle of any Riemannian manifold of constant sectional curvature carries such a structure). The theory of $(\kappa,\mu)$-spaces was soon developed, with many interesting results. In particular, we can point out the outstanding paper [@boeckx2000], where E. Boeckx classified non-Sasakian $(\kappa,\mu)$-spaces by using the invariant $I$ (depending only on the values of $\kappa$ and $\mu$) introduced by himself. He also provided examples for all possible $(\kappa,\mu)$.
Nevertheless, the theory of submanifolds of $(\kappa,\mu)$-spaces has not been developed in depth yet, even if we can find some very interesting papers about it. For example, in [@CTT], B. Cappelletti Montano, L. Di Terlizzi and M. M. Tripathi proved that any invariant submanifold of a non-Sasakian contact $(\kappa,\mu)$-space is always totally geodesic and, conversely, that every totally geodesic submanifold of a non-Sasakian contact $(\kappa,\mu)$-space such that $\mu\neq 0$ and the characteristic vector field $\xi$ is tangent to the submanifold is invariant. Motivated by these results, we consider the case of submanifolds which are normal to $\xi$. Moreover, we restrict our study to the case of Legendrian submanifolds, i.e., those with dimension $n$ in a $(2n+1)$-dimensional ambient space.
>From our point of view, a key step in continuing the analysis of submanifolds of $(\kappa,\mu)$-spaces should be to understand the behavior of the so-called $h$ operator of the ambient space with respect to the submanifold. Therefore, in this paper, we first establish in Section \[sec3\] a decomposition of that operator in its tangent and normal parts, and find its main properties. In Section \[sec4\] we present several examples of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces with $I\leq -1$. Actually, we prove in Section \[sec5\] that these examples constitute the complete local classification of these kinds of submanifolds, given by our main results Theorems \[theo51\] and \[theo52\].
Preliminaries
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Let $M$ be a $(2n+1)$-dimensional smooth manifold $M$. Then an *almost contact structure* is a triplet $(\varphi,\xi,\eta)$, where $\varphi$ is a $(1,1)$-tensor field, $\eta$ a $1$-form and $\xi$ a vector field on $M$ satisfying the following conditions $$\label{almostcontact}
\varphi^{2}=-I+\eta\otimes\xi, \; \eta(\xi)=1.$$ It follows from that $\varphi\xi=0$, $\eta\circ\varphi=0$ and that $\text{rank}(\varphi)=2n$ ([@blairbook]).
Any almost contact manifold $\left(M,\varphi,\xi,\eta\right)$ admits a *compatible metric*, i.e. a Riemannian metric $g$ satisfying $$g\left(\varphi X,\varphi Y\right)=g\left(X,Y\right)-\eta\left(X\right)\eta\left(Y\right),$$ for all vector fields $X,Y$ on $M$. It follows that $\eta=g(\cdot,\xi)$ and $g(\cdot,\varphi\cdot)=-g(\varphi\cdot,\cdot)$. The manifold $M$ is said to be an *almost contact metric manifold* with structure $\left(\varphi,\xi,\eta,g\right)$.
We can define the *fundamental $2$-form* $\Phi$ of an almost contact metric manifold by $\Phi\left(X,Y\right)=g\left(X,\varphi Y\right)$. If $\Phi=d\eta$, then $\eta$ becomes a contact form, with $\xi$ its Reeb/characteristic vector field and ${\mathcal D}=\ker(\eta)$ its corresponding contact distribution, and $M(\varphi,\xi,\eta,g)$ is called a *contact metric manifold*.
Every contact metric manifold satisfies $$\label{eq-h}
\nabla\xi=-\varphi -\varphi h,$$ where $2h$ is the Lie derivative of $\varphi$ in the direction of $\xi$, i.e. $h=\frac12 L_\xi \varphi$. The tensor field $h$ is symmetric with respect to $g$, satisfies $h\xi=0$, anticommutes with $\varphi$ and vanishes identically if and only if the Reeb vector field $\xi$ is Killing. In this last case the contact metric manifold is said to be $K$-*contact*.
An almost contact metric manifold is said to be *normal* if $N_{\varphi}:=[\varphi,\varphi]+2d\eta\otimes\xi=0$. A normal contact metric manifold is called a *Sasakian manifold*. Any Sasakian manifold is K-contact and the converse holds in dimension $3$ but not in general.
A special class of contact metric manifold is that of $(\kappa,\mu)$-spaces, first studied in [@blair95] under the name of *contact metric manifolds with $\xi$ belonging to the $(\kappa,\mu)$-distribution*. A *contact metric $(\kappa,\mu)$-space* is one satisfying the condition $$\label{kappamu}
R(X,Y)\xi=\kappa \, (\eta(Y)X-\eta(X)Y)+\mu \, (\eta(Y)hX-\eta(X)hY),$$ for some constants $\kappa$ and $\mu$. In this paper, all manifolds will be contact metric, so we will shorten “contact metric $(\kappa,\mu)$-space" to “$(\kappa,\mu)$-space".
Every $(\kappa,\mu)$-space satisfies $$\begin{aligned}
h^2&=(\kappa-1)\varphi^2, \label{eqh2} \\
(\nabla_X\varphi)Y&=g(X,Y+hY)\xi-\eta(Y)(X+hX), \label{nablaphi}\\
(\nabla_Xh)Y&=((1-\kappa)g(X,\varphi Y)-g(X,\varphi hY))\xi-\eta(Y)((1-\kappa)\varphi X+\varphi hX)-\mu\eta(X)\varphi hY. \label{nablah}\end{aligned}$$ Moreover, we have the following result:
\[theoblair\] Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa,\mu)$-space. Then $\kappa \leq 1$. If $\kappa = 1$, then $h = 0$ and $M^{2n+1}$ is a Sasakian manifold. If $\kappa < 1$, $M^{2n+1}$ admits three mutually orthogonal and integrable distributions $E(0)=\text{span}(\xi)$, $E(\lambda)$ and $E(-\lambda)$ determined by the eigenspaces of $h$, where $\lambda=\sqrt{1-\kappa}$.
As a consequence of this theorem, it was also proved in [@blair95] that the sectional curvature of a plane section $\{X,Y\}$ normal to $\xi$ is given by $$\label{sectional}
K(X,Y)=
\begin{cases}
2(1+\lambda)-\mu, \text{ for any } X,Y\in E(\lambda), \quad n>1,\\
2(1-\lambda)-\mu, \text{ for any } X,Y\in E(-\lambda), \quad n>1,\\
-(\kappa+\mu)(g(X,\varphi Y))^2, \text{ for any unit vectors } X\in E(\lambda), Y\in E(-\lambda).
\end{cases}$$
Given a contact metric manifold $M^{2n+1}(\varphi,\xi,\eta,g)$, a *$D_a$-homothetic deformation* is a change of structure tensors of the form $$\label{deformation}
\tilde\varphi= \frac1a \varphi, \; \tilde\xi=\xi, \; \tilde\eta=a \eta, \; \tilde g = a g +a(a-1) \eta \otimes \eta,$$ where $a$ is a positive constant. It is well known that $M^{2n+1}(\tilde\varphi,\tilde\xi,\tilde\eta,\tilde{g})$ is also a contact metric manifold.
It was also proved in [@blair95] that the class of $(\kappa,\mu)$-spaces remains invariant under $D_a$-homothetic deformations. Indeed, applying one of these deformations to a $(\kappa,\mu)$-space yields a new $(\tilde\kappa,\tilde\mu)$-space, where $$\tilde\kappa =\frac{\kappa+a^2-1}{a^2} , \, \tilde\mu = \frac{\mu+2a-2}{a}.$$
Many authors studied $(\kappa,\mu)$-spaces later, as can be seen in [@blairbook]. We highlight here the work of Boeckx, who gave in [@boeckx2000] an explicit writing of the curvature tensor of these spaces: $$\label{Rkappamu}
\begin{aligned}
R(X,Y)Z &=\left(1-\frac\mu2 \right)(g(Y,Z)X-g(X,Z)Y)\\
&+g(Y,Z)hX-g(X,Z)hY-g(hX,Z)Y+g(hY,Z)X\\
&+\frac{1-\frac\mu2}{1-\kappa}(g(hY,Z)hX-g(hX,Z)hY)\\
&-\frac\mu2 (g(\varphi Y,Z)\varphi X-g(\varphi X,Z)\varphi Y)\\
&+\frac{\kappa-\frac\mu2}{1-\kappa} (g(\varphi hY,Z)\varphi h X-g(\varphi hX,Z)\varphi h Y)\\
&-\eta(X)\eta(Z)\left(\left(\kappa-1+\frac\mu2 \right)Y+(\mu-1)hY\right)\\
&-\eta(Y)\eta(Z)\left(\left(\kappa-1+\frac\mu2 \right)X+(\mu-1)hX \right)\\
&-\eta(Y)\left(\left(\kappa-1+\frac\mu2 \right)g(X,Z)+(\mu-1)g(hX,Z)\right)\xi.
\end{aligned}$$ Boeckx [@boeckx2000] also classified the $(\kappa,\mu)$-spaces in terms of an invariant that he introduced: $I_M=\frac{1-\frac{\mu}2}{\sqrt{1-\kappa}}$. Indeed, he proved that if $M_1$ and $M_2$ are two non-Sasakian $(\kappa_i,\mu_i)$-spaces of the same dimension, then $I_{M_1}=I_{M_2}$ if and only if, up to a $D_a$-homothetic deformation, the two spaces are locally isometric as contact metric spaces. In particular, if both spaces are simply connected and complete, they are globally isometric up to a $D_a$-homothetic deformation.
It was also stated in paper [@boeckx2000] that “it follows that we know all non-Sasakian $(\kappa,\mu)$-spaces locally as soon as we have, for every odd dimension $2n + 1$ and for every possible value for the invariant $I$, one $(\kappa,\mu)$-space $M$ with $I_M=I$.” For $I > -1$, we have the unit tangent sphere bundle $T_1 M^n(c)$ of a space of constant curvature $c$ ($c\neq1$) for the appropriate $c$ (see [@blair95]). For $I \leq -1$, Boeckx presented in [@boeckx2000] the following examples for any possible odd dimension $2n + 1$ and value of $I$.
\[books\] Let $\mathfrak{g}$ be a $(2n+1)$-dimensional Lie algebra with basis $\{\xi,X_1,\dots,X_n,Y_1,\dots,Y_n\}$ and the Lie brackets given by $$\label{brackets-books}
\begin{array}{rclrcl}
\left[\xi,X_1\right] & = & -\displaystyle{\frac{\alpha \beta}{2}X_2-\frac{\alpha^2}{2}Y_1}, & \left[Y_i,Y_j\right] & = & 0, \quad i,j \neq 2, \\
\\
\left[\xi,X_2\right] & = & \displaystyle{\frac{\alpha \beta}{2}X_1-\frac{\alpha^2}{2}Y_2}, & \left[X_1,Y_1\right] & = & -\beta X_2+2\xi, \\
\\
\left[\xi,X_i\right] & = & -\displaystyle{\frac{\alpha^2}{2}Y_i}, \quad i\geq 3, & \left[X_1,Y_i\right] & = & 0, \quad i\geq 2, \\
\\
\left[\xi,Y_1\right] & = & \displaystyle{\frac{\beta^2}{2}X_1-\frac{\alpha\beta}{2}Y_2}, & \left[X_2,Y_1\right] & = & \beta X_1-\alpha Y_2, \\
\\
\left[\xi,Y_2\right] & = & \displaystyle{\frac{\beta^2}{2}X_2+\frac{\alpha\beta}{2}Y_1}, & \left[X_2,Y_2\right] & = & \alpha Y_1+2\xi, \\
\\
\left[\xi,Y_i\right] & = & \displaystyle{\frac{\beta^2}{2}X_i}, \quad i\geq 3, & \left[X_2,Y_i\right] & = & \beta X_i, \quad i\geq 3,\\
\\
\left[X_1,X_i\right] & = & \alpha X_i, \quad i\neq 1, & \left[X_i,Y_1\right] & = & -\alpha Y_i, \quad i\geq 3,\\
\\
\left[X_i,X_j\right] & = & 0, \quad i,j\neq 1,& \left[X_i,Y_2\right] & = & 0, \quad i\geq 3, \\
\\
\left[Y_2,Y_i\right] & = & \beta Y_i, \quad i\neq 2, & \left[X_i,Y_j\right] & = & \delta_{ij}(-\beta X_2+\alpha Y_1+2\xi), \quad i,j\geq 3,
\end{array}$$ for real numbers $\alpha$ and $\beta$. Next we define a left-invariant contact metric structure $(\varphi,\xi,\eta,g)$ on the associated Lie group $G$ as follows:
- the basis $\{\xi, X_1,\ldots, X_n, Y_1,\ldots, Y_n \}$ is orthogonal,
- the characteristic vector field is given by $\xi$,
- the one-form $\eta$ is the metric dual of $\xi$,
- the $(1, 1)$-tensor field $\varphi$ is determined by $
\varphi \xi=0, \; \varphi X_i = Y_i, \; \varphi Y_i =-X_i.
$
It can also be proved that $G$ is a $(\kappa,\mu)$-space with $$\kappa= 1-\frac{(\beta^2-\alpha^2)^2}{16}, \; \mu=2+\frac{\alpha^2+\beta^2}{2}.$$ Moreover, supposing $\beta^2 >\alpha^2$ gives us that $\lambda=\frac{\beta^2-\alpha^2}2 \neq0$ and thus the $(\kappa,\mu)$-space is not Sasakian. The orthonormal basis also satisfies that $hX_i=\lambda X_i$ and $hY_i=-\lambda Y_i$.
Finally, $I_G= -\frac{\beta^2+\alpha^2}{\beta^2-\alpha^2} \leq -1$, so for the appropriate choice of $\beta > \alpha \geq 0$, $I_G$ attains any real value smaller than or equal to $-1$.
Lastly, we will recall some formulas from submanifolds theory in order to fix our notation. Let $N$ be an $n$-dimensional submanifold isometrically immersed in an $m$-dimensional Riemannian manifold $(M,g)$. Then, the Gauss and Weingarten formulas hold: $$\begin{aligned}
\nabla_XY&=\overline\nabla_XY+\sigma(X,Y), \label{eqgauss} \\
\nabla_XV&=-A_VX+\nabla^{\perp}_XV, \label{eqwein}\end{aligned}$$ for any tangent vector fields $X,Y$ and any normal vector field $V$. Here $\sigma$ denotes the *second fundamental form*, $A$ the *shape operator* and $\nabla^{\perp}$ the *normal connection*. It is well known that the second fundamental form and the shape operator are related the following way: $$\label{eqsigma}
g(\sigma(X,Y),V)=g(A_VX,Y).$$ We denote by $R$ and $\overline R$ the curvature tensors of $M$ and $N$, respectively. They are related by Gauss and Codazzi’s equations $$\label{eqgauss2}
R(X,Y,Z,W)=\overline R(X,Y,Z,W)-g(\sigma(X,W),\sigma(Y,Z))+g(\sigma(X,Z),\sigma(Y,W)),$$ $$\label{eqcodazzi}
(R(X,Y)Z)^\perp =(\nabla _{X}\sigma)(Y,Z)-(\nabla_{Y}\sigma)(X,Z),$$ respectively, where $R(X,Y)Z^{\perp}$ denotes the normal component of $R(X,Y)Z$ and $$(\nabla _{X}\sigma)(Y,Z)=\nabla^\perp _{X}(\sigma(Y,Z))
-\sigma(\overline\nabla _{X}Y,Z)-\sigma(Y,\overline\nabla _{X}Z).$$
The submanifold $N$ is said to be *totally geodesic* if the second fundamental form $\sigma$ vanishes identically. It is said that it is *totally umbilical* if there exists a normal vector field $V$ such that $\sigma(X,Y)=g(X,Y)V$, for any tangent vector fields $X,Y$. In fact, it can be proved that, in such a case, $V$ has to be the *mean curvature* $\widetilde H=\frac1n\sum_{i=1}^n\sigma(e_i,e_i)$, where $\{e_1,\dots,e_n\}$ is a local orthonormal frame. It is clear that every totally geodesic submanifold is also totally umbilical but the converse is not true in general.
Decomposition of the $h$ operator {#sec3}
=================================
Let $N$ be a Legendrian submanifold of a $(2n+1)$-dimensional $(\kappa,\mu)$-space $M$, that is, an $n$-dimensional submanifold such that $\xi$ is normal to $N$. Therefore, $\eta(X)=0$ for any tangent vector field $X$ and so it follows from that $\varphi^2X=-X$. Moreover, it was proved in [@lotta] that $N$ is an anti-invariant submanifold, i.e., $\varphi X$ is normal for any tangent vector field $X$. Moreover, under our assumptions about the dimensions of $M$ and $N$, it holds that every normal vector field $V$ can be written as $\varphi X$, for a certain tangent vector field $X$.
Therefore, we can decompose the $h$ operator in the following way: $$\label{h1h2}
hX=h_1X+\varphi h_2X,$$ for any tangent vector field $X$, where $h_1X$ (respectively $\varphi h_2X$) denotes the tangent (resp. normal) component of $hX$.
We can prove the following properties:
Let $N$ be a Legendrian submanifold of a $(\kappa,\mu)$-space $M$. Then, $h_1$ and $h_2$ are symmetric operators that satisfy $h_1\xi=h_2\xi=0$ and equations $$\begin{aligned}
h_1^2+h_2^2&=(1-\kappa)I,\label{propo11}\\
h_1h_2&=h_2h_1.\label{propo12}\end{aligned}$$
The symmetry of $h_1$ and $h_2$ can be directly obtained from that of $h$ and the compatibility of the metric $g$. Similarly, $h\xi=0$ implies $h_1\xi=h_2\xi=0$.
Furthermore, given a tangent vector field $X$, it follows from , and the anticommutativity of $h$ and $\varphi$ that $$\label{anticom}
h\varphi X=-\varphi hX=-\varphi h_1X+h_2X.$$ Using , we have that $h^2X=(1-\kappa)X$. On the other hand, by virtue of and , we obtain $$h^2X=h(h_1X+\varphi h_2X)=h_1^2X+\varphi h_2h_1X-\varphi h_1h_2X+h_2^2X.$$ Joining both expressions for $h^2$ and identifying the tangent and normal parts give us equations and .
Let $N$ be a Legendrian submanifold of a $(\kappa,\mu)$-space $M$. Then, $h_1$ and $h_2$ satisfy $$\begin{aligned}
(\overline\nabla_X h_1)Y&=-\varphi \sigma(X,h_2Y)-h_2\varphi \sigma(X,Y), \label{nablah1}\\
(\overline\nabla_X h_2)Y&=\; \; \, \varphi \sigma(X,h_1Y)+h_1\varphi \sigma(X,Y), \label{nablah2}\end{aligned}$$ for any tangent vector fields $X,Y$.
It follows from Gauss and Weingarten formulas and that $$(\nabla_X\varphi)Y=\nabla_X\varphi Y-\varphi \nabla_XY=-A_{\varphi Y}X+\nabla^{\perp}_X\varphi Y-\varphi\overline\nabla_XY-\varphi \sigma(X,Y),$$ for any tangent vector fields $X,Y$. Therefore, by using and identifying the tangent and normal components, we obtain: $$\begin{aligned}
A_{\varphi Y}X &=-\varphi\sigma(X,Y), \label{wein1} \\
\nabla^{\perp}_X\varphi Y &=\varphi\overline\nabla_XY+g(X,Y+h_1Y)\xi. \label{wein2}\end{aligned}$$
On the other hand, using and , we have $$\nabla_X(h_1Y+\varphi h_2Y)-h(\nabla_X Y)=g(X,h_2Y)\xi,$$ from where, by virtue of Gauss and Weingarten formulas and , we deduce $$\label{wein3}
\overline\nabla_Xh_1Y+\sigma(X,h_1Y)-A_{\varphi h_2Y}X+\nabla^{\perp}_X\varphi h_2Y-h\overline\nabla_XY-h\sigma(X,Y)=g(X,h_2Y)\xi.$$ We can put $h\overline\nabla_XY=h_1\overline\nabla_XY+\varphi h_2\overline\nabla_XY$ by . Now, by using , we can write $\sigma(X,Y)=-\varphi^2\sigma(X,Y)+\eta(\sigma(X,Y))\xi$, and hence $h\sigma(X,Y)=-h\varphi^2\sigma(X,Y)=\varphi h \varphi\sigma(X,Y)$. Again, equation gives us $h\sigma(X,Y)=\varphi h_1\varphi\sigma(X,Y)-h_2\varphi\sigma(X,Y)$. Therefore, if we substitute these two expressions, together with and , in , we obtain: $$\label{wein4}
\begin{aligned}
\overline\nabla_Xh_1Y+\sigma(X,h_1Y)+\varphi\sigma(X,h_2Y)+\varphi\overline\nabla_Xh_2Y+g(X,h_2Y+h_1h_2Y)\xi& \\
-h_1\overline\nabla_XY-\varphi h_2\overline\nabla_XY
-\varphi h_1\varphi\sigma(X,Y)+h_2\varphi\sigma(X,Y)&=g(X,h_2Y)\xi.
\end{aligned}$$ By identifying the tangent and normal parts of , equations and hold.
It is clear that, if we multiply by $\xi$, then we obtain $$g(\sigma(X,h_1Y),\xi)+g(X,h_1h_2Y)=0,$$ for any tangent vector fields $X,Y$. In fact, we can prove a more general result, which will be very useful in the proof of our main theorems:
Let $N$ be a Legendrian submanifold of a $(\kappa,\mu)$-space $M$. Then, $$\label{eqsigma2}
g(\sigma(X,Y),\xi)+g(X,h_2Y)=0,$$ for any tangent vector fields $X,Y$.
It follows from Weingarten equation and from that $$g(X,\nabla_X\xi)+g(\sigma(X,Y),\xi)=0,$$ for any tangent vector fields $X,Y$. Then, it is enough to use , and to obtain .
Examples {#sec4}
========
We will present in this section some examples of totally geodesic and totally umbilical Legendrian submanifolds of the $(\kappa,\mu)$-spaces of Example \[books\]. Let us begin with the totally geodesic ones.
\[exx\] Let $M$ be a $(\kappa,\mu)$-space from Example \[books\] with invariant $I_M\leq -1$. Then, the distribution $\mathcal{D}$ spanned by $\{X_1,\dots,X_n\}$ is involutive and any integral submanifold $N$ of it is a totally geodesic submanifold of $M$. Indeed, the involutive condition can be easily checked from . In order to prove the totally geodesic one, it is enough to show that $\nabla_{X_i}X_j\in \mathcal{D}$, for any $i,j=1,\dots,n$, where $\nabla$ denotes the Levi-Civita connection on $M$. In fact, in can be directly computed that: $$\label{nablax}
\begin{aligned}
& \nabla_{X_1}X_1=\nabla_{X_1}X_2=0, \quad \nabla_{X_2}X_1=-\alpha X_2, \quad \nabla_{X_2}X_2=\alpha X_1, \\
& \nabla_{X_1}X_i=\nabla_{X_2}X_i=0, \mbox{ for any } i=3,\dots,n, \\
& \nabla_{X_i}X_1=-\alpha X_i, \quad \nabla_{X_i}X_2=0, \quad \nabla_{X_i}X_j=\delta_{ij}\alpha X_1, \mbox{ for any }i,j=3,\dots,n.
\end{aligned}$$ Moreover, since $hX_i=\lambda X_i$ for any $i=1,\dots,n$, then $TN=E(\lambda)$.
\[exy\] Let $M$ be a $(\kappa,\mu)$-space from Example \[books\] with invariant $I_M\leq -1$. Then, the distribution $\mathcal{D}$ spanned by $\{Y_1,\dots,Y_n\}$ is also involutive and any integral submanifold $N$ of it is a totally geodesic submanifold of $M$. Indeed, both conditions can be checked the same way as in Example \[exx\], by taking now into account that: $$\label{nablay}
\begin{aligned}
& \nabla_{Y_1}Y_1=\beta Y_2, \quad \nabla_{Y_1}Y_2=-\beta Y_1, \quad \nabla_{Y_2}Y_1=\nabla_{Y_2}Y_2=0, \\
& \nabla_{Y_1}Y_i=\nabla_{Y_2}Y_i=0, \mbox{ for any } i=3,\dots,n, \\
& \nabla_{Y_i}Y_1=0, \quad \nabla_{Y_i}Y_2=-\beta Y_i, \quad \nabla_{Y_i}Y_j=\delta_{ij}\beta Y_2, \mbox{ for any }i,j=3,\dots,n.
\end{aligned}$$ In this case, since $hY_i=-\lambda Y_i$ for any $i=1,\dots,n$, then $TN=E(-\lambda)$.
\[exxy\] Let $M$ be a $(\kappa,\mu)$-space from Example \[books\] with invariant $I_M\leq -1$. Then, the distribution $\mathcal{D}$ spanned by $\{X_1,Y_2,Z_3,\dots,Z_n\}$, where $Z_i$ is either $X_i$ or $Y_i$, for any $i=3,\dots,n$, is also involutive and any integral submanifold $N$ of it is a totally geodesic submanifold of $M$. Indeed, both conditions can be checked the same way as in Examples \[exx\] and \[exy\], by using now , and the following formulas: $$\label{nablaz}
\begin{aligned}
& \nabla_{X_1}Y_i=0 \mbox{ for any }i=2,\dots,n, \\
& \nabla_{Y_2}X_i=0 \mbox{ for any }i=1,3,\dots,n, \\
& \nabla_{X_i}Y_2=\nabla_{Y_i}X_1=0 \mbox{ for any } i=3,\dots,n, \\
& \nabla_{X_i}Y_j=\nabla_{Y_i}X_j=0 \mbox{ for any }i,j=3,\dots,n, \mbox{ such that } i\neq j.
\end{aligned}$$ Finally, $TN=E(\lambda)\oplus E(-\lambda)$, with $\dim{E(\lambda)}=k$ (respectively $\dim{E(-\lambda)}=n-k$), where $k-1$ (resp. $n-k-1$) is the number of $Z_i$ such that $Z_i=X_i$ (resp. $Z_i=Y_i$). Therefore, we can obtain an example for any value of $k$ from $1$ to $n-1$.
We now present the family of totally umbilical examples:
\[excd\] Let $M$ be a $(\kappa,\mu)$-space from Example \[books\] with invariant $I_M\leq -1$. Then, the distribution $\mathcal{D}$ spanned by $\{cX_1+dY_1,\dots,cX_n+dY_n\}$, with $c,d$ non-zero constants, is involutive and any integral submanifold $N$ of it is a totally umbilical submanifold of $M$. Indeed, the involutive condition can be easily checked from . In order to prove the totally umbilical one, we will first show that $\sigma(cX_i+dY_i,cX_j+dY_j)=2\delta_{ij}cd\lambda\xi$ by checking that the Levi-Civita connection on $M$ satisfies $\nabla_{cX_i+dY_i}(cX_j+dY_j)=Z+ 2\delta_{ij} cd\lambda \xi $, with $Z \in \mathcal{D}$, for any $i,j=1,\dots,n$. In fact, it can be directly computed that: $$\begin{aligned}
& \nabla_{cX_1+dY_1}(cX_1+dY_1)=\beta d(cX_2+dY_2)+2cd\lambda \xi , \quad \nabla_{cX_1+dY_1}(cX_2+dY_2) =-\beta d(cX_1+dY_1),\\
& \nabla_{cX_2+dY_2}(cX_1+dY_1)=-\alpha c(cX_2+dY_2) , \quad \nabla_{cX_2+dY_2}(cX_2+dY_2) =\alpha c (cX_1+dY_1)+2cd\lambda \xi,\\
&\nabla_{cX_1+dY_1}(cX_j+dY_j)=\nabla_{cX_2+dY_2}(cX_j+dY_j)=0, \text{ for any }j=3,\dots,n, \\
& \nabla_{cX_i+dY_i}(cX_1+dY_1)=-\alpha c (cX_i+dY_i) , \\
& \nabla_{cX_i+dY_i}(cX_2+dY_2) =-\beta d (cX_i+dY_i), \text{ for any }i=3,\dots,n,\\
&\nabla_{cX_i+dY_i}(cX_j+dY_j)=\delta_{ij}
(\alpha c (cX_1+dY_1)+\beta d (cX_2+dY_2)+2cd\lambda \xi),
\text{ for any }i,j=3,\dots,n.
\end{aligned}$$ Therefore, we can write $\sigma(cX_i+dY_i,cX_j+dY_j)=g(cX_i+dY_i, cX_j+dY_j)\frac{2 cd\lambda}{c^2+d^2} \xi$ and, since $\frac{2 cd\lambda}{c^2+d^2} \xi\neq0$, the submanifold is totally umbilical but not totally geodesic.
Finally, we observe that $cX_i+dY_i$, $i=1,\ldots,n$, is not an eigenvector of $h$.
Main results {#sec5}
============
\[theo51\] Let $N$ be a Legendrian submanifold of a $(2n+1)$-dimensional $(\kappa,\mu)$-space $M$, with $\kappa<1$ and $I_M\leq -1$. If $N$ is totally geodesic, then, up to local isometries, it must be one of the submanifolds given in Examples \[exx\], \[exy\] or \[exxy\].
Since the submanifold $N$ is totally geodesic, if follows directly from that $h_2=0$ and so $h=h_1$ and $h_1^2=(1-\kappa)I$ (see and ). By using the decomposition given by Theorem \[theoblair\], we can write $$\label{decomp}
TN=E(\lambda)\oplus E(-\lambda),$$ where $\dim(E(\lambda))=k$ and $\dim(E(-\lambda))=n-k$, for a certain $k\in\{0,\dots,n\}$.
Moreover, we deduce from that $\overline\nabla h_1=0$. Therefore, it is straightforward to check that, if $Y_{\lambda} \in E(\lambda)$, then $\overline \nabla_X Y_{\lambda} \in E(\lambda)$, for every tangent vector field $X$. Similarly, if $Y_{-\lambda} \in E(-\lambda)$, then $\overline \nabla_X Y_{-\lambda} \in E(-\lambda)$. Thus, $E(\lambda)$ and $E(-\lambda)$ are parallel and hence involutive. By virtue of Theorem 5.4 of [@KN], $N$ can be locally decomposed as $M_1\times M_2$, where $M_1$ and $M_2$ are leaves of the distributions $E(\lambda)$ and $E(-\lambda)$, respectively. Furthermore, it follows from that, if $\dim{M_1}\geq 2$ (resp. $\dim{M_2}\geq 2$), then $M_1$ (resp. $M_2$) has constant curvature $2(1+\lambda)-\mu=2\lambda(I_M+1)\leq 0$ (resp. $2(1-\lambda)-\mu=2\lambda(I_M-1)< 0$).
Recall that we have examples of submanifolds with decomposition for every value of $k$. Indeed, see Example \[exx\] for $k=n$, Example \[exy\] for $k=0$ and Example \[exxy\] for any value of $k$ from $1$ to $n-1$. Now, we will prove that any example must be one of these, up to local isometries.
Let us denote by $F:N^n\to M^{2n+1}(\kappa,\mu)$ the immersion of $N$ into $M$. Since $\kappa<1$ and $I_M\leq -1$, we can suppose that, locally, $M^{2n+1}(\kappa,\mu)$ is one of the Lie groups from Example \[books\]. Thus, it is homogeneous and we can fix a point $p_0 \in N$ such that $F(p_0)=e$, where $e$ is the neutral element of the group.
We will give the explicit details when $2 \le k \le n-2$. The other cases can be done in a similar way. We have that $N = M_1( 2 \lambda (I_M+1)) \times M_2 (2 \lambda (I_M-1))$ and we also identify $N$ with its image as the (totally geodesic) integral submanifold through $e$ of the distribution spanned by $X_1,X_3,\dots,X_{k+1}, Y_2, Y_{k+2},\dots Y_n$. We denote by $G$ the latter immersion of $N$ and we pick an orthonormal basis $\{e_1,\ldots,e_n\}$ at the point $p_0$ of $N$, with $G(p_0)=e$, such that $E_{p_0}(\lambda)=\langle e_1(p_0),\ldots,e_k(p_0) \rangle$, $E_{p_0}(-\lambda)=\langle e_{k+1}(p_0),\ldots,e_n(p_0) \rangle$ and $$\begin{aligned}
&dG(e_1(p_0))= X_1(e),\\
&dG(e_j(p_0))=X_{j+1}(e), \quad j=2,\dots,k,\\
&dG(e_{k+1}(p_0)) =Y_2(e),\\
&dG(e_j(p_0)) = Y_j(e),\quad j=k+2,\dots,n,\end{aligned}$$
Note that by construction both $$X_1(e),X_3(e),\dots,X_{k+1}(e), \varphi Y_2(e),\varphi Y_{k+2}(e),\dots, \varphi Y_n(e)$$ and $$dF(e_1(p_0)),\dots,dF(e_k(p_0)),\varphi dF(e_{k+1}(p_0)),\dots,\varphi dF(e_{n}(p_0))$$ are basis of $E_e(\lambda)$. So, in view of Theorem 3 of [@boeckx2000], there exists an isometry $H$ of $M^{2n+1}(\kappa,\mu)$ preserving the structure such that $H(e)=e$ and $H$ maps one basis of $E_e(\lambda)$ into the other one. As a consequence, we have that $H \circ F(e) = G(e)$ and $d(H \circ F)(e_i) = dG(e_i)$.
We now take a geodesic $\gamma$ in $N$ through the point $p_0$. Since $N$ is totally geodesic, both with respect to the immersions $H \circ F$ and $G$, the curves $H \circ F(\gamma)$ and $G(\gamma)$ are both geodesics in $M^{2n+1}(\kappa,\mu)$ through $e$. Since $d(H \circ F)(e_i) = dG(e_i)$, they are also determined by the same initial conditions. Therefore, both curves need to coincide, so $H \circ F(\gamma(s))= G(\gamma(s))$ for all $s$ and thus $F$ and $G$ are congruent.
\[theo52\] Let $N$ be a Legendrian submanifold of a $(2n+1)$-dimensional $(\kappa,\mu)$-space $M$, with $n\geq 3$, $\kappa<1$ and $I_M\leq -1$. If $N$ is totally umbilical (but not totally geodesic), then, up to local isometries, it must be one of the submanifolds given in Example \[excd\].
Since $N$ is totally umbilical (but not totally geodesic), then there exists a normal vector field $V \neq0$ such that $\sigma(X,Y)=g(X,Y) V$. It follows from that $g(X,Y)\eta(V)+g(X,h_2Y)=0$, for any tangent vector fields $X,Y$, and thus $$\label{totumb}
h_2Y=aY,$$ with $a=-\eta(V)$.
We will now prove that $a\neq0$. Indeed, if we suppose that $a=0$, then $h_2=0$ and, as in the proof of Theorem \[theo51\], we have that $h=h_1$, $h_1^2=(1-\kappa)I$ and $\overline{\nabla} h_1=0$. Moreover, since $h_2=0$, it is clear that $\overline{\nabla} h_2=0$ and we obtain from that $\varphi\sigma(X,h_1Y)+h_1\varphi\sigma(X,Y)=0$, which, by using that $N$ is totally umbilical, becomes $$\label{proof52}
g(X,h_1Y)\varphi V+g(X,Y)h_1\varphi V=0,$$ for any tangent vector fields $X,Y$. Let us now choose unit vector fields $X_{\lambda} \in E(\lambda)$ and $X_{-\lambda} \in E(-\lambda)$. Then, taking $X=Y=X_{\lambda}$ in implies $h_1\varphi V=-\lambda \varphi V$ and taking $X=Y=X_{-\lambda}$ in implies $h_1\varphi V=\lambda \varphi V$. Since $V\neq 0$, this yields a contradiction.
Therefore, we can suppose from now on that holds for $a\neq 0$. We deduce from equation that $$X(a) Y= \varphi \sigma(X,h_1 Y)+h_1 \varphi \sigma(X,Y)=g(X,h_1 Y)\varphi V+g(X,Y) h_1 \varphi V,$$ for every $X,Y$ tangent vector fields.
Since $\dim N \geq 3$, we can take $Y$ linearly independent from $\varphi V$ and $h_1 \varphi V$. Then we deduce from the previous equation that $X(a)=0$, for every $X$, thus $a$ is a constant. Moreover, $g(X,h_1 Y)\varphi V+g(X,Y) h_1 \varphi V=0$, for every $X,Y$ tangent vector fields. Taking unit $X=Y$, we obtain that $h_1 \varphi V=-g(X,h_1X)\varphi V$, which is only possible if $h_1=0$ or $\varphi V=0$. If $h_1=0$, then substituting in gives that $2ag(X,Y) \varphi V=0$, so again $\varphi V=0$.
In both cases, we have obtained that $\varphi V=0$, so $V$ is parallel to $\xi$ and it follows from $a=-\eta(V)$ that $V=-a \xi$ and $\sigma(X,Y)=-ag(X,Y)\xi$ holds, for every $X,Y$ tangent, where $a\neq 0$ is a constant.
Let us now recall Codazzi’s equation : $$(R(X,Y)Z)^\perp =(\nabla _{X}\sigma)(Y,Z)-(\nabla_{Y}\sigma)(X,Z).$$ The first term is the normal component of $R(X,Y)Z$, so by equation and the fact that $h_2 X=h_1 X+a \varphi X$, we can write $$\begin{aligned}
(R(X,Y)Z)^\perp &= a (g(Y,Z) \varphi X -g(X,Z) \varphi Y)\\
&+a \frac{1-\frac\mu 2}{1-\kappa} (g(h_1 Y,Z) \varphi X-g(h_1 X,Z) \varphi Y)\\
&-a \frac{\kappa-\frac\mu 2}{1-\kappa} (g(Y,Z) \varphi h_1 X-g(X,Z) \varphi h_1 Y).
\end{aligned}$$ On the other hand, $$\begin{aligned}
(\nabla _{X}\sigma)(Y,Z)
&=\nabla^\perp _{X}(\sigma(Y,Z))-\sigma(\overline\nabla _{X}Y,Z)-\sigma(Y,\overline\nabla _{X}Z)=\\
&=\nabla^\perp _{X}(-ag(Y,Z)\xi)+ag(\overline\nabla _{X}Y,Z)\xi+ag(\overline\nabla _{X}Z,X)\xi =\\
&=-ag(Y,Z)\overline\nabla _{X}^\perp \xi =ag(Y,Z) (\varphi X+\varphi h_1 X).\end{aligned}$$ Therefore, the second term of Codazzi’s equation is $$\begin{aligned}
(\nabla _{X}\sigma)(Y,Z)-(\nabla _{Y}\sigma)(X,Z)&=ag(Y,Z) (\varphi X+\varphi h_1 X)-a g(X,Z) (\varphi Y+\varphi h_1 Y)\\
&=a (g(Y,Z) \varphi X -g(X,Z) \varphi Y)+a (g(Y,Z) \varphi h_1 X -g(X,Z) \varphi h_1 Y).\end{aligned}$$ Joining both terms, and bearing in mind that $a\neq0$, we obtain $$\begin{aligned}
& \frac{1-\frac\mu 2}{1-\kappa} (g(h_1 Y,Z) \varphi X-g(h_1 X,Z) \varphi Y)=\\
&= \frac{1-\frac\mu 2}{1-\kappa} (g(Y,Z) \varphi h_1 X-g(X,Z) \varphi h_1 Y).\\
\end{aligned}$$ Since we are supposing that $I_M =\frac{1-\frac\mu 2}{\sqrt{1-\kappa}} \leq -1$, then $\frac{1-\frac\mu 2}{1-\kappa} \neq0$ and applying $\varphi$ to both terms of the previous equation gives us that $$g(h_1 Y,Z) X-g(h_1 X,Z) Y=g(Y,Z) h_1 X-g(X,Z) h_1 Y,\\$$ for every $X,Y,Z$ tangent vector fields.
Since $\dim (N) \geq 3$, we can choose $Y=Z$ unit and orthogonal to $X, h_1 X$, and we obtain that $$h_1 X=g(h_1 Y,Y)X,\\$$ and thus $h_1 X=bX$ for some function $b$.
From , we have that $a^2+b^2=1-\kappa =\lambda^2 \neq0$, and in particular that $b$ must be constant. We can also write that $a=\lambda \cos(\theta)$ and $b=\lambda \sin(\theta)$ for some constant $\theta\in [-\pi,\pi]$. Since $a\neq0$, then $\theta \neq \pm \frac{\pi}{2}$.
By Gauss equation and the fact that $h_2 X=aX$, then $$\begin{aligned}
R(X,Y,Z,W)&=\overline R(X,Y,Z,W)-g(\sigma(X,W),\sigma(Y,Z))+g(\sigma(X,Z),\sigma(Y,W))=\\
&=\overline R(X,Y,Z,W)-a^2(g(X,W)g(Y,Z)+g(X,Z)g(Y,W)),\end{aligned}$$ for every $X,Y,Z,W$ tangent vector fields.
On the other hand, we know from equation and the fact that $hX =bX+a\varphi X$, that $$\begin{aligned}
R(X,Y,Z,W)&=\left(1-\frac\mu2 +2b+b^2 \frac{1-\frac\mu2}{1-\kappa}+a^2 \frac{\kappa-\frac\mu2}{1-\kappa} \right) (g(X,W)g(Y,Z)-g(X,Z)g(Y,W)).\end{aligned}$$ Joining the last two equations, we obtain $$\begin{aligned}
\overline R(X,Y,Z,W)&=
\left(1-\frac\mu2 +2b+b^2 \frac{1-\frac\mu2}{1-\kappa}+a^2 \left(\frac{\kappa-\frac\mu2}{1-\kappa}+1 \right) \right) (g(X,W)g(Y,Z)-g(X,Z)g(Y,W))\\
&=\left(1-\frac\mu2 +2b+(a^2+b^2) \frac{1-\frac\mu2}{1-\kappa}\right) (g(X,W)g(Y,Z)-g(X,Z)g(Y,W))\\
&=2 (1 -\frac\mu2 +b ) (g(X,W)g(Y,Z)-g(X,Z)g(Y,W)).\end{aligned}$$ This means that the submanifold is a space form with constant curvature $2 (1 -\frac\mu2 +b )$. Moreover, since $I_M =\frac{1-\frac\mu 2}{\sqrt{1-\kappa}} \leq -1$ and $b=\lambda \sin (\theta) \neq \lambda$, then $1-\frac\mu 2+b < 1-\frac\mu 2+\lambda \leq0$ and the submanifold is a hyperbolic space $N=\mathbb{H}(2 (1 -\frac\mu2 +\lambda \sin(\theta) ))$.
Summing up, there exists $\theta \in [-\pi,\pi]$, $\theta \neq \pm \frac\pi2$, such that $$\label{dem2}
\begin{aligned}
N &=\mathbb{H}(2 (1 -\frac\mu2 +\lambda \sin(\theta) )),\\
h_1 X&= \lambda \sin(\theta) X,\\
h_2 X&= \lambda \cos(\theta) X,\\
\sigma(X,Y)&=-\lambda \cos(\theta)g(X,Y) \xi.
\end{aligned}$$
We have examples of submanifolds with these properties for every value of $\theta$. Indeed, Examples \[excd\] with $c=\cos(\pi/4-\theta/2)$, $d=-\sin(\pi/4-\theta/2)$ satisfy $$\begin{aligned}
\sigma(cX_i+dY_i,cX_j+dY_j)
&=2\delta_{ij}cd\lambda\xi
=-2\delta_{ij} \sin\left(\frac\pi4-\frac\theta2 \right)
\cos \left(\frac\pi4-\frac\theta2 \right)\lambda \xi=\\
&=-\delta_{ij}\sin\left(\frac\pi2-\theta \right)\lambda \xi =-\delta_{ij}\lambda \cos(\theta) \xi=\\
&=-\lambda \cos(\theta)g(cX_i+dY_i,cX_j+dY_j) \xi,
\end{aligned}$$ and the rest of conditions also hold.
Now, we will prove that any totally umbilical submanifold $N$ must be one of these, up to local isometries. Let us denote by $F:N^n\to M^{2n+1}(\kappa,\mu)$ the immersion of $N$ into $M(\kappa,\mu)$. Since $\kappa<1$ and $I_M\leq -1$, we can suppose that, locally, $M(\kappa,\mu)$ is one of the Lie groups from Example \[books\]. Thus, it is homogeneous and we can fix a point $p_0 \in N$ such that $F(p_0)=e$, where $e$ is the neutral element of the group.
We have that $N=\mathbb{H}(2 (1 -\frac\mu2 +\lambda \sin(\theta) ))$ and we can identify $N$ with its image as the (totally umbilical) integral submanifold through $e$ of the distribution spanned by $\{ cos \left(\frac{\pi}4-\frac{\theta}2 \right) X_i(e)
-\sin \left(\frac{\pi}4-\frac{\theta}2 \right) Y_i(e)$, $i=1, \dots,n \}$. We denote by $G$ this immersion of $N$ and we take an orthonormal basis $\{e_1,\ldots,e_n\}$ at the point $p_0$ of $N$ such that $$dG(e_i) = cos \left(\frac{\pi}4-\frac{\theta}2 \right) X_i(e)
-\sin \left(\frac{\pi}4-\frac{\theta}2 \right) Y_i(e), \; i=1, \dots,n.$$
On the other hand, we have that $$\begin{aligned}
h(dF(e_i))
&=dF(\lambda \sin(\theta) e_i)+\varphi dF(\lambda \cos(\theta) e_i)
= \lambda \sin(\theta) dF(e_i)+\lambda \cos(\theta) \varphi dF( e_i), \label{eq-h-dem}\\
h\varphi (d F(e_i))
&=-\varphi h(dF(e_i))=\lambda \cos(\theta) dF(e_i)-\lambda \sin(\theta) \varphi dF( e_i). \label{eq-hphi-dem}\end{aligned}$$ Therefore, using and , we can construct eigenvectors of $h$ associated with the eigenvalue $\lambda$ the following way: $$\begin{aligned}
h &\left(cos \left(\frac{\pi}4-\frac{\theta}2 \right) dF(e_i)
+\sin \left(\frac{\pi}4-\frac{\theta}2 \right) \varphi ( dF(e_i)) \right)=\\
%&=cos \left(\frac{\pi}4-\frac{\theta}2 \right) h(dF(e_i))
%+\sin \left(\frac{\pi}4-\frac{\theta}2 \right)h \varphi (dF(e_i))=\\
&=\lambda \left(
\left( cos \left(\frac{\pi}4-\frac{\theta}2 \right) \sin(\theta)
+\sin \left(\frac{\pi}4-\frac{\theta}2 \right) \cos(\theta) \right)
dF(e_i) \right.\\
&
\hspace{0.6cm} \left.
+ \left(cos \left(\frac{\pi}4-\frac{\theta}2 \right) \cos(\theta)
-\sin \left(\frac{\pi}4-\frac{\theta}2 \right) \sin(\theta) \right)
\varphi (dF( e_i)) \right)=\\
&=\lambda \left( \sin \left(\frac{\pi}4+\frac{\theta}2 \right) dF(e_i)
+ \cos \left(\frac{\pi}4+\frac{\theta}2 \right) \varphi ( dF( e_i) )\right)=\\
&=\lambda\left(cos \left(\frac{\pi}4-\frac{\theta}2 \right) dF(e_i)
+\sin \left(\frac{\pi}4-\frac{\theta}2 \right) \varphi ( dF(e_i)) \right),\end{aligned}$$ for any $i=1,\ldots,n$.
Note that, by construction, both $$\cos \left(\frac{\pi}4-\frac{\theta}2 \right) dF(e_i)
+\sin \left(\frac{\pi}4-\frac{\theta}2 \right) \varphi ( dF(e_i)), \; i=1,\dots,n$$ and $$X_1(e),\dots,X_n(e)$$ are basis of $E_e(\lambda)$. So, in view of Theorem 3 of [@boeckx2000], there exists an isometry $H$ of $M^{2n+1}(\kappa,\mu)$ preserving the structure such that $H(e)=e$ and $H$ maps one basis of $E_e(\lambda)$ into the other one. As a consequence, we have that $H \circ F(e) = G(e)$ and $d(H \circ F)(e_i) = dG(e_i)$.
We now take a geodesic $\gamma$ in $N$ through the point $p_0$. Since $N$ is totally umbilical with respect to both $H \circ F$ and $G$, then $\gamma_1=H \circ F(\gamma)$ and $\gamma_2=G(\gamma)$ are curves in $M(\kappa,\mu)$ passing through $e$ that satisfy $\nabla_{\gamma_1'} \gamma_1'=\nabla_{\gamma_2'} \gamma_2' =-\lambda \sin(\theta) \xi.$ Since $d(H \circ F)(e_i)=dG(e_i)$, they are also determined by the same initial conditions. Therefore, both curves need to coincide, so $H \circ F(\gamma(s))= G(\gamma(s))$ for all $s$ and thus $F$ and $G$ are congruent.
[99]{} D. E. Blair. *Riemannian Geometry of Contact and Symplectic Manifolds*, Second Edition. Progress in Mathematics **203**. Birkhäuser, Boston, 2010.
D. E. Blair, T. Koufogiorgos and B. J. Papantoniou. Contact metric manifolds satisfying a nullity condition. *Israel J. Math.* **91** (1995), 189–214.
E. Boeckx. A full classification of contact metric $(\kappa,\mu )$-spaces. *Illinois J. Math.* **44** (2000), 212–219.
B. Cappelletti Montano, L. Di Terlizzi and M. M. Tripathi. Invariant submanifolds of contact $(\kappa,\mu)$-manifolds. *Glasgow Math. J.* **50** (2008), 499–507.
S. Kobayashi and K. Nomizu. *Foundations of Differential Geometry*. Interscience Tracts in Pure and Applied Mathematics **15**, Volume I. Interscience, New York, 1963.
A. Lotta. Slant submanifolds in contact geometry. *Bull. Math. Soc. Roumanie* **39** (1996), 183–198.
[^1]: The first two authors are partially supported by the MINECO-FEDER grant MTM2014-52197-P. They are members of the IMUS (Instituto de Matemáticas de la Universidad de Sevilla), and of the PAIDI groups FQM-327 and FQM-226 (Junta de Andalucía, Spain), respectively.
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---
abstract: 'Geometric Berry phase can be induced either by spin-orbit coupling, giving rise to the anomalous Hall effect in ferromagnetic materials, or by chiral spin texture, such as skyrmions, leading to the topological Hall effect. Recent experiments have revealed that both phenomena can occur in topological insulator films with magnetic doping, thus providing us with an intriguing platform to study the interplay between these two phenomena. In this work, we numerically study the anomalous Hall and topological Hall effects in a four-band model that can properly describe the quantum well states in the magnetic topological insulator films by combining Landauer-Büttiker formula and the iterative Green’s function method. Our numerical results suggest that spin-orbit coupling in this model plays a different role in the quantum transport in the clean and disordered limits. In the clean limit, spin-orbit coupling mainly influences the longitudinal transport but does not have much effect on topological Hall conductance. Such behavior is further studied through the analytical calculation of scattering cross-section due to skyrmion within the four-band model. In the disordered limit, the longitudinal transport is determined by disorder scattering and spin-orbit coupling is found to affect strongly the topological Hall conductance. This sharp contrast unveils a dramatic interplay between spin-orbit coupling and disorder effect in topological Hall effect in magnetic topological insulator systems.'
author:
- 'Jian-Xiao Zhang'
- Domenico Andreoli
- Jiadong Zang
- 'Chao-Xing Liu'
bibliography:
- 'main.bib'
title: Topological Hall Effect in Magnetic Topological Insulator Films
---
Introduction
============
For electric conductors placed in an external magnetic field, the Lorentz field felt by electrons can lead to a voltage transverse to the electric current, which is known as the Hall effect [@hall1879new]. In magnetic systems, the exchange interaction between electron spin and magnetic moments can give rise to additional topological contributions to the Hall effect. In a ferromagnetic (FM) system with a strong spin-orbit coupling (SOC), the additional Hall contribution is induced by Berry phases accumulated by the adiabatic motion of quasiparticle on the Fermi surface in the momentum space and normally known as (intrinsic) anomalous Hall effect (AHE) [@nagaosa2010anomalous; @karplus1954hall]. On the other hand, when electrons propagate through chiral magnetic structures, e.g. skyrmions, in the real space, they can also feel Berry phase due to the magnetization texture, leading to the so-called “topological Hall effect (THE)” (also known as “geometric Hall effect”) [@machida2007unconventional; @kanazawa2015discretized; @neubauer2009topological; @oveshnikov2015berry; @ishizuka2018spin]. Both Hall phenomena originate from Berry phase contribution and thus are topological. An intriguing question is how to understand the topological contribution to the Hall effect in a magnetic skyrmion system with strong SOC, where the Berry phase exists in both the real and momentum spaces.
Topological insulator (TI) films with magnetic doping, dubbed “magnetic topological insulator (MTI)” below, provide an ideal platform to explore the interplay between AHE and THE. The coexistence of strong SOC and ferromagnetism in MTI films can result in a strong AHE [@jungwirth2002anomalous], and the Hall resistance can even achieve the quantized value when the chemical potential is tuned into the magnetization gap of surface states. Such phenomenon, known as the quantum anomalous Hall (QAH) effect [@yu2010quantized; @haldane1988model; @liu2008quantum; @liu2016quantum], has been experimentally observed in Cr or V doped (Bi,Sb)$_2$Te$_3$ films [@chang2013experimental]. Furthermore, the surface states in TI film can also mediate Dzyaloshinsky-Moriya (DM) interaction between magnetic moments due to spin-momentum locking [@zhu2011electrically; @ye2010spin]. As a result, chiral magnetic structures, such as skyrmion, are also possible. Indeed, recent experiments on Cr-doped-(Bi,Sb)$_2$Te$_3$/(Bi,Sb)$_2$Te$_3$ structure [@yasuda2016geometric] and Mn-doped Bi$_2$Te$_3$ [@liu2017dimensional] have observed a hump in the Hall resistance hysteresis loop at a small magnetic field. The hump structure is attributed to THE while the Hall hysteresis loop implies AHE. Therefore, the interplay between the AHE from ferromagnetism and the THE from magnetic skyrmion will be substantial to understand the electron transport phenomena in MTI films. In addition, MTI is normally highly disordered due to magnetic doping and it is not well understood how the disorder influences the THE in such strong spin-orbit coupled materials.
In this work, we numerically study the magneto-transport of MTI films with a magnetic skyrmion based on a four-band model by combining the iterative Green’s function method and the Landauer-Buttiker formalism. Our numerical results suggest that (1) both AHE and THE can coexist in our model system and the total Hall effect can be decomposed into the summation of these two effects; (2) in the clean limit, the topological Hall conductance (THC) almost remains constant but the topological Hall resistance (THR) can increase due to the reduction of longitudinal conductance when the SOC is increasing; (3) in the disorder limit, both the THC and THR are increasing with increasing SOC, while longitudinal conductance is not influenced much by SOC. In addition to numerical simulations, we also studied the scattering cross-section of a skyrmion texture analytically with the second-order Born approximation to provide a more theoretical understanding of this system. Our results are organized as the following. In Sec.II, we will describe our model Hamiltonian for the quantum well states in MTI films. In Sec.III, we will give our numerical results based on Landauer-Buttiker formalism for the model Hamiltonian and present the corresponding theoretical analysis. The calculation of scattering cross-section will be performed in Sec.IV to provide the additional theoretical understanding of the asymmetric scattering for our model Hamiltonian. The disorder effect is numerically calculated and discussed in Sec.V. The conclusion will be drawn in Sec.VI.
Model Hamiltonian
==================
The TI films can be modeled by a 3D four-band model in a quantum well (QW) with an infinite potential along the $z$ direction [@zhang2009topological; @liu2010model]. The confinement effect along the $z$ direction can be approximated by choosing $\langle k_z\rangle=0$ and $\langle k_z^2\rangle=(n_b \pi /d)^2$, where $n_b$ is an integer to label the sub-band index and $d$ is the width of the QW [@liu2010oscillatory]. As shown in the AppendixA [@appendix], we project the 3D four-band model into the subspace spanned by these QW sub-bands and obtain a 2D four-band BHZ-like model given by [$$H_0(\bm{k}) = M(\bm{k})\sigma_0\tau_z + \alpha(p_x \sigma_x + p_y \sigma_y)\tau_x,
\label{eq:h0}$$]{} on the basis $({\left|+ \uparrow\right>},{\left|+ \downarrow\right>}, {\left|- \uparrow\right>}, {\left|- \downarrow\right>})$, where $\sigma$ and $\tau$ are Pauli matrices for spin and orbital subspaces. $M(\bm{k})=M_0+Bk_\perp^2$ and $\alpha$ labels the SOC strength. The parameter $M_0=M+B_1 (n_b \pi /d)^2$ (See appendix for details) depends on the integer sub-band index $n_b$. Depending on the sub-band index $n_b$ (assuming $M<0$ and $B_1>0$), the four band model for the QW sub-bands can be in the inverted regime if $M_0 B<0$ or in the normal regime $M_0 B>0$. It should be mentioned that the Hamiltonian (\[eq:h0\]) is block-diagonal with one block set by the basis $({\left|+ \uparrow\right>}, {\left|- \downarrow\right>})$ and the other block by the basis $({\left|+ \downarrow\right>}, {\left|- \uparrow\right>})$. These two blocks are related to each other by time-reversal symmetry and are degenerate. This degeneracy will be broken when introducing ferromagnetism or magnetic skyrmion into the system. Since we focus on the transport regime dominated by these QW states in this work, we expect that multiple QW sub-bands with different $n_b$ will be present at the Fermi energy. To simplify the problem, we treat QW sub-bands in the Hamiltonian (Eq.\[eq:h0\]) with different $n_b$ independently. Thus, we may choose $M_0$ as an independent parameter and discuss below the transport behaviors for the parameter $M_0$ in different regimes. The SOC term ($\alpha$ term) couples the state ${\left|+ \uparrow\right>}$ $({\left|+ \downarrow\right>})$ to the state ${\left|- \downarrow\right>}$ $({\left|- \uparrow\right>})$ in different orbital basis, which is different from the conventional Rashba SOC where the SOC term couples different spin states in the same orbital basis. Here we adopted the Hamiltonian form (Eq.\[eq:h0\]) in Ref., which is equivalent to the more standard Hamiltonian form given in Ref. up to a unitary transformation $U=\textrm{Diag} (1,1,-i,i)$.
For MTI, the exchange interaction between electron spin and magnetic moment is given by [$$H_{\textrm{ex}}(x,y) = \bm{m} (x, y) \cdot \bm{\sigma} \tau_0,
\label{eq:Hex}$$]{} where $\bm{m}(x,y)$ represents the magnetization. The magnetic skyrmion texture can be taken into account by choosing the $\bm{m}$ configuration [$$\begin{split}
\bm m &= m_0(\sin \theta \cos \phi ,\sin \theta \sin \phi
,\cos \theta )\\
\theta &= \pi \tanh \left(\frac{\rho }{R}\right) \\
\phi &= n \varphi + \eta,
\label{eq:skyrmionmag}
\end{split}$$]{} where $m_0$ represents the magnetization strength, $\theta$ and $\phi$ label the magnetization direction, $\rho$ and $\varphi$ define the spatial polar coordinates with $(x,y)=(\rho\cos\varphi,\rho\sin\varphi)$, and $R$ is the radius of the skyrmion. The chirality of the skyrmion is characterized by the integer number $n$, which is chosen to be $+1$ (a single skyrmion) or $-1$ (a single anti-skyrmion) below. The parameter $\eta$ denotes the helicity phase, which is an irrelevant parameter. It should be pointed out that the unitary transformation $U$ should be applied to the Hamiltonian (\[eq:Hex\]) in order to be consistent with the Hamiltonian (\[eq:h0\]). However, we find the THE only depends on the chirality of skyrmion texture, which is unchanged under the transformation $U$, and thus we can still use the Hamiltonian (\[eq:Hex\]) to describe skyrmion texture. Physically, the magnetic skyrmion can be energetically stabilized by the interplay of the Zeeman coupling and DM interaction in MTI films [@koshibae2016theory]. In this study, we assume the skyrmion structure in our system (Fig.\[fig:config\](a)) and focus on the influence of skyrmion on magneto-transport.
![(a) Schematic configuration of the system. The square region in the center represents the sample, and the extended transparent edges represent semi-infinite FM leads. A skyrmion of $m=1, \eta=\pi/2, R = 0.4 L$ is shown at the origin. Cones point to the local magnetic moment direction. (b, c) Band dispersions for the Hamiltonian Eq.\[eq:hamiltonian\] for two parameter sets (i) and (ii) (see main text). A periodic boundary condition is applied on the $y$ direction. The shaded regions show the energy range for the transport calculations in Fig.\[fig:the01m1\] to \[fig:THEenhancement\]. []{data-label="fig:config"}](figs/config.png "fig:"){width="35.00000%"} ![(a) Schematic configuration of the system. The square region in the center represents the sample, and the extended transparent edges represent semi-infinite FM leads. A skyrmion of $m=1, \eta=\pi/2, R = 0.4 L$ is shown at the origin. Cones point to the local magnetic moment direction. (b, c) Band dispersions for the Hamiltonian Eq.\[eq:hamiltonian\] for two parameter sets (i) and (ii) (see main text). A periodic boundary condition is applied on the $y$ direction. The shaded regions show the energy range for the transport calculations in Fig.\[fig:the01m1\] to \[fig:THEenhancement\]. []{data-label="fig:config"}](figs/spectra.png "fig:"){width="23.00000%"} ![(a) Schematic configuration of the system. The square region in the center represents the sample, and the extended transparent edges represent semi-infinite FM leads. A skyrmion of $m=1, \eta=\pi/2, R = 0.4 L$ is shown at the origin. Cones point to the local magnetic moment direction. (b, c) Band dispersions for the Hamiltonian Eq.\[eq:hamiltonian\] for two parameter sets (i) and (ii) (see main text). A periodic boundary condition is applied on the $y$ direction. The shaded regions show the energy range for the transport calculations in Fig.\[fig:the01m1\] to \[fig:THEenhancement\]. []{data-label="fig:config"}](figs/spectrainv.png "fig:"){width="23.00000%"}
Due to the absence of translation symmetry in a system with a single skyrmion, we numerically explore magneto-transport directly in the real space. To perform such calculation, we implement the tight-binding regularization on the Hamiltonian (\[eq:h0\]) and (\[eq:Hex\]), which is given by [$$\begin{split}
\hat{H} &= \hat{H}_0 + \sum_i \Psi_i^\dagger H_{\textrm{ex}} \Psi_i ,\\
\hat{H}_0 &= \sum_i \Psi_{i}^\dagger \epsilon_i \Psi_{i} + \sum_{\langle i,j\rangle}(V_{ij} \Psi_{j}^\dagger \Psi_{i} + \text{h.c.}),
\label{eq:hamiltonian}
\end{split}$$]{} where $\Psi^\dagger_{i}$ presents $(c^\dagger_{i,+\uparrow}, c^\dagger_{i,+\downarrow}, c^\dagger_{i,-\uparrow}, c^\dagger_{i,-\downarrow})$ at the position $ i = (i_x,i_y)$. $\epsilon_i$ is the on-site energy and $V_{ij}$ is the hopping matrix between the nearest neighbors $i$ and $j$. Both $\epsilon_i$ and $V_{ij}$ are 4 by 4 matrices and their detailed forms can be related to those in the continuous model (Eq.\[eq:h0\]), as listed in the AppendixB[@appendix]. The consistency between the tight-binding Hamiltonian (\[eq:hamiltonian\]) and the continuous Hamiltonian (\[eq:h0\]) and (\[eq:Hex\]) is also discussed in the AppendixB[@appendix].
We consider a 2D square lattice with the side length $L$. The skyrmion texture is located at the center of the lattice, as shown in Fig.\[fig:config\](a). Four semi-infinite leads with the width $L$, labeled as 1 to 4 in Fig.\[fig:config\], are attached to each side of the square lattice. We adopt the recursive Green’s function method [@datta1997electronic] to evaluate the transmission coefficient $T_{pq}$ between the leads $p$ and $q$ ($p,q=1,2,3,4$). The relationship between currents and voltages is calculated using the Landauer-Büttiker formalism $$\begin{aligned}
\label{eq:LB}
I_p=e^2/h \sum_{q\neq p}\left(T_{qp}V_p-T_{pq}V_q\right).\end{aligned}$$ Due to the charge conservation of the whole system, the matrix $T$ is singular. Without loss of generality, we set $V_4=0$ and remove the corresponding column/row in $T$. To set up a Hall configuration, we consider a current flow from the lead 1 to 3, as setting $I_1 = -I_3 = I$ and $I_2=0$. Voltages of the leads are calculated through $(V_1,V_2,V_3)^\intercal = \frac{h}{e^2}T^{-1} (I_1,I_2,I_3)^\intercal$, where the matrix $T$ is the $3\times 3$ transmission matrix. The longitudinal resistance $R_{xx}$ and the Hall resistance $R_{xy}$ can be extracted by $R_{xx} = (V_1-V_3)/I$ and $R_{xy} = V_2/I$.
Numerical results and Analysis in the Clean limit
=================================================
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Fig.\[fig:the01m1\](a1) and (b1) reveal the Hall resistance as a function of the Fermi energy in different parameter regimes. Here we choose a system of length $L =80 a_0$ and Skyrmion radius $R = 0.4 L$. Hopping parameters are expressed in term of the nearest-neighbor hopping strength $t \equiv B/a_0^2$ (See AppendixB[@appendix]). Here we consider two sets of parameters, one for the trivial regime, denoted as (i), and the other for the QAH regime, denoted as (ii) below. While all the other parameters are the same ($m_0=4/3,\alpha/t=2, a_0=1, B=1$) for the parameter sets (i) and (ii), the parameter $M_0$ is chosen to be different ($M_0 = 2$ for (i) and $M_0=1$ for (ii)). To see the topological property of the full Hamiltonian with these two parameter sets, we may consider the FM case with $\bm{m}(x,y)=m_0(0,0,1)$. For the parameter set (i), we notice that both blocks of bands in the Hamiltonian (\[eq:hamiltonian\]) are in the normal regime since $(M_0\pm m_0)B>0$. In contrast, the system for the parameter set (ii) is in the QAH regime since one block is in the normal regime $(M_0+m_0)B>0$ while the other is in the inverted regime $(M_0-m_0)B<0$. The corresponding energy dispersions for a slab configuration with these two parameter sets are shown in Fig.1(b) and (c), from which one can see a full gap for the parameter set (i) and gapless chiral edge states appear in the bulk gap for the parameter set (ii). In this work, we focus on the transport behavior of the metallic regime when the Fermi energy $E_f$ crosses one valence band top ($E_f/t \in [-2, -0.7]$ for the parameter set (i) and $[-2.6,-1]$ for the parameter set (ii) in Fig.1(b) and (c)). For the purpose of the quantized conductance within the gap, the parameter set (i) and (ii) represent a comparison between a trivial and a non-trivial gap. The difference is briefly discussed in AppendixD[@appendix]. For the transport calculation, three magnetic configurations, namely ferromagnetism ($\hat m = -m_0 \hat z$), a skyrmion ($n=+1$) and an anti-skyrmion ($n=-1$) are considered and the corresponding Hall resistances $R_{\textrm{FM}}$ , $R_{\textrm{Sk},+1}$ and $R_{\textrm{Sk},-1}$ are shown by the yellow, red and blue lines in Fig.\[fig:the01m1\](a1) and (b1) for the parameter sets (i) and (ii), respectively. One can clearly see that $R_{\textrm{Sk},+1}$ is much larger than $R_{\textrm{FM}}$, while $R_{\textrm{Sk},-1}$ has the opposite sign. For the FM case, the Hall resistance only originates from the AHE, while in the skyrmion cases, both THE and AHE can contribute due to the coexistence of strong SOC and chiral magnetic structure. We expect the THE (AHE) is dependent (independent) on the chirality of the skyrmions. Therefore, we can decompose the Hall resistances $R_{\textrm{Sk},\pm 1}$ into chirality dependent part, $R_{\textrm{THE}}$, and independent part, $R_{\textrm{AHE}}$, $$R_{\textrm{Sk},n}=R_{\textrm{AHE}}+ n R_{\textrm{THE}},
\label{eq:Hall_skyrmion_decomposition}$$ where the index $n$ stands for the chirality of the skyrmion.
Based on the decomposition of Eq.\[eq:Hall\_skyrmion\_decomposition\], Fig.\[fig:the01m1\](a2) and (b2) depict $R_{\textrm{AHE}}$ (blue line) and $R_{\textrm{THE}}$ (yellow line) as a function of Fermi energy for the parameter sets (i) and (ii), respectively. In addition, $R_{\textrm{FM}}$ is shown by the red line. Fig.\[fig:the01m1\](a2) and (b2) show the following features. (1) $R_{\textrm{FM}}$ generally shows a similar behavior as the blue line of $R_{\textrm{AHE}}$ (except that the Fermi energy is close to band gap), and thus the magnetic skyrmion does not have a strong influence on the AHE in the metallic regime and validates the decomposition of the Hall resistance. (2) $R_{\textrm{AHE}}$ is much larger for the parameter set (i) than that for (ii), due to the energy range (shaded area in Fig.\[fig:config\](b,c)) is closer to band center for parameter set (ii). (3) For both parameter sets, we notice that $R_{\textrm{THE}}$ increases rapidly when the Fermi energy is tuned towards the valence band top.
We next turn to the Hall conductance with the same parameters, as shown in Fig.\[fig:theg01m1\]. Here the red, blue and yellow lines are for the Hall conductance with the FM, skyrmion ($n=+1$) and anti-skyrmion ($n=-1$) configurations in Fig.\[fig:theg01m1\](a1) and (b1) for two parameter sets. A similar decomposition $$G_{\textrm{Sk},n}=G_{\textrm{AHE}}+n G_{\textrm{THE}}
\label{eq:Hall_skyrmion_decomposition_conductance}$$ is considered and the corresponding $G_{\textrm{AHE}}$ and $G_{\textrm{THE}}$ are plotted in Fig.\[fig:theg01m1\](a2) and (b2), which show the following features. (1) The decomposition of Hall conductance also remains valid in most energy ranges, as indicated by the coincidence between $G_{\textrm{AHE}}$ and $G_{\textrm{FM}}$ in most energy ranges. (2) In contrast to Hall resistance, the Hall conductance is almost a constant in the whole metallic region for both parameter sets. (3) $G_{\textrm{AHE}}$ (or $G_{\textrm{FM}}$) is smaller for the parameter set (i) compared to that for (ii) while $G_{\textrm{THE}}$ is comparable for both parameter sets. Fig.\[fig:THEenhancement\](a) and (b) ((c) and (d)) reveal the THE contribution $R_{\textrm{THE}}$ ($G_{\textrm{THE}}$) from the decomposition of Eq.\[eq:Hall\_skyrmion\_decomposition\] (Eq.\[eq:Hall\_skyrmion\_decomposition\_conductance\]) as a function of the Fermi energy for different SOC strength $\alpha$ for both parameter sets. An enhancement of $R_{\textrm{THE}}$ is found while $G_{\textrm{THE}}$ remains almost unchanged when increasing the SOC strength or the Fermi energy for both parameter sets. $G_{\textrm{THE}}$ is only found to drop when the Fermi energy is close to the band gap (insulating regime).
![(a) and (b) show THR $R_{\textrm{THE}}$ as the function of Fermi energy and SOC strength $\alpha$, for the parameter sets (i) and (ii), respectively. (c) and (d) show THC $G_{\textrm{THE}}$ decomposed in the similar way. []{data-label="fig:THEenhancement"}](figs/THERG.png){width="50.00000%"}
To understand our numerical results, we will next present a theoretical analysis of the transport behavior of the model based on our numerical simulation of the Landauer-Buttiker formalism. The symmetry property of the transmission matrix $T_{pq}$ in the Eq.\[eq:LB\] will be first analyzed. For the FM and $n=+1$ skyrmion cases, the system respects the $C_4$ rotation symmetry, while for the anti-skyrmion with $n=-1$, the system possesses the $S_4$ improper rotation symmetry. In all cases, we find the transmission matrix elements can be characterized by three independent parameters based on the following relations [$$\begin{split}
T_{13} = T_{24} = T_{31} = T_{42} \equiv -a \\
T_{14} = T_{21} = T_{32} = T_{43} \equiv -b \\
T_{12} = T_{23} = T_{34} = T_{41} \equiv -c \\
\end{split}$$]{} where $a$, $b$ and $c$ can be understood as the probability of electronic modes going straight, turning left and turning right, respectively, after they entered the spin-textured structure from any of the leads. With this simplification, direct calculations from the Landauer-Büttiker formalism give rise to the Hall and longitudinal resistance as [$$\begin{split}
R_{xy} &= \frac{(b-c)}{2 a^2+2 a (b+c)+b^2+c^2} \approx \frac{b-c}{2 (a+b)^2} \equiv \frac{\Delta}{2 \beta^2} \\
R_{xx} &= \frac{(2 a+b+c)}{2 a^2+2 a (b+c)+b^2+c^2} \approx \frac{1}{a+b} \equiv \frac{1}{\beta},
\label{eq:resistance_analytical}
\end{split}$$]{} where we define the parameters $\beta \equiv a+b$ to be the transmission probability and $\Delta \equiv (b-c)$ to be the asymmetric scattering between the left and right directions. We further assume $\Delta\ll b\approx c$, which can be justified based on our numerical calculations for both parameter sets. The corresponding Hall and longitudinal conductance is [$$\begin{split}
G_{xy} \approx -\Delta/2 \\
G_{xx} \approx \beta.
\label{eq:conductance_analytical}
\end{split}$$]{}
![(a), (b) and (c) show $\Delta$ as a function of $E$ for FM, skyrmion and antiskyrmion cases. (d), (e) and (f) reveal the energy dependence of $\beta$ for FM, skyrmion and antiskyrmion cases. Here we consider the parameter set (i). []{data-label="fig:ABD"}](figs/ABD.png){width="50.00000%"}
![(a) and (b) show the decomposition of $\Delta$ into the THE ($\Delta_{\textrm{THE}}$) and AHE part ($\Delta_{\textrm{AHE}}$), respectively. Here we choose the parameter set (i). []{data-label="fig:DeltaDecompose"}](figs/DeltaDecompose.png){width="50.00000%"}
Eq.(\[eq:resistance\_analytical\]) and (\[eq:conductance\_analytical\]) are the basis for the analysis below. We can see that $\beta$ is related to the forward transmission and determines the longitudinal conductance $G_{xx}$ while $\Delta$ represents the asymmetry between the left and right scattering and determines the Hall conductance $G_{xy}$. In Fig.\[fig:ABD\], we demonstrate the behaviors of $\beta$ and $\Delta$ for the parameter set (i) as an example, from which we can understand the behaviors of AHE and THE.
Below we will analyze the behavior of $\Delta$ first (Fig.\[fig:ABD\] (d) - (f)). In Fig.\[fig:ABD\](d), one can see that $\Delta$ increases when increasing the SOC parameter $\alpha$ in the FM case. From Fig.\[fig:ABD\](e) and (f), we find the value of $\Delta$ is much larger when there is a skyrmion or anti-skyrmion as compared to the FM case. We may also consider a decomposition $ \Delta_{\textrm{Sk},n}=\Delta_{\textrm{AHE}}+ n \Delta_{\textrm{THE}}$ and the corresponding $\Delta_{\textrm{THE}}$ and $\Delta_{\textrm{AHE}}$ are plotted in Fig.\[fig:DeltaDecompose\](a) and (b), respectively. One can see that all the curves for $\Delta_{\textrm{THE}}$ fall into one line and thus are independent of the SOC parameter $\alpha$, while $\Delta_{\textrm{AHE}}$ increases rapidly with $\alpha$. From Eq.(\[eq:conductance\_analytical\]), we expect that $G_{xy}$ exhibits a similar behavior as $\Delta$, which was indeed revealed in Fig.\[fig:THEenhancement\](c). Therefore, we conclude that SOC mainly increases the AHE contribution, but has little influence on the THE contribution to our four-band model in the clean limit.
Next, let us analyze the behavior of transmission $\beta$. For all three spin textures, the values of $\beta$ and the dependence of $\beta$ on SOC and the Fermi energy are quite similar, and thus we would not specify the magnetic texture for the discussion below. As expected, $\beta$ is decreasing when tuning Fermi energy to the valence band top (more insulating). In addition, we find a rapid decreasing of $\beta$ when increasing the SOC parameter $\alpha$ in Fig.\[fig:ABD\]. This can be understood as the following. SOC tends to induce precession of electron spin and rotate it from the easy axis set by local magnetization. As a consequence, a strong scattering can be induced by the exchange coupling between electron spin and magnetic moments, and thus reduces transmission. It turns out that the reduction of transmission $\beta$ has a substantial influence on the behavior of Hall resistance $R_{xy}$. From Eq.\[eq:resistance\_analytical\], we can see that $R_{xy}$ depends on the ratio between $\Delta$ and $\beta^2$. Therefore, although increasing SOC does not enhance $\Delta$, it reduces $\beta$, and thus increases $R_{xy}$ as shown in Fig.\[fig:THEenhancement\]. (For a full figure of $R_{xy}$ for all SOC and spin textures, please see Appendix Fig.C1.) This analysis leads to the following conclusion for our model in the clean limit: (1) SOC does not have much influence on THE and (2) the behavior of $R_{\textrm{THE}}$ is mainly determined by the forward transmission $\beta$, rather than the asymmetric scattering $\Delta$.
Analytical results of cross section
===================================
In this section, we will provide more physical understanding on THE for our four-band model by analytically calculating the differential cross section of this system. We notice that topological surface states scattered by a magnetic skyrmion have been studied in Ref. , while we focus on bulk QW states here. Due to the presence of spin-polarized background $S_z=\hat{z}$, we can treat $\bar{H}_0=H_0-m_0\sigma_z\otimes\tau_0$ as the unperturbed Hamiltonian, and take $\hat{V}(\bm{r}) =- (\bm{m}(\bm{r})\cdot\bm{\sigma}-m_0\sigma_z)\otimes\tau_0$ as the perturbation (scattering potential), where $\bm{m}(\bm{r})$ has been defined above in Eq. (\[eq:skyrmionmag\]) and $m_0$ can be regarded as the exchange coupling strength between the conduction electron and local magnetic moment. The differential cross section of electron scattering is given by $$\left(\frac{d\sigma(\phi)}{d\phi}\right)_{\alpha\beta} = |F_{\alpha\beta}(\bm{p},\bm{p'})|^2$$ where $\phi$ is the scattering angle. $\psi_{{\bm p}'\alpha}$ and $\psi_{{\bm p}\beta}$ are eigenstates of $\bar{H}_0$ that describe the incident and scattered states respectively with ${\bm p}$ and ${\bm p}'$ is the associated momenta. $F_{\alpha\beta}=F_{\alpha\beta}(\bm{p},\bm{p'}) = \langle\psi_{\bm{p'}\alpha}|\hat{V} + \hat{V}\hat{G}_0\hat{V}|\psi_{\bm{p}\beta}\rangle$ is the scattering amplitude up to the second order Born approximation, where $\hat{G}_0$ is the Green’s function associated with unperturbed Hamiltonian $\bar{H}_0$. Asymmetric component of $\left(\frac{d\sigma(\phi)}{d\phi}\right)_{\alpha\beta}$ with respect to $\phi$, being responsible for the Hall response, arises from cross-terms between the first and second Born approximation.
A major difficulty in this calculation is the computation of $\langle\psi_{\bm{p'}\alpha}|\hat{V}\hat{G}_0\hat{V}|\psi_{\bm{p}\beta}\rangle$ due to the Bessel function-like Green’s function in 2D. Here we use the momentum representation so that $ \langle\psi_{\bm{p'}\alpha}|\hat{V}\hat{G}\hat{V}|\psi_{\bm{p}\beta}\rangle = \int d\bm{p}_1 \langle\alpha_{\bm{p'}}|V(\bm{p'}-\bm{p}_1)G(\bm{p}_1)V(\bm{p}_1-\bm{p})|\beta_{\bm{p}}\rangle$. For simplicity, we use a Bloch skyrmion configuration and let its polar angle in Eq. (\[eq:skyrmionmag\]) be $\theta=\pi\exp(-r/a)$, where $a$ is the radius of the skyrmion. In the analytical calculations, Bessel functions of $qr$ will be used throughout the whole calculation, where ${\bm q}={\bm p}-{\bm p}'$. In the small angle scattering assumption, $qr\ll1$, so that we can expand the Bessel functions with Fourier series and keep the lowest order terms. In the current calculation, we are interested in the situation where the Fermi surface intersects only with the lowest electron band of $\bar{H}$. Direct calculation shows that the asymmetric part of the corresponding differential cross section is given by $$\begin{aligned}
\left(\frac{d\sigma}{d\phi}\right)^A\propto a^4D(\varepsilon_F)m_0^3\left[\left(1-\cos\omega\right)\cos^4\frac{\omega}{2}\right.\nonumber\\
\left. + \left(1+\cos\omega\right)\sin^4\frac{\omega}{2}\right]\sin\phi\end{aligned}$$ where $D(\varepsilon_F)$ is the density of states at the Fermi energy, and $\omega=\arccos [(Bp^2+M_0-m_0)/\varepsilon_F]$. The cross-section as a function of scattering angle $\phi$ for different $m_0$ are shown in Fig.\[fig:CrossSection\]. With a small or intermediate exchange coupling strength $m_0$, we find that the asymmetric component of the differential cross section increases with the SOC parameter $\alpha$, and eventually saturate at large $\alpha$ limit, as shown in Fig.\[fig:CrossSection\](a). On the other hand, when $m_0$ is large, the influence of SOC parameter $\alpha$ becomes negligible due to the dominant role of exchange coupling in inducing THE in this regime, as shown in Fig.\[fig:CrossSection\](b), and our numerical results are consistent with the analytical result in this regime.
![Asymmetric part of the differential cross section as a function of scattering angle for (a) $m_0=5t$ and (b) $m_0=80t$. []{data-label="fig:CrossSection"}](figs/JH.png){width="50.00000%"}
Disorder effect
===============
We next examine the disorder effect on THE in MTI films, as shown in Fig.\[fig:Disorder\](a-d), which reveal new features compared to Fig.\[fig:THEenhancement\] in the clean limit. To consider the disorder effect, we introduce a spin-independent uniformly-distributed random on-site potential term $H_{\textrm{d}}=\sum_i \Psi^\dagger_i V_{\textrm{d},i}\Psi_i$ where $V_{\textrm{d},i} = \textrm{Diag} (\mathcal{V}_{+}, \mathcal{V}_{+}, \mathcal{V}_{-}, \mathcal{V}_{-})$ and $\mathcal{V}_{\pm} \in \left[-V_{\textrm{imp}}/2, V_{\textrm{imp}}/2\right]$ with $0<V_{\textrm{imp}}\leq 2t$ chosen in our calculations. All the calculations are performed with the disorder average over 160 samples. After such a sample average, the uncertainty (dictated by the error bars in Fig.\[fig:THEenhancement\]) is much smaller than its mean value. We implement a similar decomposition of Hall conductance and resistance, as specified in Eq.(\[eq:Hall\_skyrmion\_decomposition\]) and (\[eq:Hall\_skyrmion\_decomposition\_conductance\]) for each individual run, and the disorder-averaged Hall resistance ($R_{\textrm{THE}}$) and conductance ($G_{\textrm{THE}}$) from the THE contribution are revealed in Fig.\[fig:Disorder\](a, b) and (c, d) for two parameter sets (i) and (ii), respectively. Here the circle, square and diamond label different disorder strength $V_{\textrm{imp}}=1, 1.5$ and $2$ in the unit of $t$, while the green and black colors are for different SOC strength ($\alpha/t = 0$ and $\alpha/t = 2$). The solid lines show the results in the clean limit for comparison. With increasing the disorder strength, one can clearly see the decreasing of both Hall resistance $R_{\textrm{THE}}$ and conductance $G_{\textrm{THE}}$. A striking feature emerges in the Hall conductance $G_{\textrm{THE}}$ when increasing disorder strength. $G_{\textrm{THE}}$ is unchanged for different SOC strength in the clean limit, as seen by the coincidence between black and green solid lines in Fig.\[fig:Disorder\] (c) and (d). In contrast, for intermediate or strong disorder strength, $G_{\textrm{THE}}$ at a large SOC $\alpha/t = 2$ can be much larger than that at zero SOC, as shown by the green and black markers in Fig.\[fig:Disorder\]c and d. This suggests that SOC can stabilize the THE against disorder scattering. We also analyze the disorder-averaged forward transmission $\beta$ and asymmetric scattering $\Delta$, shown in Fig.\[fig:DisorderABD\](a) and (b). Interestingly, we find that with increasing disorder scattering, forward transmission $\beta$, although being reduced for both SOC strengths, becomes comparable for $\alpha/t=0$ and $\alpha/t=2$ when $V_{\textrm{imp}}$ is increased above $1.5t$. This suggests that the mean-free path of electrons is determined by disorder, rather than SOC, at this disorder strength $V_{\textrm{imp}}/t=2$. In contrast, although the asymmetric scattering parameter $\Delta$ is reduced for both SOC strengths, its reduction is much slower when the SOC $\alpha$ is strong, which can be clearly seen by the fact that the green markers ($\alpha/t=2$) are above the black markers ($\alpha/t=0$) in Fig.\[fig:DisorderABD\] (b). In contrast to the clean limit, in which SOC only enhances $R_{xy}$ but not $G_{xy}$ due to the reduction of transmission $\beta$, SOC mainly influences the asymmetric scattering $\Delta$ and thus will enhance both $R_{xy}$ and $G_{xy}$ in the disordered limit. Therefore, we conclude that the THE is stabilized by SOC in the disordered limit.
![(a) and (c) reveal $R_{\textrm{THE}}$ and $G_{\textrm{THE}}$ for the parameter set (i) under different disorder strength ($V_{\textrm{imp}}=0, 1, 1.5, 2$ in unit of $t$), respectively. Green and black colors represent $\alpha/t = 2$ and $0$ cases, respectively. (b) and (d) are the same as (a)(c), except that we choose the parameter set (ii). []{data-label="fig:Disorder"}](figs/Disorder.png "fig:"){width="50.00000%"} ![(a) and (c) reveal $R_{\textrm{THE}}$ and $G_{\textrm{THE}}$ for the parameter set (i) under different disorder strength ($V_{\textrm{imp}}=0, 1, 1.5, 2$ in unit of $t$), respectively. Green and black colors represent $\alpha/t = 2$ and $0$ cases, respectively. (b) and (d) are the same as (a)(c), except that we choose the parameter set (ii). []{data-label="fig:Disorder"}](figs/DisorderConductance.png "fig:"){width="50.00000%"}
![(a) and (b) reveal $\bar{\beta}$ and $\Delta_{\textrm{THE}}$ as a function of $E$ in both disordered and clean limit. Here the colors represent different SOC strengths (green for $\alpha/t = 2$ and black for $\alpha/t=0$), and the line, circle, square and diamond represent $V_{\textrm{imp}}=0, 1, 1.5, 2$, respectively. []{data-label="fig:DisorderABD"}](figs/DisorderABD.png){width="50.00000%"}
Discussion and Conclusion
=========================
In summary, we have studied the AHE and THE in an MTI model with skyrmion configuration and revealed how the magneto-transport behaviors in such systems are influenced by SOC, Fermi energy and disorder through numerical calculations and theoretical analysis. In particular, our calculations demonstrate the importance of disorder effect in determining the role of SOC in the THE. Given the recent experimental efforts in MTI systems [@yasuda2016geometric; @liu2017dimensional], our numerical and theoretical results will provide a physical understanding of these magneto-transport measurements and may stimulate further experimental studies. It should be pointed out that the SOC term we used here preserves inversion symmetry and thus is different from the Rashba SOC, which breaks inversion symmetry and responses for DM interaction. Including Rashba SOC in the calculation may bring new features and will require further studies, which is beyond the scope of the current work. In light of the importance of the interplay between disorder scattering and SOC, it will be important to develop a more analytical theory (such as the Boltzman equation and diagram expansion calculation [@nagaosa2010anomalous]) to take into account random scattering of multiple magnetic skyrmions or skyrmion lattice, SOC and disorder scattering, which will be another future direction. We would like to point out that although the current calculation is on a specific skyrmion texture, the presence of transverse scattering should exist for any spin textures, including random state, with net chirality[@hou2017thermally]. It will be interesting to generalize this work to the investigation of THE-AHE crossover in other chiral systems.
Acknowledgement
===============
We acknowledge the discussion with C.Z. Chang and M.H.W. Chan. C.X.L and J.X.Z acknowledge support from the Office of Naval Research (Grant No. N00014-15-1-2675 and renewal No. N00014-18-1-2793) and the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under award No. DE-SC0019064. Work at UNH was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award No. DE-SC0016424.
|
23.5cm -1cm
[ **Charged Current Neutrino Cross Section and\
Tau Energy Loss at Ultra-High Energies** ]{}
1.cm
[.5em N. Armesto[^1], C. Merino[^2], G. Parente[^3], and E. Zas[^4] ]{}
[*Departamento de Física de Partículas $\&$\
Instituto Galego de Física de Altas Enerxías\
Universidade de Santiago de Compostela\
15706 Santiago de Compostela, Spain*]{}
.5cm [**Abstract** ]{}
> We evaluate both the tau lepton energy loss produced by photonuclear interactions and the neutrino charged current cross section at ultra-high energies, relevant to neutrino bounds with Earth-skimming tau neutrinos, using different theoretical and phenomenological models for nucleon and nucleus structure functions. The theoretical uncertainty is estimated by taking different extrapolations of the structure function $F_2$ to very low values of $x$, in the low and moderate $Q^2$ range for the tau lepton interaction and at high $Q^2$ for the neutrino-nucleus inelastic cross section. It is at these extremely low values of $x$ where nuclear shadowing and parton saturation effects are unknown and could be stronger than usually considered. For tau and neutrino energies $E=10^{9}$ GeV we find uncertainties of a factor 4 for the tau energy loss and of a factor 2 for the charged current neutrino-nucleus cross section.
Introduction
============
The detection of high energy neutrinos is one of the most important challenges in Astroparticle Physics. Conventional neutrino detectors exploit the long range of muons produced by muon neutrino charged current (CC) interactions [@Halzen]. With the discovery of neutrino flavor oscillations it has been realized that also tau neutrinos reach the Earth in spite of being heavily suppressed in all postulated production mechanisms. The possibility to search for tau neutrinos by looking for tau leptons that exit the Earth, Earth-skimming neutrinos, has been shown to be particularly advantageous to detect neutrinos of energies in the EeV range [@Fargion; @Bertou]. The short lifetime of the tau lepton originated in the neutrino charged current interaction allows the tau to decay in flight while still close to the Earth surface producing an outcoming air shower in principle detectable by both fluorescence telescopes and air shower arrays [@EZas]. This same channel yields negligible contributions for other neutrino flavors. The sensitivity to tau neutrinos through the Earth-skimming channel directly depends both on the neutrino charged current cross section and on the tau range (the energy loss) which determine the amount of matter with which the neutrino has to interact to produce an emerging tau [@ICRCAuger_neutrino1; @ICRCAuger_neutrino2]. While the energy loss for muons is shared by roughly equivalent contributions from pair production, bremsstrahlung and photonuclear interactions, for tau leptons of energies above $E = 10^7$ GeV, photonuclear interactions (i.e. lepton-nucleus inelastic interactions dominated by small values of $Q^2$) are responsible for the largest and the most uncertain contribution [@Dutta2001; @Bugaev2004; @Aramo2005].
Both the neutrino cross section and the tau photonuclear energy loss are calculated from theory using structure functions which carry the information of the nucleon and nucleus structure. In order to study the uncertainties in the calculation of Earth-skimming neutrinos the same structure functions should be consistently used for both processes due to their strong correlation in the resulting tau flux. Unfortunately this is not possible since the kinematical $Q^2$ (minus the squared momentum transfer) and Bjorken-$x$ ranges that contribute to these processes are quite different, specially at EeV energies, and the available parameterizations are not entirely adequate to describe both ranges simultaneously.
The $Q^2$ scale that contributes to the tau energy loss, dominated by photon exchange, is low and moderate $Q^2$ at very low $x$, where perturbative and non perturbative QCD effects are mixed. The CC neutrino cross section is produced by $W$-boson exchange that sets the relevant scale of $Q^2$ to values up to $M_W^2~$ at low $x$, a region where perturbative QCD is expected to work. In both cases the relevant $x$ range lies well outside the regions where structure functions are measured, so one has to rely on extrapolations which contain significant uncertainties.
The charged current neutrino cross section is usually calculated using parton distribution functions which are evolved according to perturbative QCD predictions. A number of alternative parameterizations exist, some of which allow extrapolation of the uncertainties in the fitted parameters as a mean to explore some of the uncertainties associated to the calculation. In the case of photonuclear processes existing predictions at high energy arise basically from two independent approaches, the Generalized Vector Dominance (GVD) model and Regge-like models.
In this article we study the tau energy loss (see also Ref. [@nosoICRC07]) and the neutrino-nucleus cross section. Both quantities have direct implications for high energy neutrino detection, in particular for Earth-skimming tau neutrinos. Due to the large uncertainties in the existing models, the fact that none of them covers simultaneously the kinematical region relevant for both quantities, and the need of consistency in both calculations, we use and extend available models with the aim of estimating the theoretical uncertainty by considering extreme results. In this way, in the frame of the most relevant models, we cover the range of possible scenarios for the extrapolation of structure functions to the relevant $x$ and $Q^2$ range. Two important effects to be taken into account in this extrapolation of the structure functions are nuclear shadowing corrections and saturation due to partonic screening. Nuclear corrections [@nuclear] are deviations from the naive picture in which the nucleus is treated as an incoherent sum of nucleons. Saturation [@saturacion] accounts for the fact that the structure functions cannot rise indefinitely as $x$ goes to zero. Saturation effects may be included in nuclear corrections but are also present in the nucleon structure functions (although for smaller values of $x$ and/or $Q^2$). In addition to existing calculations, a new computation of the tau energy loss and the neutrino-nucleon cross section based on saturation physics [@ASW] is also presented in this work.
The result of the present analysis is an uncertainty band for both the tau-lepton energy loss by photonuclear interactions and the CC neutrino-nucleus cross section. Understanding and minimizing the uncertainties in these two calculations must be considered an important priority for high energy neutrino astrophysics.
The photonuclear tau energy loss
================================
The average energy loss per unit depth, $X$, of taus is conveniently represented by: $$\begin{aligned}
- \left<\frac{dE}{dX}\right> = a(E) + b(E) E \; ,\end{aligned}$$ where $a(E)$ is due to ionization and $b(E)$ is the sum of fractional losses due to e$^+$e$^-$ pair production, bremsstrahlung, and photonuclear interactions. The parameter $a(E)$ is nearly constant and the term $b(E) E$ dominates the energy loss above a critical energy that for tau leptons is of a few TeV. The electromagnetic contribution to the energy loss, mainly due to pair production and bremsstrahlung, is well under control, while the photonuclear interaction which dominates for tau energies exceeding $E=10^7$ GeV is affected by relatively large uncertainties.
The contribution to $b(E)$ from photonuclear interactions is obtained by integration of the lepton-nucleus differential cross section, $d\sigma^{lA}/dy$: $$\begin{aligned}
b(E) =
\frac{N_A}{A} \int dy \; y \int dQ^2 \frac{d\sigma^{lA}}{dQ^2 dy} \; ,\end{aligned}$$ where $N_A$ is Avogadro’s number, $A$ the mass number, and $y$ the fraction of energy lost by the lepton in the interaction. For the lepton-nucleus differential cross section we consider the general expression for virtual photon exchange in terms of structure functions: $$\begin{aligned}
\frac{d\sigma^{lA}}{d Q^2 dy} = \frac{4 \pi \alpha^2}{Q^4}\frac{F_2^A}{y}
\left[ 1 - y - \frac{Q^2}{4 E^2} + \left(1-2\frac{m_l^2}{Q^2}\right)
\frac{y^2 + Q^2/E^2}{2(1+R^A)} \right] \; ,\end{aligned}$$ where $E$ is the lepton energy in the lab frame, $m_l$ the lepton mass, and $\alpha$ the fine structure constant. $F_2^A$ is the structure function $F_2$ for a nuclear target $A$ which is found to be different from the mere superposition of $A$ free nucleon structure functions $F_2^p$ [@nuclear]. $R^A$ is the ratio of the longitudinal to transverse structure functions which gives a small contribution to the cross section [@Dutta2001] and is neglected for clarity of the discussion below. The variables $x$, $y$ and $Q^2$ are related by kinematics through $Q^2=2MExy$, and both $F_2$ and $R$ are functions of $x$ and $Q^2$. The contribution to the tau energy loss from neutral current and $\gamma$-$Z$ interference interactions was estimated to be small [@BM2002] and is also neglected.
The limits in the double integral of Eq. (2) are well established: $$\begin{aligned}
Q^2_{min}=\frac{y^2 m_l^2 }{1-y} \; , \;\;\;\;\;\;\;\;\;\;
Q^2_{max}=2m_pEy -2m_{\pi}m_p-m_{\pi}^2 \; , \\
y_{min}=\frac{2 m_{\pi} m_p+ m_{\pi}^2}{2 m_p E} \;, \;\;\;\;\;\;\;\;\;
y_{max}=1-\frac{m_l}{E} \; ,\end{aligned}$$ where $m_p$ and $m_{\pi}$ are the proton and pion mass, respectively.
The predictions of the photonuclear interaction cross section in the GVD Model [@BB] (BB) and in its extension to higher energies by including a perturbative component based on the color dipole model [@BS2003] (BS), have been widely used to explore muon and tau lepton propagation in matter (see for instance [@Lipari; @Bugaev2004; @Aramo2005] and references therein).
The calculations in which the $F_2$ structure function is given by a phenomenological parameterization of data based on Regge Theory appear in Refs. [@Dutta2001] (DRSS), [@BM2002] (BM), [@KLS2005] (KLS), and [@Petrukhin] (PT). For the proton structure function, $F_2^p$, DRSS (see also Ref. [@Dutta2005]) uses the ALLM model [@ALLM], while BM and KLS both consider the CKMT model [@CKMT] at low $Q^2$ matched at high $Q^2$ to perturbative QCD predictions based on different parameterizations of parton distribution functions, and PT uses the proton structure function of Ref. [@F2petrukhin]. The $F_2^p$ structure function is shown in Figs. \[figF2a\] and \[figF2b\], together with the HERA data at the lowest measured $x$ values at different $Q^2$.
In DRSS, BM, and KLS calculations the nuclear structure function is related to the proton structure function through $F_2^A= f^A A F_2^p$. At high energy only the low $x$ behavior of the nuclear correction factor $f^A$ is relevant to the calculation of $b(E)$, as we will show below (see Fig. \[figrelx\]). In the DRSS calculation the low $x$ behavior of $f^A$ freezes at the value $f^A=A^{-0.1}$ for $x<0.0014$ ($\sim 0.73$ for standard rock, $A=22$), while in the BM (and KLS) calculations $f^A$ reaches a maximal asymptotic regime $f^A=A^{-1/3}$ ($\sim 0.36$ for $A=22$) at much lower $x$ (see Fig. \[figNuc\]). Both DRSS and BM nuclear corrections are $Q^2$-independent.
In addition to the existing calculations we present a new computation of the photonuclear tau energy loss using the results of Ref. [@ASW] (ASW) which are based on the geometric scaling property [@Stasto] that all data on $\sigma^{\gamma^* p}$ and on $\sigma^{\gamma^* A}$ lie on a single universal curve in terms of the scaling variable $\tau=Q^2/Q^2_{sat}$ whose form is inspired in saturation physics (the detailed expressions leading to the ASW $F_2$ structure function are given in the Appendix). The ASW $F_2$ structure function for the proton case is plotted in Figs. \[figF2a\] and \[figF2b\] (for $x<0.01$ where this parameterization is expected to be valid). The ASW structure function $F_2$ contains mild nuclear corrections at low $x$ when compared with DRSS and BM nuclear corrections (see Fig. \[figNuc\]). Nuclear corrections in ASW depend on $Q^2$.
The photonuclear contributions to $b(E)$ computed (for standard rock $A=22$ throughout all this paper) with ALLM and with CKMT structure functions, and the same nuclear corrections [@Dutta2001], give very close results (see Fig. \[figloss\]). Although ALLM and CKMT parameterizations share a common theoretical base, with a reggeon and a pomeron component, and they are fitted to the same data sets, ALLM systematically lies above CKMT at low $x$ (see Fig. \[figF2b\]), which accounts for the difference in $b(E)$ observed in Fig. \[figloss\].
The lowest values of $b(E)$ at high energies is obtained with the ASW structure functions. Though the ASW structure function $F_2$ contains mild nuclear corrections at low $x$, saturation effects at the nucleon level are rather strong and limit the rise of $b(E)$ with energy as observed in Fig. \[figloss\]. For energies below $E=10^6$ GeV the result from the ASW structure function is higher than those from ALLM or CKMT (see Fig. \[figloss\]). This is because at low $Q^2$ the ASW structure function is significantly higher for the region $10^{-6}<x<10^{-3}$ (see Fig. \[figF2b\]) which is the relevant range for energies below $E=10^6$ GeV, as it can be deduced from Fig. \[figrelx\]. Thus the saturation-based ASW prediction lowers the energy loss rate $b(E)$ with respect to the already existing predictions by a factor 2 at $E=10^9$ GeV, and by a factor even larger at higher energies.
The BB/BS calculation gives the largest of the predicted energy loss rates up to energies of the order $E=10^7$ GeV. Above this scale the PT result exceeds all other existing predictions by at least a factor 2 already at $E=10^9$ GeV (i.e. a factor 4 with respect the ASW prediction, see Fig. \[figloss\]). Thus the PT prediction can be considered as an estimate of the upper limit of the tau energy loss at UHE. Much of the uncertainty in the tau energy loss is actually due to nuclear effects. The choice of nuclear corrections from Ref. [@Dutta2001], Ref. [@BM2002], or from Ref. [@ASW] (see Fig. \[figNuc\]), translates into differences in the calculated value of $b(E)$ (using the ALLM structure function) by a factor rising from 1.5 to 2.5 as the tau energy increases in the range $E=10^{6}$-$10^{9}$ GeV (see Fig. \[figlossnuc\]). This energy range corresponds to the region of very low $x$ where differences in the nuclear correction factor are large. In order to quantify how much different regions of $x$ and $Q^2$ contribute to $b(E)$, the dependence of $b(E)$ on the maximum value of $x$ and on the maximum value of $Q^2$ considered in the integration is shown in Figs. \[figrelx\] and \[figrelq2\].
The differential cross section $d\sigma^{\tau A}/dy$ is also a relevant quantity for high energy neutrino detection as it enters the event rate convolutions together with the neutrino flux and the experimental acceptances. Indeed it has been shown that stochastic effects of the tau energy loss distribution have significant relevance in the prediction of emerging tau rates [@Dutta2005]. The energy loss spectrum $y d\sigma^{\tau A}/dy$ obtained using both ALLM and ASW structure functions are compared in Fig. \[figdsig\]. Clearly, the energy loss spectrum calculated with ALLM is significantly harder than the one calculated with ASW. The contributions of moderate ($Q^2>1$) and low $Q^2$ ($Q^2<1$) (in a rough way corresponding respectively to hard and soft interactions) are shown separately in Fig. \[figdsig12\] for the ALLM structure function.
The charged current neutrino cross section
==========================================
The absolute value of the cross section naturally has a direct impact on the sensitivity of experiments because the event rate is directly proportional to it, but it also enters with opposite effect in the attenuation of the neutrino beam as a function of matter depth traversed, having much impact on the angular and energy distribution of the events. These two effects combine in the case of Earth-skimming tau neutrino interactions to play an important role for the rate calculation. In addition to the tau lepton photonuclear cross section we also study how the uncertainties in the $F_2$ structure function at low $x$ affect the CC neutrino deep inelastic cross section. Since in the more realistic expectations [@ReportNestor] the nuclear corrections to the CC neutrino-nucleon cross section decrease at low $x$ with increasing $Q^ 2$, becoming small at high $Q^ 2$ [@CastroPena], we will neglect them in our calculations.
The CC DIS neutrino-nucleon cross section is expressed in terms of the structure function $F_2$ as follows: $$\begin{aligned}
\frac{d\sigma_{CC}^{\nu N}}{d Q^2 dy} =
\frac{G_F^2}{4 \pi} \left(\frac{M_W^2}{M_W^2+Q^2}\right)^2
\frac{F_2^{\nu N}}{y} [ 1+(1-y)^2] \; ,\end{aligned}$$ where $E$ is the neutrino energy and $y$ the fraction of energy lost by the neutrino in the interaction. In this expression $F_L$ and $xF_3$ contributions are neglected since $F_L$ tends to zero as $Q^2$ rises and $xF_3$ deals basically with the valence partons which hardly contribute at the low $x$ values relevant for the cross section.
In order to consistenly use the structure functions from charged lepton interactions (as ALLM, CKMT, and ASW models) in neutrino interactions we must relate the electromagnetic and weak structure functions. The $F_2$ structure function for neutrino interaction is related to the $F_2$ structure function for charged lepton interactions by the ratio of the weak and electromagnetic couplings through $F_2^{\nu N}=18/5 \; F_2^{lN}$ (assuming a symmetric sea), although the kinematical regions of the two processes are different (low and moderate $Q^2\sim 0.01$-$10$ GeV$^2$ in the photonuclear case and high $Q^2 \sim M_W^2$ in the high energy CC interaction). Concerning the $x$ range the main contribution comes from low $x$ in both cases, though $x$ values are lower in the photonuclear case than in the CC interaction. For the calculation of the neutrino-nucleon cross section at high energies, we then use the structure function $F_2$ for charged lepton interaction valid up to very low $x$ and high $Q^2$, instead of following the standard approach based on parton densities.
The neutrino-nucleon cross sections from ALLM and CKMT structure functions are presented in Fig. \[fignuxsection1\]. They are clearly below predictions from modern parton densities [@Anchordoqui], since the ALLM parameterization is not consistent with high $Q^2$ experimental points (see Fig. \[figF2a\]) and CKMT is not evolved to high $Q^2$, so we have not used them to discuss the theoretical uncertainties in the estimation of the CC neutrino-nucleon cross section. Instead we have taken the parameterization of $F_2$ [*à*]{} $la$ BCDMS obtained by the SMC Collaboration [@SMC], which correctly represents the existing experimental data at high $Q^2$ (see Fig. \[figF2a\]) and provides a smooth connection at neutrino energies around $E=10^7$ GeV with the parton density prediction of the CC neutrino-nucleon cross section [@Anchordoqui] (see Fig. \[fignuxsection1\]).
We have performed three different extrapolations at low $x$ of the $F_2$ parameterization given in Ref. [@SMC], one following the ASW structure function, a second one from the phenomenological parameterization fitting low $x$ HERA data [@HERA], and the third one which corresponds to the double logarithmic approximation (DLA) in QCD from Ref. [@KOPA] (KOPA) (ASW and KOPA structure functions are valid at low $x$, $x<0.01$, i.e. at high energy).
In Fig. \[fignuxsection2\] the effect of taking the three different parameterizations of the structure function $F_2$ at low $x$ on the neutrino-nucleon cross section is shown. We see that in comparison with the prediction obtained with evolved QCD parton densities of Ref. [@Anchordoqui], both KOPA (which corresponds to the DLA of perturbative QCD) and ASW (which includes strong saturation effects) estimations are below at high energies.
On the other hand the extrapolation of the HERA based parameterization with the exponent $\lambda=0.0481 \ln(Q^2/0.292^2)$ ($F_2 \sim x^{-\lambda}$), produces an extremely fast increase of the cross section with energy (see the upper curve in Fig. \[fignuxsection2\]), since this exponent rises to values above $\lambda \sim 0.5$ when $Q^2$ becomes large. This raw extrapolation is in contradition with perturbative calculations and we do not consider it for uncertainty estimates as it is not physically motivated. Nevertheless it is considered here to explicitely show its discrepancy with pQCD. For the more realistic scenarios, when the rise of the exponent freezes to smaller values $\lambda < 0.4$, our prediction supports the result obtained in the detailed analysis of Ref. [@Anchordoqui]. The curves are shown in Fig. \[fignuxsection2\] (from up to down corresponding to $\lambda$ frozen to $\lambda=0.50$, $0.40$, and $0.38$ respectively). When considering only physically motivated extrapolations, the theoretical uncertainty at $E=10^{9}$ GeV is a factor 2.
Conclusions
===========
We estimate the uncertainties coming from the extrapolations of the existing models for proton and nucleus structure functions for tau energy loss and for CC neutrino-nucleon cross section. Both calculations must be done consistently within the same model as their effect on the tau flux produced by Earth-skimming neutrinos is correlated. The theoretical uncertainty in the tau energy loss is greater than that of the neutrino-nucleon CC cross section because the $Q^2$ region contributing to the tau energy loss cross section is lower and so are the relevant values of Bjorken-$x$. In addition the structure functions conventionally used for the calculation of the tau energy loss are not suitable to be used in the high-$Q^2$ range which is relevant for the CC neutrino-nucleon interaction. As a result systematic effects arising in the calculation of a tau neutrino bound from Earth-skimming events due to uncertainties in the structure functions turn out to be difficult to evaluate. Several extreme models allowed by extrapolation of structure functions have been explored in order to estimate ranges for these quantities.
Below energies in the $E=10^7$ GeV range the highest prediction for the photonuclear contribution to tau energy loss, $b(E)$, is provided by the BB/BS calculation. Above this energy range the PT result exceeds all other considered predictions while the lowest calculation is obtained using the ASW structure functions. The difference between the two extreme predictions reaches a factor 4 at $E=10^9$ GeV and increases as the energy rises. The BB/BS, ALLM, and CKMT calculations agree within a 30 $\%$ and go approximately parallel for all energies, which is an indication of a systematic normalization difference of the structure functions in each model. The application of much stronger nuclear shadowing (than usually considered) at low $x$ can lower the prediction of $b(E)$ with respect to the already existing calculations by a factor up to 2 at $E=10^9$ GeV.
In the case of the CC neutrino-nucleon cross section the importance of nuclear effects at high energies is expected to be small [@CastroPena]. We have also considered saturation effects in the CC neutrino-nucleon cross section by using the structure function ASW. At $E=10^{10}$ GeV, the CC neutrino-nucleon cross section calculated with the ASW structure function is found to be half of the pQCD calculation with parton densities, in rough agreement with the evaluation of saturation effects reported in Ref. [@Kutak_Kwiecinski], and also in Ref. [@Henley:2005ms] where different parameterizations of the dipole cross section containing saturation are employed. The calculation of the neutrino cross section with the ASW structure function has also been performed in Ref. [@Machado:2005af]. Though the quantitative agreement of our result with this calculation is reasonably good, some discrepancy appears due to the fact that to extend the validity of ASW to the region $x>0.01$, we have connected the ASW structure function to the parameterization of HERA data [*à*]{} $la$ BCDMS from Ref. [@SMC].
The effect of a rapid rise of the $F_2$ structure function at low $x$ in the CC neutrino-nucleon cross section has also been studied using the $x$-slope $\lambda(Q^2)$ of the $F_2$ HERA data ($F_2 \sim x^{-\lambda}$) for all $Q^2$ values. We have found that the cross section rises with energy very rapidly. At $E=10^{10}$ GeV it can become a factor 4 above the pQCD calculation with parton densities. On the other hand the logarithmic rise of the structure function $F_2$ at small $x$ predicted by the DLA-pQCD results in a slower increase of the CC neutrino-nucleon cross section with energy. At $E=10^{10}$ GeV the DLA estimation is a 20 $\%$ below the pQCD calculation with parton densities. When considering only realistic extrapolations, the theoretical uncertainty at $E=10^{9}$ GeV is a factor 2.
The obtained uncertainty for the tau energy loss is to be implemented, together with the corresponding one for the CC neutrino-nucleon cross section, both in analytical and Monte Carlo calculations of the rates of taus emerging from Earth-skimming tau neutrinos, which is currently being used to calculate high energy neutrino bounds. This task is beyond the scope of this paper.
We thank O. Blanch Bigas, M.V.T. Machado, D. Pertermann, Yu.M. Shabelski, and D.A. Timashkov for useful comments on this work. N.A. acknowledges financial support by Ministerio de Educación y Ciencia (MEC) of Spain under a Ramón y Cajal contract. This work has been supported by MEC under grants FPA2005-01963 and FPA2004-01198, by Xunta de Galicia under grant 2005 PXIC20604PN and Consellería de Educación, and by FEDER Funds.
The form of the single universal curve where all data on $\sigma^{\gamma^* p}$ and on $\sigma^{\gamma^* A}$ lie as function of the scaling variable $\tau=Q^2/Q^2_{sat}$ is motivated by saturation and given by [@ASW; @Albacete:2005ef]: $$\sigma^{\gamma^* p}(x,Q^2) \equiv
\Phi(\tau) =
\bar\sigma_0
\left[ \gamma_E + \Gamma\left(0,\xi\right) +
\ln\xi \right]\, ,
\label{eqscalf}$$ with $\gamma_E$ the Euler constant, $\Gamma\left(0,\xi\right)$ the incomplete $\Gamma$ function, and $\xi=a/\tau^b$, with $a=1.868$ and $b=0.746$ extracted from a fit to lepton-proton data. The saturation scale ${Q_{\rm sat}}^2$ is parameterized as ${Q_{\rm sat}}^2$(GeV$^{2}$) $=(\bar{x}/x_0)^{-\lambda}$ [@Golec-Biernat:1998js], where $x_0= 3.04\cdot 10^{-4}$, $\lambda=0.288$, and $\bar
x=x\,(Q^2+4m_f^2)/Q^2$ with $m_f=0.14$ GeV. The normalization is fixed by $\bar\sigma_0=40.56$ $\mu$b.
The extension to the nuclear case is done through $$\sigma^{\gamma^*A}=\frac{\pi R_A^2}{\pi R_p^2}
\sigma^{\gamma^*p}(\tau_A)
\label{eqnormal}$$ and $$Q_{\rm sat,A}^2=Q_{\rm sat,p}^2\left(\frac{A \pi R_p^2}
{ \pi R_A^2}\right)^\frac{1}{\delta} \hspace{-0.1cm}
\Rightarrow
\tau_A=\tau \; \left[\frac{\pi R_A^2}{A \pi R_p^2}\right]^\frac{1}{\delta},
\label{eqtaua}$$ where the nuclear radius is given by the usual parameterization $R_A=(1.12 A^{1/3}-0.86 A^{-1/3})$ fm, and $\delta=0.79\pm0.02$ and $\pi
R_p^2=1.55 \pm 0.02$ fm$^2$ are extracted from a fit to lepton-nucleus data. The nuclear structure function $F_2^A$ is $F_2^A(x,Q^2)=Q^2 \sigma^{\gamma^*A}/(4\pi^2\alpha)$. The ASW structure function for the proton case is recovered by taking $A=1$ in the expressions above (see Figs. \[figF2a\] and \[figF2b\]).
The functional shape of (\[eqscalf\]) is motivated by considerations in saturation physics [@ASW; @Albacete:2005ef]. From a pragmatic point of view, it provides a very good description of existing lepton-proton and lepton-nucleus data in the region $0.01<\tau,\tau_A<100$ and $x<0.01$ which for $Q^2=0.01, 0.1, 1$, and $10$ GeV$^2$ corresponds to a low $x$ limit of $\sim 10^{-5}$, $10^{-7}$, $10^{-10}$, and $10^{-13}$, respectively. For $\tau\to 0$ $F_2$ behaves like a single logarithm, so $F_2\propto \ln{1/x}$ for $x\to 0$ and $F_2/A\propto
\ln{A}/A^{1/3}$ for $A\to \infty$. Thus this model results in very large screening corrections for asymptotic values of $x$ and $A$.
[99]{}
T. K. Gaisser, F. Halzen and T. Stanev, Phys. Rept. 258 (1995) 173 \[Erratum-ibid. 271 (1996) 355\]. D. Fargion, Astrophys. J. 570 (2002) 909. X. Bertou, P. Billoir, O. Deligny, C. Lachaud, and A. Letessier-Selvon, Astropart. Phys. 17 (2002) 183. E. Zas, New J. Phys. 7 (2005) 130. O. Blanch Bigas (Pierre Auger Collaboration), Proceedings of the 30th ICRC (2007), to be published. J. Alvarez-Muñiz (Pierre Auger Collaboration), Proceedings of the 30th ICRC (2007), to be published. S.I. Dutta, M.H. Reno, I. Sarcevic, and D. Seckel, Phys. Rev. D63 (2001) 094020. E.V. Bugaev, T. Montaruli, Yu.V. Shlepin, and I. Solkalski, Astropart. Phys. 21 (2004) 491. C. Aramo [*et al.*]{}, Astropart. Phys. 23 (2005) 65. N. Armesto, C. Merino, G. Parente, and E. Zas, Proceedings of the 30th ICRC (2007), to be published. M. Arneodo, Phys. Rep. 240 (1994) 301; D. F. Geesaman, K. Saito, and A. W. Thomas, Ann. Rev. Nucl. Part. Sci. 45 (1995) 337. QCD Perspectives on Hot and Dense Matter, edited by J.-P. Blaizot and E. Iancu, Kluwer, Dordrecht, The Netherlands, 2002 (NATO Science Series, II, Mathematics, Physics, and Chemistry, Vol. 87). N. Armesto, C. Salgado, and U.A. Wiedemann, Phys. Rev. Lett. 94 (2005) 022002. A.V. Butkevich and S.P.Mikheyev, Zh. Eksp. Teor. Fiz 122 (2002) 17. L.B. Bezrukov and E.V. Bugaev, Yad. Fiz. 33 (1981) 1195. E.V. Bugaev and Yu.V. Shlepin, Phys. Rev. D67 (2003) 934027. P. Lipari and T. Stanev, Phys. Rev. D 44 (1991) 3543. K.S. Kuzmin, K.S. Lokhtin, and S.I. Sinegovsky, Int. J. Mod. Phys. A20 (2005) 6956; A.A. Kochanov, K.S. Lokhtin, and S.I. Sinegovsky, arXiv:hep-ph/0508306; K.S. Lokhtin and S.I. Sinegovsky, Russ. Phys. J. 49 (2006) 326 \[Izv. Vuz. Fiz. 49 (2006) 82\]. D.A. Timashkov and A.A. Petrukhin, Proceedings of the 29th ICRC (2005) 9, 89-92. S.I. Dutta, Y. Huang, and M.H. Reno, Phys. Rev. D72 (2005) 013005. H. Abramowicz and A. Levy, arXiv:hep-ph/9712415. A. Capella, A. Kaidalov, C. Merino, and J. Tran Thanh Van, Phys. Lett. B337 (1994) 358; A. Kaidalov, C. Merino, and D. Pertermann, Eur. Phys. J. C20 (2001) 301. A.A. Petrukhin and D.A. Timashkov, Yad. Fiz. 67 (2004) 2241 \[Phys. At. Nucl. 67 (2004) 2216\]. C. Adloff [*et al.*]{} (H1 Collaboration), Eur. Phys. J. C21 (2001) 33. J. Breitweg [*et al.*]{} (ZEUS Collaboration), Phys. Lett. B487 (2000) 53. A.M. Stasto, K. Golec-Biernat, and J. Kwiecinski, Phys. Rev. Lett. 86 (2001) 596. N. Armesto, J. Phys. G32 (2006) R367. J.A. Castro Pena, G. Parente, and E. Zas, Phys. Lett. B507 (2001) 231. L.A. Anchordoqui, A.M. Cooper-Sarkar, D. Hooper, and S. Sarkar, Phys. Rev. D74 (2006) 043008. B. Adeva [*et al.*]{} (Spin Muon Collaboration), Phys. Rev. D58 (1998) 112001. C. Adloff [*et al.*]{} (H1 Collaboration), Phys. Lett. B520 (2001) 183. A.V. Kotikov and G. Parente, Nucl. Phys. B549 (1999) 242. K. Kutak and J. Kwiecinski, Eur. Phys. J. C29 (2003) 521. E. M. Henley and J. Jalilian-Marian, Phys. Rev. D73 (2006) 094004. M. V. T. Machado, Phys. Rev. D71 (2005) 114009. J.L. Albacete, N. Armesto, J.G. Milhano, C.A. Salgado, and U.A. Wiedemann, Eur. Phys. J. C 43 (2005) 353. K. Golec-Biernat and M. Wüsthoff, Phys. Rev. D 59 (1999) 014017.
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[^1]: nestor@fpaxp1.usc.es
[^2]: merino@fpaxp1.usc.es
[^3]: gonzalo@fpaxp1.usc.es
[^4]: zas@fpaxp1.usc.es
|
---
abstract: 'Single-photon detection is an essential component in many experiments in quantum optics, but remains challenging in the microwave domain. We realize a quantum non-demolition detector for propagating microwave photons and characterize its performance using a single-photon source. To this aim we implement a cavity-assisted conditional phase gate between the incoming photon and a superconducting artificial atom. By reading out the state of this atom in single shot, we reach an internal photon detection fidelity of 71%, limited by the coherence properties of the qubit. By characterizing the coherence and average number of photons in the field reflected off the detector, we demonstrate its quantum non-demolition nature. We envisage applications in generating heralded remote entanglement between qubits and for realizing logic gates between propagating microwave photons.'
author:
- 'Jean-Claude Besse'
- Simone Gasparinetti
- 'Michele C. Collodo'
- Theo Walter
- Philipp Kurpiers
- Marek Pechal
- Christopher Eichler
- Andreas Wallraff
title: 'Single-Shot Quantum Non-Demolition Detection of Itinerant Microwave Photons'
---
[^1]
Single-photon detectors [@Hadfield2009] for itinerant fields are a key element in remote entanglement protocols [@Chou2005], in linear optics quantum computation [@Kok2007; @Knill2001], and in general to characterize correlation properties of radiation fields [@Glauber1963b]. While such detectors are well established at optical frequencies, their microwave equivalents are still under development, partly due to the much lower photon energy in this frequency band [@Gu2017]. At microwave frequencies, itinerant fields are typically recorded with linear detection schemes, analogous to optical homodyne detection [@Eichler2012]. Such detection can now be realized with high efficiency by employing near quantum limited parametric amplifiers [@Castellanos2007] and furthermore allows for a full tomographic characterization of radiation fields [@Bozyigit2011]. However, protocols such as entanglement heralding require the intrinsic nonlinearity of a single-photon detector in order to yield high purity states despite losses between the source and the detector. Such a component has therefore raised interest in the community, leading to a variety of theoretical proposals [@Romero2009; @Helmer2009b; @Peropadre2011; @Witthaut2012; @Manzoni2014; @Fan2014a; @Sathyamoorthy2014; @Koshino2015; @Sathyamoorthy2016b; @Kyriienko2016; @Koshino2016a] as well as initial experimental demonstrations in the last decade [@Gleyzes2007; @Johnson2010; @Leek2010; @Chen2011a; @Inomata2016; @Narla2016; @Kono2017a].
The first photodetection experiment in the superconducting regime that did not require photons to be stored in high-quality factor cavities [@Gleyzes2007; @Johnson2010; @Leek2010] was based on current biased Josphson junctions [@Chen2011a], but was destructive and involved a long dead time. Later, systems involving absorption into artificial atoms (and thus destruction) of traveling photons [@Inomata2016; @Narla2016] were implemented. Very recently single photons have been heralded from a non-demolition detection scheme based on a photon-qubit entangling gate, similar in spirit to this manuscript, using a strong dispersive shift in a 3D cavity [@Kono2017a].
Here we demonstrate single-shot quantum non-demolition (QND) detection of itinerant single photons in the microwave domain, based on a cavity-assisted controlled phase gate between an artificial atom and a propagating photon [@Duan2004]. We show the unconditional detection of an itinerant photonic wave packet containing a Fock state at the single photon level.
Our setup consists of a transmon-type superconducting artificial atom coupled to two resonators acting as single-mode cavities, see Fig. \[fig:jaynes\_phase\]a for a sketch and Suppl. Mat. for details. Both resonators are coupled to a Purcell filter [@Reed2010] to protect the qubit from decay into the output lines. We tune the first to second-excited state transition of the transmon, $\omega_\mathrm{ef}/(2\pi)=6135$ MHz, into resonance with the *detector* resonator. The transmon anharmonicity is $\alpha/(2\pi)=-340$ MHz, such that its ground to first-excited state transition is at $\omega_\mathrm{ge}/(2\pi)=6475$ MHz. The transmon is coupled to the detector resonator with rate $g_\mathrm{0}/(2\pi)=40$ MHz and the latter has an effective linewidth $\kappa/(2\pi)=19$ MHz. The resulting level diagram is displayed in Fig. \[fig:jaynes\_phase\]b,c. With the transmon in the ground state (panel b), photons impinging on the detector at the resonator frequency acquire a phase $\varphi_g=\pi$ as they are reflected. By contrast, with the transmon in the first excited state (panel c), photons of the same frequency are reflected without interacting with the cavity, and thus acquire no phase ($\varphi_e=0$). The *readout* resonator, at $\omega_\mathrm{ro}/(2\pi)=4800$ MHz, is used to perform high-fidelity, dispersive single-shot readout of the qubit state [@Walter2017]. We connect the input of our detector with the output to a single photon source embedded in an on-chip switch, see Ref. [@Pechal2016]. The single photon source is a transmon strongly coupled to its output port with an emission linewidth of $1.77~\rm{MHz}$. The switch is based on a combination of hybrid couplers and tunable resonators. It enables toggling between its two inputs, supplying either a coherent tone from a conventional microwave generator or a single-photon wave packet emitted by the source-qubit to the input of the detector.
Our protocol for single photon detection (Fig. \[fig:jaynes\_phase\]d) begins with a measurement of the transmon state, in order to reject those instances in which the qubit is found to be thermally excited (6% of the total traces were discarded, see Suppl. Mat. for details). We then prepare the transmon in the superposition state $(|g\rangle+|e\rangle)/\sqrt{2}$ with a $\pi/2$ pulse. This defines the time $t=0$, at which the detection window of length $T_\mathrm{w}$ begins. 20 ns later we emit a photon wave packet in the state $|\gamma\rangle = \cos(\theta/2) \left| 0 \right\rangle + \sin(\theta/2) \left| 1 \right\rangle$, a coherent superposition of vacuum and a single-photon Fock state, with a relative weight set by the preparation angle $\theta$ of the Rabi pulse applied to the source-qubit. At time $t=T_\mathrm{w}$, we apply a $-\pi/2$ pulse to the transmon, effectively completing a Ramsey sequence, and immediately measure the qubit state.
We first characterize the response of the detector by considering the phases $\varphi_\mathrm{e,g}$ acquired by a weak coherent tone reflected off the detector input, dependent on the state of the transmon (Fig. \[fig:detection\_pulsescheme\]a,b). We measure the difference $\delta\varphi=\varphi_\mathrm{g}-\varphi_\mathrm{e}$ by pulsed spectrosopy and find $\delta\varphi=\pi$ at the cavity frequency $\omega_\mathrm{ef}$, as well as at the dressed-state frequencies $\omega_\mathrm{ef}\pm \sqrt{2}g_\mathrm{0}$. In these configurations, a controlled-phase gate is realized between the qubit and a propagating photon. For a definite phase to be acquired by the photon, its spectral bandwidth is required to be smaller than the detector cavity linewidth, $\kappa$. With that condition fulfilled, the gate is not dependent on the temporal shape of the photon.
![\[fig:jaynes\_phase\] Principle of quantum non-demolition single-photon detection. (a) Sketch of the setup connecting a source of single photons or a coherent source via an on-chip switch and a circulator to the input of the detector. (b,c) Energy-level diagram of the atom-cavity system when the atom is either in the ground (b) or in the excited state (c), contrasting the harmonic ladder of a bare cavity in (b) to the Jaynes-Cummings anharmonic ladder in (c). (d) The pulse scheme consisting of a $\pi/2$ and $-\pi/2$ pulses on the detection-qubit, defining the length of the time window $T_\mathrm{w}$, as well as a pulse on the source-qubit of Rabi amplitude $\theta$. The emitted photon lineshape is sketched in red. A readout pulse is used to measure the state of the detection-qubit at the end of the protocol, as well as to preselect the single shot traces to discard thermal population. ](fig_jaynes_phase6.pdf)
![\[fig:detection\_pulsescheme\] (a) Expected phase $\varphi_\mathrm{g}$ ($\varphi_\mathrm{e}$) of a weak coherent signal upon reflection off of the cavity-atom system, when the detection-qubit is in $|g\rangle$ (green) ($|e\rangle$ (red)). (b) Phase difference $\delta\varphi$ measured in pulsed spectroscopy (blue dots), with model (solid line). The linewidth $\kappa$ of the bare cavity is indicated by the shaded area.](fig_detection_pulsescheme2.pdf)
The Ramsey sequence displayed in Fig. \[fig:jaynes\_phase\]d realizes a phase gate on the detection-qubit controlled by the presence of the photon. This process implements the photon detection. To test the fidelity of this protocol with single photons at the input, we measure the average excited state population of the detection-qubit as a function of the preparation angle of the photon source $\theta$ (Fig. \[fig:timewindow\]a, $T_\mathrm{w}=250$ ns). The scaling factor between pulse amplitude and preparation angle for the source is independently calibrated in a Rabi experiment. The data follows a sine-squared dependence, corresponding to the average photon number prepared by the source, with a reduced visibility characterized by the probability $P(e|1)=65.8\%$ of measuring the detection-qubit in the excited state when a photon is emitted and the probability $P(e|0)=5.9\%$ of measuring the detection-qubit in the excited state without emitting a photon. In the context of photon detection we refer to $P(e|1)$ as the detection efficiency and $P(g|0)$ as the dark count probability. As a performance metric we define the detection fidelity $F=1-P(g|1)-P(e|0)=P(e|1)-P(e|0)=59.9\%$ as the difference between detection efficiency and dark count probability.
To gain insight into the sources of errors in the protocol, we extract the detection efficiency and dark count probability vs. the length of detection time window, $T_\mathrm{w}$ (Fig. \[fig:timewindow\]b).
![\[fig:timewindow\] (a) Measured (averaged readout) excited state population of the detection-qubit (green dots) as a function of the preparation angle $\theta$ of the photon state $|\gamma\rangle$, for $T_\mathrm{w}=250$ ns. The red line is a fit to the expected dependence, with the detection efficiency $P(e|1)=65.8\%$ (orange) and the dark count probability $P(e|0)=5.9\%$ (blue line) as parameters. The fidelity $F=59.9\%$ is indicated. (b) Dark count probability $P(e|0)$ (blue), photon detection efficiency $P(e|1)$ (yellow), (c) their ratio $P(e|1)/P(e|0)$ (red), and the fidelity $F$ (black) vs. length of the detection time window $T_\mathrm{w}$. Theory lines take into account finite lifetime, losses and photon lineshape (see main text). The length of detection time window $T_\mathrm{w}=250$ ns is used in the remainder of the paper (vertical gray line).](fig_timewindow-ratio3.pdf)
The detection efficiency peaks at an optimal length of $T_\mathrm{w}\simeq 300$ ns, while the dark count probability monotonically increases. A first source of errors is the limited coherence $T_2^*=1.8\,\mu$s of the detection-qubit in a Ramsey experiment. A second type of error is due to loss in the components which connect the source to the detector. The measured total loss in the switch, the coaxial cables and the circulator was found to be approximately 25% (see Suppl. Mat. for calibration measurements using the nonlinear response of the source and the detector). As a result, approximately 75% of the photons emitted by the source reach the detector, leading to an overall scaling factor independent of $T_\mathrm{w}$. Finally, for short detection windows, part of the photon envelope is cut off by terminating the protocol with the $-\pi/2$ pulse, limiting the detection efficiency. The trends in Fig. \[fig:timewindow\]b (solid lines) are quantitatively explained by three sources of errors which we have characterized independently. For entanglement distribution and other heralded experiments, the ratio $P(e|1)/P(e|0)$ between detection efficiency and dark count probability directly relates to the error rate. We report this ratio, together with the fidelity $F$ vs. the length of detection time window $T_\mathrm{w}$, in Fig. \[fig:timewindow\]c. While the fidelity peaks at around the same $T_\mathrm{w}$ as the detection efficiency, the ratio still improves for shorter $T_\mathrm{w}$ as the dark count probability approaches zero. In our case, this ratio reaches up to 16, with significant variations at short $T_\mathrm{w}$ attributed to fluctuations in the low dark count probability.
The fidelity extracted from Fig. \[fig:timewindow\]a refers to an averaged readout. When performing single-shot readout in $100$ ns (see Suppl. Mat.), we find that the total fidelity of detecting single photons for $T_\mathrm{w}=250$ ns is $F=50\%$. This value agrees with the one obtained from the averaged measurements after taking into account the measured 92% readout fidelity, mainly limited by the transmon decay time $T_1=3.0~\mu$s. The infidelity is due to imperfect detection efficiency, $P(g|1)=37\%$, and dark count probability, $P(e|0)=13\%$. After accounting for the calibrated losses before the detector, the internal probability to miss a photon is $P_\mathrm{in}(g|1)=16\%$ (corresponding to a quantum detection efficiency of 0.84), so our detector has an internal fidelity of $F_\mathrm{in}=1-P_\mathrm{in}(g|1)-P(e|0)=71\%$.
To test the quantum non-demolition nature of the photon detection, we employ a linear amplification chain to measure the average photon number and coherence of the radiation field reflected off the detector. We consider two states of the detector. The “ON” state describes the operation reported up to here. In the “OFF” state, we keep the atom in the ground state and detune its frequency to be far off-resonant from the cavity. In both cases, the source emits the same radiation field. We monitor the ensemble averaged photon number ($\langle a^\dagger a \rangle$ moment) and amplitude ($\text{Re}(\langle a \rangle)$ moment, in the optimized quadrature) by integrating the time traces with a filter matched to the temporal shape of the photon [@Eichler2011]. The results are reported in Fig. \[fig:QND\], together with a model. We scale the axes globally by the separately calibrated loss.
![\[fig:QND\] Non-demolition character of the measurement. (a) Power, reported as photon number $\langle a^\dagger a\rangle$ is conserved while (b) phase, measured as optimized quadrature $\mathrm{Re}(\langle a \rangle)$, is erased. Solid lines are the expected response of a QND detector.](fig_QND.pdf)
Up to our measurment accuracy of 2%, we observe no difference in power whether the detection pulse sequence is executed or not (Fig. \[fig:QND\]a). Accordingly, we conclude that we are performing a quantum non-demolition measurement. However, the phase of the incoming photon state $|\gamma\rangle$ is randomized (Fig. \[fig:QND\]b), as quantum mechanics imposes for the conjugate variables $\{n,\phi\}$. We measure a small remaining coherence offset in the ON measurement, even without emitting any photons ($\theta=0$). We ascribe this offset to unintended driving of the $e$-$f$ transition by the first Ramsey pulse, resulting in the subsequent emission of phase-coherent radiation at the frequency $\omega_\mathrm{ef}$.
The phase difference of $\delta\varphi=\pi$ occurs for any Fock state with $n>0$ at the cavity frequency, such that we expect our detector to click for any wave packet containing $n>0$ photons. Detecting photons at one of the dressed state frequencies in the $n$^th^ manifold, and taking advantage of the photon-blockade effect generated by the Jaynes-Cummings ladder could yield an operation mode that projectively selects the Fock state with $n$ photons. This could be useful in entanglement schemes where a particular Bell state is associated to a definite photon number $n$.
We note that in principle, the protocol can be run continuously with a dead time on the order of the single shot readout time of 100 ns by using the readout result as the initial state for the next iteration. One does not need to perform active feedback, nor discard the results of an initially excited atom, but instead could simply invert the association between the measured qubit state and the presence of a photon.
A clicking detector for itinerant photons in the microwave regime, independent of their temporal shape and with internal fidelity limited by qubit coherence, adds to the circuit QED toolbox for characterization of propagating quantum radiation fields. We have demonstrated single photon detection with radiation fields at the quantum level, composed of a superposition of vacuum and an $n=1$ photon Fock state. Our device does not internally lose photons upon detection and is built with separate detection and readout lines, which provides easy access to the reflected radiation field. This allows to take advantage of the non-demolition nature of the detector and use the device as a mediator of photon-photon interactions for all photonic quantum computation [@Hacker2016b; @Kokkoniemi2017; @Koshino2016; @Wang2016a; @Hua2015]. Other applications include heralded entanglement [@Narla2016] with high rate without the need to shape the photons, or to perform Bell state analysis [@Witthaut2012].
Acknowledgments
===============
The authors thank Alexandre Blais, Christian Kraglund Andersen and Paul Magnard for useful discussions. This work is supported by the European Research Council (ERC) through the “Superconducting Quantum Networks” (SuperQuNet) project, by the National Centre of Competence in Research “Quantum Science and Technology” (NCCR QSIT), a research instrument of the Swiss National Science Foundation (SNSF), and by ETH Zurich.
Supplemental Material
=====================
Sample fabrication and cabling
------------------------------
The sample, shown in Fig. \[fig:samples\], is fabricated on a 4.3 mm x 7 mm Sapphire chip cut along c-plane. All elements except for the qubit are fabricated from a 150 nm-thick sputtered niobium film using photolithography and reactive ion etching. The transmon’s islands and Josephson junctions are fabricated in a second step using electron-beam lithography and shadow-evaporation of aluminum in an electron-beam evaporator. Both the photon detection device and the single photon source embedded in an on-chip switch [@Pechal2016suppl] are mounted at the base temperature stage (20 mK) of a dilution refrigerator, as shown in the wiring diagram in Fig. \[fig:setup\].
![\[fig:samples\] False color micrograph of the detector sample. A transmon qubit (red) is coupled to the detection cavity (green) and its Purcell filter (light blue), as well as to a readout cavity (purple) and an according Purcell filter (brown). A charge line (dark blue) allows for driving the qubit and a weakly coupled input port (orange) allows for transmission measurements through the readout resonator. Sapphire is shown in dark and Niobium in light gray.](samples.png){width="\columnwidth"}
![\[fig:setup\] Scheme of the experimental set-up. All microwave lines are inner/outer DC-blocked at the fridge flange. Source and switch are physically on the same sample. DC cabling for applying external magnetic field with two coils on the switch sample holder and one coil at the detector are not shown.](fig_setup2){width="\columnwidth"}
Calibration of photon loss
--------------------------
We distinguish internal detector inefficiencies from photon losses by measuring the attenuation constant between the single photon source and the detector. This measurement relies on operating both devices as calibrated power sources and comparing the relative power levels at room temperature.
First, we employ the photon-blockade effect of the photon source-qubit to realize a calibrated power source. We continuously drive the qubit at its transition frequency $\omega_\mathrm{ge}$ and measure the power spectral density of the inelastically scattered radiation emitted into the output port (Fig. \[fig:Mollow\]). In the limit of large drive rate $\Omega > \Gamma$ the measured spectrum features characteristic satellite peaks at detunings $\delta\approx\pm\Omega$ relative to the drive frequency [@Lang2011suppl]. This nonlinear property of the spectrum allows one to calibrate the emitted power $P_\mathrm{s}=n_\mathrm{q}\Gamma\hbar\omega_{ge}$ from a global fit of the Mollow triplets. Here, $\Gamma/(2\pi)=1.77$ MHz is the source-qubit linewidth, and $n_\mathrm{q}$ the steady state average excited state population of the qubit. We note that in the limit of large drive rate $\Omega \gg \Gamma$, the qubit is driven into a mixed state with $n_\mathrm{q}\approx1/2$. Based on this fit we obtain the ratio $G_\mathrm{s}$ of the power detected at room temperature and the power $P_\mathrm{s}$ emitted from the source. $G_\mathrm{s}=(1-L)G_d$ is composed of the photon loss $L$ from the source to the detector and the effective gain $G_d$ from the output of the photon detector to the room temperature electronics.
![\[fig:Mollow\] Measured power spectral density (PSD) of the inelastic scattering of a coherent tone resonant with the source-qubit (symbols) for various drive rates $\Omega$ in units of the linewidth $\Gamma$. Solid lines are fits to the data. Individual data sets are offset by 0.5 photons s^-1^Hz^-1^ for clarity.](fig_Mollow)
To calibrate the gain $G_d$, we operate the photon detector as a calibrated power source. For this purpose we tune the detection-qubit to its sweet spot, detuned from the detector resonator by $\Delta/(2\pi)=(\omega_\mathrm{cav}-\omega_\mathrm{ge})/(2\pi)=-676$ MHz, and populate the detector resonator using a coherent tone applied through the second port of the switch. We measure the power-dependent qubit frequency $\omega_\mathrm{q}$, which decreases linearly with applied power $P_\mathrm{in}$ (Fig. \[fig:ACStark\]). This is due to the AC Stark shift $\Delta_\mathrm{q}=\omega_\mathrm{q}-\omega_\mathrm{q}^0=2\chi n_p$ [@Schuster2005suppl]. Independently, we infer the dispersive shift $\chi/(2\pi)=\alpha g^2/(\Delta (\Delta-\alpha))/(2\pi)=-2.4$ MHz from spectroscopic measurements of the qubit anharmonicity $\alpha/(2\pi)=-340$ MHz, and the resonant qubit-cavity coupling $g_\mathrm{0}/(2\pi)=40$ MHz. This yields a calibration for the number of photons $n_\mathrm{p}$ in the detector resonator.
![\[fig:ACStark\] Stark-shifted frequency $\nu_\mathrm{q}$ of the detection-qubit (blue dots) as a function of the input power $P_\mathrm{in}$ at the generator (bottom axis), as well as the inferred photon number $n_\mathrm{p}$ in the cavity (top axis). The red curve is a linear fit to the data.](fig_ACStark)
Knowing the effective linewidth $\kappa$ of the cavity we extract the expected photon power $P_\mathrm{d}=\kappa n_\mathrm{p} \hbar \omega_\mathrm{cav}$ at the output of the detector. A comparison with the power measured at room temperature yields the effective gain $G_d$ of the amplification chain. The loss between the source-qubit and the detector resonator is thus estimated as $L_\mathrm{s-d}=1-G_\mathrm{s}/G_\mathrm{d}=0.25$.
We attribute this loss to the following main contributions. First, the circulator placed between the two chips has an insertion loss specified by the manufacturer to be 8%. Second, the finite directivity of the single-pole, double-throw switch [@Pechal2016suppl] contributes to effective losses by routing about 5% of the power to its second output, terminated by a 50 $\Omega$ load at base temperature. Third, the SMP connectors and female-female bullets used to couple the radiation from a printed circuit board (PCB) to a microwave cable also contribute to a manufacturer-specified insertion loss of 5%. Finally, the attenuation in 50 cm of CC85Cu cables connecting the two samples via the circulator amounts to approximately 2% loss [@Kurpiers2017suppl]. The total identified sources of loss add up to approximately 20%. The difference relative to the loss of 25%, extracted from the power detection measurements, could be due to factors not accounted for such as impedance mismatches or internal losses of components along the detection path, in particular at wirebonds between samples and PCBs.
Single-shot readout and detection
---------------------------------
Each experimental sequence starts with a measurement pulse, used to reject approximately 6% of all measured traces (Fig. \[fig:preselect\]), in which the qubit was initially found in the excited state. Such instances are due to residual excitations from the previous run and thermal excitations. To realize this pre-selection measurement we perform single-shot readout with the methods described in Ref. [@Walter2017suppl]. After integrating the signal we obtain an integrated quadrature amplitude $q$ for each realization which we compare to a conservatively chosen threshold value to herald the qubit ground state.
![\[fig:preselect\] Histograms of the integrated quadrature amplitude $q$ of the initial pre-selection measurement performed to reject initially excited qubit states. The solid line is a fit to a Gaussian. The dashed gray line indicates the chosen threshold. The 6% of measurement traces above the threshold have been discarded.](fig_preselect)
To characterize the qubit readout fidelity, we prepare 12,500 repetitions of each of the detection-qubit ground $|g\rangle$ and first excited states $|e\rangle$. We perform readout in 100 ns with a gated measurement pulse, obtaining the histograms shown in Fig. \[fig:shots\]a. A readout fidelity of $(91.5\pm0.3)\%$ is extracted. The errors are composed of $P(g|e)=(6.3\pm0.2)\%$ and $P(e|g)=(2.2\pm0.1)\%$. The overlap error is below 0.2%.
![\[fig:shots\] Single shots measurements. (a) Histogram of integrated quadrature amplitude $q$ for preparing the state $|g\rangle$ (blue) or $|e\rangle$ (red), as well as (b) for preparing the photon state $|0\rangle$ (blue) or $|1\rangle$ (red). In each panel the solid lines are fits to a double-Gaussian model whose individual components are indicated by dashed lines. The green areas depict the overlap error. The dashed gray line indicates the qubit-state threshold.](fig_shots3)
To obtain the single-shot photon detection fidelity, we carry out the same readout procedure after 12,500 realizations of each emitting a single photon ($|\gamma\rangle=|1\rangle$) or not ($|\gamma\rangle=|0\rangle$), in both cases performing the detection protocol. The histograms are reported in Fig. \[fig:shots\]b. This corresponds to the single-shot single photon detection fidelity of $F=(49.6\pm0.5)\%$ reported in the main text. The infidelity is due to the finite detection efficiency, $P(g|1)=(37.0\pm0.4)\%$, dominated by losses between the source and the detector, and the dark count probability, $P(e|0)=(13.4\pm0.2)\%$, dominated by the detection-qubit decoherence during the length of detection window $T_\mathrm{w}$.
[^1]: Current address: Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
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Astro2020 Science White Paper
On the Origin of the Initial Mass Function
**Thematic Areas:** $\square$ Planetary Systems ${\rlap{$\square$}{\raisebox{2pt}{\large\hspace{1pt}{\ding{51}}}}\hspace{-2.5pt}}$ Star and Planet Formation $\square$ Formation and Evolution of Compact Objects $\square$ Cosmology and Fundamental Physics ${\rlap{$\square$}{\raisebox{2pt}{\large\hspace{1pt}{\ding{51}}}}\hspace{-2.5pt}}$ Stars and Stellar Evolution ${\rlap{$\square$}{\raisebox{2pt}{\large\hspace{1pt}{\ding{51}}}}\hspace{-2.5pt}}$ Resolved Stellar Populations and their Environments ${\rlap{$\square$}{\raisebox{2pt}{\large\hspace{1pt}{\ding{51}}}}\hspace{-2.5pt}}$ Galaxy Evolution $\square$ Multi-Messenger Astronomy and Astrophysics
**Principal Authors:**
Name: Roberta Paladini Institution: Caltech-IPAC Email: paladini@ipac.caltech.edu Phone: + 1 626 395 1848
Name: Matthew Povich Institution: California State Polytechnic University, Pomona Email: mspovich@cpp.edu Phone: +1 909 869 3608
**Co-authors:** (names and institutions) Lee Armus (Caltech-IPAC), Cara Battersby (U. Connecticut), Bruce Elmegreen (IBM), Adam Ginsburg (NRAO), Doug Johnstone (NRC), David Leisawitz (NASA-GSFC), Peregrine McGehee (College of the Canyons), Sarah Sadavoy (CfA), Marta Sewilo (NASA-GSFC), Alessio Traficante (INAF), Martina Wiedner (Observatoire de Paris)\
**Abstract:** It is usually assumed that the stellar initial mass function (IMF) takes a universal form and that there exists a direct mapping between this and the distribution of natal core masses (the core mass function, CMF). The IMF and CMF have been best characterized in the Solar neighborhood. Beyond 500 pc from the Sun, in diverse environments where metallicity varies and massive star feedback may dominate, the IMF has been measured only incompletely and imprecisely, while the CMF has hardly been measured at all. In order to establish if the IMF and CMF are indeed universal and related to each other, it is necessary to: 1) perform multi-wavelength large-scale imaging and spectroscopic surveys of different environments across the Galaxy; 2) require an angular resolution of ${<}0.1''$ in the optical/near-IR for stars and ${<}5''$ in the far-IR for cores; 3) achieve far-IR sensitivities to probe 0.1 M$_{\odot}$ cores at 2–3 kpc.
Adopted forms of the IMF and the CMF
====================================
The IMF is not an observable quantity, but rather an analytical description of the mass distribution among a newly-formed stellar population (Kroupa 2013). Modern forms of the IMF adopt either a log-normal distribution at low masses and a power-law tail above 1 M$_{\odot}$ (Chabrier et al. 2003, 2005) or a continuous set of several “broken” power-laws (Kroupa 2001, 2013). Above $\sim$ 0.2 M$_{\odot}$, the Chabrier and Kroupa IMFs agree, and the integrated mass of a stellar population is the same using either IMF formalism (Chomiuk & Povich 2011). Below 0.2 M$_{\odot}$ the form of the IMF is still very uncertain and the subject of much debate.
The CMF has a shape similar to Chabrier and Kroupa IMFs but is shifted towards larger masses by a factor ${\sim}3$, which has generally been interpreted as a core-to-star conversion efficiency of ${\sim}30\%$ (see Fig. 2).
The similar shape of the IMF and CMF has led to believe that there is an intrinsic mapping between these two quantities. However, this one-to-one correspondence does not find much theoretical ground (see below).
Are the the IMF and CMF Universal?
===================================
Over the last decade, there has been growing evidence of a variable IMF, as opposed to the common assumption that the IMF of the Milky Way is universal (Kroupa 2002; Bastian et al. 2010, Fig. 1). These claims come from a wide variety of approaches, including stellar population analysis (e.g. van Dokkum $\&$ Conroy 2010; Ferreras et al. 2013), gravitational lensing (Treu et al. 2010), and dynamical models (Cappellari et al. 2012). A notable Galactic example of exceptions to a universal IMF is the Taurus Molecular Cloud, which shows an excess of 0.6–0.8 M$_{\odot}$ stars. Other examples are the massive clusters Westerlund 1 (Lim et al. 2013), Quintuplet (Hussman et al. 2012), Arches (Hosek et al. 2019), and the young nuclear star clusters (Lu et al. 2013), although these could depart from a [*[standard IMF]{}*]{} as a consequence of mass segregation.\
From the extragalactic point of view, since 2010 there has been a flurry of IMF studies focusing on early type elliptical galaxies. These studies have both found an over-abundance of high-mass stars (“top-heavy" IMF, e.g. Davé et al. 2008), and an over-abundance of low-mass stars (“bottom-heavy" IMF, e.g. van Dokkum $\&$ Conroy 2010). In all cases, it is important to keep in mind that to determine the IMF of a stellar population, one has to go over a complicated process which consists of several steps: (1) measure the Luminosity Function (LF) of a complete sample of stars that lie in a defined volume; (2) convert the LF into a present day mass function (PDMF), using a mass-magnitude relationship; and (3) correct the PDMF for the star-formation history, stellar evolution, galactic structure, cluster dynamical evolution and binarity to obtain the individual-star IMF. Each of these steps is affected by potential biases and pitfalls that can lead to highly uncertain results.
![Recent IMF estimates for 8 star forming regions. The error bars represent the Poisson error for each data point. The solid lines are the log-normal form proposed by Chabrier (2005) for the IMF, normalized to best follow the data. From Offner et al. (2014).[]{data-label="fig:offner2014"}](Offner_etal.jpg){width="55.00000%"}
The most recent CMF determinations (Fig. 2) have been obtained with Herschel (e.g., André et al. 2010; Konyves et al. 2015, Olmi et al. 2018) and ALMA (Motte et al. 2018). The Herschel data support the conclusions of early studies performed in the $\rho$-Oph and Serpens molecular clouds (Motte et al. 1998; Johnstone et al. 2000; Testi $\&$ Sargent, 1998), which suggested that the CMF can be described, similarly to the IMF, by $dN \sim M_{\rm core}^{-1.5} dM$ below 0.5 M$_{\odot}$ and by $dN \sim M_{\rm core}^{-2 - 2.5} dM$ at higher core masses. However, the recent ALMA observations in the mini Galactic starburst W43 appear to show a departure from a standard IMF, with a much shallower CMF.\
![Core mass function (histogram with error bars) of the prestellar cores identified with Herschel in Aquila (Konyves et al. 2015; André et al. 2010). The Kroupa and Chabrier IMF and the typical mass spectrum of CO clumps is shown for comparison.[]{data-label="fig:andre2014"}](Andre_etal2014.jpg){width="55.00000%"}
From an analytical standpoint, Inutsuka (2001) and Hennebelle & Chabrier (2008) applied the Press-Schechter formalism and Shadmehri & Elmegreen (2011) used the ISM power spectrum with a density cutoff to obtain the clump mass function. A big uncertainty with this method, and with the conversion of a theoretical clump mass function into a stellar mass function, is the unknown multiplicity and mass function of stars inside each clump, for which there are few observations. Often clumps contain several stars and the one-to-one correspondence between clump mass and stellar mass is lost. Numerical simulations of star formation usually get the IMF in a more dynamical process involving long-term accretion into cores along filaments (e.g., Haugb[ø]{}lle et al. 2018, Bate 2019). In these models, the instantaneous CMF and the final stellar IMF are not one-to-one either.
Importantly, the CMF has been barely measured across different environments, especially outside the Solar neighborhood. The ALMA-IMF project (PI. F. Motte), now underway, is an attempt to measure the CMF in fifteen star forming regions across the Galaxy. We also note that many current dust measurements of the CMF are somewhat uncertain, due to intrinsic challenges, such as temperature determinations and whether each core will form one star or more (for more details see Offner et al. 2014). The difficulties in assignment of emission to a single object using automated routines, and the potential bias in the resulting CMFs are discussed in Pineda et al. (2009).
Does Environment Sculpt the IMF?
================================
The question of how environment shapes the IMF can be reduced to two principal variables: metallicity and stellar feedback.
Metallicity Effects
-------------------
Metallicity sets the opacity of the cloud, which governs the minimum mass of a gravitationally bound core, the cooling rate of cores, and the maximum possible stellar mass (Eddington luminosity). Therefore, we expect metallicity to play a pivotal role in shaping the IMF. Indeed, recent observations of early-type galaxies find that their local IMFs become increasingly bottom-heavy (i.g. more lower mass stars) in those galaxies that are metal rich (Martin-Navarro et al. 2015).
Feedback Effects
----------------
According to simulations by Krumholz et al. (2016), radiative heating is the main driver of the characteristic IMF mass. These simulations show that when radiative heating increases, the efficiency of fragmentation is reduced, leading to a top-heavy IMF. Conversely, Conroy & van Dokkum (2012) suggest that a pivotal role is played by radiative ambient pressure, which is responsible for giving rise to bottom-heavy IMFs (at increasing pressure) as observed in elliptical galaxies with a history of starburst-generating mergers. An additional effect is represented by kinetic feedback. Stellar winds, protostellar outflows/jets, and ionization all likely affect the efficiency of star formation (e.g. Li $\&$ Nikamura 2006). However, it is still matter of debate how and if they ultimately affect stellar masses. For instance, it is thought that outflows slow the star formation rate (e.g. Dale $\&$ Bonnell 2008; Wang et al. 2010), but it is unclear if this has any effect on the stellar mass distribution. Likewise for ionization, several studies (e.g. Dale $\&$ Bonnell 2012; Walch et al. 2013) have shown that ionizing radiation can provide both negative or positive feedback, in the sense of suppressing or triggering star formation, but none of these have been conclusive in demonstrating the impact on the IMF.
Open Questions
==============
[****]{}
1. [To what extent can we assume the IMF is universal?]{}
2. [Does the CMF map directly on to the IMF in all environments?]{}
3. [How does environment shape the CMF and IMF?]{}
[****]{}\
1. [What physical mechanism(s) suppresses the formation of brown dwarfs? Can this lead to a better understanding of the distinction between brown dwarfs and giant planets?]{}
2. [Hierarchical collapse models and many observations suggest that giant molecular cloud (GMC) complexes make stars over an extended time period. Can we observe time-evolution in the CMF?]{}
3. [Massive stars do not seem to obey the CMF–IMF mapping. The CMF appears lognormal, not a Salpeter power-law slope at high masses. Do star formation efficiencies change at higher masses or are cloud mergers required to form the most massive stars?]{}
4. [How do binary/multiple stellar systems arise from the CMF? What determines whether gravitationally-bound cores fragment further?]{}
5. [Can we reconcile observations of bottom-heavy IMFs in elliptical galaxies that were once starbursts with top-heavy IMFs in young massive clusters (YMCs)? What are the implications for Pop III stars?]{}
Observational Goals and Recommendations
=======================================
To answer the questions above we outline the following observational goals and recommendations:
- [**[Observational Goal—IMF:]{}**]{} To achieve an accurate measurement of the IMF in diverse environments and explore potential variations with metallicity and feedback, we require observations of numerous YMCs and associations distributed at increasing distances across the Galaxy and beyond, such as Taurus ($d=180$ pc); Orion (400 pc); M17 (1.6 kpc); W3/4/5 (2.0 kpc, outer Galaxy); NGC 7538 (2.8 kpc), NGC 3603 (7 kpc), and the Large and Small Magellanic Clouds (50–60 kpc).\
[**[Recommendation:]{}**]{} While [*Gaia*]{} can provide information on distances and velocities for the stars in these star-forming complexes, we need high spatial-resolution (${<}0.1"$), wide-field imaging and spectroscopy at visual and particularly NIR wavelengths to allow stellar age and mass determinations in both unobscured and obscured regions up to several degrees wide on the sky. This type of information can be obtained by the [*Cosmological Advanced Survey Telescope for Optical and ultraviolet Research*]{} ([*CASTOR*]{}, Coté et al. 2012) and by [*WFIRST*]{}. These facilities, combined with LSST ($u,g,r,i,z, Y$) and Euclid ($R,I,Z, Y, J, H$), will be ideal for studies of the IMF, thanks to their wavelength coverage (0.15 - 0.4 $\mu$m for [*CASTOR*]{}/Visible Imager and Spectrometer and 0.4 - 2 $\mu$m for [*WFIRST*]{}/WFI) and large FOV (0.67 deg$^{2}$ for [*CASTOR*]{}/Visible Imager, and 0.25 deg$^{2}$ for [*WFIRST*]{}/WFI).
- [**[Observational Goal—CMF:]{}**]{} Along the same lines as for the IMF, we advocate for surveys of Galactic and extra-galactic GMC complexes (as described above) to investigate potential variations of the CMF with environment.\
[**[Recommendation:]{}**]{} Interferometric observations (ALMA, EVLA, SMA) will provide high-resolution observations of targeted regions. However, the [*Origins Space Telescope (OST)*]{} will be uniquely capable of performing statistical measurements of the CMF and protostellar luminosity functions in distant/obscured Galactic regions, including starburst-like environments. What makes [[*OST*]{}]{} ideal for this task is its unique imaging and mapping capabilities of the far-IR cold dust emission peak in dense, prestellar clumps and cores. This can be achieved through the combination of (1) a large FOV for efficient scanning of extended regions on the sky; (2) sufficiently high angular resolution to resolve a 0.1 pc cores at a distance of a few kpc, and (3) sufficiently high sensitivity to detect 0.1 M$_{\odot}$ cores at 2–3 kpc. The 5.9-m [[*OST*]{}]{} mirror allows achieving a resolution of ${\sim}6''$ at 50 $\mu$m, which is comparable to the angular resolution at shorter wavelengths of [*Spitzer*]{}/IRAC ($2''$), and [*Spitzer*]{}/MIPS ($6''$). The Far-Infrared Imager and Polarimeter instrument, FIP, can map 1 deg$^{2}$ of the sky in 100 hrs while achieving a 5-$\sigma$ sensitivity of ${\sim}1~\mu$Jy. We note that the baseline concept for the [[*OST*]{}]{}/FIP instrument has two bands—50 and 250 $\mu$m—but an optional upscope would add the 100 and 500 $\mu$m channels. We recommend the inclusion of these additional bands that would better constrain the peak of the cold dust emission.
Importantly, currently existing (e.g. [*HST*]{}, the Magellan telescope, etc.) or planned facilities (e.g., the [*James Webb Space Telescope*]{}) will be able to carry out imaging and multi-object spectroscopy of targeted Galactic YMCs. While such observations will be useful for IMF studies, the reach of these measurements will be limited by the small FOVs, which do not allow efficient mapping of large (i.e. of the order of deg$^{2}$) star-forming complexes across the Galaxy.
**References**\
|
INTRODUCTION
============
This work addresses a combined action of two mechanisms of resonant excitation of (classical) nonlinear oscillating systems. The first is [*parametric resonance*]{}. The second is [*autoresonance*]{}.
There are numerous oscillatory systems which interaction with the external world amounts only to a periodic time dependence of their parameters. The corresponding resonance is called [*parametric*]{} [@Landau; @Bogolubov]. A textbook example is a simple pendulum with a vertically oscillating point of suspension [@Landau]. The main resonance occurs when the excitation frequency $\omega$ is nearly twice the natural frequency of the oscillator $\omega_{0}$ [@Landau; @Bogolubov]. Applications of this basic phenomenon in physics and technology are ubiquitous.
[*Autoresonance*]{} occurs in nonlinear oscillators driven by a small [*external*]{} force, almost periodic in time. If the force is [*exactly*]{} periodic, and in resonance with the natural frequency of the oscillator, the resonance region of the phase plane has a finite (and relatively small) width [@Sagdeev; @Lichtenberg]. If instead the driving frequency is slowly varying in time (in the right direction determined by the nonlinearity sign), the oscillator can stay phase-locked despite the nonlinearity. This leads to a continuous resonant excitation. Autoresonance has found many applications. It was extensively studied in the context of relativistic particle acceleration: in the 40-ies by McMillan [@McMillan], Veksler [@Veksler] and Bohm and Foldy [@Bohm1; @Bohm2], and more recently [@Golovanivsky2; @Meerson2; @Meerson10; @Friedland1]. Additional applications include a quasiclassical scheme of excitation of atoms [@Meerson1] and molecules [@molecules], excitation of nonlinear waves [@Deutsch; @Friedland2], solitons [@Aranson; @Friedland3], vortices [@Friedland4; @Friedland5] and other collective modes [@Fajans1] in fluids and plasmas, an autoresonant mechanism of transition to chaos in Hamiltonian systems [@Yariv1; @Cohen], etc.
Until now autoresonance was considered only in systems executing [*externally*]{} driven oscillations. In this work we investigate autoresonance in a [*parametrically*]{} driven oscillator.
Our presentation will be as follows. In Section 2 we briefly review the parametric resonance in non-linear oscillating systems. Section 3 deals, analytically and numerically, with parametric autoresonance. The conclusions are presented in Section 4. Some details of derivation are given in Appendices A and B.
PARAMETRIC RESONANCE WITH A CONSTANT DRIVING FREQUENCY
======================================================
The parametric resonance in a weakly nonlinear oscillator with finite dissipation and detuning is describable by the following equation of motion [@Bogolubov; @Struble2; @Morozov]: $$\ddot{x}+2\gamma\dot{x}+
\left[1+\epsilon \cos \left((2+\delta)t\right)\right]x- \beta
x^3=0. \label{Bbegin1}$$ where the units of time are chosen in such a way that the scaled natural frequency of the oscillator in the small-amplitude limit is equal to 1. In Eq. (\[Bbegin1\]) $ \epsilon$ is the amplitude of the driving force, which is assumed to be small: $
0<\epsilon\ll1$, $ \delta \ll 1$ is the detuning parameter, $
\gamma$ is the (scaled) damping coefficient $ ( 0<\gamma\ll1 ) $ and $\beta$ is the nonlinearity coefficient. For concreteness we assume $\beta>0$ (for a pendulum $\beta =1/6$).
Working in the limit of weak nonlinearity, dissipation and driving, we can employ the method of averaging [@Bogolubov; @Sagdeev; @Rabinovich; @Drazin], valid for most of the initial conditions [@Sagdeev; @Lichtenberg]. The unperturbed oscillation period is the fast time. Putting $ x=a(t)\cos\theta(t)$ and $
\dot{x}=-a(t)\sin\theta(t)$ and performing averaging over the fast time, we arrive at the averaged equations $$\begin{aligned}
\dot{a}&=&-\gamma a+\frac{\epsilon a}{4}\sin 2\psi,\nonumber
\\
\dot{\psi}&=& -\frac{\delta}{2}-\frac{3\beta a^2}{8}
+\frac{\epsilon}{4}\cos 2\psi, \label{B6}\end{aligned}$$ where a new phase $\psi=\theta-[(2+\delta)/2] t$ has been introduced. The averaged system (\[B6\]) is an autonomous dynamical system with two degree of freedom and therefore integrable. In the conservative case $\gamma=0$ Eqs. (\[B6\]) become: $$\begin{aligned}
\dot{a}&=&\frac{\epsilon a}{4}\sin 2\psi,\nonumber
\\
\dot{\psi}&=&-\frac{\delta}{2}-\frac{3\beta
a^2}{8}+\frac{\epsilon}{4}\cos 2\psi.\label{B61}\end{aligned}$$ As $\sin 2\psi$ and $\cos 2\psi$ are periodic functions of $\psi $ with a period $\pi$, it is sufficient to consider the interval $-\pi/2<\psi\leq\pi/2 $. For small enough detuning, $
\delta<\epsilon/2 $, there is an elliptic fixed point with a non-zero amplitude:$$a_{*}=\pm\left[\frac{2\epsilon}{3\beta}\left(1-\frac{2\delta}{\epsilon}\right)\right]^{1/2};
\indent\psi_{*}=0.$$ We need to calculate the period of motion in the phase plane along a closed orbit around this fixed point (such an orbit is shown in Fig. \[close\]).
This calculation was performed by Struble [@Struble2]. For a zero detuning, $\delta=0$, Hamilton’s function (we will call it the Hamiltonian) of the system (\[B61\]) is the following:
$$H(I,\psi)=\frac{\epsilon I}{4}\cos 2\psi-\frac{3\beta
I^2}{8}=H_{0} = const., \label{help1}$$
where we have introduced the action variable $I=a^2/2$. Solving Eq. (\[help1\]) for $I$ and substituting the result into the Hamilton’s equation for $\dot{\psi}$ we obtain: $$\dot{\psi}=\mp\frac{\epsilon}{4}\left(\cos^2 2\psi-\frac{24\beta
H_{0}}{\epsilon^2}\right)^{1/2}, \label{help2}$$ where the minus (plus) sign corresponds to the upper (lower) part of the closed orbit. The period of the amplitude and phase oscillations is therefore $$T=\frac{8}{\epsilon}\int_{-\overline{\psi}}^{\overline{\psi}}\frac{d\psi}{\left(\cos^2
2\psi-\frac{24\beta H_{0}}{\epsilon^2}\right)^{1/2}},
\label{help3}$$ where $-\overline{\psi}$ and $\overline{\psi}$ are the roots of the equation $ \cos^2 2\psi=24\beta H_{0}/\epsilon^2.$ Calculating the integral, we obtain $$T=\frac{8}{\epsilon}K(m), \label{help4}$$ where $K(m)$ is the complete elliptic integral of the first kind [@Abramowitz], and $m=1-24\beta H_{0}/\epsilon^2$. This result will be used in Section 3 to establish a necessary condition for the parametric autoresonance to occur.
PARAMETRIC RESONANCE WITH A TIME-DEPENDENT DRIVING FREQUENCY: PARAMETRIC AUTORESONANCE
======================================================================================
Now let the driving frequency vary with time. This time dependence introduces an additional (third) time scale into the problem. The governing equation becomes $$\ddot{x}+ 2\gamma\dot{x} + (1+\epsilon\cos\phi)x-\beta x^3=0,
\label{B6.5}$$ where $\dot{\phi}=\nu(t)$. We will assume $\nu(t)$ to be a [*slowly*]{} decreasing function which initial value is $\nu(t=0)=2+\delta.$ Using the scale separation, we obtain the averaged equations. The averaging procedure of Section 2 can be repeated by replacing $(2+\delta)t$ by $\phi$ in all equations. There is one new point that should be treated more accurately. The averaging procedure is applicable (again, for most of the initial conditions) if there is a separation of time scales. It requires, in particular, a strong inequality $2\dot{\theta}+\nu (t)\gg 2\dot{\theta}-\nu(t)$. This inequality can limit the time of validity of the method of averaging. Let us assume, for concreteness, a linear frequency “chirp”: $$\nu(t)=2+\delta-2\mu t, \label{B7}$$ where $\mu\ll 1$ is the chirp rate. In this case the averaging procedure is valid as long as $\mu t\ll 1$.
Introducing a new phase $\psi=\theta-\phi/2 $, we obtain a reduced set of equations (compare to Eqs. (\[B6\])): $$\begin{aligned}
\dot{a}&=&-\gamma a+\frac{\epsilon a}{4}\sin 2\psi,\nonumber
\\
\dot{\psi}&=&-\frac{\delta}{2}+\mu t-\frac{3\beta
a^2}{8}+\frac{\epsilon}{4}\cos 2\psi. \label{B8}\end{aligned}$$ The first of Eqs. (\[B8\]) is typical for [*parametric*]{} resonance: to get excitation one should start from a non-zero oscillation amplitude. As we will see, the $\mu t$ term in the second of Eqs. (\[B8\]) (when small enough and of the right sign) provides a continuous phase locking, similar to the externally driven autoresonance.
Consider a numerical example. Fig. \[figexample\] shows the time dependence $a (t)$ found by solving Eqs. (\[B8\]) numerically. One can see that the system remains phase locked which allows the amplitude of oscillations to increase, on the average, with time in spite of the nonlinearity. The time-dependence of the amplitude includes a slow trend and relatively fast, decaying oscillations. These are the two time scales remaining after the averaging over the fastest time scale.
Similar to the externally-driven autoresonance, a persistent growth of the oscillation amplitude requires the characteristic time of variation of $\nu(t)$ to be much greater than the “nonlinear” period $T$ \[see Eq. (\[help4\])\] of oscillations of the amplitude: $$\left|\frac{\nu(t)}{\dot{\nu}(t)}\right|\gg T. \label{Ccrit}$$
Like its externally-driven analog, the parametric autoresonance is insensitive to the exact form of $\nu(t)$. For a given set of parameters, the optimal chirping rate can be found: too low a chirping rate means an inefficient excitation, while too high a rate leads to phase unlocking and termination of the excitation.
In the remainder of the paper we will develop an analytical theory of the parametric autoresonance. The first objective of this theory is a description of the slow trend in the amplitude (and phase) dynamics. When the driving frequency $\nu$ is constant, there is an elliptic fixed point $a_{*}$ (see Section 2). When $\nu$ varies with time, the fixed point ceases to exist. However, for a [*slowly*]{}-varying $\nu(t)$ one can define a “quasi-fixed” point $a_{*}(t)$ which is a slowly varying function of time. It is this quasi-fixed point that represents the slow trend seen in Fig. \[figexample\] and corresponds to an “ideal” phase-locking regime. The fast, decaying oscillations seen in Fig. \[figexample\] correspond to oscillations around the quasi-fixed point in the phase plane \[this phase plane is actually projection of the extended phase space ($a,\psi, t$) on the ($a,\psi$)-plane\].
In the main part of this Section we neglect the dissipation and use a Hamiltonian formalism. First we will consider excitation in the vicinity of the quasi-fixed point. Then excitation from arbitrary initial conditions will be investigated. Finally, the role of dissipation will be briefly analyzed.
For a time-dependent $\nu(t)$, the Hamiltonian becomes \[compare to Eq. (\[help1\])\]: $$H(I,\psi,t)=\frac{\epsilon I}{4}\left(\alpha(t)+\cos
2\psi\right)-\frac{3\beta I^2}{8}, \label{C1}$$ where $\alpha(t)=(4/\epsilon)(1-\nu(t)/2).$ The Hamilton’s equations are: $$\begin{aligned}
\dot{I}&=&\frac{\epsilon I}{2}\sin 2\psi,\nonumber
\\
\dot{\psi}&=&\frac{\epsilon}{4}\left(\alpha+\cos
2\psi\right)-\frac{3\beta I}{4}. \label{C2}\end{aligned}$$ Let us find the quasi-fixed point of (\[C2\]), i.e. the special autoresonance trajectory $I_{*}(t),\,\psi_{*}(t)$ corresponding to the “ideal” phase locking (a pure trend without oscillations).
Assuming a slow time dependence, we put $\dot{\psi_{*}} = 0$, that is $$\frac{\epsilon}{4}\left(\alpha+\cos 2\psi_{*}\right)-\frac{3\beta
I_{*}}{4}=0. \label{C2-1}$$ Differentiating it with respect to time and using Eqs. (\[C2\]), we obtain an algebraic equation for $\psi_* (t)$: $$2\alpha (t) \sin 2\psi_{*}+\sin 4\psi_{*} = \frac{16\mu}{\epsilon^2} .
\label{C2-2}$$ At this point we should demand that $\dot{\psi_*} (t)$, evaluated on the solution of Eq. (\[C2-2\]), is indeed negligible compared to the rest of terms in the equation (\[C2\]) for $\dot{\psi} (t)$. It is easy to see that this requires $16 \mu/\epsilon^2 \ll 1$. In this case the sines in Eq. (\[C2-2\]) can be replaced by their arguments, and we obtain the following simple expressions for the quasi-fixed point: $$\begin{aligned}
I_{*}&\simeq&\frac{\epsilon}{3\beta}\left(\alpha+1\right)\,,\nonumber
\\
\psi_{*}&\simeq&\frac{k}{\alpha+1}\,, \label{C3}\end{aligned}$$ where $k=4\mu/\epsilon^2$.
Excitation in the vicinity of the quasi-fixed point
---------------------------------------------------
Let us make the canonical transformation from variables $I$ and $\psi$ to $ \delta I=I-I_{*}$ and $\delta\psi=\psi-\psi_{*}.$ Assuming $\delta I$ and $\delta\psi$ to be small and keeping terms up to the second order in $ \delta I$ and $\delta\psi$, we obtain the new Hamiltonian: $$\begin{aligned}
H(\delta I,\delta\psi,\alpha(t))&=&
%\frac{\epsilon^2}{24\beta}(\alpha+1)^2
-\frac{\epsilon k}{\alpha+1}\delta I \delta\psi- \nonumber
\\
&-&\frac{3\beta}{8}(\delta I)^2-
\frac{\epsilon^2}{6\beta}(\alpha+1)(\delta\psi)^2\,. \label{C4}\end{aligned}$$ Here and in the following small terms of order of $k^2$ are neglected. Let us start with the calculation of the local maxima of $\delta I
(t)$ and $\delta\psi (t)$, which will be called $\delta I_{max}(t)$ and $\delta\psi_{max}(t)$, respectively. As $\alpha(t)$ is a slow function of time \[so that the strong inequality (\[Ccrit\]) is satisfied\], we can exploit the approximate constancy of the adiabatic invariant [@Landau; @Goldstein]: $$J=\frac{1}{2\pi}\oint\delta I d(\delta\psi) \simeq const. \label{C5}$$ $|J|$ is the area of the ellipse defined by Eq. (\[C4\]) with the time-dependencies “frozen”. Therefore, $$J=\frac{2}{\epsilon}\frac{H}{(\alpha+1)^{1/2}}
%\left(\frac{\epsilon^2}{24\beta}(\alpha+1)^2-H\right)
\simeq const. \label{C6}$$ This expression can be rewritten in terms of $\delta
I$ and $\delta\psi$: $$\begin{aligned}
|J| &=&\frac{2k}{(\alpha+1)^{3/2}}\delta I
\delta\psi+\frac{3\beta}{4\epsilon}\frac{1}{(\alpha+1)^{1/2}}(\delta
I)^2\nonumber
\\
&+&\frac{\epsilon}{3\beta}(\alpha+1)^{1/2}(\delta\psi)^2.
\label{C7}\end{aligned}$$ If $k=4 \mu/\epsilon^2 \ll 1$, the term with $\delta I
\delta\psi$ in (\[C7\]) can be neglected (in this approximation one has $\psi_* = 0$). Then $J$ becomes a sum of two non-negative terms, one of them having the maximum value when the other one vanishes. Therefore, $$\delta I_{max}(t)=2\left(\frac{\epsilon
J}{3\beta}\right)^{1/2}\left(\alpha+1\right)^{1/4}, \label{C8}$$ and $$\delta\psi_{max}(t)=\left(\frac{3\beta
J}{\epsilon}\right)^{1/2}\frac{1}{(\alpha+1)^{1/4}}. \label{C9}$$ Now we calculate the period of oscillations of the action and phase. Using the well-known relation [@Landau] $T=2\pi(\partial J/\partial H)$, we obtain from Eq. (\[C6\]): $$T=\frac{4\pi}{\epsilon}\frac{1}{(\alpha+1)^{1/2}}. \label{C10}$$ The period of oscillations versus time is shown in Fig. \[fiq-C3\]. The theoretical curve \[Eq. (\[C10\])\] shows an excellent agreement with the numerical solution.
Now we obtain the complete solution $\delta I(t)$ and $\delta\psi(t)$. The Hamilton’s equations corresponding to the Hamiltonian (\[C4\]) are:
$$\begin{aligned}
\dot{\delta I}&=&
\frac{\epsilon^2}{3\beta}\left(\alpha+1\right)\delta\psi+\frac{\epsilon
k}{\alpha+1}\delta I,\nonumber
\\
\dot{\delta\psi}&=&-\frac{3\beta}{4}\delta I-\frac{\epsilon
k}{\alpha+1}\delta\psi. \label{C11}\end{aligned}$$
Differentiating the second equation with respect to time and substituting the first one, we obtain a linear differential equation for $\delta\psi(t)$: $$\ddot{\delta\psi}+\omega^2(t)\delta\psi=0, \label{C12}$$ where $\omega(t)=(\epsilon/2)(\alpha(t)+1)^{1/2}$. For the linear $\nu(t)$ dependence (Eq. (\[B7\])) we have $\alpha(t)=4\mu t/\epsilon-2\delta/\epsilon$, therefore for $k\ll 1$ the criterion $\dot{\omega}/\omega^2\ll 1$ is satisfied, and Eq. (\[C12\]) can be solved by the WKB method (see, [*e.g.*]{} [@Lichtenberg]).
The WKB solution takes the form (details are given in Appendix A): $$\begin{aligned}
\delta\psi(t)&=&\left(\frac{3\beta
J}{\epsilon}\right)^{1/2}\frac{1}{(\alpha+1)^{1/4}}\nonumber
\\
& \times & \cos\left(q_{0}+\frac{(\alpha+1)^{3/2}}{3k}\right),
\label{C15}\end{aligned}$$ where the phase $q_{0}$ is determined by the initial conditions. The full solution for the phase is $\psi=\delta\psi+\psi_{*}$ and Fig. \[fig-c2\] compares it with a numerical solution of Eq. (\[C2\]). Also shown are the minimum and maximum phase deviations predicted by Eqs. (\[C9\]) and (\[C3\]). One can see that the agreement is excellent.
The solution for $\delta I (t)$ can be obtained by substituting Eq. (\[C15\]) into the second equation of the system (\[C11\]). In the same order of accuracy (see Appendix A) $$\delta I(t)=2\left(\frac{\epsilon
J}{3\beta}\right)^{1/2}\left(\alpha+1\right)^{1/4}\sin\left(q_{0}+\frac{(\alpha+1)^{3/2}}{3k}\right)\,.
\label{C19}$$
Fig. \[fig-c1\] shows the dependence of the action variable with the trend $I_*(t)$ subtracted, $\delta I (t)$, on time predicted by Eq. (\[C19\]), and found from the numerical solution. It also shows the minimum and maximum action deviations (\[C8\]). Again, a very good agreement is obtained.
Excitation from arbitrary initial conditions
--------------------------------------------
In this Subsection we go beyond the close vicinity of the quasi-fixed point and calculate the maximum deviations of the action $I$ and phase $\psi$ for arbitrary initial conditions. Again, these calculations are made possible by employing the adiabatic invariant for the general case. Correspondingly, the period of the action and phase oscillations will be also calculated.
Let us first express the maximum and minimum [*action*]{} deviations in terms of the Hamiltonian $H$ and driving frequency $\nu(t)$. Solving Eq. (\[C1\]) as a quadratic equation for $I$, we obtain: $$I_{1,2}=\frac{\epsilon}{3\beta}\left(\alpha+\cos
2\psi\right)\pm\left[ \frac{\epsilon^2}{9\beta^2}\left(\alpha+\cos
2\psi\right)^2-\frac{8H}{3\beta}\right]^{1/2}.$$ The time derivative of $I$ vanishes when $I=I_{max}$ or $I=I_{min}$. Therefore, from the first equation of the system (\[C2\]) $\psi=0$ so that $$I_{max,min}=\frac{\epsilon}{3\beta}\left(\alpha+1\right)\pm\left[
\frac{\epsilon^2}{9\beta^2}\left(\alpha+1\right)^2-\frac{8H_{up,down}}{3\beta}\right]^{1/2},
\label{C20}$$ where $H_{up, down}=H(I_{max, min},\psi=0)$.
Now we express the maximum and minimum [*phase*]{} deviations through the Hamiltonian $H$ and driving frequency $\nu(t)$. The time derivative $\dot{\psi}$ vanishes if $\psi=\psi_{max}$ or $\psi=\psi_{min}$, then the second equation of the system (\[C2\]) yields $I=(\epsilon/3\beta)(\alpha+\cos 2\psi)$. In this case the Hamiltonian (\[C1\]) becomes $H_{right,left}=(\epsilon^2/24\beta)(\alpha+\cos
2\psi_{max,min})^2$. Finally, the expression for $\psi_{max,min}$ is $$\psi_{max,min}=\pm\frac{1}{2}\arccos\left[\left(\frac{24\beta
H_{right,left}}{\epsilon^2}\right)^{1/2}-\alpha\right]\,. \label{C21}$$
Fig. \[fiq-C6\] shows a part of a typical autoresonant orbit in the phase plane. For $\nu(t)=const.$ this orbit is determined by the equation $H(I,\psi,\nu)=const.$, and it is closed. As in our case $\nu(t)$ changes with time, the trajectory is not closed. To calculate the maximum and minimum deviations of action and phase we should know the values of the Hamiltonian at 4 points of the orbit that we will call “up", “down", “left", and “right" in the following.
Knowing the values of the Hamiltonian at these 4 points, we calculate $I_{max,min}$ from Eq. (\[C20\]) and $\psi_{max,min}$ from Eq. (\[C21\]). Figs. (\[fiq-C7\]) and (\[fiq-C8\]) show these deviations for action and phase correspondingly, and the values of $I$ and $\psi$, found from numerical solution. The theoretical and numerical results show an excellent agreement.
Now we are prepared to calculate the adiabatic invariant $J(H,\nu(t))$. Its (approximate) constancy in time allows one, in principle, to find the Hamiltonian $H(t)$ at any time $t$, in particular at the points of the maximum and minimum action and phase deviations (see Fig. \[fiq-C6\]).
It is convenient to rewrite the adiabatic invariant in the following form: $$J=\frac{1}{2\pi}\oint\psi dI. \label{C22}$$
Using Eq. (\[C1\]), we can find $\psi=\psi(H,I,\alpha(t))$: $$\psi=\pm \frac{1}{2}\arccos\left(\frac{8H+3\beta I}{2\epsilon I}
-\alpha\right)\,, \label{C23}$$ so that Eq. (\[C22\]) becomes: $$J=\frac{1}{2\pi}\int^{I_{max}}_{I_{min}}\arccos\left(\frac{8H+3\beta
I}{2\epsilon I} -\alpha\right) dI, \label{C24}$$ where $I_{max}$ and $I_{min}$ are given by Eq. (\[C20\]). Notice that $H(t)$ and $\alpha(t)$ should be treated as constants under the integral (\[C24\]), see Refs. [@Landau; @Sagdeev; @Goldstein]. This integral can be expressed in terms of elliptic integrals (see Appendix B for details). For definiteness, we used the values of $H(t)$ and $\alpha(t)$ in the “up” points, see Fig. \[fiq-C6\]. We checked numerically that the adiabatic invariant $J(H(t),\alpha(t))$ is constant in our example within 0.12 per cent.
Now we calculate the period of action and phase oscillations. From the first equation of system (\[C2\]) we have: $$T=2\int^{I_{max}}_{I_{min}}\frac{dI}{(\epsilon I/2)\sin 2\psi},
\label{C26}$$ where $I_{max}$ and $I_{min}$ are given by Eq. (\[C20\]), while $\psi=\psi(I)$ is defined by (\[C23\]).
Using Eq. (\[C1\]), we obtain after some algebra: $$T=\frac{8}{3\beta}\int^{I_{max}}_{I_{min}}\frac{dI}{G(I)^{1/2}},
\label{C27}$$ where $G(I)$ is given in Appendix B, Eq. (\[C-G\]). Again, we treat $H(t)$ and $\alpha(t)$ as constants under the integral (\[C27\]), and take their values in the “right” points, see Fig. \[fiq-C6\]. The final result is:
$$T=C_{2}K(C_{3}), \label{period}$$
where $C_{2}=4(2/3\beta H\epsilon^2)^{1/4}$ and $$C_{3}=\frac{1}{2}-\frac{C_{2}^2}{16}\left[\frac{3\beta
H}{2}+\frac{\epsilon^2}{16}\left(1-\alpha^2\right)\right]\,.$$
Figure \[fiq-C10\] shows the period $T$ of the phase and action oscillations versus time obtained analytically and from numerical solution. This completes our consideration of the parametric autoresonance without dissipation.
Role of dissipation
-------------------
Now we very briefly consider the role of dissipation in the parametric autoresonance. Consider the averaged equations (\[B8\]) and assume that the detuning is zero. The non-trivial quasi-fixed point exists when the dissipation is not too strong: $\gamma < \epsilon/4$, and it is given by $$\begin{aligned}
a_{*}&=&\left(\frac{2\epsilon}{3\beta}\right)^{1/2}\left[\alpha (t) +
\left(1-\frac{16\gamma^2}{\epsilon^2}\right)^{1/2}\right]^{1/2},\nonumber
\\
\psi_{*}&=&\frac{1}{2}\arcsin\left(\frac{4\gamma}{\epsilon}+
\frac{2k}{\alpha (t) +(1-16\gamma^2/\epsilon^2)^{1/2}}\right)\,.
\label{C29}\end{aligned}$$ Again, we assume $k\ll 1$. This quasi-fixed point describes the slow trend in the dissipative case. As we see numerically, fast oscillations around the trend, $\delta a=a-a_{*}$ and $\delta\psi=\psi-\psi_{*}$ decay with time. Therefore, one can expect that the $a(t)$ will approach, at sufficiently large times, the trend $a_{*}(t)$. Fig. \[fiq-C13\] shows the time dependence of the amplitude, found by solving numerically the system of averaged equations (\[B8\]), and the amplitude trend from (\[C29\]). We can see that indeed the amplitude $a(t)$ approaches the trend $a_{*}(t)$ at large times.
Therefore, a small amount of dissipation enhances the stability of the parametric autoresonance excitation scheme. A similar result for the externally-driven autoresonance was previously known [@Yariv2].
CONCLUSIONS
===========
We have investigated, analytically and numerically, a combined action of two mechanisms of resonant excitation of nonlinear oscillating systems: parametric resonance and autoresonance. We have shown that parametric autoresonance represents a robust and efficient method of excitation of nonlinear oscillating systems. Parametric autoresonance can be extended for the excitation of nonlinear [*waves*]{}. We expect that parametric autoresonance will find applications in different fields of physics.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
This research was supported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities.
CALCULATION OF PHASE AND ACTION DEVIATIONS BY THE WKB-METHOD
============================================================
Changing the variables from time $t$ to $\alpha$, we can rewrite Eq. (\[C12\]) in the following form: $$\delta\psi^{\prime\prime}+
\left(\frac{\alpha(t)+1}{4k^2}\right)\delta\psi=0\,, \label{AP31}$$ where $^{\prime\prime}$ denotes the second derivative with respect to $\alpha$. Solving this equation by the WKB-method [@Lichtenberg], we obtain for $\delta\psi$: $$\delta\psi(t)=\frac{(2kC)^{1/2}}{(\alpha+1)^{1/4}}\cos\left(\Omega_{0}+\frac{(\alpha(t)+1)^{3/2}-1}{3k}\right),
\label{AP36}$$ where $\Omega_{0}$ and $C$ are constants to be found later. Now we obtain the solution for $\delta I$. Substituting (\[AP36\]) into the second equation of the system (\[C11\]), we obtain in the same order of accuracy: $$\begin{aligned}
\delta
I(t)&=&\frac{2\epsilon}{3\beta}\left(2kC\right)^{1/2}\left(\alpha+1\right)^{1/4}\nonumber
\\
& \times &
\sin\left(\Omega_{0}+\frac{(\alpha(t)+1)^{3/2}-1}{3k}\right).
\label{AP37}\end{aligned}$$ The constant $C$ can be expressed through the adiabatic invariant $J$, given by (\[C7\]). From Eqs. (\[AP36\]) and (\[AP37\]) we have: $$2kC=\left(\frac{3\beta}{2\epsilon}\right)^2\frac{1}{(\alpha+1)^{1/2}}\left(\delta
I\right)^2+\left(\alpha+1\right)^{1/2}(\delta\psi)^2.$$ Comparing it with (\[C7\]) we find: $ C\simeq 3\beta J/2k\epsilon.$ Substituting this value into Eqs. (\[AP36\]) and (\[AP37\]) we obtain the final expressions (\[C15\]) and (\[C19\]) for $\delta\psi(t)$ and $\delta I(t)$.
CALCULATION OF THE ADIABATIC INVARIANT
======================================
After integration by parts and some algebra, using Eqs. (\[C1\]) and (\[C20\]), we obtain the following expression for the adiabatic invariant: $$J=\frac{1}{2\pi}\int^{I_{max}}_{I_{min}}\left(\frac{I^2-
\frac{8H}{3\beta}}{G(I)^{1/2}}\right)
dI, \label{C25}$$ where $$G(I)=\left(I_{max}-I\right)\left(I-I_{min}\right)\left[\left(I+\frac{\epsilon(1-\alpha)}{3\beta}\right)^2-\frac{16D}{9\beta^2}\right],
\label{C-G}$$ and we assume $D=(\epsilon^2/16)(1-\alpha)^2-3\beta H/2<0.$ Calculation of this integral employs several changes of variable shown in the best way by Fikhtengolts [@Fikhtengolts]. Using the reduction formulas [@Abramowitz], we arrive at: $$\begin{aligned}
J=C_{1}\left[\frac{1+mm'}{(1-m)^2(1+m')}\Pi\left(\frac{m}{m-1}\backslash
k^2\right)\right.\nonumber
\\
\left.-\frac{1}{1-m}K\left(k^2\right)+\frac{m+m'}{(1-m)(1+m')}E\left(k^2\right)\right],
\label{invariant}\end{aligned}$$ where $$m=\frac{(\epsilon/3\beta)(1+\alpha)-(8H/3\beta)^{1/2}}{(\epsilon/3\beta)(1+\alpha)+(8H/3\beta)^{1/2}}>0,$$ $$m'=\frac{(\epsilon/3\beta)(1-\alpha)+(8H/3\beta)^{1/2}}{-(\epsilon/3\beta)(1-\alpha)+(8H/3\beta)^{1/2}}>0.$$ $$k^2=\frac{m}{m+m'},\indent
C_{1}=c\cdot\frac{64H}{3\beta(m+m')^{1/2}},$$ and $$\begin{aligned}
c&=&\frac{1}{2\pi}\left[\frac{\epsilon}{3\beta}\left(1+\alpha\right)+\left(\frac{8H}{3\beta}\right)^{1/2}\right]^{-1/2}\nonumber
\\
& \times &
\left[-\frac{\epsilon}{3\beta}\left(1-\alpha\right)+\left(\frac{8H}{3\beta}\right)^{1/2}\right]^{-1/2}\,.\nonumber\end{aligned}$$ Here $K$, $E$ and $\Pi$ are the complete elliptic integrals of the first, second and third kind, respectively.
L.D. Landau and E.M. Lifshits, [*Mechanics*]{} (Pergamon Press, Oxford, 1976).
N.N. Bogolubov and Y.A. Mitropolsky, [*Asymptotic Methods In The Theory of Non-linear Oscillations*]{} (Gordon and Breach Science Publishers, New York, 1961).
R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky, [*Nonlinear Physics*]{} (Harwood Academic, Switzerland, 1988).
A.J. Lichtenberg and M.A. Lieberman, [*Regular and Chaotic Dynamics*]{} (Springer-Verlag, Oxford, 1992).
E.M. McMillan, Phys. Rev. [**68**]{}, 143 (1945).
V. Veksler, J.Phys.(USSR) [**9**]{}, 153 (1945).
D. Bohm and L. Foldy, Phys. Rev. [**70**]{}, 249 (1947).
D. Bohm and L. Foldy, Phys. Rev. [**72**]{}, 649 (1947).
K.S. Golovanivsky, Phys. Scripta [**22**]{}, 126 (1980).
B. Meerson, Phys. Lett. A [**150**]{}, 290 (1990).
B. Meerson and T. Tajima, Optics Communications [**86**]{}, 283 (1991).
L. Friedland, Phys. Plasmas [**1**]{}, 421 (1994).
B. Meerson and L. Friedland, Phys. Rev. A [**41**]{}, 5233 (1990).
J.M. Yuan and W.K. Liu, Phys. Rev. A [**57**]{}, 1992 (1998).
M. Deutsch, B. Meerson, and J.E. Golub, Phys. Fluids B [**3**]{}, 1773 (1991).
L. Friedland, Phys. Plasmas [**5**]{}, 645 (1998).
I. Aranson, B. Meerson, and T. Tajima, Phys. Rev. A [**45**]{}, 7500 (1992).
L. Friedland and A. Shagalov, Phys. Rev. Lett. [**81**]{}, 4357 (1998).
L. Friedland, Phys. Rev. E [**59**]{}, 4106 (1999).
L. Friedland and A.G. Shagalov, Phys. Rev. Lett. [**85**]{}, 2941 (2000).
J. Fajans, E. Gilson, and L. Friedland, Phys. Rev. Lett. [**82**]{}, 4444 (1999); Phys. Plasmas [**6**]{}, 4497 (1999).
B. Meerson and S. Yariv, Phys. Rev. A [**44**]{}, 3570 (1991).
G. Cohen and B. Meerson, Phys. Rev. E [**47**]{}, 967 (1993).
R.A. Struble, Quart. Appl. Math. [**21**]{}, 121 (1963).
A.D. Morozov, J. Appl. Math. Mech. [**59**]{}, 563 (1995).
M.I. Rabinovich and D.I. Trubetskov, [*Oscillations and Waves in Linear and Nonlinear Systems*]{} (Kluwer Academic Publisher, Dordrecht, 1989).
P.G. Drazin, [*Nonlinear Systems*]{} (Cambridge University Press, Cambridge, 1992).
M. Abramowitz, [*Handbook of Mathematical Functions*]{} (National Bureau of Standards, Washington, 1964).
H. Goldstein, [*Classical Mechanics*]{} (Addison-Wesley, Reading, Mass., 1980).
S. Yariv and L. Friedland, Phys. Rev. E [**48**]{}, 3072 (1993).
G.M. Fikhtengolts, [*The Fundamentals of Mathematical Analysis*]{} (Pergamon Press, New York, 1965).
|
---
author:
- 'A. Manara and S. Covino M. Di Martino'
title: Visual Spectroscopy of Asteroids at San Pedro Martir
---
Introduction
============
The population of Earth–approaching asteroids has long eluded systematic study due their small sizes and low brightness. Spectroscopic observations can help to determine the surface mineralogy of Near Earth Asteroids (NEAs). Investigations of asteroid compositions can identify potential parent bodies of specific meteorites or meteorite types or objects which have experienced similar evolutionary histories. Most meteorites are asteroidal fragments ejected from their parent bodies as a consequence of impacts, and channeled into chaotic dynamical routes, associated with mean motion and secular resonances. The main problem is that approximately 73% of the meteorites that fall on Earth are classified as ordinary chondrites (consisting of grains of olivine and pyroxene thought to be only modestly altered during the formation process), which cannot be matched with the typical observed reflectance spectra of any common asteroid taxonomic type (Wetherill & Chapman 1988). The source of these bodies is still a matter of great debate. Numerical orbital dynamic simulations (Migliorini et al. 1998) show that many asteroids in the main belt are driven toward Mars–crossing orbits by numerous weak mean motion resonances; in addition, half of the Mars–crossing asteroids are injected in Earth–crossing orbits in less than 20 million years. Gladman et al. (1996) suggest that even ejecta from Mars may be consistent with the dynamical constraints imposed by the small Earth–approachers. There has been a persistent problem of finding a source body for the ordinary chondrites. Due to their dynamically short lifetimes (10–100Myr), Near–Earth asteroids must be actively replenished (Wetherill 1985, 1988). It has been argued that S(IV)–type asteroids provide the only plausible source of parent bodies, with (3) Juno, (6) Hebe and (7) Iris being the leading candidates (Gaffey et al. 1993, Broglia et al. 1994, Migliorini et al. 1997a, 1997b).
The reflectance spectra of 3 Nysa family asteroids have been measured in order to investigate the mineralogical characterization of this family. These observations belong to a systematic campaign to study the peculiar Nysa family (Zappalà et al. 1995). We have some difficulty in assessing whether the Nysa family can be considered as a unique group or as the result of the merging of two independent families, because it is known that in the region of the belt surrounding the family, there is an unusual concentration of F–type asteroids, some of them are included into the list of nominal Nysa members, while some others apparently do not belong to the family.
We carried out spectroscopic observations of Near–Earth asteroid (3908) Nyx and 3 Nysa asteroids family (2007) McCuskey, (3130) Hillary, (3384) Daliya.
Observations and data reductions
================================
Spectroscopic observations were performed at the San Pedro Martir Observatory (Mexico) using a 2.1m telescope equipped with a Boller & Chivens spectrograph and a CCD–Tektronix TK–1024 AB detector at the f7.5 focus with a dispersion of 8Å/pixel in the wavelength direction. The grating used was a 150gr/mm with a dispersion of 326Å/mm; also a GG455 filter was used (blaze 3:26). The useful spectral range is about $4800 < \lambda < 10000$.
The slit width (2“ and 2”.5) has been chosen to minimize the consequences of atmospheric differential refraction and to reduce the loss of light at both ends of the spectrum. We observed in 1996 September 3–6, but only the first night was good for the observations.
----------------------- ------------------ -------------------------------- --------------------------------- ----------- ------------------- --------------- ----------------
[**Number & Name**]{} [**UT**]{} [**$\alpha_{\mbox{2000}}$**]{} [ **$\delta_{\mbox{2000}}$**]{} [**R**]{} [**$\Delta$** ]{} [**Phase**]{} [**$m_V$** ]{}
[*1996 Sep.* ]{} [*hmsec* ]{} [*dms*]{} [*AU*]{} [*AU*]{}
(2007) McCuskey 4.39375 004739 +034201 2.641 1.723 11.3 15.8
(3130) Hillary 4.35278 225828 -134851 1.982 0.977 3.4 14.5
(3384) Daliya 4.30555 223432 -105306 2.085 1.079 2.8 15.7
(3908) Nyx 4.23333 205411 -032556 1.174 0.186 25.2 15.2
----------------------- ------------------ -------------------------------- --------------------------------- ----------- ------------------- --------------- ----------------
\[tab:uno\]
The observational circumstances are listed in Table\[tab:uno\].
Column1 gives the observed object, column2 the date of observations, column3 and 4 RA and DEC of the object, column5 and 6 Sun and Earth distance, column7 phase angle and the last column the visual apparent magnitude.
The spectral data reduction was performed using the ESO–MIDAS package and taking much care to ensure a proper calibration of the spectra. The bias level of each night was determined through an average of the many bias images taken at night. This averaged bias was then subtracted from each frame and pixel–to–pixel variations were removed by dividing the resulting image by a normalized medium flat field. The MIDAS “long” context was used to sum the pixel values within a specified aperture and to subtract the background level. Wavelength calibration was performed several times during each night using a He–Ar lamp, and spectra were corrected for airmasses by using the mean extinction curve of San Pedro Martir (Buzzoni 1994).
This correction was checked by comparing the same analog star taken at different air mass and the differences were negligible. Since each analog was observed several times we also reduced each asteroid spectrum with the solar analog taken at the same air mass (or as near as possible). Again no difference could be observed, which confirms the quality of the data. Two solar analog, 16 CygB and HD191854 (Hardorp 1978) were observed to compute reflectivities, since these are solar analogs which closely match the spectra of the Sun. The ratios between the spectra of the two solar analogs for the night of September 3th show no substantial variation. The influence of different solar analogs on the resulting spectra has also been checked, showing negligible differences. The obtained reflectance spectra are normalized at 7000Å.
Results
=======
#### Asteroids of Nysa family
The spectra of (2007) McCuskey and (3130) Hillary are similar, therefore this confirms their membership on the same group (Fig.\[fig:1\]). These spectra are representative of subgroup of objects of the Nysa family belonging to the F taxonomic class. This conclusion can be also obtained by comparing the spectral reflectance curve of these two asteroids with those (in particular asteroids 2391, 4026, 3485, 3228, 3064, 1076) shown in Cellino et al. (2000) and in Xu et al. 1995. Concerning asteroid (3384) Daliya, his spectrum (Fig.\[fig:1\]) shows a curve similar to S–type objects (Cellino et al. 2000) and therefore is a representative of the subgroup of the Nysa family belonging to S taxonomic class. In any case it is also possible that (3384) Daliya may be an interloper of the Nysa family (Zappalà et al. 1995).
#### Asteroid (3908) Nyx
This near-Earth asteroid, classified as V–Type by Tholen and Barucci (1989), has been spectroscopically studied by McFadden et al. (1989) and Luu & Jewitt (1990). Our spectrum (Fig.\[fig:2\]) confirms the strong absorption feature centered at about 9500Å. The similarity between the spectrum of 3908 Nyx and those of objects belonging to the Vesta Family (Binzel & Xu, 1993) suggests that also this asteroid is a chip of a Vesta-like parent body (Cruikshank et al. 1991).
#### Acknowledgments
We are grateful to the staff at San Pedro Martir Observatory (Mexico) for their support during the observing runs.
Broglia P., Manara A., & Farinella P. 1994, Icarus 109, 204 Binzel R.P., Xu S. 1993, Science 260, 186 Buzzoni A., 1994, INAOE Tech. Report 184 Cellino A. 2000, private comunication Cruikshank D.P., Tholen D.J., Hartmann W.K. et al. 1991, Icarus 89, 1 Gaffey M.J., Bell J.F., Hamilton Brown R., et al. 1993, Icarus 106, 573 Gladman B.J., Burns J.A., Duncan M., Lee P., & Levison H.F. 1996, Science 271, 1387 Hardorp J. 1978, A&A 63, 383 Luu J.X., Jewitt D.C. 1990, AJ 99, 1955 McFadden L.A., Tholen D.J., Veeder G.J. 1989, in Asteroids II (R.P. Binzel, T. Gehrels and M.S. Matthews Eds.) Univ. of Arizona Press, Tucson, 442 Migliorini F., Manara A., Cellino A., Di Martino M., & Zappalà V. 1997, A&A 321, 652 Migliorini F., Manara A., Scaltriti F. et al. 1997, Icarus 128, 104 Migliorini F.P., Michel A., Morbidelli D., Nesvorný D. & Zappalà V. 1998, Science 281, 2022 Tholen D.J., Barucci M.A. 1989, in Asteroids II (R.P. Binzel, T. Gehrels and M.S. Matthews Eds.) Univ. of Arizona Press, Tucson, 298 Wetherill G W. 1985, Meteoritics 20, 1 Wetherill G W. 1988, Icarus 76, 1 Wetherill G.W. & Chapman C.R. 1988, in Meteorites and the Early Solar System (J.F. Kerridge and M.S. Matthews, Eds.) Univ. of Arizona Press, Tucson, 35 Xu S., Binzel R.P., Burbine T.H. & Bus S.J. 1995, Icarus 115, 1 Zappalà V., Bendjoya Ph., Cellino A., Farinella P. & Froeschlé C., 1995, Icarus 116, 291
|
---
abstract: |
\
Prosocial punishment has been proved to be a powerful mean to promote cooperation. Recent studies have found that social exclusion, which indeed can be regarded as a kind of punishment, can also support cooperation. However, if prosocial punishment and exclusion are both present, it is still unclear which strategy is more advantageous to curb free-riders. Here we first study the direct competition between different types of punishment and exclusion. We find that pool (peer) exclusion can always outperform pool (peer) punishment both in the optional and in the compulsory public goods game, no matter whether second-order sanctioning is considered or not. Furthermore, peer exclusion does better than pool exclusion both in the optional and in the compulsory game, but the situation is reversed in the presence of second-order exclusion. Finally, we extend the competition among all possible sanctioning strategies and find that peer exclusion can outperform all other strategies in the absence of second-order exclusion and punishment, while pool exclusion prevails when second-order sanctioning is possible. Our results demonstrate that exclusion is a more powerful strategy than punishment for the resolution of social dilemmas.
author:
- Linjie Liu
- Xiaojie Chen
- Attila Szolnoki
title: Competitions between prosocial exclusions and punishments in finite populations
---
Introduction {#introduction .unnumbered}
============
Cooperation is widespread in our world, which has a fundamental role on the evolution of human civilization [@axelod_r_m06; @taylor_m87; @searle_j10; @boyd_r09; @chase_i_d80]. However, cooperation is vulnerable to be invaded by selfish individuals who are always maximizing their short-term and immediate interests. Thus how to overcome such individuals is a vital task for the emergence of cooperation in a population [@hardin_g_s68; @komorita_s_s09; @miller_d_t99]. Several mechanisms, such as spatial reciprocity, reputation, wisdom of groups, and costly punishment, have been demonstrated to be effective for cooperators to fight against defectors [@clutton_b_t02; @fu_f08; @chen_x15; @szolnoki_a12; @boyd_r10; @santos_prl05; @perc_sr15]. Staying at the last option, costly punishment has received considerable attention in the last decade because of its importance and widespread prevalence in human societies [@henrich_j06; @chen_xj15; @helbing_d10; @chen_x_j14]. By using public goods game (PGG), which is a standard metaphor of social dilemmas, many theoretical and experimental studies have shown that prosocial punishment can reduce the number of free-riders and encourage the majority of individuals to contribute to the common pool [@denant-Boemont_l07; @sigmund_k10; @shinada_m07; @traulsen_a12].
As an alternative incentive tool to prevent free-riders exploiting community effort, social exclusion can also be observed in human societies [@gruter_m86; @feinberg_m14; @molden_d_c09]. It is based on the idea that convicted offenders are denied certain rights and benefits of citizenship or membership of joint ventures [@travis_02]. Accordingly, individuals who are identified to violate the rule or jeopardize others’ common interests could be excluded from the community [@unit_s_e01; @twenge_j_m01; @byrne_d05; @masclet_d03]. In this way exclusion serves as a sort of institution to tame defectors not to exploit others. Previous studies have shown that social exclusion can increase social sensitivity [@twenge_j_m07; @dewall_c_n11; @tuscherer_t15; @eisenberger_n_i06; @bernstein_m_j12] and induce a positive impact on cooperation when partners are fixed [@maierrigaud_f_p10; @ouwerkerk_j_w05]. Recently, Sasaki and Uchida introduced peer exclusion into the PGG and established a game-theoretical model to study the evolution of social exclusion by using replicator equations in infinite populations [@sasaki_t13]. They found that peer exclusion can overcome two shortages of peer punishment: first, a rare punishing cooperator barely subverts the asocial society of free-riders; second, natural selection often eliminates punishing cooperators in the presence of non-punishing cooperators (namely, second-order free-riders). Subsequently, Li et al. [@li_k15] studied the comparison between peer exclusion and pool exclusion, and claimed that peer excluders can overcome pool excluders if the exclusion costs are small and excluders can dominate the whole population in a suitable parameters range in the presence of second-order free-riders. Note that pool excluder, similarly to pool punisher, pays a fixed, permanent cost before contributing to the public goods to maintain an institutionalized mechanism for punishing exploiters.
To summarize our present knowledge, both prosocial exclusion and prosocial punishment have been proved to be effective ways for promoting cooperation, but their systematic comparison is still missing. Indeed, the mentioned works [@sasaki_t13; @li_k15] have compared their independent impacts on the cooperation level, but the consequence of their simultaneous presence is still unexplored. It remained unclear which strategy is more evolutionary advantageous if both exclusion and punishment are simultaneously available for individuals in the population. We wonder if exclusion or punishment is a better way to curb free-riding. How does their relation change if second-order sanctioning is also possible? In the latter case non-punishing individuals or those who deny contribution to the cost of exclusion may also be punished. Furthermore, we also wonder whether peer punishment (peer exclusion) or pool punishment (pool exclusion) is more efficient individual strategy to control transgressors for a higher well-being.
Motivated by these open problems, in this study we focus on the competition between prosocial exclusion and punishment in finite populations who play the PGG. We first investigate the direct competition between pool exclusion and pool punishment, and demonstrate that pool exclusion has the evolutionary advantage over pool punishment both in the optional and in the compulsory PGG, no matter whether second-order exclusion and punishment are considered or not. We then investigate the competition between peer exclusion and peer punishment, and find that peer exclusion is evolutionarily advantageous over peer punishment both in the optional and in the compulsory PGG, independently of the choice of second-order sanctioning. Third, we study the competition between pool exclusion and peer exclusion, and observe that peer exclusion can outperform pool exclusion both in the optional and in the compulsory PGG if second-order exclusion is ignored, while the situation is reversed in the presence of second-order exclusion. Finally, we investigate the full competition of all previously mentioned strategies, such as pool exclusion, peer exclusion, pool punishment, and peer punishment. As our main observation, it turns out that peer exclusion is the most advantageous strategy in the absence of second-order exclusion and punishment, but pool exclusion outperforms other strategies when second-order sanctioning is possible.
Model {#model .unnumbered}
=====
We consider the standard PGG in a finite, well-mixed population with size $M$. In each round of the game, $N\geq2$ individuals are selected randomly from the population to form a group for participating in a one-shot game. Then, each individual in the group decides whether or not to contribute an amount of cost $c$ to the common pool. The individual who is willing to contribute is called a cooperator, and the individual who does not contribute is called a defector. In the optional PGG we also consider a third option, a strategy which gives up participating in the game, hence is called as a loner. The latter strategy has a constant payoff $\sigma$ which is not affected by others. The sum of the contributions to the common pool is multiplied by the enhancement factor $r$ ($1<r<N$), and then equally allocated among all individuals who participated in the game no matter they contributed or not. In agreement with previous works [@hauert_c02; @hauert_c07], if only one individual participates in the game then her income equals with $\sigma$.
In the second stage of the game exclusion or/and punishment is considered where both related strategies contribute $c$ to the common pool. By following Refs. [@sigmund_k10; @sasaki_t15] peer punishers impose a fine $\beta$ on each free-rider in their group at a cost $\gamma$. Accordingly, each defector will be fined an amount $\beta N_W$, where $N_W$ is the number of peer punishers in the group. Pool punishers, however, pay a permanent cost $G$ to the punishment pool beforehand. If there exist defectors in the group, they will be fined an amount $BN_V$, where $N_V$ is the number of pool punishers in the group. It simply means that the additional cost of pool punisher is independent of the number of defectors in the group, while the related cost of peer punisher is proportional to the presence of defectors. If considering second-order punishment, second-order free-riders (individuals who contribute to the game but do not bear the extra cost of punishment) will be fined the same amount [@sigmund_k10].
When exclusion is applied we follow conceptually similar protocol as for punishment. Here exclusion serves as a sort of institution to prevent defectors to exploit other group members. Hence the role of excluder can be viewed as a sentinel who alarms other group members about the danger of defectors. Evidently, such an extra effort requires additional cost which is paid by excluder player. Consequently, a peer excluder does not only contribute $c$ to the public goods game but also pay a cost $c_{E}$ after every defector in the group to prevent them collecting benefit from the public goods sharing. In stark contrast to peer exclusion, pool excluders pay a permanent cost $\delta$ to maintain the institution of exclusion which will block defectors to gain benefit from PGG in the presence of pool excluders. As previously, in case of the second-order exclusion, second-order free-riders (individuals who do not take the extra cost of exclusion) will also be excluded.
In order to study the evolutionary dynamics, we use the so-called pairwise comparison rule with the mutation-selection process [@szabo_pr07; @traulsen_a07]. According to this protocol at each time step a randomly chosen player $i$ may change her strategy. We consider the possibility of mutation, hence the player adopts a randomly chosen available strategy with probability $\mu$. Alternatively, which happens with probability $1-\mu$, a player $i$ tries to imitate a randomly chosen player $j$ with a probability $$\begin{aligned}
f(\Pi_{j}-\Pi_{i})=\frac{1}{1+\exp^{-\kappa(\Pi_{j}-\Pi_{i})}} \,\,.\end{aligned}$$ Here $\Pi_{i}$ and $\Pi_{j}$ are the collected payoffs of the mentioned players $i$ and $j$, while $\kappa$ characterizes the intensity of selection. In the $\kappa \rightarrow \infty$ strong imitation limit the more successful player $j$ always succeeds in enforcing her strategy to player $i$, but never otherwise. On the other hand, $\kappa \rightarrow 0$ indicates the so-called weak selection limit where strategy adoption becomes random independently of the payoff values. In between these extremes, at a finite value of $\kappa$, it is likely that a better performing player $j$ is imitated, but it is still not impossible to adopt her strategy when performing worse.
In the following we consider four different scenarios when punishment and exclusion compete and we compute the resulting stationary distribution of all available strategies. We suppose a well-mixed finite population where all players interact with each other randomly. To make comparison with previous works easier we have adopted notations for variables by earlier works [@sigmund_k10; @sasaki_t13]. Accordingly, let $X$ denote the number of cooperators who contribute to the public pool, but do not bear the cost of punishment or exclusion; $Y$ the number of defectors who contribute neither to PGG nor to the sanctions; $Z$ the number of loners; $V$ the number of pool punishers; $W$ the number of peer punishers; $F$ the number of pool excluders; and $E$ the number of peer excluders. The whole size of population is denoted by $M$ and $N$ randomly chosen individuals are offered to form a group and establish a joint enterprise. In the next section we present the results of the more complex optional PGG while further details and results for the simplified compulsory PGG game are summarized in the Supplementary Information (SI).
Results {#results .unnumbered}
=======
Competition between pool exclusion and pool punishment {#competition-between-pool-exclusion-and-pool-punishment .unnumbered}
------------------------------------------------------
We first study the direct competition between pool exclusion and pool punishment in the optional PGG. In this scenario, there are five available strategies in the population fulfilling the constraint $X+Y+Z+F+V=M$. We assume that $0<\sigma<rc-c-\delta$ and $0<\sigma<rc-c-G$, which ensure that a punisher or excluder can get higher profit than a loner if there is more than one participant in the group. In the absence of second-order exclusion and punishment only defectors are sanctioned by punishment or/and exclusion. In Fig. \[fig1\](a) we plot the long-run frequencies for each strategy which determine the stationary distribution of all available strategies in dependence of imitation strength $\kappa$. We find that for $\kappa<10^{-4}$ the frequencies of the five competing strategies are identical due to the practically random imitation process. As we increase the strength of imitation then all the five strategies can survive and coexist. More precisely, in the perfect imitation limit the system evolves towards a homogeneous state, where the flips between almost homogeneous states are triggered by rare mutations. (A representative trajectory of evolution can be seen in Fig. 1 in Ref. [@sigmund_k10].) In this way the presented frequencies of stationary states are calculated from the time average of frequencies for competing strategies. As Fig. \[fig1\](a) suggests pure cooperators form the highest portion who can enjoy the benefit of exclusion and punishment without paying their costs. Interestingly, the second largest population is formed by pool-excluders followed by defectors and loners, while pool-punishers can make up the smallest fraction, or can be detected with the smallest probability. This result suggests that pool exclusion is more effective against defection and has an evolutionary advantage over pool-punishment strategy. In particular, in the strong imitation ($\kappa \rightarrow \infty$) limit the long-run frequencies in the $[X, Y, Z, F, V]$ subpopulations are $\frac{3}{8}, \frac{3}{16}, \frac{1}{8}, \frac{4M+5}{16M+32}$, and $\frac{M+5}{16M+32}$, respectively (for further details, see Section $1$ in SI).
The comparison of stationary strategies in Fig. \[fig1\](a) emphasizes that it is better to cooperate but also to avoid the additional cost of sanctions. Needless to say, if everyone chose this option then we would face the original dilemma. To minimize this undesired consequence of “second-order free-riders" we may penalize those who refuse participating in the sanctioning process. In particular, when second-order exclusion and punishment are considered, we assume that pool-excluders drive out all strategies who do not contribute to the exclusion pool. In parallel, pool-punishers will also punish those who do not bear the additional cost of punishment pool. By applying this scenario, we observe that the long-run frequencies in the $[X, Y, Z, F, V]$ subpopulations are $[0, 0, 0, 1, 0]$ for $\kappa>10^{-3}$, as shown in Fig. \[fig1\](b). In other words, when second-order sanctioning is allowed pool-excluders prevail and all other strategies extinct during the evolutionary process. To answer our original question both panels plotted in Fig. \[fig1\] highlight that pool-exclusion is a more advantageous strategy than pool-punishment independently of second-order sanctioning is considered or not. We should stress that our observation remains valid for a broad range of parameter interval including the level of the punishment fine or the group size. Furthermore, conceptually identical conclusion can be made if the compulsory PGG is assumed where loner strategy cannot compete (the related results are presented in Sec. $2$ of SI).
****
Competition between peer exclusion and peer punishment {#competition-between-peer-exclusion-and-peer-punishment .unnumbered}
------------------------------------------------------
In this subsection, we investigate the competition between peer punishment and peer exclusion in the optional PGG. According to this scenario there are also five available strategies in the population whose fractions fulfill $X+Y+Z+E+W=M$. In the absence of second-order punishment and exclusion, we assume that $0<c_{E}<\frac{rc-c}{N-1}$, which ensures that a single peer excluder can invade a group of all defectors. In Fig. \[fig2\](a) we present the long-run frequencies of the five strategies as a function of the imitation strength $\kappa$. As expected, for weak selection, when $\kappa<10^{-4}$, the frequencies of the five strategies are practically identical because of the random strategy updating. By increasing the imitation strength $\kappa$ peer excluder strategy becomes gradually dominant and occupies the majority of the population. Peer punisher strategy can only reach the second best position in the rank of strategies. In the strong imitation limit the long-run frequencies for the $[X, Y, Z, E, W]$ subpopulations are $[\frac{6}{5M+23}, \frac{3}{5M+23},
\frac{2}{5M+23}, \frac{3M+6}{5M+23}, \frac{2M+6}{5M+23}]$ (see Section. $3$ of SI). This suggests that the number of peer excluders is about $1.5$ times larger in time average than the second best peer punishers for large population size, hence demonstrating the superiority of the former strategy.
If second-order sanctioning is allowed then the relation of sanctioning strategies becomes even more unambiguous. Interestingly, in this case excluders do not only ostracize pure cooperators but also punishers who refuse to contribute to the cost of exclusion. But the penalty works also in the reversed direction because punishers lower the payoff of both cooperation and exclusion strategies. The consequence of this mutual sanctioning is summarized in Fig. \[fig2\](b), which suggests that peer exclusion prevails and gives no space for any other strategies. This observation supports our previous conclusion about the effectiveness of exclusion that is not restricted to pool strategies, but is still valid for peer strategies. We stress that this conclusion remains unchanged if we release the restriction for the value of $c_E$, which means that for $c_E>\frac{rc-c}{N-1}$ peer exclusion has still evolutionary advantage over peer punishment. The border within this observation is valid can be extended further because peer exclusion outperforms peer punishment in the compulsory PGG, no matter whether second-order sanctioning is applied or not (for more details see Section $4$ in SI).
****
Competition between pool exclusion and peer exclusion {#competition-between-pool-exclusion-and-peer-exclusion .unnumbered}
-----------------------------------------------------
In the following subsection we compare the peer- and pool-exclusion strategies in the optional PGG, which are proved to be more effective than their punishing mates in the previously studied cases. Here, there are five available strategies in the population whose fractions fulfill the constraint $X+Y+Z+F+E=M$. For their proper comparison we assume that their costs remain below the previously established limit, that is $0<c_{E}<\frac{rc-c}{N-1}$ and $ 0<\sigma<rc-c-\delta$. First, we consider the case when second-order exclusion is not allowed, hence both excluder strategies penalize pure defectors only. Fig. \[fig3\](a) illustrates that peer excluder strategy becomes dominant as we gradually increase the imitation strength. All the other strategies can share a reasonable portion only at an intermediate value $\kappa$. If the imitation strength exceeds the threshold $\kappa
>10^{-1}$ then the long-run frequencies of defectors, loners, and pool excluders are close to zero, and only cooperators can coexist with peer excluders. In particular, the fractions of $[X, Y, Z, F,
E]$ strategies in the strong imitation ($\kappa\rightarrow \infty$) limit are $[\frac{9}{6M+25}, \frac{3}{6M+25}, \frac{2}{6M+25},
\frac{2}{6M+25}, \frac{6M+9}{6M+25}]$ (further details can be seen in Section $5$ of SI). These results suggest that peer excluder strategy is able to dominate the whole population in the absence of second-order exclusion.
Interestingly, the outcome of evolutionary trajectory is completely reversed if second-order exclusion is considered. In this case, in strong agreement with a previous work where peer- and pool-punisher strategies were compared [@sigmund_k10], pool excluders are capable to crowd out peer excluders. The result of this competition is summarized in Fig. \[fig3\](b) where the long-run frequencies for each strategy are plotted. In the strong imitation limit the victory of pool excluders is total, yielding $[0, 0, 0, 1, 0]$ fractions for $X,Y,Z,F$, and $E$ strategies respectively. As in the previous cases, these results remain valid if the compulsory PGG is played. Here, in the absence of loners, peer-excluders dominate when second-order exclusion is not considered yielding $[\frac{4}{3M+11},
\frac{2}{3M+11}, \frac{1}{3M+11}, \frac{3M+4}{3M+11}]$ values for the competing $X, Y, F$, and $E$ strategies in the strong imitation limit (details can be found in Sec. 6 of SI). When second-order exclusion is possible then the pool excluder strategy prevails in close agreement with the result of optional PGG.
****
Competition between prosocial exclusions and punishments {#competition-between-prosocial-exclusions-and-punishments .unnumbered}
--------------------------------------------------------
The pair comparison of competing strategies may provide a first guide about their relations, but the presence of a third party could be a decisive factor, which may completely rearrange the ranks of competitors. To clarify this possibility in the following we explore the simultaneous competitions of all previously studied strategies. Namely, we consider an optional PGG where seven strategies, namely pure cooperator, defector, loner, peer excluder, peer punisher, pool excluder, and pool punisher are present. As in the previous cases, we first consider the option when second-order sanctions are not applied hence only defectors suffer from the presence of excluders and punishers. Fig. \[fig4\](a) summarizes our results, which suggest that “peer-sactioning" strategies are the most effective, but more importantly, peer excluders can dominate the population. In this way the dominance of peer excluders over peer punishers is not disturbed by the presence of other sanctioning strategies such as pool excluders or pool punishers. As Fig. \[fig4\](a) shows all the other strategies become irrelevant in the strong imitation limit. In particular, the long-run frequencies of $X,Y,Z,E,W,F$ and $V$ subpopulations are $\frac{45}{163+35M}$, $\frac{15}{163+35M}$, $\frac{6}{163+35M}$, $\frac{20M+45}{163+35M}$, $\frac{15M+45}{163+35M}$, $\frac{18M+7}{(163+35M)(3M+2)}$, and $\frac{3M+7}{(163+35M)(3M+2)}$, respectively (further details can be seen in Section $7$ of SI). This result suggests that only the sanctioning strategies survive in the large population limit where the majority of individuals are peer excluders in most of the time.
In the next logical step we consider the case when second-order sanctioning is possible. This option offers an extremely complex food-web between competing strategies, because practically all sanctioning strategies try to beat all the others. For instance, pool excluders ostracize not only defectors and simple cooperators, but also peer excluders, peer punishers, and pool punishers. In this “almost everybody beats everybody else" battle the final victor is pool excluder strategy. This case is plotted in Fig. \[fig4\](b) where the resulting fractions of the strategies are $[0, 0, 0, 0, 0,
1, 0]$ in the strong imitation limit (further details can be seen in Sec. 7 of SI).
To close this section we briefly summarize the results of the compulsory PGG where 6 competing strategies remain. The details of the calculation can be found in Sec. 8 of SI. In the absence of second-order exclusion and punishment, we find that the behaviour is conceptually similar to the one we observed for the optional PGG. Here peer excluders and peer punishers perform the best, but all the other strategies survive at intermediate strength of imitation. In the strong imitation limit the resulting fractions of $X,Y,E,W,F$, and $V$ strategies are $[\frac{6}{5M+22}, \frac{3}{5M+22},
\frac{3M+6}{5M+22}, \frac{2M+6}{5M+22}, \frac{3M+1}{(5M+22)(3M+2)},
\frac{1}{(5M+22)(3M+2)}]$, which suggests that only sanctioning strategies survive in the large-population limit. When second-order exclusion and punishment are possible then we get back the result obtained previously for the optional PGG: only pool excluders survive for strong enough imitation strength.
Discussion {#discussion .unnumbered}
==========
Penalizing free-riders whose behaviour threaten the collective efforts seems almost inevitable. But which sanctioning tool shall we apply to reach our goal efficiently? To punish them by lowering their payoffs or to deny their rights to enjoy the benefit of public goods? The answer could be even more complicated because both peer and pool sanctioning can be used. While peer punishers and peer excluders invest an extra cost only in the presence of defectors, pool punisher and pool excluder strategies apply a permanent effort to maintain the sanctioning institutions. Based on previous works both punishment and exclusion seem to be appropriate methods [@sigmund_k10; @abdallah_s14], but their systematic comparison has not been done yet.
In this work, we have thus studied the competitions between costly punishments and exclusions in finite populations playing the PGG by using different scenarios. For a fair comparison we have applied equally high cost of punishment and exclusion. We have found that peer exclusion is always favored by natural selection when it competes with peer punishment both in the optional and in the compulsory PGG, independently of second-order punishment and exclusion are considered or not. Conceptually similar findings have been obtained for pool exclusion when it directly competes with pool punishment. Furthermore, when peer exclusion competes with pool exclusion, peer exclusion wins in the absence of second-order exclusion, while pool exclusion prevails when second-order exclusion is applied. Lastly, we have also explored the most complex option when all four sanctioning methods compete with the pure cooperator, defector, and loner strategies. In the latter case peer exclusion is proved to be the most viable tool in the absence of second-order punishment and exclusion, while pool exclusion prevails when second-order sanctioning is allowed. To sum up, the systematic comparison of sanctioning strategies highlights that exclusion is always a more effective way to control free-riders than punishment, but the absence or the presence of second-order sanctioning could be a decisive factor, because the former condition supports peer exclusion while the latter option helps pool exclusion strategy to prevail.
We would like to stress that our finding is robust and remains valid in a broad range of model parameters (some representative plots are given in Sec. $9$ of SI). For instance, if we increase the punishment fine by fixing the cost of punishment then the superiority of exclusion is still not in danger. In general, if the fine is not unrealistically high and the cost of exclusion does not exceed the cost of punishment then exclusion strategy always performs better similarly to the cases we discussed earlier. Indeed, we have verified that in the absence of second-order sanctioning the exclusion strategy still has an evolutionary advantage over the punishment strategy, no matter an enhanced fine value applied by peer and pool punishers. To give an example, the outcomes remain conceptually intact when the punishment fine exceeds eight times the punishment cost. But if second-order sanctioning is applied at such severe punishment then the advantage of excluders diminishes because their payoff becomes negative, which implies the victory of punishers. However, we should note that applying such a severe punishment is not an attractive feature when humans qualify potential social partners [@rockenbach_pnas11]. We have also considered different group sizes and found that it has no significant role in the competition of sanctioning strategies (this is demonstrated clearly in Fig. S7 of SI).
In order to provide a convenient framework for studying the competitions between costly exclusions and punishments, we focused on the option when free-riders are always exiled in the presence of excluders who have to bear the related cost. A further step could be when this sanction is not perfect and exclusion happens in a probabilistic manner [@sasaki_t13; @li_k15]. Indeed, previous works emphasized the value of probabilistic sanctioning [@szolnoki_jtb13; @chen_x_j14], which opens promising avenue for future studies. Our work can be also extended where the error of perception, i.e. defectors are identified with some ambiguity, or the error of punishment or exclusion are also considered. In the latter cases innocent players are punished or excluded from the joint venture by mistake. To consider anti-social punishment and anti-social exclusion may also open interesting research avenue to explore the effectiveness of exclusion [@hauser_jtb14; @dossantos_prsb15; @szolnoki_prsb15]. Lastly, we note that our calculation is restricted to the simplest, well-mixed population because of the extremely high number of competing strategies. However, it is a frequently discussed fact that in structured populations, where interaction topology is considered, the evolutionary outcomes could be significantly different from those presented for mean-field systems [@masuda_prsb07; @perc_m13; @pinheiro_njp12; @szolnoki_pre11; @szolnoki_prx13; @chen_pre15]. Therefore, we expect similar exciting new observations from related efforts which will hopefully make our understanding more accurate.
[54]{}
Axelrod, R. M. . (Basic books, New York, 2006).
Taylor, M. . (Cambridge University Press, Cambridge, UK, 1987).
Searle, J. . (Oxford University Press, London, UK, 2010).
Boyd, R. & Richerson, P. J. . , 3281-3288 (2009).
Chase, I. D. . , 827-857 (1980).
Hardin, G. . , 1243-1248 (1968).
Komorita, S. S. & Lapworth, C. W. Cooperative choice among individuals versus groups in an N-person dilemma situation , 487 (1982).
Miller, D. T. . , 1053 (1999).
Santos, F. C. & Pacheco, J. M. . , 098104 (2005).
Perc, M. & Szolnoki, A. . , 11027 (2015).
Clutton-Brock, T. Breeding together: kin selection and mutualism in cooperative vertebrates. , 69-72 (2002).
Fu, F., Hauert, C., Nowak, M. A. & Wang, L. Reputation-based partner choice promotes cooperation in social networks. , 026117 (2008).
Chen, X., Sasaki, T. & Perc, M. Evolution of public cooperation in a monitored society with implicated punishment and within-group enforcement. , 17050 (2015).
Szolnoki, A., Wang, Z. & Perc, M. . , 576-576 (2012).
Boyd, R., Gintis, H. & Bowles, S. . , 617-620 (2010).
Henrich, J., McElreath, R., Barr, A. & Lesorogol, C. Costly punishment across human societies. , 1767-1770 (2006).
Chen, X., Sasaki, T., Brännström, [Å]{}. & Dieckmann, U. . , 20140935 (2015).
Helbing, D., Szolnoki, A., Perc, M., & Szab[ó]{}, G. . , 083005 (2010).
Chen, X., Szolnoki, A. & Perc, M. . , 083016 (2014).
Denant-Boemont,L., Masclet, D. & Noussair, C. N. . , 145-167 (2007).
Sigmund, K., De Silva, H., Traulsen, A. & Hauert, C. . , 861-863 (2010).
Shinada, M. & Yamagishi, T. . , 330-339 (2007).
Traulsen, A., Röhl, T., & Milinski, M. . , 3716-3721 (2012).
Gruter, M. & Masters, R. D. . , 149-158 (1986).
Feinberg, M., Willer, R., & Schultz, M. Gossip and ostracism promote cooperation in groups. , 656-664 (2014).
Molden, D. C., Lucas, G. M., Gardner, W. L., Dean, K., & Knowles, M. L. . , 415 (2009).
Travis, J. . (The New Press, London, New York, 2002).
Unit, S. E. & Britain, G. . (Cabinet Office, London, 2001).
Twenge, J. M., Baumeister, R. F., Tice, D. M. & Stucke, T. S. . , 1058 (2001).
Byrne, D. . (McGraw-Hill Education, London, UK, 2005).
Masclet, D. . , 867-887 (2003).
Bernstein, M. J. & Claypool, H. M. . , 113-130 (2012).
DeWall, C. N., Deckman, T., Pond, R. S. & Bonser, I. . , 1281-1314 (2011).
Tuscherer, T., Sacco, D. F., Wirth, J. H., Claypool, H. M., Hugenberg, K. & Wesselmann, E. D. . , 280-293 (2015).
Twenge, J. M., Baumeister, R. F., DeWall, C. N., Ciarocco, N. J. & Bartels, J. M. . , 56 (2007).
Eisenberger, N. I., Jarcho, J. M., Lieberman, M. D. & Naliboff, B. D. . , 132-138 (2006).
Maier-Rigaud, F. P., Martinsson, P. & Staffiero, G. . , 387-395 (2010).
Ouwerkerk, J. W., Kerr, N. L., Gallucci, M. & Van Lange, P. A. . , 321-332 (2005).
Sasaki, T. & Uchida, S. . , 20122498 (2013).
Li, K., Cong, R., Wu, T. & Wang, L. . , 042810 (2015).
Hauert, C., De Monte, S., Hofbauer, J. & Sigmund, K. Volunteering as red queen mechanism for cooperation in public goods games. , 1129-1132 (2002).
Hauert, C., Traulsen, A., Brandt, H., Nowak, M. A. & Sigmund, K. . , 1905–1907 (2007).
Sasaki, T., Uchida, S. & Chen, X. . , 8917 (2015).
Szabó, G. & Fáth, G. . , 97–216 (2007).
Traulsen, A., Pacheco, J. M. & Nowak, M. A. . , 522-529 (2007).
Abdallah, S., Sayed, R., Rahwan, I., LeVeck, R. L., Cebrian, M, Rutherford, A. & Fowler, J. H. . , 20131044 (2014).
Rockenbach, B. & Milinski, M. . , 18307–18312 (2011).
Szolnoki, A. & Perc, M. . , 34–41 (2013).
Hauser O. P., Nowak, M. A. & Rand, D. G. , 163–171 (2014).
dos Santos, M. , 20141994 (2015).
Szolnoki, A. & Perc M. , 20151975 (2015).
Masuda, N. . , 1815–1821 (2007).
Perc, M., Gómez-Gardeñes, J., Szolnoki, A., Floría, L. M. & Moreno, Y. . , 20120997 (2013).
Chen, X., Szolnoki, A. & Perc, M. . , 012819 (2015).
Szolnoki, A. & Perc, M. . , 041021 (2013).
Pinheiro, F. L., Santos, F. C. & Pacheco, J. M. . , 073035 (2012).
Szolnoki, A., Szab[ó]{}, G. & Czak[ó]{}, L. . , 046106 (2011).
**Acknowledgments**\
This research was supported by the National Natural Science Foundation of China (Grants No. 61503062) and the Fundamental Research Funds of the Central Universities of China.
\
**Author contributions**\
The authors designed and performed the research as well as wrote the paper.
\
**Competing financial interests**\
The authors declare no competing financial interests.
|
---
abstract: 'One of the predictions of quantum gravity phenomenology is that, in situations where Planck-scale physics and the notion of a quantum spacetime are relevant, field propagation will be described by a modified set of laws. Descriptions of the underlying mechanism differ from model to model, but a general feature is that electromagnetic waves will have non-trivial dispersion relations. A physical phenomenon that offers the possibility of experimentally testing these ideas in the foreseeable future is the propagation of high-energy gamma rays from GRBs at cosmological distances. With the observation of non-standard dispersion relations within experimental reach, it is thus important to find out whether there are competing effects that could either mask or be mistaken for this one. In this letter, we consider possible effects from standard physics, due to electromagnetic interactions, classical as well as quantum, and coupling to classical geometry. Our results indicate that, for currently observed gamma-ray energies and estimates of cosmological parameter values, those effects are much smaller than the quantum gravity one if the latter is first-order in the energy; some corrections are comparable in magnitude to the second-order quantum gravity ones, but they have a very different energy dependence.'
author:
- Luca Bombelli
- Oliver Winkler
date: 29 Mar 2004
title: |
Comparison of QG-Induced Dispersion\
with Standard Physics Effects
---
Introduction {#introduction .unnumbered}
------------
One of the heuristic predictions from quantum gravity phenomenology that has been attracting a considerable amount of attention is the possibility that fields propagating in a quantum spacetime may exhibit dispersion, due to their interaction with the fluctuating, possibly discrete quantum geometry. In general the dispersion relation in such scenarios, whether they be motivated by loop quantum gravity, spin foams, strings, or non-commutative geometry, is obtained from a modified mass shell equation of the form [@Ame97; @Ame04] c\^2p\^2 = E\^2\[1 + (E/E\_[QG]{})\^+ \], \[mom-en\] where $\alpha$ and $\beta$ are model-dependent constants, with $\beta = 1$ or 2 considered to be the likely values. This corresponds to propagation in a medium with an effective frequency-dependent phase velocity and index of refraction $n(\omega) = c/v_{\rm ph}(\omega) = cp/E$, which gives n\_[QG]{} = 1 + (E/E\_[QG]{})\^+ = 1 + (/\_[QG]{})\^+ \[disprel\] The effect is expected to be very small, but it may become observable in the case of gamma rays from distant GRBs, propagating over cosmological distances of billions of light years. For such photons, we can assume the high end of the energy spectrum to be somewhat higher than 1 MeV [@ZM], which means that $E/E_{\rm QG}$ is slightly larger than $10^{-22}$, if we take the quantum gravity energy scale to be the Planck energy, $E_{\rm QG} = E_{\rm P} = \sqrt{\hbar c^5/G} \simeq 10^{19}$ GeV. Observing this effect would be extremely interesting, but the smallness of the estimated numbers implies that this may be experimentally possible [@Ame97] in the near future only if the leading order term in Eq \[disprel\] corresponds to $\beta = 1$.
Observationally, bounds on $n-1 \sim \Delta c/c$ around $10^{-20}$, close to the $\beta = 1$ range, have already been obtained [@Sch; @Gha] and data have already been used to set bounds on parameters for Lorentz symmetry violating models [@PS]. Improved techniques may soon allow us to make more general statements about first order Planck-scale effects; as the search for quantum gravity effects on photon dispersion is pushed toward smaller orders of magnitude, it is becoming increasingly important not just to improve the observational tools, but also to be able to distinguish this effect from other actual physical effects that could either mask it or be mistaken for it (“theoretical noise"). One should therefore examine systematically other possible mechanisms, such as QED vacuum effects or couplings to other forms of classical matter, quantum fields, and geometry which can produce dispersion in gamma rays.
In this letter, we will consider effects from standard physics only. Classically, there are two possibly relevant interactions, the electromagnetic and the gravitational ones. We will begin by estimating the contribution to dispersion by the plasma effect due to electrons in the galactic and intergalactic medium, treated as free particles because of the gamma-ray frequencies involved. Then we will consider the general relativistic coupling of photons to gravity by scattering off the curved geometry, both by multiple scattering from particles and local gravitating objects of any size, and by scattering off the global geometry. Finally, we look at non-trivial vacuum effects in QED as a possible source of dispersion, by gathering results from the existing literature and applying them to our situation. In each case, the first goal is to compare the deviation of $n$ from 1 produced by the effect under consideration with the quantum gravity estimate. If the two contributions turned out to be of similar orders of magnitude in the relevant energy range, we might still be able to discriminate between them if the energy dependences are different; therefore the next important point would be to compare the frequency dependences.
A few parameter values will be needed for our estimates. In addition to a reference gamma-ray energy that we will take to be $E_\gamma = 1$ MeV, corresponding to a frequency $\omega_\gamma \approx 1.5 \times 10^{21}$ rad/s, the main obervational values we will use are [@WMAP] the baryon density $\rho_{\rm b} \approx 4.2 \times 10^{-28}$ kg/m$^3$, equivalent to a number density $N_{\rm b} \sim 0.25$ m$^{-3}$, the total fractional energy density $\Omega_{\rm tot} = 0.02 \pm 0.02$, and the Hubble parameter $H_0 \approx 71$ km/s/Mpc.
Dispersion Due to Classical Electromagnetic Interactions {#dispersion-due-to-classical-electromagnetic-interactions .unnumbered}
--------------------------------------------------------
Cosmological gamma ray photons interact electromagnetically with charged particles they encounter, mainly electrons and protons. A detailed treatment of this interaction would consider the states of these particles (bound or free) and take into account various effects. However, the particles that most affect the photons’ propagation are the lighter ones, and at the gamma-ray energies we are considering, much higher than their binding energies, the electrons can be effectively treated, for our purposes, as free particles.
As part of our underlying model, we will assume that space is homogeneous, aside from localized electromagnetic or gravitational scatterers, and we will characterize these scatterers only by their average properties, reducible to the numbers listed at the end of the previous section. Photons propagating in intergalactic space will then see a medium that we treat as a uniform plasma of free electrons, which responds to an electromagnetic wave by absorbing and re-radiating energy. As an extended distribution, the electrons produce a cumulative effect which can be described, in the high-frequency approximation, by the dispersion relation [@Jac] n = 1- [NZe\^22\_0\^2]{}, where as an approximate value for the product $NZ$ of the effective atom number density and the effective atomic number we will use the baryon number density.
With those parameter values we can estimate the effect for our reference 1-MeV photon, n - 1 1.8 10\^[-40]{}, a value greater than the expected quantum gravity one for $\beta = 2$. Notice however that the value is very approximate. Furthermore, the correction term has an $\omega^{-2}$ dependence, which means that in the infinite-frequency limit there is no dispersion (geometric optics approximation), contrary to the quantum gravity case, in which high-frequency photons probe the small-scale geometry better and produce a larger effect [@Ame97].
Dispersion from Classical Gravitational Multiple Scattering {#dispersion-from-classical-gravitational-multiple-scattering .unnumbered}
-----------------------------------------------------------
In general relativity, one potential source of dispersion is geometrical scattering off objects in an extended random distribution or cluster, a “gas" of gravitating masses. Such an effect was briefly considered in the early 1970’s by Peters [@Pet]. In Peters’ treatment, the scatterer distribution is modeled by (the continuum limit of) a sum of weak-field approximations to Schwarzschild metrics centered at locations ${\bf r}_i$, represented by a scattering potential ([**r**]{}) = -\_i[m\_i|[**r**]{}-[**r**]{}\_i|]{}. When a plane wave is incident on a layer of such a “gas", the superposition of waves diffracted by individual objects gives rise to the effective dispersion, since each single diffraction pattern is $\omega$-dependent; this is analogous to what happens with ordinary dispersion in the atmosphere, although the scattering mechanism from individual scatterers is different in that case. Peters considered scalar, electromagnetic and gravitational waves. While for the (minimally coupled) scalar and gravitational wave cases he found the dispersion relation n = 1 + [2\^2]{}, in the electromagnetic case he found that, to first order in the scatterer masses $m_i$, there was [*no*]{} frequency-dependent phase shift for a plane wave after crossing the layer of scatterers. However, the approximations he used, both in modeling the situation and in treating the quantities encountered in the calculations (some of which were divergent, as one might expect from scattering from a Coulomb-type potential), lead us to believe that his results are not very conclusive, and suggest that we do not yet discard the effect. A more cautious approach may be not to rely on the validity of Peters’ calculations, but to simply notice that, independently of any model, one can estimate the size of the dispersion effect on purely dimensional grounds. Since to first order $n-1$ will be proportional to $\GN\rho$, we conclude that, [*if*]{} there is a gravitational multiple scattering effect on the index of refraction, in terms of orders of magnitude it will be at most n - 1 \~[\_[matter]{}\^2]{}, in agreement with Peters’ results for the other types of waves, where this time for $\rho$ we use the average density of all gravitating matter, roughly 6 times larger than $\rho_{\rm b}$. Notice that this estimate, as can be seen in Peters’ calculations [@Pet], depends only on the average $\rho$, and not on details such as the masses and sizes of individual scatterers.
If we again estimate the effect for our 1-MeV photon, we get n - 1 7.5 10\^[-80]{}; thus, not only do we obtain an inverse $\omega$-dependence, consistent with what we expect from a classical effect, but the magnitude of the departure of $n$ from 1 is such that it would not compete with the quantum gravity effect even for $\beta = 3$.
Dispersion from Scattering off the Global Curvature — “Wave Tails" {#dispersion-from-scattering-off-the-global-curvature-wave-tails .unnumbered}
------------------------------------------------------------------
In addition to the average densities used in previous sections, a homogeneous cosmological model is characterized by a global spatial geometry and expansion rate. When waves propagate in a curved spacetime, they can scatter off the global curvature, a phenomenon that is usually described in terms of the formation of tails, or non-validity of the Huygens principle, rather than in terms of a modified index of refraction. It is known, for example, that a necessary condition for the validity of the Huygens principle in 4-dimensional spacetime is that the geometry be that of an Einstein space [@Gol], and that tails generically form in the propagation of fields, both near isolated objects [@Man] and in a cosmological setting [@FG]. It would be useful therefore to analyze the latter effect in more detail in terms of modified effective dispersion relations.
The recent WMAP observational data on the microwave background are consistent with, indeed can be taken to be an indication of, a vanishing overall spatial curvature, in which case the universe on “average" is a $k = 0$ Robertson-Walker space; such spaces are conformally flat, and since the classical Maxwell equations are conformally invariant, photons cannot develop tails from their propagation in the overall geometry. A more careful analysis (motivated by the fact that in some situations cosmological tails can be strong [@Nol]) would take into account the actual bounds on the spatial curvature; in this paper, we will limit ourselves to a simple comment. We can obtain a bound for the radius of curvature $R$ of space using the relationship R = [cH\_0]{}[1|1-\_[tot]{}|\^[1/2]{}]{} between cosmological parameters and spatial geometry, and the WMAP data [@WMAP]. Specifically, an indicative lower bound on $R$ can be obtained from the upper limit of the error bar on $\Omega_{\rm tot}$, giving R 3.010\^4 = 9.310\^[26]{}. Any classical propagation effect for photons in curved spacetime that depends only on $R$ and the wavelength will give a contribution to $n-1$ which is at most of the order of $\lambda/R$, with $\lambda \approx c/(\omega/2\pi)$. In our case, $\lambda \approx 10^{-12}$ m, and we conclude that the contribution to dispersion would be at most n-1 10\^[-39]{}.
Dispersion from QED Effects {#dispersion-from-qed-effects .unnumbered}
---------------------------
If we take into account the fact that a better description of gamma-ray propagation consists in treating it as taking place on the background of some (homogeneous) QED state, several effects arise which can modify their dispersion relations [@Sho]. These effects do not show up for light propagating in the QED vacuum for flat spacetime with no other background fields, but in a cosmological setting the speed of propagation of light can be affected by a background electromagnetic field, by the overall spacetime curvature (through vacuum polarization), and by the cosmic microwave background radiation (through photon-photon interaction). (Higher-derivative gravity theories also give rise to modified, dispersive photon propagation [@AB], but we will not consider those here.)
For high-frequency electromagnetic waves propagating in a weak background magnetic field, the main parameter which determines whether the effect is dispersive or not is the number [@TE] := [eB\^2\^2\^3c\^4]{} . Here, $B$ is the magnitude of the magnetic field, assumed to be constant, and $\theta$ the angle between $\vec B$ and the direction $\hat k$ of propagation. Dispersion occurs only if $\lambda > 1$. While the magnetic field in intergalactic space is not very well known [@Gio], we can take $B \approx 10^{-7}$ G as an indicative value, at least for clusters of galaxies (it is probably smaller outside, and the fact that it is not really constant will also decrease the effective value of $\lambda$ and the overall effect); as a further overestimate, let us set $\sin\theta = 1$. We then get $\lambda \approx 4.3\times 10^{-60}\,\omega$(rad/s); even for the highest frequency $\gamma$-rays, $\lambda \ll 1$, in which case the effective index of refraction (for both polarizations) is n - 1 = [C4]{} ([eB\^2c\^2]{})\^[2]{}, where $\alpha$ is the fine structure constant and $C$ a known polarization-dependent number of order 1. This result is independent of $\omega$, so the effect is non-dispersive (with our parameter values, this contribution to $n - 1$ is of the order of $10^{-37}$).
Let us now consider photon-photon interactions in a thermal vacuum at temperature $T$, with the aim of taking into account the effect of the CMB on propagating photons. This contribution to the index of refraction has been calculated in the low-energy situation in which electron pair creation can be neglected [@Bar], where it has been shown to give the non-dispersive result n - 1 (k\_[B]{}T/c\^2)\^4, \[therm1\] and in the high-energy limit ($\hbar\omega \gg \me c^2$), where one obtains the dispersive relation [@LPT] n - 1 (k\_[B]{}T/)\^2 \^2([c\^2]{}[k\_[B]{}T c\^2]{}), \[therm2\] but the resulting values are smaller than that from Eq \[therm1\]. Although a better estimate may eventually be necessary, we will therefore take as an indicative value of the size of the effect the one in Eq \[therm1\], which equals $4.7 \times 10^{-43}$ at today’s temperature of 2.7 K (and $6.5 \times 10^{-31}$ at the recoupling temperature 3000 K). Therefore, we have again a value that may be somewhat larger than the one from second-order quantum gravity effects but decreases with energy.
In fact, Eq \[therm1\] can be considered as a special case of a more general result for low-energy photons in modified QED vacua [@LPT], in which $T^4$ is replaced by a quantity proportional to the excess energy density of the modified vacuum with respect to the standard one. On the one hand, as the authors pointed out, one gets a criterion for identifying situations with superluminal phase velocities. On the other hand, one may use this pattern as motivation for assuming that different modified QED vacuum effects behave in similar ways, becoming dispersive at wavelengths smaller than the Compton wavelength but giving contributions to $n - 1$ that can be bounded by extrapolating the low-energy expressions. This assumption has its limitations [@DG], and more work on the actual values of various effective QED contributions to $n$ in the intermediate-energy range, and their interplay [@Gie], is necessary.
Finally, for high-frequency electromagnetic waves propagating in a curved spacetime, the known general results [@Sho] can be summarized in the curvature-dependent “effective light cone" in momentum space given by ($c = \GN = \hbar = 1$) g\_[ab]{}k\^ak\^b - [8\^2]{}F([k\^m \_[m]{} \^2]{}) T\_[ab]{}k\^ak\^b + [1\^2]{}G([k\^m \_[m]{}\^2]{}) C\_[abcd]{} k\^ak\^ca\^ba\^d = 0, where all indices are spacetime indices, $k^m \nabla_{\!m}$ denotes a covariant derivative along the null geodesic with tangent vector $k^a$, the functions $F$ and $G$ are [*in principle*]{} known, and $a^a$ is the polarization vector. If we take a Friedmann-Robertson-Walker metric to be a good description for the purpose of calculating the effective index of refraction, then the Weyl tensor $C_{abcd}$ vanishes because the space is conformally flat, but the phase velocity depends on $\omega$ because of the non-trivial dependence of $F$ on its argument. To find an actual expression for $n(\omega)$, even if we use the approximation that $k^a$ is constant along these null geodesics, the calculation is reduced to that of $F(\me^{-2}k^m\nabla_{\!m})\cdot T_{ab}$, for which intermediate-energy results in FRW spacetime are not available, to our knowledge. Therefore, we will again use the $C_{abcd} = 0$ low-energy, non-dispersive expression [@Sho] g\_[ab]{}k\^ak\^b - [2245\^2]{}T\_[ab]{}k\^ak\^b = 0, v\_[ph]{} = 1 + [1145]{} [\^2\^2c\^4]{} (where in the last equation all constants have been restored and we have used the matter-dominated universe approximation of vanishing pressure), to bound the desired value. If we set $\rho = \rho_{\rm matter}$, our estimate then is |n-1| < 5 10\^[-82]{}; clearly, our conclusions regarding this effect are not very sensitive to minor improvements in the approximations used.
Comments {#comments .unnumbered}
--------
To summarize, none of the effects we considered gives a contribution to $n - 1$ for $\gamma$-rays of energies in the MeV range that can compete with a quantum gravity effect as described by Eq \[disprel\] with $\beta = 1$, i.e., of order $10^{-22}$ in this energy range. It seems plausible that tighter observational bounds of this magnitude will be available in the not-too-distant future [@SUV], and we should therefore consider looking for second-order effects in $E/E_{\rm P}$.
A few possible sources of dispersion have not been included in our discussion so far. An obvious one is a possible photon mass. Most current bounds on $m_\gamma$ are around $10^{-16}$ eV [@Jac], although some much tighter bounds exist from the galactic magnetic field; even with that value, and treating the photon like a relativistic massive particle, a 1-MeV photon has $n - 1 \approx m_\gamma^2c^4/2E^2 < 10^{-44}$, a bound similar to (but somewhat smaller than) others we obtained above that have an inverse $\omega$ dependence and may compete in magnitude with $\beta = 2$ quantum gravity effects. Another interesting effect, of a somewhat different kind, may arise from the possible multi-valued nature of dispersion relations of the type (\[mom-en\]), which could manifest itself in birefringence [@Leh].
A less obvious (and less easy to evaluate) additional mechanism is related to the presence of inhomogeneities in the universe. Using cosmological models characterized just by the average values of physical quantities is appropriate for many purposes, but is not always a good approximation. In general, qualitatively new phenomena may appear when considering local fluctuations (see, e.g, Ref [@HS]); for example, our argument concerning multiple gravitational scattering breaks down as soon as we consider scatterers of finite size and mass, which provide additional dimensional parameters, and a general feature of local inhomogeneities is that their effect may cancel out [*on average*]{} but not as far as fluctuations are concerned. There is probably no immediate need to obtain results for these effects, but while ideas and techniques are being developed to look for second-order quantum gravity effects, one should also look into ways of tightening the bounds we listed above, and filling in our omissions.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work was supported in part by NSF grant number PHY-0010061 to the University of Mississippi. We are grateful to H Gies, R Lehnert, and S Sarkar for helpful comments, and LB would like to thank Perimeter Institute for hospitality.
[999]{}
Amelino-Camelia G et al 1998 Potential sensitivity of gamma-ray burster observations to wave dispersion in vacuo [*Nature*]{} [**393**]{} 763–765, [astro-ph/9712103]{}
Amelino-Camelia G 2004 A perspective on quantum gravity phenomenology [gr-qc/0402009]{}, and references therein
Zhang B and Meszaros P 2003 Gamma-ray bursts: Progress, problems & prospects [astro-ph/0311321]{}, and references therein
Schaefer B E 1999 Severe limits on variations of the speed of light with frequency [*Phys. Rev. Lett.*]{} [**82**]{} 4964–4966, [astro-ph/9810479]{}
Gharibyan V 2003 Possible observation of photon speed energy dependence [hep-ex/0303010]{}
Pérez A and Sudarsky D 2003 Comments on challenges for quantum gravity [gr-qc/0306113]{}
Bennett C L et al 2003 First year WMAP observations: Preliminary maps and basic results [*Astrophys. J. Suppl.*]{} [**148**]{} 1, and [astro-ph/0302207]{}
Jackson J D 1998 [*Classical Electrodynamics*]{} 3rd ed (John Wiley & Sons)
Peters P C 1974 Index of refraction for scalar, electromagnetic, and gravitational waves in weak gravitational fields [*Phys. Rev.*]{} D [**9**]{} 2207–2218
Goldoni R 1977 A necessary condition for the validity of Huygens’ principle on a curved spacetime [*J. Math. Phys.*]{} [**18**]{} 2125–2128
Mankin R et al 2001 A new approach to electromagnetic wave tails on a curved spacetime [*Phys. Rev.*]{} D [**63**]{}: 063003, [gr-qc/0008047]{}
Faraoni V and Gunzig E 1999 Tales of tails in cosmology [*Int. J. Mod. Phys.*]{} D [**2**]{} 177–188, [astro-ph/9902262]{}
Nolan B 1997 A characterisation of strong wave tails in curved space-times [*Class. Quantum Grav.*]{} [**14**]{} 1295–1308, [gr-qc/9703031]{}
Shore G M 2003 Quantum gravitational optics [*Contemp.Phys.*]{} [**44**]{} 503–521, [gr-qc/0304059]{}
Accioly A and Blas H 2001 Gravitational rainbow [*Phys. Rev.*]{} D [**64**]{}: 067701, [gr-qc/0107003]{}
Tsai W and Erber T 1975 Propagation of photons in homogeneous magnetic fields: Index of refraction [*Phys. Rev.*]{} D [**12**]{} 1132–1137
Giovannini M 2003 The magnetized universe [astro-ph/0312614]{}
Barton G 1990 Faster-than-$c$ light between parallel mirrors: The Scharnhorst effect rederived [*Phys. Lett.*]{} [**B237**]{} 559–562
Latorre J I, Pascual P and Tarrach R 1995 Speed of light in non-trivial vacua [*Nucl. Phys.*]{} [**B437**]{} 60–82, [hep-th/9408016]{}
Dittrich W and Gies H 1998 Light propagation in non-trivial QED vacua [*Phys. Rev.*]{} D [**58**]{}: 025004, [hep-ph/9804375]{}; see also [*Springer Tracts in Modern Physics*]{} 166 (2000) 1–241
Gies H 2000 QED effective action at finite temperature: Two-loop dominance [*Phys. Rev.*]{} D [**61**]{}: 085021, [hep-ph/9909500]{}
Sudarsky D, Urrutia L and Vucetich H 2002 Observational bounds on quantum gravity signals using existing data [*Phys. Rev. Lett.*]{} [**89**]{}: 231301, [gr-qc/0204027]{}
Lehnert R 2003 Threshold analyses and Lorentz violation [*Phys. Rev.*]{} D [**68**]{}: 085003, and references therein
Hu B-L and Shiokawa K 1998 Wave propagation in stochastic spacetimes: Localization, amplification and particle creation [*Phys. Rev.*]{} D [**57**]{} 3474–3483, [gr-qc/9708023]{}
|
---
abstract: |
Metric learning methods have been shown to perform well on different learning tasks. Many of them rely on target neighborhood relationships that are computed in the original feature space and remain fixed throughout learning. As a result, the learned metric reflects the original neighborhood relations. We propose a novel formulation of the metric learning problem in which, in addition to the metric, the target neighborhood relations are also learned in a two-step iterative approach. The new formulation can be seen as a generalization of many existing metric learning methods. The formulation includes a target neighbor assignment rule that assigns different numbers of neighbors to instances according to their quality; ‘high quality’ instances get more neighbors. We experiment with two of its instantiations that correspond to the metric learning algorithms LMNN and MCML and compare it to other metric learning methods on a number of datasets. The experimental results show state-of-the-art performance and provide evidence that learning the neighborhood relations does improve predictive performance.
Learning, Neighborhood Learning
author:
- Jun Wang
- Adam Woznica
- Alexandros Kalousis
bibliography:
- 'LNML.bib'
title: Learning Neighborhoods for Metric Learning
---
Introduction
============
The choice of the appropriate distance metric plays an important role in distance-based algorithms such as $k$-NN and $k$-Means clustering. The Euclidean metric is often the metric of choice, however, it may easily decrease the performance of these algorithms since it relies on the simple assumption that all features are equally informative. Metric learning is an effective way to overcome this limitation by learning the importance of difference features exploiting prior knowledge that comes in different forms. The most well studied metric learning paradigm is that of learning the Mahalanobis metric with a steadily expanding literature over the last years [@xing2003dml; @schultz2004learning; @globerson2006mlc; @davis2007itm; @nguyen2008metric; @weinberger2009distance; @lu2009geometry; @guillaumin2009you; @wang2011]. Metric learning for classification relies on two interrelated concepts, similarity and dissimilarity constraints, and the target neighborhood. The latter defines for any given instance the instances that should be its neighbors and it is specified using similarity and dissimilarity constraints. In the absence of any other prior knowledge the similarity and dissimilarity constraints are derived from the class labels; instances of the same class should be similar and instances of different classes should be dissimilar.
The target neighborhood can be constructed in a *global* or *local* manner. With a global target neighborhood all constraints over all instance pairs are active; [*all*]{} instances of the same class should be similar and [*all*]{} instances from different classes should be dissimilar [@xing2003dml; @globerson2006mlc]. These admittedly hard to achieve constraints can be relaxed with the incorporation of slack variables [@schultz2004learning; @davis2007itm; @nguyen2008metric; @lu2009geometry]. With a local target neighborhood the satisfiability of the constraints is examined within a local neighborhood [@goldberger2005nca; @weinberger2006dml; @nguyen2008metric; @weinberger2009distance]. For any given instance we only need to ensure that we satisfy the constraints that involve that instance and instances from its local neighborhood. The resulting problem is considerably less constrained than what we get with the global approach and easier to solve. However, the appropriate definition of the local target neighborhood becomes now a critical component of the metric learning algorithm since it determines which constraints will be considered in the learning process. [@weinberger2009distance] defines the local target neighborhood of an instance as its $k$, same-class, nearest neighbors, under the Euclidean metric in the original space. Goldberger et al. [@goldberger2005nca] initialize the target neighborhood for each instance to all same-class instances. The local neighborhood is encoded as a soft-max function of a linear projection matrix and changes as a result of the metric learning. With the exception of [@goldberger2005nca], all other approaches whether global or local establish a target neighborhood prior to learning and keep it fixed throughout the learning process. Thus the metric that will be learned from these fixed neighborhood relations is constrained by them and will be a reflection of them. However, these relations are not necessarily optimal with respect to the learning problem that one is addressing.
In this paper we propose a novel formulation of the metric learning problem that includes in the learning process the learning of the local target neighborhood relations. The formulation is based on the fact that many metric learning algorithms can be seen as directly maximizing the sum of some quality measure of the target neighbor relationships under an explicit parametrization of the target neighborhoods. We cast the process of learning the neighborhood as a linear programming problem with a totally unimodular constraint matrix [@sierksma2002linear]. An integer 0-1 solution of the target neighbor relationship is guaranteed by the totally unimodular constraint matrix. The number of the target neighbors does not need to be fixed, the formulation allows the assignment of a different number of target neighbors for each learning instance according to the instance’s quality. We propose a two-step iterative optimization algorithm that learns the target neighborhood relationships and the distance metric. The proposed neighborhood learning method can be coupled with standard metric learning methods to learn the distance metric, as long as these can be cast as instances of our formulation.
We experiment with two instantiations of our approach where the Large Margin Nearest Neighbor (LMNN) [@weinberger2009distance] and Maximally Collapsing Metric Learning (MCML) [@globerson2006mlc] algorithms are used to learn the metric; we dub the respective instantiations LN-LMNN and LN-MCML. We performed a series of experiments on a number of classification problems in order to determine whether learning the neighborhood relations improves over only learning the distance metric. The experimental results show that this is indeed the case. In addition, we also compared our method with other state-of-the-art metric learning methods and show that it improves over the current state-of-the-art performance.
The paper is organized as follows. In section \[sec:relatedWork\], we discuss in more detail the related work. In Section \[sec:LNML\] we present the optimization problem of the Learning Neighborhoods for Metric Learning algorithm (LNML) and in Section \[sec:optAlgorithm\] we discuss the properties of LNML. In Section \[sec:ML\] we instantiate our neighborhood learning method on LMNN and MCML. In Section \[sec:experiments\] we present the experimental results and we finally conclude with Section \[sec:discussion\].
Related Work {#sec:relatedWork}
============
The early work of Xing et al., [@xing2003dml], learns a Mahalanobis distance metric for clustering that tries to minimize the sum of pairwise distances between similar instances while keeping the sum of dissimilar instance distances greater than a threshold. The similar and dissimilar pairs are determined on the basis of prior knowledge. Globerson & Roweis, [@globerson2006mlc] introduced the Maximally Collapsing Metric Learning (MCML). MCML uses a stochastic nearest neighbor selection rule which selects the nearest neighbor $\mathbf x_j$ of an instance $\mathbf x_i$ under some probability distribution. It casts the metric learning problem as an optimization problem that tries to minimize the distance between two probability distributions, an ideal one and a data dependent one. In the ideal distribution the selection probability is always one for instances of the same class and zero for instances of different class, defining in that manner the similarity and dissimilarity constraints under the global target neighborhood approach. In the data dependent distribution the selection probability is given by a soft max function of a Mahalanobis distance metric, parametrized by the matrix $\mathbf M$ to be learned. In a similar spirit Davis et al., [@davis2007itm], introduced Information-Theoretic Metric Learning. ITML learns a Mahalanobis metric for classification with similarities and dissimilarities constraints that follow the global target neighborhood approach. In ITML all same-class instance pairs should have a distance smaller than some threshold and all different-class instance pairs should have a distance larger than some threshold. In addition the objective function of ITML seeks to minimize the distance between the learned metric matrix and a prior metric matrix, modelling like that prior knowledge about the metric if such is available. The optimization problem is cast as a distance of distributions subject to the pairwise constraints and finally expressed as a Bregman optimization problem (minimizing the LogDet divergence). In order to be able to find a feasible solution they introduce slack variables in the similarity and dissimilarity constraints.
The so far discussed metric learning methods follow the global target neighborhood approach in which all instances of the same class should be similar under the learned metric, and all pairs of instances from different classes dissimilar. This is a rather hard constraint and assumes that there is a linear projection of the original feature space that results in unimodal class conditional distributions. Goldberger et al., [@goldberger2005nca], proposed the NCA metric learning method which uses the same stochastic nearest neighbor selection rule under the same data-dependent probability distribution as MCML. NCA seeks to minimize the soft error under its stochastic nearest neighbor selection rule. It uses only similarity constraints and the original target neighborhood of an instance is the set of all same-class instances. After metric learning some, but not necessarily all, same class instances will end up having high probability of been selecting as nearest neighbors of a given instance, thus having a small distance, while the others will be pushed further away. NCA thus learns the local target neighborhood as a part of the optimization. Nevertheless it is prone to overfitting, [@yang2007regularized], and does not scale to large datasets. The large margin nearest neighbor method (LMNN) described in [@weinberger2006dml; @weinberger2009distance] learns a distance metric which directly minimizes the distances of each instance to its local target neighbors while keeping a large margin between them and different class instances. The target neighbors have to be specified prior to metric learning and in the absence of prior knowledge these are the $k$ same class nearest neighbors for each instance.
Learning Target Neighborhoods for Metric Learning {#sec:LNML}
=================================================
Given a set of training instances $\{(\mathbf x_1,y_1),(\mathbf x_2,y_2)$ $,\ldots,(\mathbf x_n,y_n)\}$ where $\mathbf x_i\in \mathbb {R}^d$ and the class labels $ y_i\in \{1,2,\ldots,c\}$, the Mahalanobis distance between two instances $\mathbf x_i$ and $\mathbf x_j$ is defined as: $$D_{\mathbf M}(\mathbf x_i,\mathbf x_j)=(\mathbf x_i-\mathbf x_j)^T{\mathbf M}(\mathbf x_i-\mathbf x_j)$$ where $\mathbf M$ is a Positive Semi-Definite (PSD) matrix ($\mathbf{M} \succeq 0$) that we will learn.
We can reformulate many of the existing metric learning methods, such as [@xing2003dml; @schultz2004learning; @globerson2006mlc; @nguyen2008metric; @weinberger2009distance], by explicitly parametrizing the target neighborhood relations as follows: $$\begin{aligned}
\label{ML}
\min_{\mathbf { {M,\Xi}}}&&\sum_{ij,i \neq j,y_i=y_j}\mathbf P_{ij}\cdot F_{ij}({\mathbf {M,\Xi}}) \\\nonumber
s.t. && \mbox{ constraints of the original problem }\nonumber\end{aligned}$$ The matrix $\mathbf P, \mathbf P_{ij} \in \{0,1\}$, describes the target neighbor relationships which are established prior to metric learning and are not altered in these methods. $\mathbf {P}_{ij}=1$, if $\mathbf x_{j}$ is the target neighbor of $\mathbf x_{i}$, otherwise, $\mathbf {P}_{ij}=0$. Note that the parameters $\mathbf P_{ii}$ and $\mathbf P_{ij}: y_i \neq y_j$ are set to zero, since an instance $\mathbf x_i$ cannot be a target neighbor of itself and the target neighbor relationship is constrained to same-class instances. This is why we have $i \neq j, y_i=y_j$ in the sum, however, for simplicity we will drop it from the following equations. $F_{ij}({\mathbf {M,\Xi}})$ is the term of the objective function of the metric learning methods that relates to the target neighbor relationship $\mathbf P_{ij}$, $\mathbf M$ is the Mahalanobis metric that we want to learn, and $\mathbf \Xi$ is a set of other parameters in the original metric learning problems, e.g. slack variables. Regularization terms on the $\mathbf {M}$ and $\mathbf \Xi$ parameters can also be added into Problem \[ML\] [@schultz2004learning; @nguyen2008metric].
The $F_{ij}({\mathbf {M,\Xi}})$ term can be seen as the ’quality’ of the target neighbor relationship $\mathbf P_{ij}$ under the distance metric $\mathbf M$; a low value indicates a high quality neighbor relationship $\mathbf P_{ij}$. The different metric learning methods learn the $\mathbf M$ matrix that optimizes the sum of the quality terms based on the a priori established target neighbor relationships; however, there is no reason to believe that these target relationships are the most appropriate for learning.
To overcome the constraints imposed by the fixed target neighbors we propose the Learning the Neighborhood for Metric Learning method (LNML) in which, in addition to the metric matrix $\mathbf M$, we also learn the target neighborhood matrix $\mathbf P$. LNML has as objective function the one given in Problem \[ML\] which we now optimize also over the target neighborhood matrix $\mathbf P$. We add some new constraints in Problem \[ML\] which control for the size of the target neighborhoods. The new optimization problem is the following: $$\begin{aligned}
\label{LNML}
{\min_{\mathbf{M,\Xi,P}}} &&\sum_{ij}\mathbf P_{ij}\cdot F_{ij}(\mathbf M, \Xi) \\ \nonumber
s.t.
&& \sum_{i,j}{\mathbf {\mathbf P}}_{ij}=K_{av}*n \\\nonumber
&& {K_{max}} \geq \sum_{j}{\mathbf{\mathbf P}}_{i,j} \geq {K_{min}} \\\nonumber
&& 1 \geq {\mathbf {\mathbf P}}_{ij} \geq 0 \\ \nonumber
&& \mbox{constraints of the original problem} \nonumber\end{aligned}$$ $K_{min}$ and $K_{max}$ are the minimum and maximum numbers of target neighbors that an instance can have. Thus the second constraint controls the number of target neighbor that $\mathbf x_i$ instance can have. $K_{av}$ is the average number of target neighbor per instance. It holds by construction that ${K_{max}} \geq {K_{av}} \geq {K_{min}}$. We should note here that we relax the target neighborhood matrix so that its elements $\mathbf P_{ij}$ take values in $[0,1]$ (third constraint). However, we will show later that a solution $\mathbf P_{ij} \in \{0,1\}$ is obtained, given some natural constraints on the $K_{min}$, $K_{max}$ and $K_{av}$ parameters.
Target neighbor assignment rule
-------------------------------
Unlike other metric learning methods, e.g. LMNN, in which the number of target neighbors is fixed, LNML can assign a different number of target neighbors for each instance. As we saw the first constraint in Problem \[LNML\] sets the average number of target neighbors per instance to $K_{av}$, while the second constraint limits the number of target neighbors for each instance between $K_{min}$ and $K_{max}$. The above optimization problem implements a target neighbor assignment rule which assigns more target neighbors to instances that have high quality target neighbor relations. We do so in order to avoid overfitting since most often the ’good’ quality instances defined by metric learning algorithms [@globerson2006mlc; @weinberger2009distance] are instances in dense areas with low classification error. As a result the geometry of the dense areas of the different classes will be emphasized. How much emphasis we give on good quality instances depends on the actual values of ${K_{min}}$ and ${K_{max}}$. In the limit one can set the value of ${K_{min}}$ to zero; nevertheless the risk with such a strategy is to focus heavily on dense and easy to learn regions of the data and ignore important boundary instances that are useful for learning.
Optimization {#sec:optAlgorithm}
============
Properties of the Optimization Problem
--------------------------------------
We will now show that we get integer solutions for the $\mathbf P$ matrix by solving a linear programming problem and analyze the properties of Problem \[LNML\].
\[integer\] Given $\mathbf{{M, \Xi}}$, and ${K_{max}} \geq {K_{av}} \geq {K_{min}}$ then $\mathbf{{P}}_{ij} \in \{0,1\}$, if $K_{min}$, $K_{max}$ and $K_{av}$ are integers.
Given $\mathbf{{M}}$ and $\mathbf{{\Xi}}$, $F_{ij}(\mathbf {M,\Xi})$ becomes a constant. We denote by ${\mathbf p}$ the vectorization of the target neighborhood matrix $\mathbf P$ which excludes the diagonal elements and $\mathbf P_{ij}:y_i \neq y_j$, and by $\mathbf f$ the respective vectorized version of the $F_{ij}$ terms. Then we rewrite Problem \[LNML\] as: $$\begin{aligned}
\label{integer-opt}
\min_{{\mathbf p}} & & {\mathbf p}^T{\mathbf f} \nonumber \\
s.t. & &(\underbrace{K_{max},\cdots,K_{max}}_n, K_{av}*n)^T \geq \mathbf A \mathbf p \geq\nonumber\\ &&{(\underbrace{K_{min},\cdots,K_{min}}_n,K_{av}*n)^T}\nonumber\\
& & 1 \geq \mathbf p_i \geq 0 \end{aligned}$$ The first and second constraints of Problem \[LNML\] are reformulated as the first constraint in Problem \[integer-opt\]. $\mathbf A$ is a $(n+1) \times (\sum_{c_l}n^2_{c_l}-n)$ constraint matrix, where $n_{c_l}$ is the number of instances in class $c_l$ $${\mathbf{A}}=\left[
\begin{array}{cccc}
\mathbf 1 & \mathbf 0&\cdots &\mathbf 0\\
\mathbf 0 & \mathbf 1&\cdots &\mathbf 0\\
\vdots & \vdots & \ddots & \vdots \\
\mathbf 0 & \mathbf 0&\cdots &\mathbf 1\\
\mathbf 1 & \mathbf 1&\cdots &\mathbf 1
\end{array}
\right]$$ where $\mathbf 1$ ($\mathbf 0$) is the vector of ones (zeros). Its elements depends on the its position in the matrix $\mathbf A$. In its $i$th column, all $\mathbf 1$ ($\mathbf 0$) vectors have $n_i-1$ elements, where $n_i$ is the number of instances of class $c_j$ with $c_j=y_{p_i}$. According to the sufficient condition for total unimodularity (Theorem 7.3 in [@sierksma2002linear]) the constraint matrix $\mathbf A$ is a totally unimodular matrix. Thus, the constraint matrix $\mathbf B=[\mathbf{I, -I, A, -A}]^T$ in the following equivalent problem also is a totally unimodular matrix (pp.268 in [@schrijver1998theory]). $$\begin{aligned}
\label{integer-B}
\min_{\mathbf p} && \mathbf p^T \mathbf f \nonumber\\
s.t.&&\mathbf B {\mathbf p} \leq {\mathbf e}\nonumber\\
&&e=(\underbrace{1,\cdots,1}_{\sum_{c_l}n^2_{c_l}-n},\underbrace{0,\cdots,0}_{\sum_{c_l}n^2_{c_l}-n},\underbrace{K_{max},\cdots,K_{max}}_n,\nonumber\\
&&K_{av}*n,\underbrace{-K_{min},\cdots,-K_{min}}_n,-K_{av}*n)^T \end{aligned}$$ Since $\mathbf e$ is an integer vector, provided $K_{min}$, $K_{max}$, and $K_{av}$, are integers, and the constraint matrix $\mathbf B$ is totally unimodular, the above linear programming problem will only have integer solutions (Theorem 19.1a in [@schrijver1998theory]). Therefore, for the solution $\mathbf p$ it will hold that $\mathbf p_i \in \{0,1\}$ and consequently $\mathbf P_{ij} \in \{0,1\}$.
Although the constraints to control the size of the target neighborhood are convex, the objective function in Problem \[LNML\] is not jointly convex in $\mathbf P$ and $(\mathbf M,\mathbf \Xi)$. However, as shown in Lemma \[integer\], the binary solution of $\mathbf P$ can be obtained by a simple linear program if we fix $(\mathbf{M,\Xi})$. Thus, Problem \[LNML\] is individually convex in $\mathbf P$ and $(\mathbf M,\mathbf \Xi)$, if the original metric learning method is convex; this condition holds for all the methods that can be coupled with our neighborhood learning method [@xing2003dml; @schultz2004learning; @globerson2006mlc; @nguyen2008metric; @weinberger2009distance].
Optimization Algorithm
----------------------
Based on Lemma \[integer\] and the individual convexity property we propose a general and easy to implement iterative algorithm to solve Problem \[LNML\]. The details are given in Algorithm \[algo:LNML\]. At the first step of the $k$th iteration we learn the binary target neighborhood matrix $\mathbf P^{(k)}$ under a fixed metric matrix $\mathbf M^{(k-1)}$ and $\mathbf \Xi^{(k-1)}$, learned in the $k-1$th iteration, by solving the linear programming problem described in Lemma \[integer\]. At the second step of the iteration we learn the metric matrix $\mathbf M^{(k)}$ and $\mathbf{{\Xi}}^{(k)}$ with the target neighborhood matrix $\mathbf P^{(k)}$ using as the initial metric matrix the $\mathbf M^{(k-1)}$. The second step is simply the application of a standard metric learning algorithm in which we set the target neighborhood matrix to the learned $\mathbf P^{(k)}$ and the initial metric matrix to $\mathbf M^{(k-1)}$. The convergence of proposed algorithm is guaranteed if the original metric learning problem is convex [@bezdek2002some]. In our experiment, it most often converges in 5-10 iterations.
$\mathbf{{X}}$, $\mathbf{{Y}}$, $\mathbf{{M}}^0$,$\mathbf{{\Xi}}^0$, $K_{min}$, $K_{max}$ and $K_{av}$ $\mathbf{{M}}$ $\mathbf{{P}}^{(k)}$=LearningNeighborhood($\mathbf X,\mathbf Y,\mathbf{{M}}^{(k-1)},\mathbf{{\Xi}}^{(k-1)}$) by solving Problem \[integer-opt\] $(\mathbf{{M}}^{(k)},\mathbf{\Xi}^{(k)})$=MetricLearning($\mathbf{{M}}^{(k-1)}$,$\mathbf{P}^{(k)}$) $k:=k+1$
Instantiating LNML {#sec:ML}
==================
In this section we will show how we instantiate our neighborhood learning method with two standard metric learning methods, LMNN and MCML, other possible instantiations include the metric learning methods presented in [@xing2003dml; @schultz2004learning; @nguyen2008metric].
Learning the Neighborhood for LMNN
----------------------------------
The optimization problem of LMNN is given by: $$\begin{aligned}
\label{LMNN-opt-prob}
\min_{\mathbf{ M, \xi}} &&\sum_{ij} \mathbf {P}_{ij}\{(1-\mu) D_{\mathbf {M}}(\mathbf x_i,\mathbf x_j)
+\mu \sum_{l}(1-\mathbf{Y}_{il})\xi_{ijl}\} \\ \nonumber
s.t. & & D_{\mathbf M}(\mathbf x_i,\mathbf x_l)-D_{\mathbf M}(\mathbf x_i,\mathbf x_j) \geq 1-\xi_{ijl}\\ \nonumber
& & \xi_{ijl}>0\\ \nonumber
& & \mathbf M \succeq 0\end{aligned}$$ where the matrix $\mathbf Y, \mathbf Y_{ij} \in \{0,1\},$ indicates whether the class labels $y_i$ and $y_j$ are the same ($\mathbf Y_{ij}=1$) or different ($\mathbf Y_{ij}=0$). The objective is to minimize the sum of the distances of all instances to their target neighbors while allowing for some errors, this trade off is controlled by the $\mu$ parameter. This is a convex optimization problem that has been shown to have good generalization ability and can be applied to large datasets. The original problem formulation corresponds to a fixed parametrization of $\mathbf P$ where its non-zero values are given by the $k$ nearest neighbors of the same class.
Coupling the neighborhood learning framework with the LMNN metric learning method results in the following optimization problem: $$\begin{aligned}
\label{LNLMNN}
\min_{\mathbf{M,P,\xi}}&&\sum_{ij}\mathbf P_{ij}\cdot F_{ij}(\mathbf M, \xi) \\ \nonumber
=\min_{\mathbf M, \mathbf P,\xi} & & \sum_{ij}{\mathbf {P}}_{ij}\{(1-\mu) D_{\mathbf {M}}(\mathbf x_i,\mathbf x_j)
+\mu \sum_{l}(1-\mathbf{Y}_{il})\xi_{ijl}\}\\ \nonumber
s.t. & & {K_{max}} \geq \sum_{j}{\mathbf{P}}_{i,j} \geq {K_{min}} \\ \nonumber
& & \sum_{i,j}{\mathbf {P}}_{ij}=K_{av}*n\\ \nonumber
& & 1 \geq {\mathbf {P}}_{ij} \geq 0 \\ \nonumber
& & D_{\mathbf {M}}(\mathbf x_i,\mathbf x_l)-D_{\mathbf {M}}(\mathbf x_i,\mathbf x_j) \geq 1-\xi_{ijl}\\ \nonumber
& & \xi_{ijl}>0\\ \nonumber
& & {\mathbf {M}} \succeq 0 \nonumber\end{aligned}$$ We will call this coupling of LNML and LMNN LN-LMNN. The target neighbor assignment rule of LN-LMNN assigns more target neighbors to instances that have small distances from their target neighbors and low hinge loss.
Learning the Neighborhood for MCML
----------------------------------
MCML relies on a data dependent stochastic probability that an instance $\mathbf x_j$ is selected as the nearest neighbor of an instance $\mathbf x_i$; this probability is given by: $$\begin{aligned}
\label{stochastic-rule}
p_\mathbf{M}(j|i)=\frac{e^{-D_\mathbf {M}(\mathbf x_i,\mathbf x_j)}}{ Z_i}=
\frac{e^{-D_\mathbf M(\mathbf x_i,\mathbf x_j)}}{\sum_{k \ne i}{e^{-D_\mathbf M(\mathbf x_i,\mathbf x_k)}}}, & i \ne j\nonumber\\\end{aligned}$$ MCML learns the Mahalanobis metric that minimizes the KL divergence distance between this probability distribution and the ideal probability distribution $p_0$ given by: $$\begin{aligned}
\label{ideal distribution}
p_0(j|i)=\frac{\mathbf P_{ij}}{\sum_{k}{\mathbf P_{ik}}}, & p_0(i|i)=0
\end{aligned}$$ where $\mathbf P_{ij}=1$, if instance $\mathbf x_j$ is the target neighbor of instance $\mathbf x_i$, otherwise, $\mathbf P_{ij}=0$. The optimization problem of MCML is given by: $$\begin{aligned}
\label{MCML}
\min_{\mathbf M}&& \sum_{i}KL[p_0(j|i)|p_\mathbf M(j|i)] \\ \nonumber
=\min_{\mathbf M}&& \sum_{i,j}{\mathbf P_{ij}} \frac{(D_\mathbf M(\mathbf x_i,\mathbf x_j)+\log Z_i)}{\sum_{k}{\mathbf P_{ik}}} \\ \nonumber
s.t.&&\mathbf M \succeq 0 \nonumber\end{aligned}$$ Like LMNN, this is also a convex optimization problem. In the original problem formulation the ideal distribution is defined based on class labels, i.e. $\mathbf P_{ij}=1$, if instances $\mathbf x_i$ and $\mathbf x_j$ share the same class label, otherwise, $\mathbf P_{ij}=0$. The neighborhood learning method cannot learn directly the target neighborhood for MCML, since the objective function of the latter cannot be rewritten in the form of the objective function in Problem \[LNML\], due to the denominator $\sum_{k}{\mathbf{P}}_{ik}$. However, if we fix the size of the neighborhood to $\sum_{k}{\mathbf{P}}_{i,k}=K_{av}=K_{min}=K_{max}$ the two methods can be coupled and the resulting optimization is given by: $$\begin{aligned}
\label{LNMCML}
\min_{\mathbf{M,P}}&&{\sum_{ij}\mathbf P_{ij}\cdot F_{ij}(\mathbf M)} \\ \nonumber
=\min_{\mathbf {M,P}} &&\sum_{i,j}{\mathbf P_{ij}}\frac{(D_\mathbf M(\mathbf x_i, \mathbf x_j)+\log Z_i)}{K_{av}}\\ \nonumber
s.t. && \sum_{j}{\mathbf{\mathbf P}}_{i,j} = {K_{av}} \\ \nonumber
&& \mathbf M \succeq 0\end{aligned}$$ We will dub this coupling of LNML and MCML as LN-MCML. The original MCML method follows the global approach in establishing the neighborhood, with LN-MCML we get a local approach in which the neighborhoods are of fixed size $K_{av}$ for every instance.
Experiments {#sec:experiments}
===========
With the experiments we wish to investigate a number of issues. First, we want to examine whether learning the target neighborhood relations in the metric learning process can improve predictive performance over the baseline approach of metric learning with an apriori established target neighborhood. Second, we want to acquire an initial understanding of how the parameters $K_{min}$ and $K_{max}$ relate to the predictive performance. To this end, we will examine the predictive performance of LN-LMNN with two fold inner Cross Validation (CV) to select the appropriate values of $K_{min}$ and $K_{max}$, method which we will denote by LN-LMNN(CV), and that of LN-LMNN, with a default setting of $K_{min}=K_{max}=K_{av}$. Finally, we want to see how the method that we propose compares to other state of the art metric learning methods, namely NCA and ITML. We include as an additional baseline in our experiments the performance of the Euclidean metric (EucMetric). We experimented with twelve different datasets: seven from the UCI machine learning repository, Sonar, Ionosphere, Iris, Balance, Wine, Letter, Isolet; four text mining datasets, Function, Alt, Disease and Structure, which were constructed from biological corpora [@kalousis2007stability]; and MNIST [@MNIST], a handwritten digit recognition problem. A more detailed description of the datasets is given in Table \[datasets\].
Since LMNN is computationally expensive for datasets with large number of features we applied principal component analysis (PCA) to retain a limited number of principal components, following [@weinberger2009distance]. The datasets to which this was done were the four text mining datasets, Isolet and MNIST. For the two latter 173 and 164 principal components were respectively retained that explain 95% of the total variance. For the text mining datasets more than 1300 principal components should be retained to explain 95% of the total variance. Considering the running time constraints, we kept the 300 most important principal components which explained 52.45%, 47.57%, 44.30% and 48.16% of the total variance for respectively Alt, Disease, Function and Structure. We could experiment with NCA and MCML on full tranining datasets only with datasets with a small number of instances due to their computational complexity. For completeness we experimented with NCA on large datasets by undersampling the training instances, i.e. the learning process only involved 10% of full training instances which was the maximum number we could experiment for each dataset. We also applied ITML on both versions of the larger datasets, i.e. with PCA-based dimensionality reduction and the original ones.
0.15in
-0.1in
For ITML, we randomly generate for each dataset the default $20c^2$ constraints which are bounded repectively by the 5th and 95th percentiles of the distribution of all available pairwise distances for similar and dissimilar pairs. The slack variable $\gamma$ is chosen form $\{10^i\}^4_{i=-4}$ using two-fold CV. The default identity matrix is employed as the regularization matrix. For the different instantiations of the LNML method we took care to have the same parameter settings for the encapsulated metric learning method and the respective baseline metric learning. For LN-LMNN, LN-LMNN(CV) and LMNN the regularization parameter $\mu$ that controls the trade-off between the distance minimization component and the hinge loss component was set to 0.5 (the default value of LMNN). For LMNN the default number of target neighbors was used (three). For LN-LMNN, we set $K_{min}=K_{max}=K_{av}=3$, similar to LMNN. To explore the effect of a flexible neighborhood, the values of the $K_{min}$ and $K_{max}$ parameters in LN-LMNN(CV) were selected from the sets $\{1,4,3\}$ and $\{2,5,3\}$ respectively, while $K_{av}$ was fixed to three. Similarly for LN-MCML we also set $K_{av}=3$. The distance metrics for all methods are initialized to the Euclidean metric. As the classification algorithm we used 1-Nearest Neighbor.
We used 10-fold cross validation for all datasets to estimate classification accuracy, with the exception of Isolet and MNIST for which the default train and test split was used. The statistical significance of the differences were tested with McNemar’s test and the p-value was set to 0.05. In order to get a better understanding of the relative performance of the different algorithms for a given dataset we used a ranking schema in which an algorithm A was assigned one point if it was found to have a significantly better accuracy than another algorithm B, 0.5 points if the two algorithms did not have a significantly different performance, and zero points if A was found to be significantly worse than B. The rank of an algorithm for a given dataset is simply the sum of the points over the different pairwise comparisons. When comparing $N$ algorithms in a single dataset the highest possible score is $N-1$ while if there is no significant difference each algorithm will get $(N-1)/2$ points.
Results
-------
The results are presented in Table \[results\]. Examining whether learning also the neighborhood improves the predictive performance compared to plain metric learning, we see that in the case of LN-MCML, and for the five small datasets for which we have results, learning the neighborhood results in a statistically significant deterioration of the accuracy in one out of the five datasets (balance), while for the remaining four the differences were not statistically significant. If we now examine LN-LMNN(CV), LN-LMNN and LMNN we see that here learning the neighborhood does bring a statistically significant improvement. More precisely, LN-LMNN(CV) and LN-LMNN improve over LMNN respectively in six (two small and four large) and four (two small and two large) out of the 12 datasets. Moreove, by comparing LN-LMNN(CV) and LN-LMNN, we see that learning a flexible neighborhood with LN-LMNN(CV) improves significantly the performance over LN-LMNN on two datasets. The low performance of LN-MCML on the balance dataset was intriguing; in order to take a closer look we tried to determine automatically the appropriate target neighborhood size, $K_{av}$, by selecting it on the basis of five-fold inner cross validation from the set $K_{av}= \{3,5,7,10,20,30\}$. The results showed that the default value of $K_{av}$ was too small for the given dataset, with the average selected size of the target neighborhood at 29. As a result of the automatic tunning of the target neighborhood size the predictive performance of LN-MCML jumped at an accuracy of 93.92% which represented a significant improvement over the baseline MCML for the balance dataset. For the remaining datasets it turned out that the choice of $K_{av}=3$ was a good default choice. In any case, determining the appropriate size of the target neighborhood and how that affects the predictive performance is an issue that we wish to investigate further. In terms of the total score that the different methods obtain the LN-LMNN(CV) achieves the best in both the small and large datasets. It is followed closely by NCA in the small datasets and by LN-LMNN in the large datasets.
-0.1in
Conclusion and Future Work {#sec:discussion}
==========================
We presented LNML, a general Learning Neighborhood method for Metric Learning algorithms which couples the metric learning process with the process of establishing the appropriate target neighborhood for each instance, i.e. discovering for each instance which same class instances should be its neighbors. With the exception of NCA, which cannot be applied on datasets with many instances, all other metric learning methods whether they establish a global or a local target neighborhood do that prior to the metric learning and keep the target neighborhood fixed throughout the learning process. The metric that is learned as a result of the fixed neighborhoods simply reflects these original relations which are not necessarily optimal with respect to the classification problem that one is trying to solve. LNML lifts these constraints by learning the target neighborhood. We demonstrated it with two metric learning methods, LMNN and MCML. The experimental results show that learning the neighborhood can indeed improve the predictive performance. The target neighborhood matrix $\mathbf P$ is strongly related to the similarity graphs which are often used in semi-supervised learning [@JebaraICML2009], spectral clustering [@Luxburg2007] and manifold learning [@RoweisSaulLLE2000]. Most often the similarity graphs in these methods are constructed in the original space, which nevertheless can be quite different from true manifold on which the data lies. These methods could also profit if one is able to learn the similarity graph instead of basing it on some prior structure.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was funded by the Swiss NSF (Grant 200021-122283/1). The support of the European Commission through EU projects DebugIT (FP7-217139) and e-LICO (FP7-231519) is also gratefully acknowledged.
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---
abstract: 'An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ has exactly three zeros at $\pm 1$ and $0$, is balanced and odd, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks like the one dimensional, heteroclinic solution. In this paper we consider the family of four ended solutions whose ends are almost parallel at $\infty$. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended solutions we can classify all such solutions. Thus we show that four end solutions form, up to rigid motions, a one parameter family. This family contains the saddle solution, for which the angle between the nodal lines is $\frac{\pi}{2}$ as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).'
address:
- 'Micha[ł]{} Kowalczyk, Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile.'
- 'Y. Liu - Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile.'
- 'Frank Pacard, Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France et Institut Universitaire de France'
author:
- 'Micha[ł]{} Kowalczyk'
- Yong Liu
- Frank Pacard
title: 'The classification of four end solutions to the Allen-Cahn equation on the plane'
---
Introduction
============
Some entire solutions to the Allen-Cahn equation in ${\mathbb R}^2$
-------------------------------------------------------------------
This paper deals with the problem of classification of the family of four end solutions (precise definition will follow) to the Allen-Cahn equation: $$\Delta u=F^{\prime}\left( u\right) \text{ in }\mathbb{R}^{2}.\label{AC}$$ The function $F$ is a smooth, double well potential, which means that we assume the following conditions for $F:$ $F$ is even, nonnegative and has only two zeros at $\pm1$. Moreover $F^{\prime\prime}\left( 1\right) \neq0$ and $F^{\prime
}\left( t\right) \neq0,t\in\left( 0,1\right)$. We also suppose $F''(0)\neq 0$.
It is know that (\[AC\]) has a solution whose nodal set is a straight line, it will be called [*a planar solution*]{}. It is simply obtained by taking the unique, odd, heteroclinic solution connecting $-1$ to $1$: $$\begin{aligned}
\label{heteroclinic}H''=F'(H), \quad H(\pm \infty)=\pm 1, \quad H(0)=0,\end{aligned}$$ and letting $u(x,y)=H(ax+by+c)$ for some constants $a,b, c$ such that $a^2+b^2=1$. We note that if, say $a>0$ then $\partial_x u=aH'>0$. De Giorgi conjecture says that if $u$ is any smooth and bounded solution of (\[AC\]) such that $\partial_{\mathtt e} u>0$ for certain fixed direction ${\mathtt e}$ then in fact $u$ must be a planar solution. Indeed this conjecture holds in ${\mathbb R}^N$, $N\leq 8$ ([@MR1637919] when $N=2$, [@MR1775735] when $N=3$, and [@MR2480601], for $4\leq N\leq 8$ under additional limit condition), while a counterexample can be given when $N\geq 9$ [@dkp_dg]. It is worth mentioning that the De Giorgi conjecture is a direct analog of the famous Bernstein conjecture in the theory of minimal surfaces.
In order to proceed with the statement of our results we will define the family of four ended solutions of (\[AC\]), which is a particular example of a more general family of $2k$ ended solutions [@dkp-2009]. Intuitively, a four ended solution $u$ is characterized by the fact that its nodal set $N(u)$ is asymptotic at infinity to four half lines, and along each of this half lines it looks locally like the heteroclinic solution. To describe this precisely we introduce the set $\Lambda_4$ of [oriented and ordered four affine lines]{} in $\mathbb R^2$. Thus $\Lambda_4$ consists of $4$- tuples $(\lambda_1, \dots, \lambda_4)$ such that each $\lambda_j$ can be uniquely written as $$\lambda := r_j \, {\tt e}_j^\perp + \mathbb R \, {\tt e}_j ,$$ for some $r_j\in \mathbb R$ and some unit vector ${\tt e}_j =(\cos\theta_j, \sin\theta_j)\in S^1$, which defines the orientation of the line. Recall that we denote by $\perp$ the rotation of angle $\pi/2$ in $\mathbb R^2$. Observe that the affine lines are oriented and hence we do not identify the line corresponding to $(r_j, \theta_j)$ and the line corresponding to $(-r_j, \theta_j +\pi)$. Additionally we require these lines are ordered, which means: $$\theta_1 < \theta_2 < \theta_3 < \theta_{4} < 2\, \pi + \theta_1 .$$ For future purpose we denote by $$\label{min angle}
\theta_\lambda : = \frac{1}{2} \, \min \{ \theta_{2}- \theta_1 , \theta_{3} - \theta_{2} ,\theta_4-\theta_3, 2 \, \pi + \theta_1 - \theta_{4} \} ,$$ the half of the minimum of the angles between any two consecutive oriented affine lines $\lambda_1, \ldots, \lambda_4$.
Assume that we are given a $4$-tuple of oriented affine lines $\lambda = (\lambda_1, \ldots, \lambda_4)$ . It is easy to check that for all $R > 0 $ large enough and for all $j=1, \ldots, 4$, there exists $s_j \in \mathbb R$ such that :
- The point ${\tt x}_j: = r_j \, {\tt e}^\perp_{j} + s_j \, {\tt e}_j$ belongs to the circle $\partial B_R$, with $R > 0$.\
- The half lines $$\lambda_j^+ := {\tt x}_j + \mathbb R^+ \, {\tt e}_j ,
\label{eq:halfline}$$ are disjoint and included in $\mathbb R^2\setminus B_R$.\
- The minimum of the distance between two distinct half lines $\lambda^+_i$ and $\lambda_j^+$ is larger than $4$.
The set of half affine lines $\lambda_1^+, \ldots, \lambda_{4}^+$ together with the circle $\partial B_R$ induce a decomposition of $\mathbb R^2$ into $5$ slightly overlapping connected components $$\mathbb R^2 = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_{4} ,$$ where $$\Omega_0 : = B_{R+1},$$ and where, for $j=1, \ldots, 4$, $$\begin{aligned}
\label{decomp 1}
\Omega_j : = \left\{ {\tt x} \in \mathbb R^2 \, : \, |{\tt x}| > R-1 \quad \mbox{and} \quad {\rm dist} ({\tt x}, \lambda_j^+) < {\rm dist} ({\tt x}, \lambda_i^+) +2 , \quad \forall i \neq j \,\right\} ,\end{aligned}$$ where $ {\rm dist} ({\tt x}, \lambda_j^+)$ denotes the distance to $\lambda_j^+$. Observe that, for all $j=1, \ldots, 4$, the set $\Omega_j$ contains the half line $\lambda_j^+$.
We define ${\mathbb I}_0, {\mathbb I}_1, \ldots , {\mathbb I}_4$, a smooth partition of unity of $\mathbb R^2$ which is subordinate to the above decomposition of $\mathbb R^2$. Hence $$\sum_{j=0}^4 \mathbb I_j \equiv 1,$$ and the support of $\mathbb I_j$ is included in $\Omega_j$, for $j=0, \ldots, 4$. Without loss of generality, we can also assume that ${\mathbb I}_0\equiv 1$ in $$\Omega'_0 := B_{R -1},$$ and ${\mathbb I}_j\equiv 1$ in $$\Omega'_j := \left\{ {\tt x} \in \mathbb R^2 \, : \, |{\tt x}| > R+1 \quad \mbox{and} \quad {\rm dist} ({\tt x}, \lambda_j^+) < {\rm dist} ({\tt x}, \lambda_i^+) - 2 , \quad \forall i \neq j \,\right\} ,$$ for $j=1, \ldots, 4$. Finally, we assume that $$\|{\mathbb I}_{j}\|_{\mathcal C^2(\mathbb R^2)} \leq C .$$
We now take $\lambda = (\lambda_1, \ldots, \lambda_4 ) \in \Lambda_4$ with $\lambda_j^+ = {\tt x}_j + \mathbb R^+ \, {\tt e}_j$ and we define $$\begin{aligned}
u_\lambda ({\tt x}): = \sum_{j=1}^{4} (-1)^j \, {\mathbb I}_j ({\tt x}) \, H (({\tt x}-{\tt x}_j)\cdot {\tt e}^\perp_j).
\label{def w}\end{aligned}$$ Observe that, by construction, the function $u_\lambda$ is, away from a compact, asymptotic to copies of planar solutions whose nodal set are the half affine lines $\lambda_1^+, \ldots, \lambda_{4}^+$. A simple computation shows that $u_\lambda$ is not far from being a solution of (\[AC\]) in the sense that $\Delta \, u_\lambda - F'(u_\lambda)$ is a function which decays exponentially to $0$ at infinity (this uses the fact that $\theta_\lambda >0$).
In this paper we are interested in four ended solutions of (\[AC\]) which means that they are asymptotic to a function $u_\lambda$ for some choice of $\lambda \in \Lambda_4$. More precisely, we have the :
Let $\mathcal S_4$ denote the set of functions $u$ which are defined in $\mathbb R^2$ and which satisfy $$u - u_\lambda \in W^{2,2} \, (\mathbb R^2) ,
\label{ass w}$$ for some $\lambda \in \Lambda_4$. We also define the [decomposition operator]{} $\mathcal J$ by $$\begin{array}{rcccllll}
\mathcal J : & \mathcal S_{4} & \longrightarrow & W^{2,2} (\mathbb R^2) \times \Lambda_4 \\[3mm]
& u &\longmapsto & \left( u- u_\lambda , \lambda \right) .
\end{array}$$ The topology on $\mathcal S_4$ is the one for which the operator $\mathcal J$ is continuous (the target space being endowed with the product topology). We define the set of four ended solutions of the Allen-Cahn equation $\mathcal M_4$ to be the set of solutions $u$ of (\[AC\]) which belong to $\mathcal S_4$. \[de:001\]
The set $\mathcal M_4$ is nonempty. Indeed, it is known that $\left( \ref{AC}\right) $ has a saddle solution $U,$ which is bounded and symmetric: $$U\left( x,y\right) =U\left( x,-y\right) =U\left( -x,y\right) .$$ Moreover, the nodal set of $U$ is exactly the lines $y=\pm x.$ Along these two lines, $U$ converges exponentially fast to the heteroclinic solution. In addition in [@MR2557944] it is shown that there exists a small number $\varepsilon_0$ such that for all $0<\theta$, with $\tan\theta< {\varepsilon}_0$ there exists a four ended solution with corresponding angles of the half lines $\lambda^+_j$, $j=1,\dots, 4$ given by $$\theta_1=\theta, \quad\theta_2=\pi-\theta, \quad\theta_3=\theta+\pi, \quad \theta_4=2\pi-\theta.$$ Observe that the fact that $\theta$ is small implies that the ends of this solution are almost parallel and their slopes given by $\pm{\varepsilon}$ are small as well. Clearly, by symmetry, it is easy to see that there exist also solutions with parallel ends whose angles are given by: $$\theta_1=\pi/2-\theta, \quad\theta_2=\pi/2+\theta, \quad\theta_3=-\theta+3\pi/2, \quad \theta_4=3\pi/2+\theta.$$ In this case we would have $\tan\theta>\frac{1}{{\varepsilon}_0}$. Clearly, any four ended solution can be translated and rotated, yielding another four ended solution. In fact, by a result of Gui [@2011arXiv1102.4022G] we know that any $u\in {\mathcal M}_4$ is, modulo rigid motions, and a multiplication of a solution by $-1$, even in its variables, and monotonic in $x$ in the set $x>0$, and in $y$ in the set $y<0$ i.e.: $$\begin{aligned}
\label{even}
u(x,y)=u(-x,y)=u(x,-y), \quad u_x(x,y)>0,\quad x>0, \quad u_y(x,y)>0, \quad y<0.\end{aligned}$$ Thus, when studying four ended solutions, it is natural to consider the set ${\mathcal M}_4^{even}\subset {\mathcal M}_4$, consisting precisely of functions satisfying (\[even\]). With each such function $u$ we may associate in a unique way the angle that the component of its nodal set in the first quadrant makes with the $x$-axis. Thus we can define [*[the angle map]{}*]{}: $$\label{def thetau}
\begin{aligned}
&\theta\colon {\mathcal M}_4^{even}\to (0,\frac{\pi}{2}),\\
&u\longmapsto \theta.
\end{aligned}$$ In principle the value of the angle map is not enough to identify in a unique way a solution to (\[AC\]) in ${\mathcal M}_4^{even}$. However for solutions with almost parallel ends we have the following:
\[teo uniqueness\] There exists a small number ${\varepsilon}_0$ such that for any two solutions $u_1, u_2\in {\mathcal M}_4^{even}$ satysfying $\tan\theta(u_1)=\tan(\theta_2)=m$, and either $m<{\varepsilon}_0$ or $m>\frac{1}{{\varepsilon}_0}$, we have necessarily $u_1\equiv u_2$.
This result gives in some sense classification of the subfamily of the family of four end solutions which contains solutions with almost parallel ends. It says that this subfamily consists precisely of the solutions constructed in [@MR2557944]. Let us explain the importance of this statement from the point of view of classification of all four end solutions. We will appeal to the following theorem proven in [@partI]:
\[part I\] Let $M$ be any connected component of $ {\mathcal M}_4^{even}$ . Then the angle map $\theta\colon M\to (0, \frac{\pi}{2})$ is surjective.
Consider for example the connected component $M_0\subset {\mathcal M}_{4}^{even}$ which contains the saddle solution $U$. Theorem \[part I\] implies that $U$ can be deformed along $M_0$ to a solution with the value of the angle map arbitrarily close to $0$ or to $\frac{\pi}{2}$, thus yielding a solution in the subfamily of the solutions with almost parallel ends. But these solutions are uniquely determined by the value of the angle map, which follows from the uniqueness statement in Theorem \[teo uniqueness\]. As a result we obtain the following classification theorem:
\[classification\] Any solution $u\in {\mathcal M}_{4}^{even}$ belongs to $M_0$ and is a continuous deformation of the saddle solution $U$.
To appreciate further this result let us mention that in [@partI] we prove a more general statement regarding arbitrary connected components in the moduli space of solutions ${\mathcal M}_4^{even}$. To explain this we consider the following map $$\begin{aligned}
P\colon {\mathcal M}_4^{even}&\to (0, \frac{\pi}{2})\times (-1,1)\subset {\mathbb R}^2,\\
u&\longmapsto (\theta(u), u(0)).\end{aligned}$$ Then, according to Theorem 1 in [@partI], the image of any connected component $M\subset {\mathcal M}_4^{even}$ under this map $P(M)$ is an embedded, smooth curve in $(0, \frac{\pi}{2})\times (-1,1)$. Thus, Theorem \[classification\] implies that the set $P({\mathcal M}_4^{even})$ consists of a single embedded curve in $(0, \frac{\pi}{2})\times (-1,1)$.
We observe that according to the conjecture of De Giorgi in two dimensions any bounded solution $u$ which is monotonic in one direction must be one dimensional and equal to $u({\tt x})=H({\tt a}\cdot {\tt x}+b)$. In the language of multiple end solutions, this solution has [*two*]{} (heteroclinic, planar) ends. Theorem \[classification\] gives on the other hand the classification of the family of solutions with [*four*]{} planar ends. Since the number of ends of a solution to (\[AC\]) must be even, the family of four ended solutions is the natural object to study. In this context, one may wonder if it is possible to classify solutions to (\[AC\]) assuming for instance that the nodal sets of $u_x$, and $u_y$ have just one component. This question is beyond the scope of this paper, however since partial derivatives of four ended solution satisfy this assumption it seems reasonable to conjecture that a result similar to Theorem \[classification\] should hold in this more general setting. We should mention here that it is in principle possible to study the problem of classification of solutions assuming for example that their Morse index is $1$. This is natural since the Morse index of $u$ and the number of the nodal domains of $u_x$ and $u_y$ are related. We recall here that the heteroclinic is stable, and from [@MR2110438] we know that in dimension $N=2$ stability of a solution implies that it is necessarily a one dimensional solution (for the related minimality conjecture, see for example [@MR2480601] and [@wei_pacard] and the reference therein). We expect that in fact the family of four ended solutions should should contain all multiple end solutions with Morse index $1$ [@partIII] (the Morse index of the saddle solution is $1$ [@MR1363002]).
Let us now explain the analogy of Theorem \[classification\] with some aspects of the theory minimal surfaces in ${\mathbb R}^3$. In 1834, Scherk discovered an example of singly-periodic, embedded, minimal surface in ${\mathbb R}^3$ which, in a complement of a vertical cylinder, is asymptotic to $4$ half planes with angle $\frac{\pi}{2}$ between them. This surface, after a rigid motion, has two planes of symmetry, say $\{x_2=0\}$ plane and $\{x_1=0\}$, and it is periodic, with period $1$ in the $x_3$ direction. If $\theta$ is the angle between the asymptotic end of the Scherk surface contained in $\{x_1>0, x_2>0\}$ and the $\{x_2=0\}$ plane by then $\theta=\frac{\pi}{4}$. This is the so called second Scherk’s surface and it will denoted here by $S_{\frac{\pi}{4}}$. In 1988 Karcher [@MR958255] found Scherk surfaces other than the original example in the sense that the corresponding angle between their asymptotic planes and the $\{x_2=0\}$ plane can be any $\theta\in (0, \frac{\pi}{2})$. The one parameter family $\{S_\theta\}_{\{0<\theta<\frac{\pi}{2}\}}$ of these surfaces is the family of Scherk singly periodic minimal surfaces. Thus, accepting that the saddle solution of the Allen-Cahn equation $U$ corresponds to the Scherk surface $S_{\frac{\pi}{4}}$ Theorem \[part I\] can be understood as an analog of the result of Karcher. We note that, unlike in the case of the Allen-Cahn equation, the Scherk family is given explicitly, for example it can be represented as the zero level set of the function: $$\begin{aligned}
F_\theta(x_1,x_2, x_3)=\cos^2\theta\cosh\big(\frac{x_1}{\cos\theta}\big)-\sin^2\theta\cosh\big(\frac{x_2}{\cos\theta}\big)-\cos x_3.\end{aligned}$$ From this it follows immediately that the angle map in this context $S_\theta \mapsto \theta$ is a diffeomorphism. A corresponding result for the family ${\mathcal M}_4^{even}$ is of course more difficult since no explicit formula is available in this case.
We will explore further the analogy of our result with the theory of minimal surfaces in ${\mathbb R}^3$, now in the context of the classification of the four ended solutions in Theorem \[classification\]. The corresponding problem can be stated as follows: if $S$ is an embedded, singly periodic, minimal surface with $4$ Scherk ends, what can be said about this surface ? It is proven by Meeks and Wolf [@MR2276776] that $S$ must be one of the Scherk surfaces $S_\theta$ described above (similar result is proven in [@MR2262839] assuming additionally that the genus of $S$ in the quotient ${\mathbb R}^3/{\mathbb Z}$ is $0$). The key results to prove this general statement are in fact the counterparts of Theorem \[teo uniqueness\] and Theorem \[part I\].
We now sketch the basic elements in the proofs of Theorem \[teo uniqueness\]. First of all let us explain the existence result in [@MR2557944]. The point of departure of the construction is the following Toda system $$\begin{aligned}
{{c_0}} \, q_1'' &= - e^{\sqrt {F''(1)} ( q_{1}- q_{2}) },\\
{{c_0}}\, q_2'' &= e^{\,\sqrt {F''(1)} (q_{1}- q_2)},
\end{aligned}
\label{toda 02}$$ for which $q_1<0<q_2$ and $q_1(x)=-q_2(x)$, as well as $q_j(x)=q_j(-x)$, $j=1,2$. Here $c_0$ is a fixed constant depending on $F$ only (when $F(u)=\frac{1}{4}(1-u^2)^2$, then $c_0=\frac{\sqrt{2}}{24}$). Any solution of this system is asymptotically linear, namely: $$q_{j}(x)=(-1)^j (m |x|+b)+{\mathcal O}(e^{\, -2\sqrt{F''(1)}m |x|}), \quad x\to \infty,$$ where $m>0$ is the slope of the asymptotic straight line in the first quadrant. On the other hand, given that we only consider solutions whose trajectories are symmetric with respect to the $x$-axis, the value of the slope $m$ determines the unique solution of (\[toda 02\]). When the asymptotic lines become parallel then $m\to 0$ or $m\to \infty$. By symmetry it suffices to consider the case $m\to 0$ and in this paper we will denote small slopes by $m={\varepsilon}$ and the corresponding solutions by $q_{{\varepsilon}, j}$. Note that if by $q_{1,j}$ we denote a solution with slope $m=1$ then $$q_{{\varepsilon},j}(x)=q_{1, j}({\varepsilon}x)+\frac{(-1)^{j}}{\sqrt{F''(1)}}\log\frac{1}{{\varepsilon}} .$$ Then, the existence result in [@MR2557944] implies that given a small ${\varepsilon}$, there exists a four ended solution $u$ to (\[AC\]) whose nodal set $N(u)$ is close to the trajectories of the Toda system given by the graphs $y=q_{{\varepsilon}, j}(x)$. Although we do not use directly this result in the present paper but the idea of relating solutions of the Toda system and the four ended solutions of (\[AC\]) that comes from [@MR2557944] is very important. In fact, what we want to achieve is to parametrize the manifold of four ended solutions with almost parallel ends using corresponding solutions of the Toda system as parameters. To do this in Section \[nodal asymp\] we obtain a very precise control of the nodal sets of the four ended solutions. The key observation is that in every quadrant the nodal set $N(u)$ of any four ended solution is a bigraph, and if we assume that the slope of its asymptotic lines is small then it is a graph of a smooth function, both in the lower and in the upper half plane. We have then $$N(u)=\{(x,y)\in {\mathbb R}^2\mid y=f_{{\varepsilon}, j}(x), \quad j=1,2, \quad f_{{\varepsilon}, 1}(x)<0, \quad f_{{\varepsilon},2}(x)=-f_{{\varepsilon}, 1}(x)\},$$ for any $u\in {\mathcal M}_{4}^{even}$, with ${\varepsilon}=\tan\theta(u)$. Our main result in Section \[nodal asymp\] says that for each ${\varepsilon}$ small $$f_{{\varepsilon}, 1}(x)-q_{{\varepsilon}, 1}(x)={\mathcal O}({\varepsilon}^\alpha e^{\, -\tau{\varepsilon}|x|}),$$ with some positive constants $\alpha, \tau$. Next, we define (Section \[toda aprox\]) a suitable approximate four ended solution based on the solution of the Toda system with slope ${\varepsilon}$. To explain this by $\widetilde\Gamma_{{\varepsilon}, 1}$ we denote the graph of the function $y=q_{{\varepsilon}, 1}(x)$, which is contained in the lower half plane. In a suitable neighborhood of the curve $\widetilde\Gamma_{{\varepsilon}, 1}$ we introduce Fermi coordinates ${\tt x}=(x, y)\mapsto (x_1, y_1)$, where $y_1$ denotes the signed distance to $\widetilde\Gamma_{{\varepsilon}, 1}$, and $x_1$ is the $x$ coordinate of the projection of the point ${\tt x}$ onto $\widetilde\Gamma_{{\varepsilon}, 1}$. With this notation we write locally the solution $u$, with ${\varepsilon}=\tan \theta(u)$ in the form $$u({\tt x})=H(y_1-h_{{\varepsilon}}(x_1))+\phi.$$ This definition is suitably adjusted to yield a globally defined function. Then it is proven in Section \[toda aprox\] that $h_{\varepsilon}\colon {\mathbb R}\to {\mathbb R}$ and $\phi\colon{\mathbb R}^2\to {\mathbb R}$ are small functions, of order ${\mathcal O}({\varepsilon}^\alpha)$ in some weighted norms.
Finally in Section \[last step\] we consider two solution $u_1$, and $u_2$ such that $\tan\theta(u_1)=\tan\theta(u_2)={\varepsilon}$. We use the results of the previous section to prove that the function $u_1-u_2$ is Lipschitz and its norm can be controlled by the norms of $h_{1, {\varepsilon}}-h_{2, {\varepsilon}}$ and $\phi_1-\phi_2$ multiplied by a small constant. But on the other hand, using an argument similar to Lyapunov-Schmidt reduction we show that the difference $h_{1,{\varepsilon}}-h_{2, {\varepsilon}}$ is controlled $u_1-u_2$, which eventually implies that $h_{1, {\varepsilon}}=h_{2, {\varepsilon}}$, and thus yields uniqueness.
Preliminaries {#sec prelim}
=============
In this section we collect some facts about the Allen-Cahn equation which will be used later on.
Refined asymptotics theorem for four ended solutions {#summary moduli}
----------------------------------------------------
Let $H(x)$ be the heteroclinic solution on the Allen-Cahn equation and let us denote $\alpha_0=\sqrt{F''(1)}$. It is known that we have asymptotically: $$H(x)=1-Ae^{\,-\alpha_0 x}+{\mathcal O}(e^{\,-2\alpha_0 x}), \quad H'(x)=A\alpha_0 e^{\,-\alpha_0 x}+{\mathcal O}(e^{\,-2\alpha_0 x}),\quad x\to \infty,$$ with similar estimates when $x \to-\infty$.
We consider the linearized operator $$\begin{aligned}
L_0\phi=-\phi''+F''(H)\phi.\end{aligned}$$ It is known that the principal eigenvalue of this operator $\mu_0=0$ and the corresponding eigenfunction is $H'$. In general, the operator $L_0$ has, possibly infinite, discrete spectrum $0<\mu_1<\dots\leq \alpha_0^2$, and the continuous spectrum which is $[\alpha_0^2,\infty)$, $\alpha_0=\sqrt{F''(1)}$. It may also happen that $L_0$ has just one eigenvalue, $\mu_0=0$ and the continuous spectrum, in which case we will set $\mu_1=\alpha_0^2$.
Next, we recall some facts about the moduli space theory developed in [@dkp-2009]. We will mostly use this theory in the case of four end solutions, thus we will restrict the presentation to this situation only. We keep the notations introduced above. Thus we let $$\lambda = (\lambda_1, \ldots, \lambda_{4} ) \in \Lambda_{4},$$ we write $ \lambda_j^+ = {\tt x}_j + \mathbb R^+ \, {\tt e}_j$ as in (\[eq:halfline\]). We denote by $\Omega_0, \ldots , \Omega_{4}$ the decomposition of $\mathbb R^2$ associated to this $4$ half affine lines and $\mathbb I_0, \ldots, \mathbb I_{4}$ the partition of unity subordinate to this partition. Given $\gamma, \delta \in \mathbb R$, we define a [*weight function*]{} $\Gamma_{\gamma, \delta}$ by $$\Gamma_{\gamma, \delta} ( {\tt x} ) : = {\mathbb I}_0 ({\tt x}) + \sum_{j=1}^{4} \, {\mathbb I}_j ({\tt x}) \, e^{{\gamma} \, ({\tt x} - {\tt x}_j) \cdot {\tt e}_j } \, \left( \cosh ( ({\tt x} - {\tt x}_j) \cdot {\tt e}_j^\perp ) \right)^{{\delta}} ,
\label{weight}$$ so that, by construction, $\gamma$ is the rate of decay or blow up along the half lines $\lambda_j^+$ and $\delta$ is the rate of decay or blow up in the direction orthogonal to $\lambda_j^+$.
With this definition in mind, we define the weighted Lebesgue space $$L_{\gamma, \delta}^2 (\mathbb R^2) : = \Gamma_{\gamma, \delta} \, L^2 (\mathbb R^2) ,
\label{l2weight}$$ and the weighted Sobolev space $$W_{\gamma, \delta}^{2,2} (\mathbb R^2) : = \Gamma_{\gamma, \delta} \, W^{2,2} (\mathbb R^2).
\label{S2weight}$$ Observe that, even though this does not appear in the notations, the partition of unity, the weight function and the induced weighted spaces all depend on the choice of $\lambda \in \Lambda_{4}$.
Our first result shows that, if $u$ is a solution of (\[AC\]) which is close to $u_\lambda$ (in $W^{2,2}$ topology) then $ u -u_\lambda$ tends to $0$ exponentially fast at infinity.
\[refined asymp\] Assume that $u \in \mathcal S_{4}$ is a solution of (\[AC\]) and define $\lambda \in \Lambda_{4}$, so that $$u - u_\lambda \in W^{2,2} (\mathbb R^2) .$$ Then, there exist $ \delta \in (0, \alpha_0 )$, $\alpha_0=\sqrt{F''(1)}$, and $\gamma > 0$ such that $$u - u_\lambda \in W^{2,2}_{-\gamma, -\delta} (\mathbb R^2) .
\label{eq:refined-2}$$ More precisely, $ \delta > 0$ and $ \gamma > 0$ can be chosen so that $$\begin{aligned}
\label{bargammadelta}
\gamma \in (0, \sqrt{\mu_1}) , \qquad \gamma^2 + \delta^2 < \alpha_0^2 \qquad \mbox{and} \qquad \alpha_0 > \delta + \gamma \, \cot \theta_\lambda ,\end{aligned}$$ where $ \theta_\lambda$ is equal to the half of the minimum of the angles between two consecutive oriented affine lines $\lambda_1, \ldots, \lambda_{4}$ (see (\[min angle\])) and $\mu_1$ is the second eigenvalue of the operator $L_0$ (or $\mu_1=\alpha_0^2$ if $0$ is the only eigenvalue).
We observe here that it is well known that for any solution of (\[AC\]) the following is true: if by $N(u)$ we will denote the nodal set of $u$ and by ${\mathrm d}(N(u), {\tt x})$ the distance of ${\tt x}$ to $N(u)$ then $$\begin{aligned}
\label{exp est 1}
|u({\tt x})^2-1|+|\nabla u({\tt x})|+|D^2 u({\tt x})|\leq C e^{\,-\beta{\mathrm d}(N(u), {\tt x})},\end{aligned}$$ where $\beta>0$. This type of estimate is relatively easy to obtain using a comparison argument. On the other hand, the estimate (\[eq:refined-2\]) is non trivial.
The balancing formulas
----------------------
We will now describe briefly the balancing formulas for $4$ ended solutions in the form they were introduced in [@dkp-2009]. Assume that $u$ is a solution of (\[AC\]) which is defined in $\mathbb R^2$. Assume that $X$ and $Y$ are two vector fields also defined in $\mathbb R^2$. In coordinates, we can write $$X = \sum_j X^j \partial_{x_j}, \qquad \qquad Y = \sum_j Y^j \partial_{x_j},$$ and, if $f$ is a smooth function, we use the following notations $$X(f) : = \sum_j X^j \, \partial_{x_j} f, \qquad \qquad \nabla f : = \sum_j \partial_{x_j} f \, \partial_{x_j} ,$$ $$\mbox{\rm div} \, X : = \sum_i \partial_{x_i} X^i ,$$ and $${\rm d}^* X : = \frac{1}{2} \sum_{i,j} ( \partial_{x_i} X^j + \partial_{x_j} X^i ) \, {\rm d}x_i \otimes {\rm d}x_j ,$$ so that $${\rm d}^* X \, (Y,Y) = \sum_{i,j} \partial_{x_i} X^j \, Y^i \, Y^j .$$
We claim that :
\[Balancing formula\] The following identity holds $$\begin{array}{rlllll}
\displaystyle \mbox{\rm div} \left( \left( \frac12 | \nabla u |^2 + F(u) \right) X - X(u) \nabla u \right) = \displaystyle \left( \frac12 | \nabla u|^2 + F(u) \right) \mbox{\rm div} \, X - {\rm d}^* X (\nabla u ,\nabla u) .
\end{array}$$ \[le:PI\]
This follows from direct computation.
Translations of $\mathbb R^2$ correspond to the constant vector field $$X : = X_0$$ where $X_0$ is a fixed vector, while rotations correspond to the vector field $$X : = x \, \partial_y - y \, \partial_x .$$ In either case, we have $\mbox{\rm div} \, X =0$ and ${\rm d}^* \, X =0$. Therefore, we conclude that $$\mbox{\rm div} \, \left( \left( \frac12 \, | \nabla u |^2 + F(u) \right) \, X - X(u) \, \nabla u \right) = 0 ,$$ for these two vector fields. The divergence theorem implies that $$\int_{\partial \Omega} \left( \left( \frac12 \, | \nabla u |^2 + F(u) \right) \, X - X(u) \, \nabla u \right) \cdot \nu \, ds = 0 ,
\label{eq:lsr}$$ where $\nu$ is the (outward pointing) unit normal vector field to $\partial \Omega$.
To see how this identity is applied let us fix a unit vector ${\tt e}\in {\mathbb R}^2$ and let $X={\tt e}$. For any $s\in {\mathbb R}$ we consider a straight line $L_s=\{{\tt x}\in {\mathbb R}^2\mid {\tt x}=s {\tt e}+t{\tt e}^\perp, t\in {\mathbb R}\}$. Then we get: $$\begin{aligned}
\label{hamilt 1}
\int_{L_s} \big[\frac{1}{2}|\nabla u|^2-|\nabla u\cdot{\tt e}|^2+F(u)\big]\,dS=const.,\end{aligned}$$ for any $4$ end solution $u$ of (\[AC\]), as long as the direction of $L_s$ does not coincide with that of any end, i.e. ${\tt e}\neq {\tt e_j}$, $j=1, \dots, 4$. In a particular case ${\tt e}=(0,1)$ we get a [*Hamiltonian identity*]{} [@MR2381198]: $$\begin{aligned}
\label{hamilt 2}
\int_{y=s} \big[\frac{1}{2}(\partial_x u)^2-\frac{1}{2}(\partial_y u)^2 + F(u)\big]\,dx=const. . \end{aligned}$$
Summary of the existence result for small angles in [@MR2557944] {#exists small eps}
----------------------------------------------------------------
To state the existence result in precise way, we assume that we are given an even symmetric solution of the [Toda system]{} (\[toda 02\]) represented by a pair of functions $q_1(t)<0<q_2(t)$, where $q_1(t)=-q_2(t)$ as well as $q_1(t)=q_1(-t)$. In addition let us assume that the slope of $q_1$ at $\infty$ is $-1$. Then, asymptotically we have: $$q_{j}(x)=(-1)^j (|x|+b)+{\mathcal O}(e^{\, -2\sqrt{F''(1)}|x|}), \quad x\to \infty,$$ Given $\varepsilon>0$, we define the the vector valued function ${\bf q}_{\varepsilon}$, whose components are given by $$q_{j, \varepsilon} (x) : = q_{j}(\varepsilon \, x )
+\frac{(-1)^j}{\sqrt {F''(1)}} \log \frac{1}{\varepsilon} \, .
\label {falpha}$$ It is easy to check that the $q_{j,\varepsilon}$ are again solutions of (\[toda 02\]).
Observe that, according to the description of asymptotics the functions $q_j$, the graphs of the functions $q_{j,\varepsilon}$ are asymptotic to oriented half lines with slope ${\varepsilon}$ at infinity. In addition, for $\varepsilon >0$ small enough, these graphs are disjoint and in fact their mutual distance is given by $ \frac{2}{\sqrt{F''(1)}} \, \log\frac{1}{\varepsilon} + \mathcal O (1)$ as $\varepsilon$ tends to $0$.
It will be convenient to agree that $\chi^+$ (resp. $\chi^-$) is a smooth cutoff function defined on $\mathbb R$ which is identically equal to $1$ for $x >1$ (resp. for $x < -1$) and identically equal to $0$ for $x < -1$ (resp. for $x > 1$) and additionally $\chi^-+\chi^+\equiv 1$. With these cutoff functions at hand, we define the $4$ dimensional space $$D : = {\rm Span} \, \{x \longmapsto \chi^\pm (x) , \, x\longmapsto x \, \chi^\pm (x) \} \, ,
\label{D}$$ and, for all $\mu \in (0,1)$ and all $\tau \in \mathbb R$, we define the space ${\mathcal C}^{2, \mu}_\tau (\mathbb R)$ of $\mathcal C^{2 , \mu}$ functions $r$ which satisfy $$\| r\|_{\mathcal C^{\ell , \mu}_\tau (\mathbb R)} : = \|Ê (\cosh x)^{\tau} \, r \|_{\mathcal C^{\ell , \mu} (\mathbb R )} \, < \infty \, .$$ Keeping in mind the above notations, we have the :
\[teo1\] For all $\varepsilon>0$ sufficiently small, there exists an entire solution $u_\varepsilon$ of the Allen-Cahn equation ${(\ref{AC})}$ whose nodal set is the union of $2$ disjoint curves ${\widetilde}\Gamma_{1, \varepsilon }, {\widetilde}\Gamma_{2, \varepsilon }$ which are the graphs of the functions $$x \longmapsto q_{j,\varepsilon } (x) + r_{j, \varepsilon} (\varepsilon \, x) \, ,$$ for some functions $r_{j, \varepsilon } \in \mathcal C^{2, \mu}_{\tau}
(\mathbb R) \oplus D$ satisfying $$\| r_{j, \varepsilon} \|_{\mathcal C^{2, \mu}_{\tau } (\mathbb R) \oplus D}
\leq C \, \varepsilon^{\alpha} \, .$$ for some constants $C , \alpha , \tau , \mu >0$ independent of $\varepsilon >0$.
In other words, given a solution of the Toda system, we can find a one parameter family of $4$-ended solutions of [(\[AC\])]{} which depend on a small parameter $\varepsilon > 0$. As $\varepsilon$ tends to $0$, the nodal sets of the solutions we construct become close to the graphs of the functions $q_{j ,\varepsilon }$.
Going through the proof, one can be more precise about the description of the solution $u_\varepsilon$. If $\Gamma \subset \mathbb R^2$ is a curve in $\mathbb R^2$ which is the graph over the $x$-axis of some function, we denote by $\mbox{Y} \, ( \cdot, \Gamma ) $ the signed distance to $\Gamma$ which is positive in the upper half of $\mathbb R^2 \setminus \Gamma$ and is negative in the lower half of $\mathbb R^2 \setminus \Gamma$. Then we have the :
The solution of [(\[AC\])]{} provided by Theorem \[teo1\] satisfies $$\| e^{ {\varepsilon} \, \hat \alpha \, |{\bf x}|} \, (u_\varepsilon - u_\varepsilon^*) \|_{L^\infty (\mathbb R^2)}
\leq C \, \varepsilon^{\bar \alpha}Ê\, ,$$ for some constants $C, \bar \alpha , \hat \alpha > 0$ independent of $\varepsilon$, where ${\bf x}=(x,y)$ and $$u_\varepsilon^* : = \sum_{\red{j=1}}^k (-1)^{j+1} \, H\big( {\rm Y} ( \cdot,
{\widetilde}\Gamma_{j , \varepsilon}) \big) - \frac 12 ( (-1)^{k} +1 ) \,.
\label{eq:bua}$$
The nodal sets of solutions {#nodal asymp}
===========================
We know [@partI] that the angle map on any connected component of the moduli space $\mathcal{M}_4$ of four end solutions is surjective, and that in particular it contains solutions whose nodal lines are almost parallel ($\theta(u)\approx 0$ or $\frac{\pi}{2}-\theta(u)\approx 0$). We recall also that, after a rigid motion, any four ended solution is even symmetric [@2011arXiv1102.4022G] and thus we will always consider solutions in ${\mathcal M}^{even}_4$, which in particular satisfy (\[even\]). To any solution $u\in {\mathcal M}^{even}_4$ we can associate the value of the angle map $\theta(u)$, which by definition is the asymptotic angle at $\infty$ between the nodal set of $u$ in the first quadrant and the $x$-axis. Finally, by $N(u)$ we will denote in this paper the nodal set of $u\in{\mathcal M}^{even}_4$. Because of (\[even\]), restricted to each quadrant, this set is a bigraph, and restricted to a half plane it is a graph. We are interested in solutions whose nodal lines are almost parallel at $\infty$ and, by symmetry, we can restrict our considerations to the case $\theta(u)\to 0$. In this case the nodal set consists of two components, one of them is a graph of a smooth function in the lower half plane and the other, which is symmetric with respect to the $x$-axis, is contained in the upper half plane. It is convenient to introduce a parameter ${\varepsilon}=\tan\theta(u)$, which is small ${\varepsilon}\to 0$, when $\theta(u)\to 0$.
Basic properties of solutions with almost parallel ends
-------------------------------------------------------
It is expected that when the angle between the ends becomes smaller, i.e. $\theta(u)\to 0$ or $\theta(u)\to \frac{\pi}{2}$, then the distance between the nodal sets becomes larger. This is indeed the case and could be seen from lemma \[lema unique 1\] to follow. With no loss of generality we will restrict our considerations to the case $\theta(u)\to 0$. In the sequel by $Q_1$ we will denote the first quadrant in ${\mathbb R}^2$.
\[lema unique 1\] Suppose $u_{n}$ is a sequence of four end solutions such that $\theta({u_{n})}\rightarrow0,$ $p_{n}\in N\left( u_{n}\right) \cap\partial Q_{1}.$ Then $\left\vert p_{n}\right\vert \rightarrow+\infty,$ as $n\rightarrow+\infty.$ Moreover $p_n\in \{x=0\}$.
To show that $p_n\to \infty$ we suppose to the contrary that $p_n\to p^*$, $|p^*|<\infty$ . We know that the sequence $\{u_n\}$ converges in ${\mathcal C}_{loc}^2({\mathbb R}^2)$ to a solution $u^*$ of the Allen-Cahn equation. Since $|p^*|<\infty$, by Theorem 4.4 of [@MR2381198] we know that if $u^*_x>0$, $x>0$, $u^*_y<0$, $y>0$, then $u^*$ must be a solution to (\[AC\]) whose nodal set in the first quadrant is asymptotically a straight line with positive slope equal to $\tan \theta^*\neq 0$. To show that $u^*_x>0$, $x>0$ and $u^*_y<0$, $y>0$, one may use an maximum principle based argument, similar to the one in the Claim 1 below, we leave the details to the reader. Although a priori $u^*$ is not a four ended solution in the sense of Definition \[de:001\], it can also be proven, using the De Giorgi conjecture in dimension two and the Refined Asymptotic theorem (Proposition \[refined asymp\]), that $u^*\in\mathcal{M}_4^{even}$ (we will not rely on this fact in the argument given below). For future purpose we recall that the nodal set in the first quadrant is a graph i.e. $N(u_n)\cap Q_1=\{(x,y)\mid y=f_n(x)\}$, where $f_n$ is a smooth function.
To proceed we need the following:
[**[Claim 1]{}.**]{} [*Let $\{u_n\}$ be a sequence of solutions as described above and let $x_n\to \infty$, as $n\to \infty$. There exists a subsequence $\{x_{n_k}\}$ such that $f_{n_k}(x_{n_k})\to \infty$, as $k\to \infty$.* ]{}
To prove this claim we argue by contradiction. We assume that for some constant $A_0>0$, we have that $$f_n(x_n)\leq A_0, \quad \forall n.$$ Since $f_n'(x)=-\frac{u_{n,x}(x, f_n(x))}{u_{n,y}(x,f_n(x))}>0$ therefore, passing to the limit as $n\to \infty$, we have $u_n\to u^*$, at least along a subsequence in $\mathcal C^2_{loc}({\mathbb R}^2)$, where $u^*$ is an even solution such that $u^*_x\geq 0$, in $x>0$, $u^*_y\leq 0$, in $y>0$. We will show that in fact $u^*_x>0$ when $x>0$ and $u^*_y<0$, when $y>0$. If $u_x^*(x^*, y^*)=0$, for some $(x^*, y^*)$ then, using the maximum principle, we get $u^*_x\equiv 0$. The same is true for $u^*_y$. Consequently either $u^*$ is an even, monotone, one dimensional solution, which is impossible, or $u^*=u_0$, where $u_0=\pm 1$ or $0$. In the latter case we consider the quantity $$E_R(u)= \frac{1}{R}\int_{B_R}\frac{1}{2}|\nabla u|^2+F(u), \quad R>0.$$ This quantity is known to be monotone in $R$ and, since $|p^*|<\infty$ we have that $$c\leq E_R(u_n)\leq C,$$ for certain constants $c, C>0$. But since $E_R(u_n)\to E_R(u^*)$ we see that $u^*$ can not be a constant.
In summary $u^*$ is an even and monotone solution in ${\mathbb R}^2$, whose $0$ level set $N(u^*)$ is contained in the strip $|y|\leq A_0$. But by Theorem 4.4 of [@MR2381198] this is impossible. This ends the proof of the claim.
Now we have the following:
[**[Claim 2]{}.**]{} [*For each sufficiently small $\delta$ there exists a $r>0$ such that for all $r<x_{1}<x_2$ we have $|f'_n(x_1)-f'_n(x_2)|<\delta$.*]{}
Accepting this we get that $\lim_{n\to \infty}\tan \theta(u_n)=\tan\theta^*\neq 0$, which is a contradiction. It remains then to prove the claim.
Arguing by contradiction we suppose that there exist sequence $r_{k}\rightarrow+\infty$ and $x_{1,n_{k}},x_{2,n_{k}}>r_{k},$ such that$$\label{contra 1}
\left\vert f_{n_{k}}^{\prime}\left( x_{1,n_{k}}\right)
-f_{n_{k}}^{\prime}\left( x_{2,n_{k}}\right) \right\vert \geq {\delta}.$$ Then, at least for one of the points $x_{i, n_k}$ we have (passing to a subsequence if necessary): $$\label{lem one 1}
|f'(x_{i, n_k})|>\frac{\delta}{2}.$$ We will call this point $x^*_{1, n_k}$. Next we chose $x^{*}_{2, n_k}$ such that $|f'_{n_k}(x^{*}_{2, n_k})|=\frac{\delta}{4}$ and $$\frac{\delta}{4}\leq |f'_{n_k}(x)|, \quad x\in [x^*_{1, n_k}, x^{*}_{2,n_k}].$$ Such point clearly exists when $k$ is sufficiently large since $f'_{n_k}(x)\to \tan(\theta(u_{n_k}))$, as $x\to \infty$.
Consider two lines $L_{1,n_{k}}$ and $L_{2,n_{k}}$ with slopes $-1$ passing though the points $(x^*_{i,n_{k}},f_{n_{k}}(x^*_{i,n_{k}}))$, $i=1,2$, respectively. Note that since the nodal lines $N(u_{n_{k}})$ are bigraphs, which are eventually asymptotically straight lines with positive slopes, therefore the lines $L_{i,n_{k}}$ must be transversal to $N(u_{n_{k}})$ at their points of intersection. Next, consider the domain $\Omega_{n_{k}}\subset Q_{1}$ bounded by the lines $L_{i,n_{k}}$, $i=1,2$, and the vector field $X=\left( 0,1\right) .$ The balancing formula (\[eq:lsr\]) tells us $$\int_{\partial\Omega_{n_{k}}}\left( \left( \frac{1}{2}\left\vert \nabla
u_{n_{k}}\right\vert ^{2}+F\left( u_{n_{k}}\right) \right) X-X\left(
u_{n_{k}}\right) \nabla u_{n_{k}}\right) \cdot\nu dS=0.$$ Now we estimate the various boundary integrals involved in the above integral. First, note that the integral over the segment $\partial\Omega_{n_k}\cap\{x=0\}$ is automatically $0$ by the choice of the vector field $X$ and the evenness of $u_{n_k}$.
Next, we will show that as $r_{k}\rightarrow+\infty,$ $$\int_{\partial\Omega_{n_{k}}\cap\left\{ y=0\right\} }\left( \left(
\frac{1}{2}\left\vert \nabla u_{n_{k}}\right\vert ^{2}+F\left( u_{n_{k}}\right) \right) X-X\left( u_{n_{k}}\right) \nabla u_{n_{k}}\right)
\cdot\nu dS\rightarrow0. \label{small}$$ To this end let $\bar x_{i, n_k}$, $i=1,2$ be two points such that $\partial \Omega_{n_k}\cap\{y=0\}=\{(x, 0)\mid x\in [\bar x_{1, n_k}, \bar x_{2, n_k}] \}$. Then, by elementary geometry we get, with some constant $c>0$: $$\mathrm{dist}\,(N(u_{n_k})\cap Q_1, {\tt x})\geq \min\left\{(x-x^*_{1,n_k}), c \left(\frac{\delta}{4}(x-x^*_{1,n_k})+f(x^*_{1,n_k})\right)\right\}, \quad {\tt x}=(x, 0), \quad x \in [\bar x_{1,n_k}, x^*_{2, n_k}].$$ On the other hand, when $x\in [x^*_{2,n_k}, \bar x_{2, n_k}]$ then $$\begin{aligned}
\mathrm{dist}\,(N(u_{n_k})\cap Q_1, {\tt x})&\geq c\left( \frac{\delta}{4}(x^*_{2,n_k}-x^*_{1,n_k})+f_{n_k}(x^*_{1,n_k})\right), \\
( \bar x_{2, n_k}-x^*_{2,n_k})&\leq C \left(\frac{\delta}{4}(x^*_{2,n_k}-x^*_{1,n_k})+f_{n_k}(x^*_{1,n_k})\right).\end{aligned}$$ Additionally we know that $$\left\vert u_{n_{k}}^{2}\left({\tt x}\right) -1\right\vert +\left\vert \nabla
u_{n_{k}}\left( {\tt x}\right) \right\vert \leq Ce^{-\beta\mathrm{dist}\,(N(u_{n_k})\cap Q_1, {\tt x})}, \quad{\tt x}=(x,0) \in\partial\Omega_{n_{k}}.
\label{de}$$ Above, the constants $c>0$ and $C>0$ can be chosen independent on $n$. Using this, the Claim 1, and the fact that $\bar x_{1, n_k}-x^*_{1,n_k}$ is proportional to $f_{n_k}(x^*_{1,n_k})$, we conclude (\[small\]).
Now we will estimate the integrals along the lines $\partial\Omega_{n_k}\cap L_{i, n_k}$. For this purpose it is convenient to denote $${\tt e}_{i,n_k}=(\cos\alpha_{i,n_k}, \sin\alpha_{i,n_k}), \quad {\tt e}^\perp_{i, n_k}=(\sin\alpha_{i,n_k}, -\cos\alpha_{i, n_k}), \quad \mbox{where}\ \alpha_{i,n_{k}}=\arctan f_{n_{k}}^{\prime}\left( x^*_{i,n_{k}}\right),
i=1,2.$$ Let us consider the sequence of functions $v_{i, k}(x,y)=u(x^*_{i, n_k}+x, f_{n_k}(x^*_{i, n_k})+y)$. By the De Giorgi conjecture in dimension two we have: $$v_k(x,y)-H({\tt e}^\perp_{i, n_k}\cdot (x,y))\to 0, \quad \mbox{in}\ {\mathcal C}^2_{loc}({\mathbb R}^2).$$ In other words, locally around $\left( x^*_{i,n_{k}},f_{n_{k}}\left( x^*_{i,n_{k}}\right) \right) $, the function $u_{n_k}$ converges to $$H\left({\tt e}^\perp_{i, n_k}\cdot(x-x^*_{i,n_{k}}, y-f_{n_{k}}( x^*_{i,n_{k}})\right).$$ On the other hand, again by (\[exp est 1\]), we know that on the segment $\partial\Omega_{n_{k}}\cap L_{i,n_{k}}$, $$|u_{n_{k}}^{2}\left({\tt x}\right) -1|+|\nabla u_{n_{k}}\left({\tt x}\right)
|\leq Ce^{-\beta|x^*_{i,n_{k}}-x|}, \quad {\tt x}=(x,y).$$ These facts, after some calculations, yield $$\int_{\partial\Omega_{n_{k}}\cap L_{i,n_{k}}}\left( \left( \frac{1}{2}\left\vert \nabla u_{n_{k}}\right\vert ^{2}+F\left( u_{n_{k}}\right)
\right) X-X\left( u_{n_{k}}\right) \nabla u_{n_{k}}\right) \cdot\nu
dS=\left( -1\right) ^{i+1}\sin\alpha_{i,n_{k}}\mathtt{e}_{F}+o\left(
1\right) ,$$ where $o\left( 1\right) $ is a term goes to $0$ as $r_{k}\rightarrow
+\infty,$ while $${\tt e}_{F}=\int_{\mathbb R}(H')^2.$$ Combining all the above estimates, we infer $$\sin\alpha_{1,n_{k}}-\sin\alpha_{2,n_{k}}=o\left( 1\right) ,$$ which is a contradiction.
It remains to show
Now, $N(u)\cap Q_1$ is a graph of a monotonically increasing, even function hence the above proposition asserts that as $\theta(u)\to 0$, the distance between the nodal set of $u$ and the $x$ axis will go the infinity as well.
A refinement of the asymptotic behavior of the nodal set
--------------------------------------------------------
Let $u$ be a four-end solution with small angle $\theta({u}).$ We denote $\varepsilon=\tan\theta(u)$ and use for simplicity ${\varepsilon}$ as a small parameter. To obtain more precise information about this solution, our first step is to define a good approximate solution. As we will see later, the fact that there is a true solution around the approximate one restricts the possible behavior of the nodal set and enables us the analyze the equation satisfied by the nodal line.
The nodal set $N\left( u\right) $ in the lower half plane is the graph $\Gamma$ of a function $y=f\left( x\right) $, $y\in {\mathbb R}$. Strictly speaking the function $f$ depends on $u$ but we will not indicate this dependence. Using the De Giorgi conjecture in $2$ dimensions it is not difficult to show that $\left\Vert f^{\prime}\left( x\right) \right\Vert _{{\mathcal C}^{1}({\mathbb R})}\to 0$ as $\theta({u})\to 0$. Indeed, let us assume that there exist a sequence of solutions $u_n$, with $\theta(u_n)\to 0$, and a sequence of points ${\tt x}_n\in N(u_n)\cap Q_1$, such that $$\label{f prime 2}
{\tt x}_n=(x_n, f(x_n)),\quad \mbox{and}\ |f'(x_n)|+|f''(x_n)|>\delta, \quad \mbox{with some} \ \delta>0.$$ Then we define: $$v_n({\tt x})=u_n({\tt x}_n+{\tt x}).$$ Using the monotonicity of $u_n$ we get that $v_{n,y}>0$, for $y<-f(x)$ and then the De Giorgi conjecture implies that $v_n$ converges in ${\mathcal C}^2_{loc}({\mathbb R}^2)$ to the heteroclinic solution which contradicts (\[f prime 2\]). For future references we observe finally that in general the graph of $N(u)\cap Q_1$ is at least a ${\mathcal C}^3({\mathbb R})$ function and,
To fix attention we will always work with the solution whose nodal lines have a small slope ${\varepsilon}=\tan\theta(u)$ at $\infty$. This means that the these lines are asymptotically parallel, as ${\varepsilon}\to 0$, to the $x$ axis and one of them is contained in the lower half plane and the other in the upper half plane. We know that they are symmetric with respect to the $x$ axis. In the sequel it will be convenient to denote the component of the nodal set $N(u)$ in the lower half plane by $\Gamma_{{\varepsilon},1}$, and the one in the upper half plane by $\Gamma_{{\varepsilon},2}$. The nodal lines are bigraphs, and consequently we have $\Gamma_{{\varepsilon},i}=\{y=f_{{\varepsilon},i}(x)\}$.
To introduce the functional analytic tools used in this paper we introduce a weight function$$W_{a}({\tt x}) =\left( \cosh x\right) ^{a}, \quad {\tt x}=(x,y), \quad a>0.$$ For $\ell =0,1,2,$ let $\mathcal{C}_{a}^{\ell,\mu}( {{\mathbb R}}^{2}) =W_{a}^{-1}{\mathcal C}^{\ell,\mu}( {{\mathbb R}}^{2}) $, endowed with the weighted norm$$\| \phi\| _{{\mathcal C}_{a}^{\ell,\mu}\left( {{\mathbb R}}^{2}\right)
}:=\sup_{{\tt x}\in {\mathbb R}^2}W_{a}({\tt x})\|\phi\|_{{\mathcal C}^{\ell,\mu}(B({\tt x}, 1)) }.$$ Likewise, we let $W_a(x)=(\cosh x)^a$ and define the weighted space $\mathcal{C}^{\ell, \mu}_a({\mathbb R})$ by: $$\|f\|_{{\mathcal C}_{a}^{\ell,\mu}\left( {{\mathbb R}}\right)
}:=\sup_{{x}\in {\mathbb R}}W_{a}({x})\|f\|_{{\mathcal C}^{\ell,\mu}(({x}-1, x+1)) }.$$ In what follows we will measure the size of various functions involved in the $\mathcal{C}_{a}^{2,\mu}( {{\mathbb R}}^{2})$, and in the ${{\mathcal C}_{a}^{2,\mu}\left( {{\mathbb R}}\right)
}$ norm, where $a, \mu>0$. Mostly we will have $a\sim {\varepsilon}$.
Let us recall that a four end solution $u$ is asymptotic to a model solution $u_{\lambda}$ defined in the introduction. Using the Proposition \[refined asymp\] we know that $u-u_\lambda\in W^{2,2}_{{\varepsilon}\bar\tau, \delta}({\mathbb R}^2)$ with some $\bar \tau >0$ and $\delta>0$, which can be chosen independent on ${\varepsilon}$. We claim that from this it follows that in fact $u-u_\lambda\in \mathcal C_{{\varepsilon}\tau}^{2,\mu}({\mathbb R}^2)$, with some $\tau>0$. To see this we let ${\varepsilon}$ be fixed and ${\tt e}$ be the asymptotic direction of the end of $u$ with ${\varepsilon}=\tan\theta(u)$ in $Q_1$. By definition, taking $R$ large, we have that $$\Gamma_{\gamma, \delta}({\tt x})\sim
\left(\cosh (({\tt x}-{\tt x}_{{\varepsilon}, 1})\cdot {\tt e})\right)^\gamma\left(\cosh(({\tt x}-{\tt x}_{{\varepsilon}, 1})\cdot {\tt e}^\perp)\right)^\delta\sim (\cosh x)^{\gamma +\delta{\mathcal O}({\varepsilon})}(\cosh y)^{\delta+\gamma\mathcal O({\varepsilon})}, \quad {\tt x}\in Q_1\setminus B_R.$$ Taking $\gamma=\bar\tau{\varepsilon}$ and $\delta$ small we get the claim. In addition, we know that outside of a large compact set in the first quadrant we have: $$u({\tt x})=H(({\tt x}-{\tt x}_{{\varepsilon}, 1})\cdot {\tt e}^\perp)+ {\mathcal O}(e^{\,-{\varepsilon}\bar\tau |x|-\delta|y|}).$$ We can use this and the fact that along the nodal set we have $u({\tt x})=0$ to show that $f''_{{\varepsilon}, 2}\in \mathcal{C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})$. Indeed, we get from the above that, with some constant ${\mathcal A}_{\varepsilon}$ we have: $$\label{veeee}
0=H(f_{{\varepsilon}, 2}(x)-{\varepsilon}x-{\mathcal A}_{\varepsilon})+ {\mathcal O}(e^{\,-\tau{\varepsilon}x}), \quad x\to \infty,$$ from which it follows that $$\label{first f}
\|f_{{\varepsilon}, 2}-{\varepsilon}|x|-{\mathcal A}_{\varepsilon}\|_{\mathcal C^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}+\|f'_{{\varepsilon}, 2}-{\varepsilon}\mathrm{sign}\,(x)\|_{\mathcal C^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}+\|f''_{{\varepsilon}, 2}\|_{\mathcal C^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}<\infty.$$
Fermi coordinates near the nodal lines
--------------------------------------
We will now describe neighborhoods of the nodal lines $\Gamma_{{\varepsilon}, i}$, $i=1,2$, where one can define the Fermi coordinates of ${\tt x}\in {\mathbb R}^2$ as the unique $(x_i, y_i)$ such that: $${\tt x}=(x_i, f_{{\varepsilon}, i}(x_i))+y_i n_{{\varepsilon}, i}(x_i), \quad n_{{\varepsilon}, i}(x)=\frac{(-1)^i(f_{{\varepsilon}, i}'(x), -1)}{\sqrt{1+\left(f_{{\varepsilon}, i}'(x)\right)^2}}.$$ We will first find a large, expanding neighborhood of $\Gamma_{{\varepsilon}, i}$ in which that map ${\tt x}\mapsto (x_i, y_i)$ is a diffeomorphism. Because of symmetry it suffices to consider a neighborhood of $\Gamma_{{\varepsilon}, 1}$. We define the projection of a point ${\tt x}\in {\mathbb R}^2$ onto $\Gamma_{{\varepsilon}, 1}$ by: $$\pi_{{\varepsilon},1}({\tt x})=(\pi_{{\varepsilon}, 1}^x({\tt x}), \pi_{{\varepsilon}, 1}^y({\tt x)}):=(x_1, f_{{\varepsilon}, 1}(x_1)), \quad \mbox{whenever}\ {\tt x}=(x_1, f_{{\varepsilon}, 1}(x_1))+y_1n_{{\varepsilon}, 1}(x_1), \quad y_1=\mathrm{dist}\,({\tt x}, \Gamma_{{\varepsilon}, 1}).$$ Note that in general the projection $\pi_{{\varepsilon},1}({\tt x})$ is a multivalued function.
We fix a small number $\theta\in (0,1)$ and let ${m}_{\varepsilon}\colon {\mathbb R}\to {\mathbb R}_+$ to be a function such that for each $x\in {\mathbb R}$, $m_{\varepsilon}(x)$ represents the largest number for which the following properties are satisfied simultaneously:
- $$m_{\varepsilon}(x) \leq\frac{\theta}{\left\vert f_{{\varepsilon}, 1}^{\prime\prime}\left(
x\right) \right\vert },$$
- The projection function $\pi_{{\varepsilon}, 1}$ is a well defined single-valued function for any ${\tt x}\in {\mathbb R}^2$ such that $$\|{\tt x }-\pi_{{\varepsilon}, 1}({\tt x})\| <(m_{\varepsilon}\circ\pi_{{\varepsilon}, 1}^x)({\tt x}).$$
We can regard the function ${m}_{\varepsilon}$ as the measure of the size of the maximal neighborhood of $\Gamma_{{\varepsilon}, 1}$, where the Fermi coordinate could be defined. In fact, conditions (i)–(ii) guarantee that the change of variables given by the Fermi coordinates is a diffeomorphism in a neighborhood of $\Gamma_{{\varepsilon}, 1}$ determined by (ii). To state the next result we let $\tau<\alpha_0=\sqrt{F''(1)}$ be a positive constant.
\[lemma 3.3\] For each $A>0$, and for each sufficiently small ${\varepsilon}$ we have the following estimate:
If $m_{\varepsilon}\left( x\right) =\frac{\theta}{{\left\vert f_{{\varepsilon}, 1}^{\prime\prime}\left(
x\right) \right\vert }},$ then $$e^{\,-m_{\varepsilon}\left( x\right) }\leq \exp\{-\frac{\theta}{|f_{{\varepsilon}, 1}''(x)|}\}
\leq
\exp\{-\frac{\theta(\cosh x)^{\tau{\varepsilon}}}{2\|f''_{{\varepsilon}, 1}\|_{{\mathcal C}^0_{{\varepsilon}\tau}({\mathbb R})}}\}\exp\{-\frac{\theta}{2\|f''_{{\varepsilon}, 1}\|_{{\mathcal C}^0({\mathbb R})}}\}
$$ The desired estimate follows from this. If for some $x_0\in {\mathbb R}$ we have $m_{\varepsilon}\left( x_0\right) <\frac{\theta}{{\left\vert f_{{\varepsilon}, 1}^{\prime\prime}\left(
x_0\right) \right\vert }},$ then there is an ${\tt x}_0$, with $\pi_{{\varepsilon},1}({\tt x}_0)=(x_0, y_0)$, and ${\tt x}_{1}=\left( x_{1},f_{{\varepsilon}, 1}(x_1)\right) $ and ${\tt x}_{2}=\left( x_{2},f_{{\varepsilon}, 1}(x_2)\right),$ on $\Gamma_{{\varepsilon}, 1}$, with $x_{1}<x_{2}$, and such that, denoting $r:=m_{\varepsilon}(x_0)$ we would have $$\|{\tt x}_0-{\tt x}_1\|=\|{\tt x}_0-{\tt x}_2\|=r\leq \frac{\theta}{{|f''_{{\varepsilon}, 1}(x_0)|}},$$ i.e. ${\tt x}_{j}$, $j=1,2$ lie on the circle $S_r$ with center at ${\tt x}_0$. For convenience we fix an orientation of $S_r$ which agrees with the orientation of the segment of $\Gamma_{{\varepsilon},1}$ between the points ${\tt x}_1$ and ${\tt x}_2$.
We claim, that the arc of $S_r$, between ${\tt x}_{1}$ and ${\tt x}_{2}$ is the graph of a function $y=g\left( x\right)
,x\in\left[ x_{1},x_{2}\right]$. Indeed, we recall that $\Gamma_{{\varepsilon}, 1}$ is a bigraph, and so a graph, and the points ${\tt x}_i$ lie on $\Gamma_{{\varepsilon}, 1}\cap S_r$. If there were a vertical line $L$ intersecting the arc of $S_r$ between ${\tt x}_1$ and ${\tt x}_2$ more then once then, since the segment of $\Gamma_{{\varepsilon}, 1}$ between these points lies outside of $S_r$, except at ${\tt x}_1$ and ${\tt x}_2$, we would conclude that necessarily $L$ intersects $\Gamma_{{\varepsilon}, 1}$ more then once. This is a contradiction.
An elementary calculation yields$$\min_{x\in\left[ x_{1},x_{2}\right] }\left\vert g^{\prime\prime}\left(
x\right) \right\vert \geq \frac{1}{r}
$$ On the other hand, $$\left\vert g^{\prime}\left( x_{2}\right) -g^{\prime}\left( x_{1}\right)
\right\vert =\left\vert f_{{\varepsilon},1}^{\prime}\left( x_{2}\right) -f^{\prime}_{{\varepsilon},1}\left(
x_{1}\right) \right\vert .$$ Therefore, one can find a point ${\tt x}_{3}=\left( x _{3},y_{3}\right) \in\Gamma_{{\varepsilon}, 1}$ which satisfies $$\left\vert f^{\prime\prime}_{{\varepsilon},1}\left( x_{3}\right) \right\vert \geq\min
_{x\in\left[ x_{1},x_{2}\right] }\left\vert g^{\prime\prime}\left(
x\right) \right\vert \geq \frac{1}{r}
$$ Summarizing, we see that there exists $x_{3}\in\left( x_0-2m_{\varepsilon}\left( x_0\right) ,x_0+2m_{\varepsilon}\left( x_0\right) \right) \,\ $such that$$\label{meps xxx}
m_{\varepsilon}\left( x_0\right)=r \geq\frac{1}{\left\vert f^{\prime\prime}\left(
x_{3}\right) \right\vert }.$$ This implies, $$e^{\,-\frac{1}{2}m_{\varepsilon}(x_0)}\leq e^{\,-\frac{1}{4}|x_0-x_3|}\leq e^{\,-2A{\varepsilon}|x_0|} e^{\,2A{\varepsilon}|x_3|},$$ as long as $A{\varepsilon}\leq \frac{1}{8}$. But then it follows: $$e^{\,-\frac{1}{2}m_{\varepsilon}(x_0)}e^{\,2A{\varepsilon}|x_0|}\leq \exp({\varepsilon}\tau |x_3|)^{\frac{2A}{\tau}}\leq \Big(\frac{m_{\varepsilon}(x_0)}{\theta}\|f''_{{\varepsilon}, 1}\|_{\mathcal C^0_{{\varepsilon}\tau}({\mathbb R})}\big)^\frac{2A}{\tau}.$$ We claim that from this it follows: $$e^{\,-\frac{1}{2}m_{\varepsilon}\left( x_0\right) }(\cosh x_0)^{A \varepsilon}\leq {\widetilde}C_{A, \tau, \theta} \exp\{\frac{A}{\tau}\log\|f_{{\varepsilon}, 1}''\|
_{\mathcal {C}_{\varepsilon\tau}^{0}\left( \mathbb{R}\right) }\}.$$ This estimate is easy to obtain when $m_{\varepsilon}(x_0)\leq 1$. When $m_{\varepsilon}(x_0)>1$ then the estimate follows from the following simple observation: $$\forall t>1 \quad -t-\frac{2A}{\tau}\log t\leq y\Longrightarrow -t\leq -\frac{1}{2}y-\frac{4A}{\tau}\log\frac{2A}{\tau}.$$ Next we observe that from (\[meps xxx\]) we have as well: $$e^{\,-m_{\varepsilon}(x_0)}\leq \exp\{-\frac{1}{\|f_{{\varepsilon}, 1}''\|_{\mathcal{C}^0({\mathbb R})}}\}.$$ From this estimate (\[est dve 1\]) follows if we write: $$e^{\,-m_{\varepsilon}(x)}(\cosh x)^{A{\varepsilon}}=e^{\,-\frac{1}{2}m_{\varepsilon}(x)}e^{\,-\frac{1}{2}m_{\varepsilon}(x)}(\cosh x)^{A{\varepsilon}}.$$ The proof of the lemma is thus completed.
Based on the result of the lemma we will define a smooth function function $\mathrm{d}_{\varepsilon}$ satisfying property (\[est dve 1\]) and such that ${\mathrm d}_{\varepsilon}(x)\leq m_{\varepsilon}(x)$. To this end we fix $A> 4\tau$ and take ${\varepsilon}\ll 1$ small such that $$C_{A, \tau, \theta}^{\frac{\tau}{A}}\leq \frac{1}{\sqrt{\|f_{{\varepsilon}, 1}''\|_{{\mathcal C}^0({\mathbb R})}}}.$$ Then, from (\[est dve 1\]) it follows: $$m_{\varepsilon}(x)\geq A{\varepsilon}\log(\cosh x)+{\frac{A}{2\tau}}\log\frac{1}{\|f_{{\varepsilon}, 1}''\|_{{\mathcal C}^0_{{\varepsilon}\tau}({\mathbb R})}}
$$ From this we see that the Fermi coordinates of $\Gamma_{{\varepsilon}, 1}$ are well defined in a tubular neighborhood $\mathcal{{O}}_1$ of $\Gamma_{{\varepsilon}, 1}$ defined by $${\mathcal {O}}_1=\{{\tt x}\in {\mathbb R}^2\mid \|{\tt x}-\pi_{{\varepsilon}, 1}({\tt x})\|\leq ({\mathrm d}_{\varepsilon}\circ \pi^x_{{\varepsilon}, 1})({\tt x})\},$$ Moreover, $\partial\mathcal{{O}}_1$ is smooth and has bounded curvature. Letting $\left( x_{1},y_{1}\right) $ be the Fermi coordinate of $\Gamma_{{\varepsilon},1}$ in $\mathcal{{O}}_1$, $y_{1}$ being the signed distance to $\Gamma_{{\varepsilon},1},$ positive in the upper part of $\mathbb{R}^{2}\setminus\Gamma_{{\varepsilon},1}$. The coordinate transformation is a diffeomorphism $\left( x_{1},y_{1}\right) \mapsto
\left( x,y\right) $ between the Fermi coordinates and the Euclidean coordinate and it is given explicitly by $$\begin{aligned}
\label{fermi 50}
\begin{aligned}
x & =x_{1}-\frac{f_{{\varepsilon},1}^{\prime}\left( x_{1}\right) }{\sqrt{1+\left(
f_{{\varepsilon},1}^{\prime}\left( x_{1}\right) \right) ^{2}}}y_{1},\\
y & =f_{{\varepsilon},1}\left( x_{1}\right) +\frac{y_{1}}{\sqrt{1+\left( f_{{\varepsilon},1}^{\prime}\left(
x_{1}\right) \right) ^{2}}}.
\end{aligned}\end{aligned}$$ Similarly, for the graph of $y=f_{{\varepsilon}, 2}(x)=-f_{{\varepsilon},1}\left( x\right)$, which is the symmetric image of $\Gamma_{{\varepsilon},1}$ with respect to the $x$ axis in the upper half plane one can associate a Fermi coordinate $\left( x_{2},y_{2}\right)\in {\mathbb R}\times (-{\mathrm d}_{\varepsilon},{\mathrm d}_{\varepsilon}),$ with in ${\mathcal {O}}_2$, which is the symmetric image of ${\mathcal {O}}_1$ defined above.
Now, the change of coordinates $(x_1,y_1)\mapsto (x, y)$ is a diffeomorphism in $\mathcal{O}_1$ (respectively $(x_2,y_2)\mapsto (x,y)$ is a diffeomporhpism in the corresponding neighborhood $\mathcal {O}_2$). We will use $\mathbf{{x}}_{{\varepsilon}, i}:\left( x_{i},y_{i}\right) \rightarrow\left( x,y\right) $ to denote this diffeomorphism. For any function $w\colon {\mathcal{O}}_i\to {\mathbb R}$ we will also define its pullback by $\mathbf {x}_{{\varepsilon}, i}$ by setting $(\mathbf{{x}}_{{\varepsilon}, i}^*w)\left(
x_{i},y_{i}\right) =w\circ\mathbf{{x}}_{{\varepsilon}, i}\left( x_{i},y_{i}\right) $.
Let ${\tt x}=(x,y)\in \mathcal {O}_i$ and let $(x_i, y_i)$ be the Fermi coordinates of this point. In what follows we will need to compare the values of $\mathrm{d}_{\varepsilon}(x_i)$ with $\mathrm{d}_{\varepsilon}(x)$. Note that we can write: $$\log\cosh x_i=\log\cosh\big(x+\mathcal{O}(\|f'_{{\varepsilon}, i}\|_{\mathcal {C}^0({\mathbb R})})y_i\big),$$ which, since $\|f'_{{\varepsilon}, i}\|_{\mathcal {C}^0({\mathbb R})}\to 0$, as ${\varepsilon}\to 0$, implies: $$|\log\cosh x_i-\log\cosh x|\leq o(1) |y_i|\leq o(1)\mathrm {d}_{\varepsilon}(x_i).$$ Then, by definition of the function $\mathrm{d}_{\varepsilon}$ it follows: $$\label{ddd 1}
\mathrm{d}_{\varepsilon}(x_i)=(1+o({\varepsilon}))\mathrm{d}_{{\varepsilon}}(x).$$
Another relation involving the Fermi coordinates that we will need is the following: $$\label{ddd 2}
|y_1|+|y_2|\geq 2|f_{{\varepsilon},1}(x)|\big(1+\mathcal{O}(\|f'_{{\varepsilon}, 1}\|^2_{\mathcal{C}^0({\mathbb R})})\big).$$ This estimate follows from the explicit formulas for the Fermi coordinates and elementary geometry.
Asymptotic profile of a solution near its nodal line {#sec 33}
====================================================
An approximate solution of (\[AC\])
-----------------------------------
We will define now an approximate solution to (\[AC\]) which accounts accurately for the asymptotic the behavior of the true solution as ${\varepsilon}\to 0$. We will use the nodal lines $\Gamma_{{\varepsilon}, i}$ as the point of departure and will base our construction on the neighborhoods $\mathcal{O}_i$, which are expanding as $x\to \infty$.
To be precise, we let ${\eta}_i$ be a cutoff function satisfying ${\eta}_i\left( {\tt x}\right)
=0,{\tt x}\not \in \mathcal{{O}}_i$ and ${\eta}_i\left( {\tt x}\right) =1$ for the point ${\tt x}\in\mathcal{{O}}_i$ such that $\mathrm{dist}\,( {\tt x},\partial\mathcal{
{O}}_i) >1.$ Moreover, ${\eta}_i$ could be chosen in such a way that $\left\Vert {\eta}_i\right\Vert _{{\mathcal C}^{3}({\mathbb R}^2)}\leq C.$ We will use $\left(
x_{i},y_{i}\right)$ to denote the Fermi coordinates associated to $\Gamma_{{\varepsilon}, i}$, $i=1,2$. Finally, we will introduce an unknown function $h_{\varepsilon}\colon {\mathbb R}\to {\mathbb R}$, which a priori is of class $\mathcal{C}^3({\mathbb R})$ , and we let $$\begin{aligned}
({\mathbf x}_{{\varepsilon}, 1}^*{H}_{{\varepsilon}, 1})\left(x_1,y_1\right) &=({\mathbf {x}}_{{\varepsilon}, 1}^*{\eta}_1)H\left(y_{1}-{h}_{\varepsilon}(x_1)\right)
+\left( 1-{\mathbf x}_{{\varepsilon}, 1}^* {\eta}_1\right) \frac{H\left( y_{1}-{h}_{\varepsilon}(x_1)\right)
}{\left\vert H\left( y_{1}-{h}_{{\varepsilon}}(x_1)\right) \right\vert },
\\
\quad {H}_{{\varepsilon}, 2}\left( x,y\right) & ={H}_{{\varepsilon}, 1}\left( x,-y\right) ,\quad \bar
{u}_{\varepsilon}={H}_{{\varepsilon}, 1}-{H}_{{\varepsilon}, 2}-1.
\end{aligned}
\label{def hve}$$ The function ${h}_{\varepsilon}$ is called the modulation function and we will show (Proposition \[prop gamma\]) that it can be defined through the orthogonality condition: $$\int_{\mathbb{R}}\big(\mathbf{{x}}_{{\varepsilon}, i}^*\left(u-\bar{u}_{\varepsilon}\right)
{\rho}_{{\varepsilon}, i}{H}_{{\varepsilon}, i}^{\prime}\big)dy_{i}=0,\quad \forall x_{i}\in\mathbb{R},$$ where $$({\mathbf x}_{{\varepsilon}, i}^* H'_{{\varepsilon}, i})(x_i, y_i)= ({\mathbf {x}}_{{\varepsilon}, i}^*{\eta}_i)H'\left(y_{i}-(-1)^{i+1}{h}_{\varepsilon}(x_i)\right), \quad i=1,2,$$ and smooth cutoff functions $\rho_{{\varepsilon}, i}$ are defined through a smooth cutoff function $\rho$ by: $$({\tt x}_{{\varepsilon}, i}^* \rho_{{\varepsilon}, i})(x_i, y_i)=\rho(x_i, y_i-(-1)^{i+1}{h}_{\varepsilon}(x_i)),$$ where $$\rho(s,t)=\begin{cases}
1, \quad |t|\leq \frac{1}{2}\mathrm{d}_{\varepsilon}(s),\\
0<\rho<1, \quad \frac{1}{2} \mathrm{d}_{\varepsilon}(s)<t<\frac{3}{4}\mathrm{d}_{\varepsilon}(s),\\
0 \quad\mbox{othewise}.
\end{cases}$$ Note that because of the definition of the function ${\mathrm d}_{\varepsilon}$ we can assume that $|\nabla \rho_{{\varepsilon}, i}({\tt x})|=\mathcal{O}(\frac{1}{{\varepsilon}|{\tt x}|})$, ${\varepsilon}|\tt x|\gg 1$ with similar estimates for higher order derivatives.
The proof of existence of the modulation function ${h}_{\varepsilon}$ will be given later on but anticipating it we observe that due to the exponential decay in $x$ of the functions involved, we have ${h}_{\varepsilon}\in \mathcal {C}_{\varepsilon\tau
}^{2,\mu}\left( \mathbb{R}\right)$ and in fact we will show $$\label{estim modulation}
\|h_{\varepsilon}\|_{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R})}\leq C{\varepsilon}^2.$$
If we let $\hat{\phi} = u-\bar{u}_{\varepsilon}$ then we have: $$L_{\bar u_{\varepsilon}} \phi=E(\bar u_{\varepsilon})-P(\phi), \quad E(\bar u_{\varepsilon})=\Delta \bar u_{\varepsilon}-F'(\bar u_{\varepsilon}).$$ Our first result is the following:
\[estim hat phi\] There exist constants $\tau\in (0, \alpha_0)$, $\mu>0$ such that the following estimate holds: $$\|{\phi}\| _{C_{\varepsilon\tau}^{2,\mu}\left(
\mathbb{R}^{2}\right) }\leq C\varepsilon^{2}.$$
The proof of this Proposition, which is based on the a priori estimates for linear operator $L_{\bar u_{\varepsilon}}$ in weighetd spaces and careful estimates of the error of the approximation $E(\bar u_{\varepsilon})$ is postponed for now and will be given in section \[proof [prop gamma]{}\].
Precise asymptotic of the nodal lines
-------------------------------------
The point of this section is to describe precise, and in particular uniform as ${\varepsilon}\to 0$, estimates for the weighted norm $\|f''_{{\varepsilon}, i}\|_{\mathcal{C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}$. Our curve of reference will be given by a solution of the Toda system: $$\begin{aligned}
{{c_0}} \, q_1'' &= - e^{\,\alpha_0 ( q_{1}- q_{2}) },\\
{{c_0}}\, q_2'' &= e^{\,\alpha_0(q_{1}- q_2)},
\end{aligned}
\label{toda 2}$$ for which $q_1<0<q_2$ and $q_1(x)=-q_2(x)$, as well as $q_j(x)=q_j(-x)$, $j-1,2$ (c.f. (\[toda 02\])). Such solution is determined by one parameter only, and in fact we only need to solve $$\begin{aligned}
\label{toda 3}
{{c_0}}\, q_1''=-2e^{\, 2\alpha_0 q_1},\end{aligned}$$ in the class of even functions. It is easy to see that solutions of (\[toda 3\]) form a one parameter family, and each solution of this family has asymptotically linear behavior. In fact this family can be parametrized by the slope of this straight line. To describe this family precisely let us consider the unique solution $U_0(x)$, whose slope at $\infty$ is $-1$. Asymptotically, as $|x|\to \infty$, we have $$U_0(x)=-|x|+{b}_0+\mathcal{O}(e^{\,-2\alpha_0|x|}),$$ where $b_0$ is a fixed constant. Then we have $$q_{{\varepsilon}, 1}(x)=U_0({\varepsilon}x)-\frac{1}{\alpha_0}\log\frac{1}{{\varepsilon}}.$$ Thus, given the nodal line $\Gamma_{{\varepsilon},1}$ of a solution $u$, with ${\varepsilon}=\tan\theta(u)$, by $q_{{\varepsilon},1}$ we will denote the solution of (\[toda 3\]) whose slope at infinity is ${\varepsilon}$. Respectively we set $$q_{{\varepsilon},2}=- q_{{\varepsilon},1}.$$ We will denote by $\widetilde \Gamma_{{\varepsilon},1}$ the curve $y=q_{{\varepsilon},1}(x)$ in the lower half plane and by $\widetilde \Gamma_{{\varepsilon},2}$ the graph of $y=q_{{\varepsilon},2}$. The hope is that the nodal set in the lower half plane of a function $u$, with ${\varepsilon}=\tan\theta(u)$ small, and $\widetilde\Gamma_{{\varepsilon},1}$ should be close to each other. To quantify this is the objective of the next result.
\[prop gamma\] Let $u$ be a four end solution of (\[AC\]) such that ${\varepsilon}=\tan \theta(u)$ is small and let $\Gamma_{{\varepsilon},1}$ be the nodal line of this solution on the lower half plane, given as a graph of the function $y=f_{{\varepsilon},1}(x)$, and let $h_{\varepsilon}\in {\mathcal C}^{2,\mu}({\mathbb R})$ be the modulation function described above. There exist $\alpha>0$ and a constant $j_{\varepsilon}$, where $|j_{\varepsilon}|\leq C{\varepsilon}^\alpha$, such that the following estimates hold for the function $\chi_{{\varepsilon},1}=f_{{\varepsilon},1}+h_{\varepsilon}+j_{\varepsilon}-q_{{\varepsilon},1}$: $$\label{first estimate}
\begin{aligned}
\left\Vert \chi_{{\varepsilon},1}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}\left({\mathbb R}\right)} & \leq C\varepsilon^{\alpha},\\
\left\Vert \chi_{{\varepsilon},1}^{\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}\left({\mathbb R}\right)} & \leq C\varepsilon^{1+\alpha
},\\
\left\Vert \chi_{{\varepsilon},1}^{\prime\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}\left({\mathbb R}\right)} & \leq C\varepsilon
^{2+\alpha}.
\end{aligned}$$ Similar statements hold for the nodal line $\Gamma_{{\varepsilon},2}$ of $u$ in the upper half plane, with $\chi_{{\varepsilon},2}=f_{{\varepsilon},2}-h_{\varepsilon}-j_{\varepsilon}-q_{{\varepsilon},2}$, replacing $\chi_{{\varepsilon},1}$.
This proposition is the main technical tool needed to prove the uniqueness and. Its proof is quite involved and we will postpone it for now proceeding directly to the proof of Theorem \[teo uniqueness\].
Proofs of Proposition \[estim hat phi\] and Proposition \[prop gamma\] {#proof {prop gamma}}
======================================================================
We recall that by definition $h_{\varepsilon}$ is required to be such that the following orthogonality condition are satisfied $$\label{orto h}
\int_{\mathbb{R}}\big(\mathbf{x}_{{\varepsilon},i}^{\ast}\left( u-\bar{u}_{\varepsilon}\right) \rho_{i,{\varepsilon}}H_{{\varepsilon},i}^{\prime}\big)(x_i,y_i)dy_{i}=0,\quad \forall x_{i}\in\mathbb{R}, \quad i=1,2.$$ We will refer to $h_{\varepsilon}$ as the [*modulation function*]{}, and we keep in mind that $h_{\varepsilon}$ is required to be small. Our first objective is to show that the modulation function $h_{\varepsilon}$ indeed exists.
\[exists h\] For each sufficiently small ${\varepsilon}$ there exists a function $h_{\varepsilon}\in {\mathcal C}^3({\mathbb R})$ such that (\[orto h\]) holds.
To find $h_{\varepsilon}$ such that the orthogonality (\[orto h\]) condition is satisfied, we first replace the function $h_{\varepsilon}$ in the definition of the functions $H_{{\varepsilon},1}$ and $H_{{\varepsilon},2}$ be two undetermined, bounded functions $h_{{\varepsilon},1}$ and $h_{{\varepsilon},2}.$ More precisely, given a function $h_{{\varepsilon},2}$ in suitable function space, we have a function $H_{{\varepsilon},2}$ which in the Fermi coordinate $\left( x_{2},y_{2}\right) $ is equal to $-H\left( y_{2}+h_{2,{\varepsilon}}\left( x_{i}\right) \right)$, at least near $\Gamma_{{\varepsilon}, 2}$. Given this, we want to find the function $h_{{\varepsilon},1},$ corresponding to the modulation of the nodal line $\Gamma_{{\varepsilon},1}$, such that for the resulting approximate solution $H_{{\varepsilon},1}$, the orthogonal condition (\[orto h\]) is satisfied for $i=1$. Note that so far the orthogonal condition for $i=2$ still may not be hold. However, if it happens that $h_{{\varepsilon},2}=h_{{\varepsilon},1}=h_{\varepsilon}$ then, by symmetry, the orthogonal condition is also satisfied for $i=2$ and this will yield the desired modulation function.
To find a $h_{{\varepsilon},2}$ such that $h_{{\varepsilon},1}=h_{{\varepsilon},2},$ we will use a fixed point argument. For brevity we will assume a priori that $h_{{\varepsilon},2}\in {\mathcal C}^0({\mathbb R})$, thus yielding $h_{\varepsilon}\in {\mathcal C}^0({\mathbb R})$. Generalizing the argument to get $h_{\varepsilon}\in {\mathcal C}^3({\mathbb R})$ is straightforward.
Obviously, $$\int_{\mathbb{R}}({\mathbf x}_{{\varepsilon},1}^*\bar{u}_{\varepsilon}\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime})(x_1, y_1)dy_{1}=-\int_{\mathbb{R}}\left[{\mathbf x}^*_{{\varepsilon},1} \left(H_{{\varepsilon},2}+1\right) \rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime}\right](x_1, y_1)dy_{1}.$$ This identity suggests to consider the function $$k_{\varepsilon}(s, x_1) :=\int_{\mathbb{R}}\rho\left( x_1, y_{1}\right) H^{\prime
}\left( y_{1}\right) \big[{\mathbf x}_{{\varepsilon},1}^*(u+H_{{\varepsilon},2}+1)\big]( x_{1},y_{1}+s) dy_{1}, \quad s\in {\mathbb R}.$$ Note that the orthogonality condition (\[orto h\]) is equivalent to $k_{\varepsilon}(s, x_1)=0$ with $s=h_{{\varepsilon},1}(x_1)$, which follows by changing variables $y_1\mapsto y_1+h_{{\varepsilon},1}(x_1)$ in the integral expression.
We have $$\begin{aligned}
\partial_s k_{\varepsilon}\left( s, x_1\right) &=\int_{\mathbb{R}}\rho\left(x_1, y_{1}\right) H^{\prime}\left( y_{1}\right)\partial_{y_1} \big[{\mathbf x}_{{\varepsilon},1}^*(u+H_{{\varepsilon},2}+1)\big]( x_{1},y_{1}+s) dy_{1}\\
&=\underbrace{\int_{\mathbb{R}}\rho\left(x_1, y_{1}\right) H^{\prime}\left( y_{1}\right)\partial_{y_1} \big[{\mathbf x}_{{\varepsilon},1}^*(H_{{\varepsilon},1}+H_{{\varepsilon},2}+1)\big]( x_{1},y_{1}+s) dy_{1}}_{l_{\varepsilon}(s,x_1)}
\\
&\quad +\underbrace{\int_{\mathbb{R}}\rho\left( x_1, y_{1}\right) H^{\prime}\left( y_{1}\right)\partial_{y_1} \big[{\mathbf x}_{{\varepsilon},1}^*(u-H_{{\varepsilon},1})\big]( x_{1},y_{1}+s) dy_{1}}_{m_{\varepsilon}(s, x_1)}\end{aligned}$$ Then, we see that for each $\delta>0$ there exists an $a>0$ such that $l_{\varepsilon}(s,x_1)>\delta$ for $s\in (-a,a)$ uniformly in small ${\varepsilon}$ and $x_1$.
Since $u$ converges locally, as ${\varepsilon}\to 0$, to the heteroclinic solution, for each sufficiently small ${\varepsilon}$ we claim that it holds, $$\begin{aligned}
|m_{\varepsilon}(s, x_1)|\leq \frac{\delta}{2}, \quad \forall x_1\in {\mathbb R}.\end{aligned}$$ The proof of this claim based on the De Giorgi conjecture is left to the reader.
Then it is seen that $\partial_s k_{\varepsilon}\left(s, x_1\right) >\delta/2$ for $s\in\left(
-a,a\right) ,$ and $x_1\in {\mathbb R}$ where $a$ is small but independent of $\varepsilon.$ We will prove that taking $a$ smaller if necessary we may assume $k_{\varepsilon}\left( a, x_1\right) >0$ and $k_{\varepsilon}\left( -a, x_1\right) <0$ for $\varepsilon$ small enough. Indeed let us write: $$\begin{aligned}
k_{\varepsilon}(s, x_1)&=\underbrace{\int_{\mathbb{R}}\rho\left(x_1, y_{1}\right) H^{\prime}\left( y_{1}\right) H(y_{1}+s) dy_{1}}_{k_0(s)}\\
&\quad +
\underbrace{\int_{\mathbb{R}}\rho\left( x_1, y_{1}\right) H^{\prime}\left( y_{1}\right) \big[{\mathbf x}_{{\varepsilon},1}^*(H_{{\varepsilon},2}+1)\big]( x_{1},y_{1}+s) dy_{1}}_{g_{1,{\varepsilon}}(s,x_1)}\\
&\quad +
\underbrace{\int_{\mathbb{R}}\rho\left( x_1, y_{1}\right) H^{\prime}\left( y_{1}\right) \big[{\mathbf x}_{{\varepsilon},1}^*(u-H_{{\varepsilon},1})\big]( x_{1},y_{1}+s) dy_{1}}_{g_{2,{\varepsilon}}(s,x_1)}.\end{aligned}$$ Then we see that $$\label{hhh 1}
k_0(s)=s\int\rho(x_1,y_1)\big(H'(y_1)\big)^2\,dy_1+k_1(s), \quad k_1(s)\sim s^2,$$ while $$\label{hhh 2}
g_{1,{\varepsilon}}(s,x_1)=o(1) e^{\,-\sqrt{2}( |h_{{\varepsilon},2}|+|s|)}, \quad g_{2,{\varepsilon}}(s,x_1)=o(1), \quad\mbox{as}\ {\varepsilon}\to 0.$$ Therefore for fixed $h_{{\varepsilon},2},$ the existence of $h_{{\varepsilon},1}$ which fulfills the orthogonal condition (\[orto h\]) follows immediately. The above argument implies that for any $h_{{\varepsilon},2}\in {\mathcal C}^{0}\left( \mathbb{R}\right) ,\left\Vert h_{{\varepsilon},2}\right\Vert_{{\mathcal C}^0({\mathbb R})} < a,$ we have a nonlinear map $T$ defined by $h_{{\varepsilon},2}\mapsto h_{{\varepsilon},1}.$ The map $T$ satisfies $$TB\left( 0,a\right) \subset B\left( 0,a\right), \quad B(0,a)=\{h\in {\mathcal C}^0({\mathbb R})\mid \|h\| _{{\mathcal C}^0({\mathbb R})} < a\}.$$ The proof that $T$ is a contraction map is standard and is omitted. At the end we obtain the existence of a fixed point $h_{\varepsilon}$. To prove its regularity we note that the fact that we have $\partial_s k_{\varepsilon}(s, x_1)=l_{\varepsilon}(x_1s)+m_{\varepsilon}(s,x_1)$, allows to use the implicit function theorem and thus the regularity follows in a straightforward manner. This ends the proof.
\[corollary orto 1\] The modulation function $h_{\varepsilon}$ satisfies: $$\|h_{\varepsilon}\|_{\mathcal C^{2,\mu}({\mathbb R})}=o(1), \quad {\varepsilon}\to 0.$$ We also have $h_{\varepsilon}\in \mathcal{C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R})$.
The fact that $\|h_{\varepsilon}\|_{\mathcal {C}^0({\mathbb R})}\to 0$ as ${\varepsilon}\to 0$ follows from (\[hhh 1\])–(\[hhh 2\]). Then the same can be shown for the higher order derivatives. Once the existence of small $h_{\varepsilon}$ is established one can use again (\[hhh 1\])–(\[hhh 2\]) and the fact that a priori $u\in \mathcal {C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)$ to show that $h_{\varepsilon}\in \mathcal {C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R})$.
Let us write $u=\bar{u}_{\varepsilon}+\phi$, where $u$ is a solution. Let us denote the linearization of the Allen-Cahn equation around $\bar u$ by $L_{\bar u_{\varepsilon}}:=-\Delta + F''(\bar u_{\varepsilon})$. Then $\phi$ satisfies $$L_{\bar{u}_{\varepsilon}}\phi
=\Delta\bar{u}_{\varepsilon}-F^{\prime}\left( \bar{u}_{\varepsilon}\right) -P\left( \phi\right) .
\label{ubar}$$ Here $$P\left( \phi\right) =F^{\prime}\left( \bar{u}_{\varepsilon}+\phi\right) -F^{\prime
}\left( \bar{u}_{\varepsilon}\right) -F^{\prime\prime}\left( \bar{u}_{\varepsilon}\right) \phi\sim\phi^2,$$ is a higher order term in $\phi$. Note that our definition of $\bar u_{\varepsilon}$ and the construction of the function $h_{\varepsilon}$ implies that $\phi=u-\bar u_{\varepsilon}$ satisfies the orthogonality condition (\[orto h\]). Our strategy to get suitable estimate for $\phi$ relies on the a priori estimates for the operator $L_{\bar{u}}$, taking into account this orthogonality condition.
To carry out the analysis, we will study the error term $E\left( \bar
{u}_{\varepsilon}\right) :=\Delta\bar{u}_{\varepsilon}-F^{\prime}\left( \bar{u}_{\varepsilon}\right) .$ First we consider the projection of $E\left( \bar{u}_{\varepsilon}\right) $ onto the two dimensional space $K=\mathrm{span}\,\{H_{{\varepsilon},i}^{\prime}\rho_{{\varepsilon},i}, i=1,2\}$, which we will denote by $E(\bar u_{\varepsilon})^\parallel$. We will also set $E(\bar u_{\varepsilon})^\perp=E(\bar u_{\varepsilon})-E(\bar u_{\varepsilon})^\parallel$. Explicitly $E(\bar u_{\varepsilon})^\perp$ is given through its pullback by the Fermi coordinates of $\Gamma_{{\varepsilon}, i}$ as: $${\mathbf x}_{{\varepsilon},i}^*E\left( \bar{u}_{\varepsilon}\right)^{\perp}:={\mathbf x}_{{\varepsilon},i}^*E\left( \bar{u}_{\varepsilon}\right)
-\sum\limits_{i=1}^{2}c_{{\varepsilon}}(x_i)\big({\mathbf x}_{{\varepsilon},i}^*\rho_{{\varepsilon},i}H_{{\varepsilon},i}^{\prime}\big)\int_{\mathbb{R}}\big[
{\mathbf x}_{{\varepsilon},i}^*E(\bar{u}_{\varepsilon})\rho_{{\varepsilon},i}H_{{\varepsilon},i}^{\prime}\big]dy_{i},$$ where $$c_{{\varepsilon}}(s)=\left( \int_{\mathbb{R}}\rho^{2}(s,t)\left( H^{\prime}(t)\right) ^{2}\,dt\right)
^{-1}\approx \left(\int_{\mathbb R}(H')^2\right)^{-1}.$$ The main idea in what follows is that the size of the function $f_{{\varepsilon}, 1}$ and its derivatives should be controlled by $E(\bar u_{\varepsilon})^\parallel$, while the size of $u-\bar u_{\varepsilon}=\phi$ is controlled by $E(\bar u_{\varepsilon})^\perp$. Of course, both projections of the error $E(\bar u_{\varepsilon})$ are coupled, in the sense that the dependence on $f_{{\varepsilon}, 1}$ and $\phi$ appears in both of them, but this coupling is relatively weak.
Recall the expression of Laplace operator in the Fermi coordinate of $\Gamma_{i,{\varepsilon}}$: $$\Delta=\frac{1}{A_{i}}\partial_{x_{i}}^{2}+\partial_{y_{i}}^{2}+\frac{1}{2}\frac{\partial_{y_{i}}A_{i}}{A_{i}}\partial_{y_{i}}-\frac{1}{2}\frac{\partial_{x_{i}}A_{i}}{A_{i}^{2}}\partial_{x_{i}}, \label{laplacian}$$ where $$A_{i}=1+\left( f^{\prime}_{{\varepsilon}, i}\left( x_{i}\right) \right) ^{2}+2y_{i}\frac{\left( -1\right) ^{i}f^{\prime\prime}_{{\varepsilon}, i}\left( x_{i}\right) }{\sqrt{1+\left( f^{\prime}_{{\varepsilon}, i}\left( x_{i}\right) \right) ^{2}}}+y_{i}^{2}\frac{\left( f^{\prime\prime}_{{\varepsilon}, i}\left( x_{i}\right) \right) ^{2}}{\left(
1+\left( f^{\prime}_{{\varepsilon}, i}\left( x_{i}\right) \right) ^{2}\right) ^{2}}.$$ Using these formulas, we can write down the expression of the error $E\left( \bar{u}_{\varepsilon}\right) .$ Because of symmetry, it suffices to carry out the calculation in the lower half plane. Observe that, $$\begin{aligned}
& -F^{\prime}\left( H_{{\varepsilon}, 2}\right) -F^{\prime}\left( H_{{\varepsilon}, 1}-H_{{\varepsilon}, 2}-1\right) \\
& =-F^{\prime}\left( H_{{\varepsilon}, 2}\right) -F^{\prime}\left( H_{{\varepsilon}, 1}\right)
+F^{\prime\prime}\left( H_{{\varepsilon}, 1}\right) \left(H_{{\varepsilon}, 2}+1\right)
+{\mathcal O}\left((H_{{\varepsilon}, 2}+1)^{2}\right)\\
& =-F^{\prime}\left( H_{{\varepsilon}, 1}\right) -\left( F^{\prime\prime}\left(
1\right) -F^{\prime\prime}\left( H_{{\varepsilon}, 1}\right) \right) \left(
H_{{\varepsilon}, 2}+1\right) +{\mathcal O}\left( ( H_{{\varepsilon}, 2}+1)^2\right) .\end{aligned}$$ The same calculation as in formula (5.65) in [@MR2557944] yields that in the portion of the lower half plane where ${\mathrm{dist}}\,(\Gamma_{{\varepsilon}, i}, {\tt x})\leq (\mathrm {d}_{\varepsilon}\circ\pi_{{\varepsilon}, i}^x)({\tt x}) -1$, for both $i=1,2$, we have $$\begin{aligned}
\begin{aligned}
E\left( \bar{u}_{\varepsilon}\right) & =\left( \frac{1}{2}\frac{\partial_{y_{1}}A_{1}}{A_{1}}-\frac{h_{\varepsilon}^{\prime\prime}\left( x_{1}\right) }{A_{1}} +\frac{1}{2}\frac{\partial_{x_{1}}A_{1}}{A_{1}^{2}}h_{\varepsilon}^{\prime}\left( x_{1}\right) \right) H_{{\varepsilon}, 1}^{\prime}\\
& \quad -\left( \frac{1}{2}\frac{\partial_{y_{2}}A_{2}}{A_{2}}+\frac
{h_{\varepsilon}^{\prime\prime}\left( x_{2}\right) }{A_{2}}-\frac{1}{2}\frac
{\partial_{x_{2}}A_{2}}{A_{2}^{2}}h_{\varepsilon}^{\prime}\left( x_{2}\right) \right)
H_{{\varepsilon},2}^{\prime}\\
& \quad +\left( \frac{\left( h_{\varepsilon}^{\prime}\left( x_{1}\right) \right) ^{2}}{A_{1}}H^{\prime\prime}\left( y_{1}-h_{\varepsilon}\left( x_{1}\right) \right)
-\frac{\left( h_{\varepsilon}^{\prime}\left( x_{2}\right) \right) ^{2}}{A_{2}}H^{\prime\prime}\left( y_{2}+h_{\varepsilon}\left( x_{2}\right) \right) \right)\\
&\quad -\left(
F^{\prime\prime}\left( 1\right) -F^{\prime\prime}\left( H_{{\varepsilon}, 1}\right)
\right) \left( H_{{\varepsilon}, 2}+1\right)+{\mathcal O}\left( (H_{{\varepsilon}, 2}+1)^2\right) .
\end{aligned}
\label{eu}$$ In fact this formula generalizes in the set ${\mathrm {dist}}\,(\Gamma_{{\varepsilon}, 1}, {\tt x})<(\mathrm{d}_{\varepsilon}\circ\pi_{{\varepsilon}, 1}^x)({\tt x})-1$, ${\mathrm {dist}}\,(\Gamma_{{\varepsilon}, 2}, {\tt x})>({\mathrm d}_{\varepsilon}\circ\pi_{{\varepsilon}, 2}^x)({\tt x})$, if we set $H_{{\varepsilon}, 2}=-1$. In the intermediate region where $(\mathrm {d}_{\varepsilon}\circ \pi_{{\varepsilon}, 1}^x)({\tt x})-1\leq {\mathrm{dist}}\,(\Gamma_{{\varepsilon}, 1}, ({\tt x})<(\mathrm {d}_{\varepsilon}\circ \pi_{{\varepsilon}, 1}^x)({\tt x})$ we control $E(\bar u_{\varepsilon})$ by $Ce^{\,-\sqrt{2} (\mathrm {d}_{\varepsilon}\circ \pi_{{\varepsilon}, 1}^x)({\tt x})}$. Finally, in the set where ${\mathrm {dist}}\,(\Gamma_{{\varepsilon}, i}, {\tt x})>(\mathrm {d}_{\varepsilon}\circ\pi_{{\varepsilon}, i}^x)({\tt x})$ the error is $0$, since $\bar u_{\varepsilon}=\pm 1$ in this set. These is the basis of the proof of the next lemma.
\[Eu\] For any $\mu\in\left( 0,1\right)$, the following estimate holds$$\|E(\bar{u}_{\varepsilon})^{\perp}\|
_{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}=o\left( \left\Vert f_{{\varepsilon},1}^{\prime\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}+\left\Vert h_{\varepsilon}^{\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}+\left\Vert h_{\varepsilon}^{\prime\prime
}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\right) +\mathcal{O}\left( \|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}\right), \label{E}$$ where we have denoted $\varDelta_{\varepsilon}=\mathcal{O}(\|f_{{\varepsilon}, 1}'\|^2_{\mathcal{C}^0({\mathbb R})})$.
Therefore to prove $\left( \ref{E}\right) ,$ it suffices to check the expression $\left( \ref{eu}\right) $ for $E\left( \bar{u}_{\varepsilon}\right)$, which applies whenever at least for one $i$ it holds ${\mathrm {dist}}\,(\Gamma_{{\varepsilon}, i}, {\tt x})<\mathrm{d}_{\varepsilon}(x_i)-1$, as we have pointed out above. This means that we only need to consider the subset of the lower half plane were $|y_i|=\mathrm{dist}\,(\Gamma_{{\varepsilon}, i}, {\tt x})\leq \mathrm {d}_{\varepsilon}(x_i)-1$ for at least one $i$. Thus we may focus on studying the formula (\[eu\]).
Projecting the $E(\bar u_{\varepsilon})$ on $K$, and using formula (\[eu\]), we get for instance the following term in $E\left( \bar{u}_{\varepsilon}\right) ^{\perp}$: $$T_{1}:=\frac{\partial_{y_{1}}A_{1}}{A_{1}}H_{{\varepsilon}, 1}^{\prime}-c_{{\varepsilon}}\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}\int_{\mathbb{R}}\frac{\partial_{y_{1}}A_{1}}{A_{1}}\rho
_{{\varepsilon},1}\left( H_{{\varepsilon}, 1}^{\prime}\right) ^{2}dy_{1}.$$ Recall that $$\frac{\partial_{y_{1}}A_{1}}{A_{1}}=-2\frac{f_{{\varepsilon},1}^{\prime\prime}\left(
x_{1}\right) }{A_{1}\sqrt{1+\left( f^{\prime}_{{\varepsilon}, 1}\left( x_{1}\right) \right)
^{2}}}+2\frac{y_{1}\left( f^{\prime\prime}_{{\varepsilon}, 1}\left( x_{1}\right) \right)
^{2}}{A_{1}\left( 1+\left( f^{\prime}_{{\varepsilon}, 1}\left( x_{1}\right) \right)
^{2}\right) ^{2}}.$$ Substituting this into the expression of $T_{1}$ results in $$\begin{aligned}
T_{1} & =\frac{\partial_{y_{1}}A_{1}}{A_{1}}H_{{\varepsilon}, 1}^{\prime}+\frac{2c_{{\varepsilon}}\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}f_{{\varepsilon}, 1}^{\prime\prime}\left( x_{1}\right) }{\sqrt{1+\left(
f_{{\varepsilon}, 1}^{\prime}\left( x_{1}\right) \right) ^{2}}}\int_{\mathbb{R}}\frac{\rho
_{{\varepsilon}, 1}\left( H_{{\varepsilon}, 1}^{\prime}\right) ^{2}}{A_{1}}dy_{1}\\
& -\frac{2c_{{\varepsilon}}\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}\left( f_{{\varepsilon}, 1}^{\prime\prime}\left(
x_{1}\right) \right) ^{2}}{\left( 1+\left( f_{{\varepsilon}, 1}^{\prime}\left( x_{1}\right)
\right) ^{2}\right) ^{2}}\int_{\mathbb{R}}\frac{y_{1}\rho_{{\varepsilon}, 1}\left(
H_{{\varepsilon}, 1}^{\prime}\right) ^{2}}{A_{1}}dy_{1}.\end{aligned}$$ Another observation we make is that when we estimate $\mathcal{C}^0_{{\varepsilon}\tau}({\mathbb R})$ norms we need to take into account the relation between the Fermi variables $(x_1, y_1)$ and the euclidean coordinates $(x,y)$ of a point ${\tt x}\in \mathcal{O}_1$. To see this let us consider a typical term that appears in $T_1$: $$|(\cosh x)^{{\varepsilon}\tau} f''_{{\varepsilon}, 1}(x_1)|\leq C e^{\,{\varepsilon}\tau|x_1-x|}\|f''_{{\varepsilon}, 1}\|_{\mathcal{C}^0_{{\varepsilon}\tau}{({\mathbb R})}}\leq C\exp\{{\varepsilon}\tau|y_1|\mathcal{O}(\|f'_{{\varepsilon}, 1}\|_{\mathcal{C}^{0}({\mathbb R})})\}\|f''_{{\varepsilon}, 1}\|_{\mathcal{C}^0_{{\varepsilon}\tau}{({\mathbb R})}}.$$ Any term of this form is additionally multiplied by $o(1)H_{{\varepsilon}, 1}'$ or $o(1)H''_{{\varepsilon}, 1}$ thus yielding a term of the order $o(\|f''_{{\varepsilon}, 1}\|_{\mathcal{C}^0_{{\varepsilon}\tau}{{\mathbb R}}})$.
Finally, it is important to notice that although it appears at first that $T_1$ carries a term of order $\mathcal{O}(\|f''_{{\varepsilon}, 1}\|_{\mathcal{C}^0_{{\varepsilon}\tau}{{\mathbb R}}})$, there is a cancellation between the first and the second term (the one containing the integral) in $T_1$. In estimating this term it is important to use the properties of the cut off function $\rho_{{\varepsilon}, 1}$.
Now, using the fact that $f_{{\varepsilon},1}^{\prime}$ and $f_{{\varepsilon}, 1}^{\prime\prime}$ are of order $o\left( 1\right)$, as ${\varepsilon}\to 0$, and the definition of the cutoff function $\rho_{{\varepsilon}, 1},$ we conclude $$\left\Vert T_{1}\right\Vert _{{\mathcal C}^{0,\mu}({\mathbb R}^2)}=o\left( \left\Vert f_{{\varepsilon}, 1}^{\prime\prime
}\right\Vert _{{\mathcal C}^{0,\mu}({\mathbb R})}\right) .$$ Similar estimates hold for the terms involving $h_{\varepsilon}^{\prime\prime}\left(
x_{1}\right) .$ Regarding to the terms involving $h_{\varepsilon}^{\prime}\left(
x_{1}\right) ,h_{\varepsilon}^{\prime}\left( x_{2}\right) ,h_{\varepsilon}^{\prime\prime}\left(
x_{2}\right) $, we note that they are all multiplied by a small order term. Furthermore, the norms of $\left( H_{{\varepsilon}, 2}+1\right) H_{{\varepsilon}, 1}^{\prime}$ and ${\mathcal O}( \left(
H_{{\varepsilon}, 2}+1\right) ^{2}) $ are controlled by $Ce^{\,-\sqrt{2}(|y_1|+|y_2|)}$. To estimate terms of this from we use the expressions for the Fermi coordinates of $\Gamma_{{\varepsilon}, i}$ to arrive at the following lower bound: $$|y_1|+|y_2|\geq 2(1+\varDelta_{\varepsilon})|f_{{\varepsilon}, 1}(x)|.$$ This ends the proof.
Observe that there are terms involving $h_{\varepsilon}$ which appear in the right hand side of $\left( \ref{E}\right) .$ This somewhat complicates the situation. However, since the Fermi coordinates are defined using the nodal line, we have the following
It holds \[hf1\]$$\left\Vert h_{\varepsilon}\right\Vert _{C^{2,\mu}_{{\varepsilon}\tau}\left( \mathbb{R}\right) }\leq
C\left\Vert \phi\right\Vert _{C^{2,\mu}_{{\varepsilon}\tau}\left( \mathbb{R}^{2}\right)
}+C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}
\label{hf}$$
We first recall that in the set $\mathrm{dist}\, (\Gamma_{{\varepsilon}, 1}, {\tt x})<\mathrm{d}_{\varepsilon}({\tt x})-1$, $$({\mathbf x}_{{\varepsilon}, 1}^*u)\left(
x_{1},y_{1}\right) =H\left( y_{1}-h_{\varepsilon}\left( x_{1}\right) \right) -({\mathbf x}_{{\varepsilon}, 1}^*H_{{\varepsilon}, 2})\left(
x_{1},y_{1}\right) -1+({\mathbf x}_{{\varepsilon}, 1}^*\phi)\left( x_{1},y_{1}\right). \label{hfu0}$$ Letting $y_{1}=0$ in the above identity and using that $x_1=x$, we have: $$\left\vert ({\mathbf x}_{{\varepsilon}, 1}^*H_{{\varepsilon}, 2})\left( x_{1},y_{1}\right) +1\right\vert \leq\begin{cases}
C\exp\big\{-2\sqrt{2}(|f_{{\varepsilon},1}(x)|(1+\varDelta_{\varepsilon})\big\}, \quad (x,f_{{\varepsilon}, 1}(x))\in \mathcal{O}_2,\\
0, \quad (x,f_{{\varepsilon}, 1}(x))\notin \mathcal{O}_2.
\end{cases}
$$ Then from $({\mathbf x}_{{\varepsilon}, 1}^*u)\left(
x_{1},0\right)=0$ one gets$$\|h_{\varepsilon}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0}({\mathbb R})} \leq C\|\phi\|
_{{\mathcal C}^{0}_{{\varepsilon}\tau}({\mathbb R}^2)}+C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}.$$ This gives us a ${\mathcal C}^{0}$ estimate. To estimate the ${\mathcal C}^{1}$ norm of $h_{\varepsilon}$, we differentiate the relation $\left( \ref{hfu0}\right) $ with respect to $x_{1}$ and let $y_{1}=0$ in the resulting equation. Then we find that$$-H^{\prime}\left( -h_{\varepsilon}\left( x_{1}\right) \right) h_{\varepsilon}^{\prime}\left(
x_{1}\right) -\frac{\partial }{\partial x_{1}}({\mathbf x}_{{\varepsilon}, 1}^* H_{{\varepsilon}, 2})+\frac{\partial}{\partial x_{1}}({\mathbf{x}}_{{\varepsilon}, 1}^*\phi)=0, \label{d1}$$ from which the ${\mathcal C}^1_{{\varepsilon}\tau}$ estimate follows. Similarly, we could differentiate the equation $\left( \ref{hfu0}\right) $ twice with respect to $x_{1}$ and let $y_{1}=0$ to estimate $h_{{\varepsilon}}''$. Corresponding estimates for the Holder norm are also straightforward.
To proceed, we need the following [*a priori*]{} estimate: The proof is by contradiction and it is essentially the same as that of Proposition 5.1 in [@MR2557944] and consists of the following steps: first an [*a priori*]{} estimate is proven for a solution of the following problem: $$-\Delta \varphi+F''(\bar u_{\varepsilon})=f, \quad\mbox{in}\ {\mathbb R}^2,$$ where $\varphi$ satisfies the orthogonality condition (\[ort xxx\]). In fact, using the fact that the heteroclinic solution in ${\mathbb R}$ is neutrally stable one can prove that $\varphi$ satisfies an estimate of the form claimed in the Proposition. This type of argument can be found for example in [@dkp_dg].
Second, we project the equation on the functions of the form $\rho_{{\varepsilon},j}H'_{{\varepsilon}, j}$, $j=1,2$ and arrive at the following expressions $$\int_{\mathbb R}{\tt x}^*_{{\varepsilon}, j}\big(\rho_{{\varepsilon},j}H'_{{\varepsilon}, j}[-\Delta\varphi+F^{\prime\prime}\left( \bar{u}_{\varepsilon}\right) \varphi]\big)(x_j, y_j)\,dy_j-\int_{\mathbb R}{\tt x}^*_{{\varepsilon}, j}\big(\rho_{{\varepsilon},j}H'_{{\varepsilon}, j}f\big)(x_j,y_j)\,dy_j=\sum_{i=1,2}\kappa_{i,{\varepsilon}}\int_R{\tt x}^*_{{\varepsilon}, j}\big(\rho_{{\varepsilon},i}H'_{{\varepsilon}, i}\big)^2(x_j,y_j)\,dy_j.$$ After an integration by parts and some calculations we can prove using the above identity that the $\mathcal{C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})$ norm of the functions $\kappa_{i,{\varepsilon}}$ can be controlled by $o(1)\|\varphi\|_{\mathcal{C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}+C\|f\|_{\mathcal{C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}$. From this and the first step the assertion of the Proposition follows. We omit the details.
With this result at hand, now we could prove
\[fif\] Let $\phi=u-\bar u_{\varepsilon}$ be the solution of (\[ubar\]). The following estimate is true:$$\left\Vert \phi\right\Vert _{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}\leq o\big(\left\Vert
f_{{\varepsilon}, 1}^{\prime\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\big)+C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}. \label{f1}$$
We will use Proposition \[fif\]. Thus we write: $$-\Delta\phi+F''(\bar u_{\varepsilon})\phi=E(\bar u_{\varepsilon})^\perp+P(\phi)+E(\bar u_{\varepsilon})^\parallel.$$ Because of Proposition \[apriori\] to control the size of the function $\phi$ it suffices to control the size of $E(\bar u_{\varepsilon})^\perp$ (which we already do by Lemma \[Eu\]) and the size of $P(\phi)$.
Next we observe that $P\left(
\phi\right) $ is essentially quadratic in $\phi$, and therefore it is not difficult to show $$\left\Vert P\left( \phi\right)
\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}=o\left( \left\Vert \phi\right\Vert
_{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}\right) .$$ Collecting all these estimates, we conclude (\[f1\]).
Roughly speaking, the above result indicates that we can control $\phi$ by $\|e^{\,-2\sqrt{2}|f_{1,{\varepsilon}}|}\|_{\mathcal{C}^0_{{\varepsilon}\tau}({\mathbb R})}$ and the second derivative of $f_{{\varepsilon}, 1}$. However, this is not quite enough for our later purpose. Remark that for the solution constructed in [@MR2557944], the corresponding error is roughly speaking controlled by $C\varepsilon^{2}.$ On the other hand, intuitively, $\|f_{{\varepsilon}, 1}-{\varepsilon}|x|\|_{\mathcal{C}^0({\mathbb R})}\sim \log\frac{1}{{\varepsilon}}$. This indeed would be true if $f_{{\varepsilon},1}$, $f_{{\varepsilon}, 2}$ were solutions of the Toda system with the asymptotic slope ${\varepsilon}$ at $\pm \infty$. For now we will show:
\[fi\] The following estimate holds: $$\begin{aligned}
\left\Vert \phi\right\Vert _{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}\leq C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}.\end{aligned}$$
Let us consider the integral $\int_{\mathbb{R}}\left( {\mathbf x}_{{\varepsilon}, 1}^* E\left( \bar{u}_{\varepsilon}\right) \rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime} \right)dy_{1}.$ We will show below (Step 1) that on one hand its ${\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})$ norm is controlled by $o\left( \left\Vert \phi\right\Vert
_{C^{0,\mu}_{{\varepsilon}\tau}}\right) .$ On the other hand (Step 2) we will show that this integral is controlled by $f''_{{\varepsilon},1}$. Then the proof of the Lemma will follow by combining this with the previous estimates.
[**Step 1.**]{} We claim that the integral $\int_{\mathbb{R}}\left( {\mathbf x}_{{\varepsilon}, 1}^* E\left( \bar{u}_{\varepsilon}\right) \rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime} \right)dy_{1}$ is controlled by $o\left( \left\Vert \phi\right\Vert
_{C^{0,\mu}_{{\varepsilon}\tau}}\right) .$ Clearly it is sufficient to estimate $E(\bar u_{\varepsilon})^\parallel$. We will show $$\left\Vert c_{{\varepsilon}}\big({\mathbf x}_{{\varepsilon},1}^*\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime}\big) \int_{\mathbb{R}}\big[{\mathbf x}_{{\varepsilon}, 1}^* E(\bar{u}_{\varepsilon})\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}\big]dy_{1}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}=o\left( \left\Vert
\phi\right\Vert _{{\mathcal C}_{{\varepsilon}\tau}^{2,\mu}({\mathbb R}^2)}\right) . \label{ep}$$ In fact, $$\begin{aligned}
\int_{\mathbb{R}}\big[{\mathbf x}_{{\varepsilon}, 1}^* E(\bar{u}_{\varepsilon})\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}\big]dy_{1} & =\int_{\mathbb{R}}\big[{\mathbf x}_{{\varepsilon}, 1}^*( -\Delta\phi
+F^{\prime\prime}( \bar{u}_{\varepsilon}) \phi)\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}\big]dy_{1}\ \\
& +\int_{\mathbb{R}}\big[{\mathbf x}_{{\varepsilon}, 1}^* P(\phi)\rho
_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}\big]dy_{1}.\end{aligned}$$ To handle the first term appearing in the right hand side we write $\Delta_{(x_1, y_1)}=\partial_{x_1}^2+\partial_{y_1}^2$ and: $$\begin{aligned}
T_{2} & :=\underbrace{\int_{\mathbb{R}}( -\Delta_{\left( x_{1},y_{1}\right)
}{\mathbf x}_{{\varepsilon}, 1}^*\phi+F^{\prime\prime}(H) {\mathbf x}_{{\varepsilon}, 1}^* \phi)({\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime})dy_{1}}_{T_{21}}\\
& + \underbrace{\int_{\mathbb{R}}\big[ - ({\mathbf x}_{{\varepsilon}, 1}^* \Delta\phi)
-\Delta_{(x_{1},y_{1})}({\mathbf x}_{{\varepsilon}, 1}^* \phi)\big]+\big[({\mathbf x}_{{\varepsilon}, 1}^* F^{\prime\prime}\left( \bar{u}_{\varepsilon}\right)\phi)
-F^{\prime\prime}\left( H\right){\mathbf x}_{{\varepsilon}, 1}^* \phi\big] ({\mathbf x}_{{\varepsilon}, 1}^* \rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime})dy_{1}}_{T_{22}}.\end{aligned}$$ Using that from $\int_{\mathbb{R}}\big[{\mathbf x}_{{\varepsilon}, 1}^*\phi\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime}\big]dy_{1}=0,$ it follows $\partial_{x_1}\int_{\mathbb{R}}\big[{\mathbf x}_{{\varepsilon}, 1}^*\phi\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime}\big]dy_{1}=0$, we get$$\begin{aligned}
\ T_{21}& =-2\int_{\mathbb{R}}\frac{\partial({\mathbf x}_{{\varepsilon}, 1}^*\phi)}{\partial x_{1}}\frac{\partial\left( {\mathbf x}_{{\varepsilon}, 1}^* \rho_{{\varepsilon},1}H_{{\varepsilon}, 1}^{\prime}\right) }{\partial x_{1}}-\int_{\mathbb{R}}({\mathbf x}_{{\varepsilon}, 1}^*\phi)\frac{\partial^{2}\left( {\mathbf x}_{{\varepsilon}, 1}^* \rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime
}\right) }{\partial x_{1}^{2}}\\
&\quad +\int_{\mathbb{R}}({\mathbf x}_{{\varepsilon}, 1}^*\phi)\left( \frac{\partial^{2}\left({\mathbf x}_{{\varepsilon}, 1}^*\rho
_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime}\right) }{\partial y_{1}^{2}}+ F^{\prime\prime
}\left( H\right) ({\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime})\right) \\
& =-2\int_{\mathbb{R}}\frac{\partial{\tt x}^*_{{\varepsilon}, 1}\phi}{\partial x_{1}}\frac{\partial\left( {\mathbf x}_{{\varepsilon}, 1}^* \rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime}\right) }{\partial x_{1}}-\int_{\mathbb{R}}({\mathbf x}_{{\varepsilon}, 1}^*\phi)\frac{\partial^{2}\left( {\mathbf x}_{{\varepsilon}, 1}^* \rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime
}\right) }{\partial x_{1}^{2}}\\
& \quad +\int_{\mathbb{R}}({\mathbf x}_{{\varepsilon}, 1}^*\phi)\left( \frac{\partial^{2}({\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon},1})}{\partial
y_{1}^{2}}({\mathbf x}_{{\varepsilon}, 1}^*H_{{\varepsilon}, 1}^{\prime})+2\frac{\partial({\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon},1})}{\partial y_{1}}\frac{\partial ({\mathbf x}_{{\varepsilon}, 1}^*H_{{\varepsilon},1}^{\prime})}{\partial y_{1}}\right) .\end{aligned}$$ Due to the presence of the derivatives of ${\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon}, 1}$ with respect to $x_1$, $y_1$ and also the presence of $H_{{\varepsilon},1}^{\prime}$ in each term, we now obtain that $$\|c_{{\varepsilon}}\big({\mathbf x}_{{\varepsilon},1}^*\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime}\big)T_{21}\|_{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}=o\left( \left\Vert
\phi\right\Vert _{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}\right) . \label{i1}$$ On the other hand, $$\begin{aligned}
T_{22}& =-\int_{\mathbb{R}}\left\{ \left( \frac{1}{A_{1}}-1\right)
\partial_{x_{1}}^{2}({\mathbf x}_{{\varepsilon}, 1}^*\phi)+\frac{1}{2}\frac{\partial_{y_{1}}A_{1}}{A_{1}}\partial_{y_{1}}({\mathbf x}_{{\varepsilon}, 1}^*\phi)-\frac{1}{2}\frac{\partial_{x_{1}}A_{1}}{A_{1}^{2}}\partial_{x_{1}}({\mathbf x}_{{\varepsilon}, 1}^*\phi)\right\} ({\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon},1}H_{{\varepsilon}, 1}^{\prime})\\
& \quad +\int_{\mathbb{R}}\big[({\mathbf x}_{{\varepsilon}, 1}^* F^{\prime\prime}\left( \bar{u}_{\varepsilon}\right)\phi)
-F^{\prime\prime}\left( H\right){\mathbf x}_{{\varepsilon}, 1}^* \phi\big] ({\mathbf x}_{{\varepsilon}, 1}^* \rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime})dy_{1}
$$ Let ${\tt x}=(t_1,z_1)$, ${\tt x}_2=(t_2,z_2)\in \mathcal{O}_1$ be given. Observe that if ${\tt x}^*_{1}=\left( s_{1},y_{1}\right) ,{\tt x}^*_{2}=\left( s_{2},y_{1}\right) $ are the Fermi coordinates with respect to $\Gamma_{{\varepsilon}, 1},$ of ${\tt x}_1$, and ${\tt x}_2$ respectively, then $$c(|s_1-s_2|^2+|y_1-y_2|^2)^{\mu/2}\leq (|{t}_{1}-{t}_{2}|^2+|z_1-z_2|^2)^{\mu/2} \leq C(| s_{1}-s_{2}|^2+|y_1-y_2|^2)^{\mu/2} .$$ Now, we will make use of the fact that $1-\frac{1}{A_1}$, $\frac{\partial_{y_{1}}A_{1}}{A_{1}},\frac{\partial_{x_{1}}A_{1}}{A_{1}^{2}}$, $\big[{\mathbf x}_{{\varepsilon}, 1}^*F^{\prime\prime}\left( \bar
{u}_{\varepsilon}\right) -F^{\prime\prime}\left( H\right)\big] $ are small terms. Let us write for example: $$\left( \frac{1}{A_{1}}-1\right) \partial_{x_{1}}^{2}({\mathbf x}_{{\varepsilon}, 1}^*\phi
\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime})\left( x_{1},y_{1}\right)={\widetilde}\phi(x_1, y_1) H'(y_1), \quad c_{{\varepsilon}}\big({\mathbf x}_{{\varepsilon},1}^*\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime}\big)(x_1,y_1)={\widetilde}H^{'}(x_1,y_1).$$ Abusing slightly the notation one has for instance $$\begin{aligned}
& \sup_{\|{\tt x}_{1}-{\tt x}_{2}\| \leq 1}\frac{1}{\|{\tt x}_{1}-{\tt x}_{2}\|^\mu}
\left| {\widetilde}H'(s_1,y_1) \int_{\mathbb R}{\widetilde}\phi(s_1, {\widetilde}y)H'({\widetilde}y)\,d{\widetilde}y-{\widetilde}H'(s_2, y_2)\int_{\mathbb R}{\widetilde}\phi(s_2, {\widetilde}y)H'({\widetilde}y)\,d{\widetilde}y \right|
\\
& \leq
\sup_{\|{\tt x}^*_{1}-{\tt x}^*_{2}\| \leq 1}\frac{1}{\|{\tt x}_{1}-{\tt x}_{2}\|^\mu}
\left| {\widetilde}H'(s_1,y_1) \int_{\mathbb R}{\widetilde}\phi(s_1, {\widetilde}y)H'({\widetilde}y)\,d{\widetilde}y-{\widetilde}H'(s_2, y_2)\int_{\mathbb R}{\widetilde}\phi(s_2, {\widetilde}y)H'({\widetilde}y)\,d{\widetilde}y \right|
& =o\left( \left\Vert \phi\right\Vert _{{\mathcal C}^{2,\mu}({\mathbb R}^2)}\right) ,\end{aligned}$$ which leads to $$\left\Vert c_{{\varepsilon}}\big({\mathbf x}_{{\varepsilon},i}^*\rho_{{\varepsilon},i}H_{{\varepsilon},i}^{\prime}\big) \int_{\mathbb{R}}\left( \frac{1}{A_{1}}-1\right) \partial_{x_{1}}^{2}({\mathbf x}_{{\varepsilon}, 1}^*\phi\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime})dy_{1}\right\Vert _{{\mathcal C}^{0,\mu}({\mathbb R}^2)}=o\left(
\left\Vert \phi\right\Vert _{{\mathcal C}^{2,\mu}({\mathbb R}^2)}\right) .$$ Other terms apperaing in the definition of $T_{22}$ can be checked similarly whence we obtain$$\left\Vert c_{{\varepsilon}}\big({\mathbf x}_{{\varepsilon},1}^*\rho_{{\varepsilon},1}H_{{\varepsilon},i}^{\prime}\big)T_{22}\right\Vert _{{\mathcal C}^{0,\mu}({\mathbb R}^2)}=o\left( \left\Vert
\phi\right\Vert _{{\mathcal C}^{0,\mu}({\mathbb R}^2)}\right) .$$ This together with $\left( \ref{i1}\right) $ tells us$$\left\Vert c_{{\varepsilon}}\big({\mathbf x}_{{\varepsilon},i}^*\rho_{{\varepsilon},i}H_{{\varepsilon},i}^{\prime}\big)T_{2}\right\Vert _{{\mathcal C}^{0,\mu}({\mathbb R}^2)}=o\left( \left\Vert \phi\right\Vert
_{{\mathcal C}^{2,\mu}({\mathbb R}^2)}\right) .$$ The estimate (\[ep\]) follows from this in a straightforward way.
[**Step 2.**]{} We claim that the weighted norm of the integral $\int_{\mathbb{R}}\left( {\mathbf x}_{{\varepsilon}, 1}^* E\left( \bar{u}_{\varepsilon}\right) \rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime} \right)dy_{1}$ is controlled by $f''_{{\varepsilon},1}$. To do this we will now check more closely the above integral using the definition of $\bar{u}_{\varepsilon},$ these calculations are actually similar as in the proof Lemma \[Eu\]. We see that one term appearing in the integral is $$\frac{1}{2}\int_{\mathbb{R}}\frac{\partial_{y_{1}}A_{1}}{A_{1}}({\mathbf x}_{{\varepsilon}, 1}^* \rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime2})dy_{1}.$$ We will concentrate on this term since the ${\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})$ norm of other terms can be estimated by ${\mathcal O}(\|h_{\varepsilon}\| _{C^{2,\mu}_{{\varepsilon}\tau}({\mathbb R})})
+C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}$, as we have seen in the proof of Lemma \[Eu\]. Plugging in the formula for $A_{1}$ into the above integral, one gets $$\begin{aligned}
\frac{1}{2}\int_{\mathbb{R}}\frac{\partial_{y_{1}}A_{1}}{A_{1}}({\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon},1}H_{{\varepsilon},1}^{\prime2})dy_{1} & =\int_{\mathbb{R}}\frac{1}{A_{1}}\left( y_{1}\frac{\left( f_{{\varepsilon}, 1}^{\prime\prime}\left( x_{1}\right) \right) ^{2}}{\left(
1+\left( f_{{\varepsilon}, 1}^{\prime}\left( x_{1}\right) \right) ^{2}\right) ^{2}}-\frac{f_{{\varepsilon}, 1}^{\prime\prime}\left( x_{1}\right) }{\sqrt{1+\left( f_{{\varepsilon}, 1}^{\prime
}\left( x_{1}\right) \right) ^{2}}}\right) ({\mathbf x}_{{\varepsilon}, 1}^*\rho_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime2})\\
& =-\frac{1}{c_{\varepsilon}}f_{{\varepsilon}, 1}^{\prime\prime}\left( x_{1}\right) +T_{4},\end{aligned}$$ where $T_{4}$ is a function such that $$\left\Vert T_{4}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}=o\left( \left\Vert f_{{\varepsilon}, 1}^{\prime\prime
}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\right) .$$ Consequently,$$\begin{aligned}
\left\Vert f_{{\varepsilon}, 1}^{\prime\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})} &\leq C\left\Vert
\int_{\mathbb{R}}( {\mathbf x}_{{\varepsilon}, 1}^* E\left( \bar{u}_{\varepsilon}\right)\rho
_{{\varepsilon}, 1}H_{{\varepsilon}, 1}^{\prime})dy_{1}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\\
&\quad +{\mathcal O}\left( \left\Vert
h_{\varepsilon}\right\Vert _{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R})}\right) +o\left( \left\Vert f_{{\varepsilon}, 1}^{\prime\prime
}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\right) +C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}.\end{aligned}$$ We then could apply Lemma $\ref{fif}$ to get$$\left\Vert f_{{\varepsilon}, 1}^{\prime\prime}\right\Vert _{{\mathcal C}^{0,\mu}({\mathbb R})}={\mathcal O}\left(\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}\right) +{\mathcal O}\left( \left\Vert h_{\varepsilon}\right\Vert _{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R})}\right) . \label{eq1}$$ This together with (\[hf\]), and $\left( \ref{f1}\right) $ implies that $$\left\Vert f_{{\varepsilon}, 1}^{\prime\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\leq C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}. \label{eq2}$$ As a consequence: $$\left\Vert \phi\right\Vert _{{\mathcal C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R}^2)}\leq C\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}.$$
Now we will proceed to estimate the quantity $e^{\,-\sqrt{2}|f_{{\varepsilon}, 1}(0)|}$. To this end, we first need to obtain some exponential decay estimate of $\phi$ along the $y$ axis away from $\Gamma_{{\varepsilon}, 1}$. Note that $E\left( \bar{u}_{\varepsilon}\right) $ decays exponentially the direction transversal to the nodal line $\Gamma_{{\varepsilon}, 1}$. Indeed, using (\[eu\]) and the exponential decay of $H\pm 1$, $H'$ and $H''$ one can show: $$\label{eu expo 1}
|E(\bar u_{\varepsilon})({\tt x})|\leq Ce^{\,-\sqrt{2}|f_{{\varepsilon}, 1}(0)|} e^{\,-\sqrt{2}\mathrm{{{\rm dist}\, }}\,({\tt x}, \Gamma_{{\varepsilon}, 1})}, \quad {\tt x}=(x, y), \quad y\leq 0.$$
We write the equation for $\phi$ in the form $$-\Delta\phi+\big[F^{\prime\prime}\left( \bar{u}_{\varepsilon}\right)+\frac{P(\phi)}{\phi}\big] \phi=E\left( \bar
{u}_{\varepsilon}\right).$$ For any constant $0<\iota_0<\sqrt{2}$ there exists $r_0$ large and ${\varepsilon}$ small, such that: $$F^{\prime\prime}\left( \bar{u}_{\varepsilon}({\tt x})\right)+\frac{P(\phi({\tt x}))}{\phi({\tt x})}\geq (\sqrt{2}-\iota_0)^2, \quad \mathrm{dist}\,({\tt x}, \Gamma_{{\varepsilon}, 1})>r_0, \quad {\tt x}=(x,y), \quad y\leq 0.$$ Using barriers we then obtain: $$|\phi({\tt x})|\leq C_{r_0}e^{\,-\sqrt{2}|f_{{\varepsilon}, 1}(0)|} e^{\,-(\sqrt{2}-\iota_0)\mathrm{dist}\, ({\tt x}, \Gamma_{{\varepsilon}, 1})},
\label{expf}$$ in the lower half plane. Note that this estimate is in some sense precise only along the $y$ axis, since in reality we expect that $\phi({\tt x})\sim e^{\,-(2\sqrt{2}-\iota_0)\mathrm{dist}\, ({\tt x}, \Gamma_{{\varepsilon}, 1})}$. This estimate can be bootstrapped using elliptic estimates to get a similar estimate for the derivatives of the function $\phi$.
Let us go back to the Toda system (\[toda 2\])–(\[toda 3\]) and recall that by $q_{{\varepsilon}, 1}(x)<0<q_{{\varepsilon}, 2}(x)$ we have denoted the solution of this system whose slope at $\infty$ is ${\varepsilon}$ (this means the tangent of the asymptotic angle between the line $y=q_{{\varepsilon}, 2}(x)$ and the $x$ axis in the first quadrant). We note that the curve $\widetilde \Gamma_{{\varepsilon}, 1}=\{y=q_{{\varepsilon}, 1}(x)\}$ is contained in the lower half plane. In what follows we use $\alpha,\beta$ to denote general positive constants independent of $\varepsilon.$
\[est at 0\] There exists $\alpha_1>0$ such that $\left\vert f_{{\varepsilon}, 1}\left( 0\right) -q_{{\varepsilon}, 1}\left(
0\right) \right\vert \leq C\varepsilon^{\alpha_1}.$
The idea of the proof is to relate the asymptotic behavior of $u$ along vertical straight lines, as ${\varepsilon}\to 0$, using the Hamiltonian identity: $$\label{ham 23}
\int_{\mathbb R}\left\{ \frac{1}{2}u_{y}^{2}\left( 0,y\right)-\frac{1}{2}u_{x}^{2}\left( 0,y\right)
+F\left( u\left( 0,y\right) \right) \right\} dy=\int_{\mathbb R}\left\{ \frac{1}{2}u_{y}^{2}\left( x,y\right)-\frac{1}{2}u_{x}^{2}\left( x,y\right)
+F\left( u\left( x,y\right) \right) \right\} dy, \quad \forall x,$$ and in particular take $x\to \infty$ on the right hand side of (\[ham 23\]). Indeed, using the asymptotic behavior of a four ended solution it is not hard to show that: $$\lim_{x\to \infty}\int_{\mathbb R}\left\{ \frac{1}{2}u_{y}^{2}\left( x,y\right)-\frac{1}{2}u_{x}^{2}\left( x,y\right)
+F\left( u\left( x,y\right) \right) \right\} dy=2{\mathbf{e}}_F \cos\theta(u), \quad {\mathbf e}_F=\int_{\mathbb R}\frac{1}{2}(H')^2+F(H), \quad {\varepsilon}=\tan\theta(u).$$ Since $u$ is an even function of $x$ we also have $u_x(0,y)=0$ and thus it follows from (\[ham 23\]): $$\int_{\mathbb R}\left\{ \frac{1}{2}u_{y}^{2}\left( 0,y\right)
+F\left( u\left( 0,y\right) \right) \right\} dy= 2{\mathbf{e}}_F \cos\theta(u).$$ We will now calculate the left hand side of the above identity.
Recall that the heteroclinic solution has the following asymptotic behavior, which can also be differentiated: $$H\left( s\right) =1-\mathbf{a}_{F}e^{-\sqrt{2}s}+O\left( e^{-2\sqrt{2}s}\right) ,\text{ as }s\rightarrow+\infty.$$ Denote $t=f_{{\varepsilon}, 1}\left( 0\right) +h_{{\varepsilon}}\left( 0\right) .$ Let $\eta_1$ be the cut off function appearing in the definition of the approximate solution (\[def hve\]). Along the $y$-axis it holds $(x_1, y_1)=(0,y)$, where $(x_1,y_1)$ are the Fermi coordinates of $\Gamma_{{\varepsilon}, 1}$ and then, abusing the notation slightly we can write $$\begin{aligned}
u\left( 0,y\right) &=\underbrace{H\left( y-t\right) -H\left( y+t\right)
-1+\phi\left( 0,y\right)}_{u_0(y)}\\
&\quad + \underbrace{(1-\eta_1(0,y))\Big[\frac{H\left( y-t\right)}{|H\left( y-t\right)}-H\left( y-t\right)\Big]-
(1-\eta_1(0,y))\Big[\frac{H\left( y+t\right)}{|H\left( y+t\right)|}-H\left( y+t\right)\Big] }_{\psi(y)}.\end{aligned}$$ We observe that for all $\sigma>0$, $$|\psi(y)|\leq C^{\,-(\sqrt{2}-\sigma)|y|} e^{\,-\sigma \mathrm{d}_{\varepsilon}(0)}, \label{est psi}$$ and by the definition of $\mathrm{d}_{\varepsilon}(0)$ $$\mathrm{d}_{\varepsilon}(0)\geq \frac{1}{2}\Big(\frac{\theta}{2{\|f_{{\varepsilon}, 1}''\|_{{\mathcal C}^0({\mathbb R})}}}\Big)\geq Ce^{\,{2\sqrt{2}}|f_{{\varepsilon}, 1}(0)|},$$ where the last estimate follows form (\[517\]). Then we find, taking $\sigma=\frac{3\sqrt{2}}{4}$ above, $$\int_{-\infty}^{0}\left\{ \frac{1}{2}u_{y}^{2}\left( 0,y\right)
+F\left( u\left( 0,y\right) \right) \right\} dy= \int_{-\infty}^{0}\left\{ \frac{1}{2}u_{0,y}^{2}\left(y\right)
+F\left( u_0\left(y\right) \right) \right\} dy+o(e^{\,-3|f_{{\varepsilon}, 1}(0)|}).$$ Now we calculate $$\begin{aligned}
& \int_{-\infty}^{0}\left\{ \frac{1}{2}u_{0,y}^{2}\left(y\right)
+F\left( u_0\left(y\right) \right) \right\} dy\nonumber\\
& =\underbrace{\int_{-\infty}^{0}\left\{ \frac{1}{2}\left( H^{\prime}\left(
y-t\right) \right) ^{2}+F\left( H\left( y-t\right) \right) \right\}
dy}_{I_1}\nonumber\\
& +\underbrace{\int_{-\infty}^{0}\left\{ H^{\prime}\left( y-t\right) \left(
\partial_{y}\phi-H^{\prime}\left( y+t\right) \right) +F^{\prime}\left(
H\left( y-t\right) \right) \left( \phi-H\left( y+t\right) -1\right)
\right\} dy}_{I_2}\nonumber\\
& +\underbrace{\frac{1}{2}\int_{-\infty}^{0}\left( \partial_{y}\phi-H^{\prime}\left(
y+t\right) \right) ^{2}+F^{\prime\prime}\left( H\left( y-t\right)
\right) \left( \phi-H\left( y+t\right) -1\right) ^{2}dy}_{I_3}\nonumber\\
& +\mathcal{O}\left( \int_{-\infty}^{0}\left( \phi-H\left( y+t\right) -1\right)
^{3}dy\right) \label{dis1}$$ We note that $$t=f_{{\varepsilon}, 1}(0)+h_{{\varepsilon}}(0)=f_{{\varepsilon}, 1}(0)+{\mathcal O}(e^{\,-2\sqrt{2}|f_{{\varepsilon}, 1}(0)|(1+o(1))})<0.$$ The first term on the right hand side of $\left( \ref{dis1}\right) $ is equal to $$\begin{aligned}
I_1& =\int_{-\infty}^{-t}\left\{ \frac{1}{2}\left( H^{\prime}\right)
^{2}+F\left( H\right) \right\} dy\\
& =\mathbf{e}_{F}-\int_{-t}^{+\infty}\left\{ \frac{1}{2}\left( H^{\prime
}\right) ^{2}+F\left( H\right) \right\} dy\\
\ & =\mathbf{e}_{F}-\int_{-t}^{+\infty}2\mathbf{a}_{F}^{2}e^{-2\sqrt{2}y}dy+{\mathcal O}\left( e^{\,-4\sqrt{2}|t|}\right) \\
& =\mathbf{e}_{F}-\frac{\sqrt{2}}{2}\mathbf{a}_{F}^{2}e^{\,-2\sqrt{2}|t|}+O\left(
e^{\,-3\sqrt{2}|t|}\right) .\end{aligned}$$ We observe that, after an integration by parts, the second term is equal to$$I_2=H^{\prime}\left( -t\right) \left( \phi\left(0\right) -H\left(
t\right) -1\right) =-\sqrt{2}\mathbf{a}_{F}^{2}e^{\,-2\sqrt{2}|t|}+{\mathcal O}\left(
e^{\,-\frac{5}{2}\sqrt{2}|t|}\right) .$$ As to the last term, one has $$\begin{aligned}
I_3 & ={\mathcal O}\left( e^{\,-4|t|}\right) +\frac{1}{2}\int_{-\infty}^{0}\left( H^{\prime
}\left( y+t\right) \right) ^{2}+F^{\prime\prime}\left( H\left(
y-t\right) \right) \left( H\left( y+t\right) -1\right) ^{2}dy\\
& ={\mathcal O}\left( e^{\,-4|t|}\right) +\frac{\sqrt{2}\mathbf{a}_{F}^{2}}{4}e^{\,-2\sqrt
{2}|t|}+\frac{\mathbf{a}_{F}^{2}}{2}\int_{-\infty}^{0}F^{\prime\prime}\left(
H\left( y-t\right) \right) e^{\,-2\sqrt{2}|y+t|}dy\\
& ={\mathcal O}\left( e^{\,-4|t|}\right) +\frac{\sqrt{2}\mathbf{a}_{F}^{2}}{2}e^{\,-2\sqrt
{2}|t|}.\end{aligned}$$ Therefore, we get that $$I_0:=\int_{\mathbb R}\left\{ \frac{1}{2}u_{y}^{2}\left( 0,y\right)
+F\left( u\left( 0,y\right) \right) \right\} dy=2\mathbf{e}_{F}-2\sqrt{2}\mathbf{a}_{F}^{2}e^{\,-2\sqrt{2}|f_{{\varepsilon}, 1}\left(
0\right) +h_{\varepsilon}\left( 0\right)| }+{\mathcal O}\left( e^{\,-3|f_{{\varepsilon}, 1}\left( 0\right)|
}\right) .$$ According to the Hamiltonian identity (\[ham 23\]), $$\begin{aligned}
I_0 =2\mathbf{e}_{F}\cos\theta({u}).\end{aligned}$$ Now, let $u_{\varepsilon}$ with $\varepsilon=\tan\theta({u})$ be a solution constructed in [@MR2557944] whose nodal line in the lower half plane is given by the curve $y=q_{{\varepsilon},1}(x)+r_{{\varepsilon},1}({\varepsilon}x)$, where $q_{{\varepsilon}, 1}$ is the solution of the Toda system whose asymptotic angle at $\infty$ is ${\varepsilon}$, and $r_{{\varepsilon},1}(x)$ satisfies, as we stated in section \[exists small eps\], with some $\alpha>0$ $$\|r_{{\varepsilon},1}\|_{\mathcal C^{2, \mu}_{\tau } (\mathbb R) \oplus D}
\leq C \, \varepsilon^{\alpha}$$ We recall that since we are working in the class of even function $|r_{{\varepsilon}, 1}(x)|\leq C{\varepsilon}^\alpha$, which implies that $r_{{\varepsilon}, 1}$ is a bounded, small function. Now, the Hamiltonian identity (\[ham 23\]) can be used for $u_{\varepsilon}$ as well and by a computation we get $$2\mathbf{e}_{F}\cos\theta({u}_{\varepsilon})=2\mathbf{e}_{F}-2\sqrt{2}\mathbf{a}_{F}^{2}e^{\,-2\sqrt{2}|q_{{\varepsilon}, 1}\left( 0\right) +r_{{\varepsilon}, 1}(0)|}+\mathcal{O}(e^{\,-3|q_{{\varepsilon}, 1}\left( 0\right) +r_{{\varepsilon}, 1}(0)|})$$ where $r_{{\varepsilon}, 1}(0)={\mathcal O}\left( \varepsilon^{\alpha}\right) .$ Therefore, $$I_0=2\mathbf{e}_{F}-2\sqrt{2}\mathbf{a}_{F}^{2}e^{\,-2\sqrt{2}|q_{{\varepsilon},1}\left( 0\right)
+r_{{\varepsilon},1}(0)|}+\mathcal{O}(e^{\,-3|q_{{\varepsilon}, 1}\left( 0\right) +r_{{\varepsilon}, 1}(0)|}).$$ That is, $$e^{\,-2\sqrt{2}|f_{{\varepsilon}, 1}\left(
0\right) +h_{\varepsilon}\left( 0\right)| }+{\mathcal O}\left( e^{\,-3|f_{{\varepsilon}, 1}|\left( 0\right)|
}\right) =e^{\,-2\sqrt{2}|q_{{\varepsilon}, 1}\left( 0\right) +r_{{\varepsilon}, 1}(0)|}+\mathcal{O}(e^{\,-3|q_{{\varepsilon}, 1}\left( 0\right) +r_{{\varepsilon}, 1}(0)|})
$$ This yields$$f_{{\varepsilon}, 1}\left( 0\right) +h_{\varepsilon}\left( 0\right) +{\mathcal O}\left( e^{\,-\left( 3-2\sqrt
{2}\right)| f_{{\varepsilon}, 1}\left( 0\right) +h_{\varepsilon}\left( 0\right)| }\right)
=q_{{\varepsilon},1}\left( 0\right) +{\mathcal O}\left( \varepsilon^{\alpha}\right) .$$ Since $q_{{\varepsilon}, 1}\left( 0\right) -\frac{\sqrt{2}}{2}\log\varepsilon={\mathcal O}\left( 1\right)
,$ we get$$f_{{\varepsilon}, 1}\left( 0\right) +h_{{\varepsilon}}\left( 0\right) =\frac{\sqrt{2}}{2}\log\varepsilon
+{\mathcal O}\left( 1\right) ,$$ which leads to $$f_{{\varepsilon}, 1}\left( 0\right) +h_{{\varepsilon}}\left( 0\right) -q_{{\varepsilon}, 1}\left( 0\right) ={\mathcal O}\left(
\varepsilon^{\alpha}\right),$$ as claimed. This ends the proof.
Now we are in position to prove Proposition \[prop gamma\]. As we will see the proof of Proposition \[estim hat phi\] will be obtained as an intermediate step
Our goal is to show estimate (\[first estimate\]) and this will be done in few steps. For brevity let us denote $p_{{\varepsilon}, 1}=f_{{\varepsilon}, 1}+h_{{\varepsilon}}$, so that $\chi_{{\varepsilon}, 1}=p_{{\varepsilon}, 1}-q_{{\varepsilon}, 1}$.
[**Step 1.**]{} We first claim that if $I_a:=\left( -a,a\right) $ is an interval where $$|p_{{\varepsilon}, 1}\left( x\right)|<2\left\vert \log\varepsilon
\right\vert ,\quad | p_{{\varepsilon}, 1}''\left( x\right) | <C_\varsigma\varepsilon^{2-\varsigma},\quad x\in I_a
\label{q}$$ (such an interval clearly exists, which can be seen by combining (\[517\]) and Lemma \[est at 0\]), then $p_{{\varepsilon},1}$ satisfies a non homogeneous Toda equation in $I_a,$ i.e., $$\label{t23}
\bar c_0p_{{\varepsilon}, 1}^{\prime\prime}\left( x\right) =-2e^{\,2\sqrt
{2}p_{{\varepsilon}, 1}\left( x\right) }+\lambda_{{\varepsilon}, 1}\left( x\right) ,\quad x\in I_a,$$ where $$\label{lambda 1}
\left\Vert \lambda_{{\varepsilon}, 1}\right\Vert _{C^{0,\mu}\left( I_a\right) }\leq
C\varepsilon^{2+\beta_{1}},$$ for some constant $\beta_{1}>0.$ Note that the solution $q_{{\varepsilon}, 1}$ of the Toda system does satisfy $\left(
\ref{q}\right) $ in the interval $\left( \frac{-|\log\varepsilon|}{\varepsilon
},\frac{|\log\varepsilon|}{\varepsilon}\right)$, and in fact we will show that $a=\mathcal{O}(\frac{|\log{\varepsilon}|}{{\varepsilon}})$. Thus we loss no generality assuming a priori $a<3\frac{|\log{{\varepsilon}}|}{{\varepsilon}}$.
To begin the proof of the claim we observe that for ${\tt x}=(x,y)$ such that $\pi_{{\varepsilon}, 1}^x({\tt x})=x_1\in I_a,$ we have$$\begin{aligned}
\label{alpha 23}
\begin{aligned}
\left\vert x_{1}-x_{2}\right\vert & \leq C\varepsilon^{\alpha},\\
\left\vert y_{1}-y_{2}+2f_{{\varepsilon}, 1}\left( x_{1}\right) \right\vert & \leq
C\varepsilon^{\alpha},
\end{aligned}\end{aligned}$$ with some $\alpha>0$, where $(x_i, y_i)$ denote the Fermi coordinate of ${\tt x}$ around $\Gamma_{{\varepsilon}, i}$ for $x\in I_a$ (whenever these coordinates are defined). Using this, $\left( \ref{ep}\right)$, and (\[soft est\]) we can calculate $\int_{\mathbb{R}}\mathbf{x}_{1}^{\ast}\left( E\left( \bar{u}\right) \rho_{1}H_{1}^{\prime
}\right) dy_{1}$ as Lemma $\ref{fi}$ to get: $$\label{t24}
\begin{aligned}
\bar c_0\left( 1+{\mathcal O}_{{\mathcal C}^{0,\mu}(I_a)}\left( \varepsilon^{\alpha}\right) \right) f_{{\varepsilon}, 1}^{\prime\prime
}\left( x_1\right) +\left( 1+{\mathcal O}_{{\mathcal C}^{0,\mu}(I_a)}\left( \varepsilon^{\alpha}\right) \right)
h_{\varepsilon}^{\prime\prime}\left( x_1\right) &=-2e^{2\sqrt
{2}\left( f_{{\varepsilon}, 1}+h_{\varepsilon}\right) \left( x_1\right) }\left( 1+{\mathcal O}_{{\mathcal C}^{0,\mu}(I_a)}\left( \varepsilon
^{\alpha}\right) \right) \\
&\quad +{\mathcal O}_{{\mathcal C}^{0,\mu}(I_a)}\left( \varepsilon^{2+\alpha}\right),
\end{aligned}$$ with some constant $\alpha>0$. (For details we refer the reader to [@MR2557944], where similar calculations can be found). This relation then leads to the claim (\[t23\]).
[**Step 2.**]{} Let us set $\hat \chi_{{\varepsilon}, 1}\left( x\right) =p_{{\varepsilon}, 1}\left( x\right) -q_{{\varepsilon}, 1}\left( x\right)$. Note that, possibly taking the interval $I_a$ smaller we may assume that $\hat\chi_{{\varepsilon},1}$ and $\hat\chi_{{\varepsilon}, 1}''$ are small in this interval. This follows by Lemma \[est at 0\], and (\[q\]), which holds for $q_{{\varepsilon}, 1}''$ as well. Now we will show the following local version of (\[first estimate\]): $$\label{first estimate A}
\begin{aligned}
\left\Vert\hat \chi_{{\varepsilon},1}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}\left(I_a\right)} & \leq C\varepsilon^{\alpha},\\
\left\Vert \hat\chi_{{\varepsilon},1}^{\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}\left(I_a\right)} & \leq C\varepsilon^{1+\alpha
},\\
\left\Vert \hat\chi_{{\varepsilon},1}^{\prime\prime}\right\Vert _{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}\left(I_a\right)} & \leq C\varepsilon
^{2+\alpha}.
\end{aligned}$$
As long as $\hat\chi_{{\varepsilon}, 1}\left( x\right)$ is small we get, $$\begin{aligned}
\bar c_0\hat\chi^{\prime\prime}_{{\varepsilon}, 1} & =-4\sqrt{2}e^{2\sqrt{2}q_{{\varepsilon}, 1}}\hat\chi_{{\varepsilon}, 1}+O\left( \chi_{{\varepsilon}, 1}^{2}\right) e^{-2\sqrt{2}q_{{\varepsilon}, 1}}+\lambda_{{\varepsilon}, 1}\left( x\right) \\
& =-4\sqrt{2}e^{2\sqrt{2}q_{{\varepsilon}, 1}}\hat\chi_{{\varepsilon}, 1}+\lambda_{{\varepsilon}, 2}\left(
x\right),\end{aligned}$$ where $$\label{lambda 2}
\lambda_{{\varepsilon}, 2}=\lambda_{{\varepsilon}, 1}+{\mathcal O}(\chi_{{\varepsilon}, 1}^2)e^{-2\sqrt{2}q_{{\varepsilon}, 1}}.$$ Let $\{\varsigma_{{\varepsilon}, i}$, $i=1,2\}$, be a fundamental set of the linearized Toda equation: $$\bar c_0\varsigma_{\varepsilon}^{\prime\prime}\left( x\right) =-4\sqrt{2}e^{\,2\sqrt{2}q_{{\varepsilon},1}\left( x\right) }\varsigma_{{\varepsilon}}\left( x\right)
,$$ with $\varsigma_{{\varepsilon}, 1}$ odd, $\varsigma_{{\varepsilon}, 2}$ even, $\varsigma_{{\varepsilon}, 1}\left(
0\right) =1$ and $\varsigma_{{\varepsilon}2}^{\prime}\left( 0\right) =\varepsilon$ and $\left\vert \varsigma_{{\varepsilon}, i}^{\prime}\right\vert \leq C\varepsilon.$ Note that although $\varsigma_{{\varepsilon}, 1}$ and $\varsigma_{{\varepsilon},2}$ can be explicitly expressed in terms of $q_{{\varepsilon}, 1}$ and its derivatives their exact formulas are not needed here. What we should keep in mind is that $\varsigma_{{\varepsilon}, 1}$ is bounded and that $\varsigma_{{\varepsilon}, 2}(x)\sim {\varepsilon}|x|$. The variation of parameters formula yields$$\begin{aligned}
\hat\chi_{{\varepsilon}, 1}\left( x\right) & =\frac{\varsigma_{{\varepsilon}, 2}\left( x\right) }{\varepsilon
}\int_{0}^{x}\varsigma_{{\varepsilon}, 1}\left( s\right) \lambda_{{\varepsilon}, 2}\left( s\right)
ds-\frac{\varsigma_{{\varepsilon}, 1}\left( x\right) }{\varepsilon}\int_{0}^{x}\varsigma_{{\varepsilon}, 2}\left( s\right) \lambda_{{\varepsilon}, 2}\left( s\right) ds\\
& +\left( p_{{\varepsilon}, 1}\left( 0\right) -q_{{\varepsilon}, 1}\left( 0\right) \right) \varsigma
_{{\varepsilon}, 1}\left( x\right)\end{aligned}$$ and $$\begin{aligned}
\hat\chi_{{\varepsilon}, 1}^{\prime}\left( x\right) & =\frac{\varsigma_{{\varepsilon}, 2}^{\prime}\left(
x\right) }{\varepsilon}\int_{0}^{x}\varsigma_{{\varepsilon}, 1}\left( s\right) \lambda
_{{\varepsilon}, 2}\left( s\right) ds-\frac{\varsigma_{{\varepsilon}, 1}^{\prime}\left( x\right)
}{\varepsilon}\int_{0}^{x}\varsigma_{{\varepsilon}, 2}\left( s\right) \lambda_{{\varepsilon}, 2}\left(
s\right) ds\\
& +\left( p_{{\varepsilon}, 1}\left( 0\right) -q_{{\varepsilon}, 1}\left( 0\right) \right) \varsigma
_{{\varepsilon}, 1}^{\prime}\left( x\right) .\end{aligned}$$ Then by Lemma \[est at 0\], estimate (\[lambda 1\]), and the estimate for $\lambda_{{\varepsilon}, 2}$ in (\[lambda 2\]), one has in the interval $I_a$: $$\label{r1}
\begin{aligned}
\left\vert\hat \chi_{{\varepsilon}, 1}\left( x\right) \right\vert & \leq\frac{1}{\varepsilon}\left\vert \varsigma_{{\varepsilon}, 2}\left( x\right) \right\vert \int_{0}^{
x}\left\vert \varsigma_{{\varepsilon}, 1}\left({s}\right) \lambda
_{{\varepsilon}, 2}\left({s}\right) \right\vert ds
+\frac{1}{\varepsilon}\left\vert \varsigma_{{\varepsilon},1}\left( x\right)
\right\vert \int_{0}^{x}\left\vert \varsigma_{{\varepsilon}, 2}\left(
{s}\right) \lambda_{{\varepsilon}, 2}\left({s}\right)
\right\vert ds\\
&\quad + |p_{{\varepsilon}, 1}(0) -q_{{\varepsilon}, 1}(0)||\varsigma_{{\varepsilon}, 1}(x)|\\
& \leq C|a|^2(\|\lambda_{{\varepsilon},1}\|_{{\mathcal C}^{0,\mu}(I_a)}+{\varepsilon}^2{\mathcal O}(\|\chi_{{\varepsilon},1}\|^2_{{\mathcal C}^{0,\mu}(I_a)})+C\varepsilon^{\alpha}
\\
& < C|a|^2{\varepsilon}^{2+\beta_1}+C\varepsilon^{\alpha_{1}}+C|a|^2{\varepsilon}^2{\mathcal O}(\|\chi_{{\varepsilon},1}\|^2_{{\mathcal C}^{0,\mu}(I_a)}).
\end{aligned}$$ From this the ${\mathcal C}^0(I_a)$ estimate for $\hat\chi_{{\varepsilon},1}$ follows immediately. It is also evident that we can take $a=\frac{|\log{\varepsilon}|}{{\varepsilon}}$. Additionally, in this same interval, due to the fact that $|\varsigma'_{{\varepsilon}, i}|\leq C{\varepsilon}$, $i=1,2$, we have, $$|\hat\chi_{{\varepsilon}, 1}^{\prime}\left( x\right)| \leq
C|a|^2{\varepsilon}^{3+\beta_1}+C\varepsilon^{1+\alpha_{1}}+C|a|^2{\varepsilon}^3{\mathcal O}(\|\chi_{{\varepsilon},1}\|^2_{{\mathcal C}^{0,\mu}(I_a)})
\label{r2}$$ from which we obtain the ${\mathcal C}^1(I_a)$ estimate for $\hat\chi_{{\varepsilon}, 1}$, with $a=|\log{\varepsilon}|/{\varepsilon}$. The argument leading finally to the full statement (\[first estimate A\]) is clear. [**Step 3.**]{} Next, we will prove that $\mathrm{dist}(\Gamma_{{\varepsilon}, 1}, \widetilde\Gamma_{{\varepsilon}, 1})\to 0$ as ${\varepsilon}\to 0$. For this we should consider the function $p_{{\varepsilon}, 1}$ outside the interval $I_{|\log{\varepsilon}|/{\varepsilon}}$
Since we have proven (\[lambda 1\]) already, it suffices to show that $$\mathrm{dist}\, \left( \Gamma_{{\varepsilon}, 1}\cap \{|x|>\frac{\left\vert \log\varepsilon\right\vert }{\varepsilon}\},\widetilde\Gamma_{1, {\varepsilon}}\cap \{|x|>\frac{\left\vert \log\varepsilon\right\vert }{\varepsilon}\}\right) \rightarrow 0.$$
Let the asymptotic line of $u$ in the fourth quadrant to be $y=-\varepsilon
x-\mathcal{A}_{\varepsilon}.$ Suppose $\left[ a_{{\varepsilon}},+\infty\right) $ is a maximal subinterval of $[\frac{\left\vert \log\varepsilon\right\vert }{\varepsilon},+\infty)$ where $\left\vert f_{{\varepsilon}, 1}\left( x\right) +\left( \varepsilon x+\mathcal{A}_{\varepsilon}\right)
\right\vert \leq1.$ This interval is not empty and possibly $a_{\varepsilon}\geq |\log\varepsilon|/{\varepsilon}$. We wish to show that in fact $a_{\varepsilon}= |\log\varepsilon|/{\varepsilon}$. To argue by contradiction let us assume that there exists a $\delta>0$, independent on ${\varepsilon}$ and such that $$\sup_{x>a_{{\varepsilon}}}|f\left( x\right) +\left( \varepsilon
x+\mathcal{A}_{\varepsilon}\right) |>\delta.$$ Next, we let $x_{\varepsilon}\in \left[ a_{{\varepsilon}},+\infty\right) $ be such that $$\left\vert f\left( x_{\varepsilon}\right) +\left( \varepsilon
x_{\varepsilon}+\mathcal{A}_{\varepsilon}\right) \right\vert=\sup_{x>a_{{\varepsilon}}}|f\left( x\right) +\left( \varepsilon
x+\mathcal{A}_{\varepsilon}\right)|$$ since $$| f_{{\varepsilon}, 1}\left( x\right) +\left( \varepsilon x+\mathcal{A}_{\varepsilon}\right)| \rightarrow0,\text{ as } x
\rightarrow \infty.$$ therefore $x_{\varepsilon}<\infty$ is well defined.
Consider the domain $$\Omega_{L}:=\left\{ \left( x,y\right)\mid y<0,\quad x>x_{{\varepsilon}},\quad y>\frac{x}{\varepsilon
}-L\right\} .$$ Here $L>{\varepsilon}x_{\varepsilon}$ is large and indeed we will finally let it go to $+\infty.$ We use the balancing formula in this domain and with the vector field $X:=\left(
f_{{\varepsilon}, 1}\left( x_{{\varepsilon}}\right) -y,x-x_{{\varepsilon}}\right) $. This formula tells us that $$\int_{\partial\Omega_{L}}\left\{ \left( \frac{1}{2}\left\vert \nabla
u\right\vert ^{2}+F\left( u\right) \right) X-X\left( u\right) \nabla
u\right\} \cdot \nu dS=0.$$ Let us estimate the relevant boundary integrals. First, $$\begin{aligned}
& \int_{\partial\Omega_{L}\cap\left\{ y=0\right\} }\left\{ \left(
\frac{1}{2}\left\vert \nabla u\right\vert ^{2}+F\left( u\right) \right)
X-X\left( u\right) \nabla u\right\} \cdot \nu dS\\
& =\int_{x_{{\varepsilon}}}^{L/{\varepsilon}}\left( \frac{1}{2}u_{x}^{2}+F\left( u\right)
\right) \left( x-x_{{\varepsilon}}\right) dx\to \int_{x_{{\varepsilon}}}^{\infty}\left( \frac{1}{2}u_{x}^{2}+F\left( u\right)
\right) \left( x-x_{{\varepsilon}}\right) dx, \quad \mbox{as}\ L\to \infty.\end{aligned}$$ To estimate this integral let us recall that, by symmetry, we have for ${\tt x}=(x,y)$ $y\leq 0$, with some $\kappa>0$ $$|u({\tt x})^2-1| +|\nabla u({\tt x})|\leq C e^{\,-\kappa \mathrm{dist}\,(\Gamma_{{\varepsilon}, 1}, {\tt x})}.$$ Since for $x\geq a_{\varepsilon}$ the distance between $\Gamma_{{\varepsilon}, 1}$ and the line $\ell_{\varepsilon}=\{y=-({\varepsilon}x+{\mathcal A}_{\varepsilon})\}$ is bounded therefore we have as well $$|u({\tt x})^2-1| +|\nabla u({\tt x})|\leq C e^{\,-\kappa \mathrm{dist}\,(\ell_{{\varepsilon}}, {\tt x})}, \quad {\tt x}=(x,y), \quad x\geq a_{\varepsilon}.$$ Now using this, and the fact that $$|\varepsilon x_{{\varepsilon}}+\mathcal{A}_{\varepsilon}|\geq |f_{{\varepsilon}, 1}(0)|-1\geq C|\log{\varepsilon}|,$$ we deduce that as $\varepsilon\rightarrow 0,$$$\int_{\partial\Omega_{L}\cap\left\{ y=0\right\} }\left\{ \left( \frac
{1}{2}\left\vert \nabla u\right\vert ^{2}+F\left( u\right) \right)
X-X\left( u\right) \nabla u\right\} \cdot \nu dS\rightarrow 0.$$ On the other hand, using the asymptotic behavior of $u$ in the lower half plane, we get: $$u=\bar u_{\varepsilon}+o(1)e^{\,-\kappa \mathrm{dist}\,(\Gamma_{{\varepsilon}, 1}, {\tt x})},\quad ({\mathbf x}_{{\varepsilon}, 1}^*\bar u_{\varepsilon})(x_1,y_1)=H(y_1-h_{\varepsilon}(x_1))\eta_1(x_1, y_1)+o(1) e^{\,-\sqrt{2} |y_1|},$$ where $(x_1, y_1)$ are the Fermi coordinates of the point ${\tt x}$. Since on the line $\{x=x_{\varepsilon}\}$ we have $X=(f_{{\varepsilon}, 1}(x_{\varepsilon})-y, 0)$ therefore:
$$\int_{\partial\Omega_{L}\cap\left\{ x=x_{{\varepsilon}}\right\} }\left\{ \left(
\frac{1}{2}\left\vert \nabla u\right\vert ^{2}+F\left( u\right) \right)
X-X\left( u\right) \nabla u\right\} \cdot \nu dS=o\left( 1\right).$$ Finally, we compute $$\begin{aligned}
\left\vert \int_{\partial\Omega_{L}\cap\left\{ y=\frac{x}{\varepsilon
}-L\right\} }\left\{ \left( \frac{1}{2}\left\vert \nabla u\right\vert
^{2}+F\left( u\right) \right) X-X\left( u\right) \nabla u\right\} \cdot
\nu dS\right\vert
=\frac{\left\vert f_{{\varepsilon}, 1}\left( x_{{\varepsilon}}\right) +\varepsilon x_{{\varepsilon}}+\mathcal{A}_{\varepsilon}\right\vert }{\sqrt{1+\varepsilon^{2}}}+o\left( 1\right) .\end{aligned}$$ Collecting all these estimates, we conclude $$\left\vert f_{{\varepsilon}, 1}\left( x_{{\varepsilon}}\right) +\varepsilon x_{{\varepsilon}}+\mathcal{A}_{\varepsilon}\right\vert
=o\left( 1\right) .$$ But then we must have $a_{\varepsilon}=\frac{|\log {\varepsilon}|}{{\varepsilon}}$, and consequently, in the interval $[\frac{\left\vert \log\varepsilon\right\vert
}{\varepsilon},+\infty),$$$\left\vert f_{{\varepsilon}, 1}\left( x\right) +\varepsilon x+\mathcal{A}_{\varepsilon}\right\vert =o\left(
1\right) .$$ This implies that outside this interval, $\Gamma_{{\varepsilon}, 1}$ is close to a straight line, which combined with the estimates (\[first estimate\]) yields the desired result. Indeed, at $x^*=\frac{|\log{\varepsilon}|}{{\varepsilon}}$ we have $q_{{\varepsilon}, 1}(x^*)=f_{{\varepsilon}, 1}(x^*)+o(1)$, $q'_{{\varepsilon}, 1}(x^*)=f'_{{\varepsilon}, 1}(x^*)+o(1)$ and $q_{{\varepsilon}, 1}(x)=-{\varepsilon}|x|-\mathcal{\tilde A}_{\varepsilon}+o(1)$, $x>x^*$. But then $\mathcal{A}_{\varepsilon}=\mathcal{\tilde A}_{\varepsilon}+o(1)$. This ends the proof of Step 3.
[**Step 4.**]{} At this point we can use what we have just proven in Step 2 and Step 3 to get $$f_{{\varepsilon}, 1}(x)=\frac{\sqrt{2}}{2}\log\frac{1}{{\varepsilon}}+{\varepsilon}|x|+\mathcal{O}_{\mathcal{C}^0({\mathbb R})}(1), \quad |x|\gg 1.$$ As a consequence $$\label{ppp 1}
\|\exp\{-2\sqrt{2}|f_{{\varepsilon},1}|(1+\varDelta_{\varepsilon})\}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}\leq C{\varepsilon}^2,$$ from which we find: $$\|\phi\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R}^2)}\leq C{\varepsilon}^2, \quad\|f''_{{\varepsilon}, 1}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}\leq C{\varepsilon}^2, \quad \|h_{\varepsilon}\|_{\mathcal{C}_{{\varepsilon}\tau}^{0,\mu}({\mathbb R})}\leq C{\varepsilon}^2.$$ Using this information to calculate $\int_{\mathbb{R}}\mathbf{x}_{1}^{\ast}\left( E\left( \bar{u}\right) \rho_{1}H_{1}^{\prime
}\right) dy_{1}$ as Lemma $\ref{fi}$ we obtain: $$\label{t24}
\begin{aligned}
\bar c_0\left( 1+{\mathcal O}_{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\left( \varepsilon^{\alpha}\right) \right) f_{{\varepsilon}, 1}^{\prime\prime
}\left( x_1\right) +\left( 1+{\mathcal O}_{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\left( \varepsilon^{\alpha}\right) \right)
h_{\varepsilon}^{\prime\prime}\left( x_1\right) & =-2e^{2\sqrt
{2}\left( f_{{\varepsilon}, 1}+h_{\varepsilon}\right) \left( x_1\right) }\left( 1+{\mathcal O}_{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\left( \varepsilon
^{\alpha}\right) \right) \\
&\quad +{\mathcal O}_{{\mathcal C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})}\left( \varepsilon^{2+\alpha}\right),
\end{aligned}$$ with some constant $\alpha>0$. At this point we repeat the argument in Step 2 to obtain (\[first estimate\]). This ends the proof of Proposition \[prop gamma\]. Now, the assertion of Proposition \[estim hat phi\] is contained in (\[ppp 1\]).
Uniqueness of solutions with almost parallel nodal lines {#sec teo uniqueness}
========================================================
Parametrization of the family of solutions of (\[AC\]) by the trajectories of the Toda system {#toda aprox}
---------------------------------------------------------------------------------------------
Let us consider the curve $\{y=q_{{\varepsilon},i}(x)\}=\widetilde \Gamma_{{\varepsilon}, i}$. When $i=1$ it is contained in the lower half plane, when $i=2$ is is contained in the upper half plane and we have actually $q_{{\varepsilon}, 1}(x)=-q_{{\varepsilon}, 2}(x)$. With this curve we will associate the Fermi coordinates $({\widetilde}{x}_{i}, {\widetilde}{y}_{i})$: $${\tt x}=({\widetilde}x_i, q_{{\varepsilon}, i}({\widetilde}{x_i}))+{\widetilde}y_{i}{\widetilde}n_{{\varepsilon}, i}({\widetilde}x_i), \quad {\widetilde}n_{{\varepsilon}, i}(x)=\frac{(-1)^i(q_{{\varepsilon}, i}'(x), -1)}{\sqrt{1+q_{{\varepsilon}, i}'(x)^2}}.$$ The change of variables $(\widetilde x_i, \widetilde y_i)\mapsto {\tt x }=(x,y)$ is a diffeomorphism in a neighborhood $\widetilde {\mathcal O}_{i}$ of $\widetilde\Gamma_{{\varepsilon}, i}$. We denote this diffeomorphism by ${\widetilde}{\tt x}_{{\varepsilon}, i}$, so that $${\widetilde}{\tt x}_{{\varepsilon}, i}({\widetilde}x_i, {\widetilde}y_i)={\tt x}\in {\widetilde}{\mathcal O}_{i}.$$ For any function $w\colon {\widetilde}{\mathcal O}_{i}\to {\mathbb R}$ by ${\widetilde}{\tt x}^*_{{\varepsilon},i} w$ we denote its pullback by ${\widetilde}{\tt x}_{{\varepsilon}, i}$: $$({\widetilde}{\tt x}^*_{{\varepsilon},i} w)({\widetilde}x_i, {\widetilde}y_i)=(w\circ {\widetilde}{\tt x}_{{\varepsilon}, i}) ({\widetilde}x_i, {\widetilde}y_i).$$
To describe more precisely the neighborhood ${\widetilde}{\mathcal O}_{i}$ we define the projection function ${\widetilde}\pi_{{\varepsilon}, i}\colon {\mathbb R}^2\to {\widetilde}\Gamma_{{\varepsilon}, i}$: $${\widetilde}\pi_{{\varepsilon},1}({\tt x})=({\widetilde}\pi_{{\varepsilon}, 1}^x({\tt x}),{\widetilde}\pi_{{\varepsilon}, 1}^y({\tt x)}):=({\widetilde}x_i, q_{{\varepsilon}, i}({\widetilde}x_i)), \quad \mbox{whenever}\ {\tt x}=({\widetilde}x_i, q_{{\varepsilon}, i}({\widetilde}x_1))+ {\widetilde}y_i n_{{\varepsilon}, i}({\widetilde}x_i), \quad {\widetilde}y_i=\mathrm{dist}\,({\tt x}, {\widetilde}\Gamma_{{\varepsilon}, i}).$$
Using basic properties (linear growth, scaling) of the trajectories of the solutions of the Toda system it is not hard to show that there exits a positive constant $c$ such that we can define $$\widetilde{\mathcal O}_{i}:=\{{\tt x}\mid \|{\widetilde}\pi_{{\varepsilon}, i}({\tt x})-{\tt x}\| \leq c\log {\varepsilon}^{-1}+ c{\varepsilon}\log (\cosh {\widetilde}\pi_{{\varepsilon}, i}^x({\tt x}))\},$$ c.f. [@MR2557944]. In particular, we can chose $c$ large, so that $\widetilde\Gamma_{{\varepsilon}, i}\in \widetilde {\mathcal O}_{j}$, $i, j=1,2$. For future reference we set $${\widetilde}{\mathrm d}_{\varepsilon}(x)=c\log {\varepsilon}^{-1}+ c{\varepsilon}\log (\cosh {x}).$$
With these preparations, we would like to write locally any solution $u$, with $\tan\theta(u)={\varepsilon}$ small, in Fermi coordinates with respect to $\widetilde\Gamma_{{\varepsilon}, i}$. To this end we will construct a suitable approximation of $u$ in ${\widetilde}{\mathcal O}_{i}$ based on the fact that the true solution is locally close to the heteroclinic. By symmetry we may focus on the case $i=1$, namely consider the lower half plane. We chose a solution $u$, whose nodal line $\Gamma_{{\varepsilon}, 1}$ in the lower half plane is a graph of $y=f_{{\varepsilon}, 1}(x)$ and chose the solution of the Toda system $q_{{\varepsilon}, 1}(x)$ such that the assertions of the Proposition \[prop gamma\] are satisfied. We let ${\widetilde}\eta$ to be a smooth cut off function equal to $1$ in ${\widetilde}{\mathcal O}_{1}\cap\{\mathrm{dist}({\tt x}, \partial{\widetilde}{\mathcal O}_1)>1\}$ and equal to $0$ in ${\mathbb R}^2\setminus {\widetilde}{\mathcal O}_{1}$. A reasonable ansatz for an approximate solution is built by defining the function ${\widetilde}H_{{\varepsilon}, 1}$: $$({\widetilde}{\tt x}_{{\varepsilon}, 1}^*{\widetilde}{H}_{{\varepsilon}, 1})\left( {\widetilde}x_1,{\widetilde}y_1\right) :=({\widetilde}{\tt x}_{{\varepsilon}, 1}^*{\widetilde}{\eta})\left({\widetilde}x_1, {\widetilde}y_1\right)
H\left( {\widetilde}y_{1}-{\widetilde}g_{\varepsilon}\left( {\widetilde}x_{1}\right) \right) +\left( 1-({\widetilde}{\tt x}_{{\varepsilon}, 1}^*{{\widetilde}\eta})\left({\widetilde}x_1, {\widetilde}y_1\right) \right) \frac{H\left( {\widetilde}y_{1}-{\widetilde}g_{\varepsilon}\left(
{\widetilde}x_{1}\right) \right) }{\left\vert H\left( {\widetilde}y_{1}-{\widetilde}{g}_{\varepsilon}\left(
{\widetilde}x_{1}\right) \right) \right\vert },$$ which is extended to the whole ${\mathbb R}^2$ by $\pm 1$, setting ${\widetilde}{H}_{{\varepsilon}, 2}\left( x,y\right) =-{\widetilde}{H}_{{\varepsilon}, 1}\left( x,-y\right)$, and finally defining $${\widetilde}{{u}}_{\varepsilon}:={\widetilde}{H}_{{\varepsilon}, 1}-{\widetilde}{H}_{{\varepsilon}, 2}-1.
\label{wtueps}$$ Note that the function ${\widetilde}g_{\varepsilon}$ has not been specified so far. It turns out that in order to have a good approximation of $u$ by ${\widetilde}u$ we should impose the following orthogonality condition: $$\int_{\mathbb{R}}\left [{\widetilde}{\tt x}_{{\varepsilon}, i}^{\ast}( u-{\widetilde}{u}_{\varepsilon})
{\widetilde}\rho_{{\varepsilon}, i}{\widetilde}{H}_{{\varepsilon}, i}^{\prime}\right]({\widetilde}x_i, {\widetilde}y_i)\,d{\widetilde}y_{i}=0,\quad \forall\, {\widetilde}x_i, \quad i=1,2.
\label{wt orto cond}$$ and smooth cutoff functions ${\widetilde}\rho_{{\varepsilon}, i}$ are defined through a smooth cutoff function ${\widetilde}\rho$ by: $$({\widetilde}{\tt x}_{{\varepsilon}, i}^* {\widetilde}\rho_{{\varepsilon}, i})({\widetilde}x_i, {\widetilde}y_i)={\widetilde}\rho({\widetilde}x_i, {\widetilde}y_i-(-1)^{i+1}{\widetilde}{g}_{\varepsilon}({\widetilde}x_i)),$$ where $${\widetilde}\rho(s,t)=\begin{cases}
1, \quad |t|\leq \frac{1}{2}{\widetilde}{\mathrm{d}}_{\varepsilon}(s),\\
0<\rho<1, \quad \frac{1}{2} {\widetilde}{\mathrm{d}}_{\varepsilon}(s)<t<\frac{3}{4}{\widetilde}{\mathrm{d}}_{\varepsilon}(s),\\
0 \quad\mbox{othewise}.
\end{cases}$$ c.f. definition of $\hat\rho_{{\varepsilon}, i}$ above, while ${\widetilde}{H}_{{\varepsilon}, i}^{\prime}$ is defined by $$\begin{aligned}
\left({\widetilde}{\tt x}^*_{{\varepsilon}, i}{\widetilde}{H}_{i}^{\prime}\right) \left( {\widetilde}x_{i}, {\widetilde}y_{i}\right) & =H^{\prime}\left(
{\widetilde}y_{i}-\left( -1\right) ^{i}{\widetilde}{g}_{\varepsilon}\left( {\widetilde}x_{i}\right) \right) .\end{aligned}$$ Note that because of the definition of the function ${\widetilde}{\mathrm {d}}_{\varepsilon}$ we can assume that $|\nabla {\widetilde}\rho_{{\varepsilon}, i}({\tt x})|=\mathcal{O}(\frac{1}{{\varepsilon}|{\tt x}|})$, ${\varepsilon}|\tt x|\gg 1$ with similar estimates for higher order derivatives. Changing variables $X_i={\widetilde}x_i$, $Y_i= {\widetilde}y_i-(-1)^i{\widetilde}g_{\varepsilon}({\widetilde}x_i)$ one gets the following, equivalent form of (\[wt orto cond\]): $$\label{l1}
\int_{\mathbb{R}}{\widetilde}\rho\left(X_i, Y_{i}\right) H^{\prime}\left( Y_{i}\right)
\left({\widetilde}{\tt x}_{{\varepsilon}, i}^{\ast}( u-{{\widetilde}{u}}_{\varepsilon}) \right) \left(
X_{i},Y_{i}+{\widetilde}{g}_{\varepsilon}\left( X_{i}\right) \right) dY_{i}=0,\quad \forall X_i, \quad i=1,2.$$ To show the existence of the function ${\widetilde}g_{\varepsilon}$ one can use the argument similar to the one in Lemma \[exists h\]. However, since the set $\{y=q_{{\varepsilon}, i}(x)\}$ does not coincide with the nodal set of the solution the function ${\widetilde}g_{\varepsilon}$ does not decay exponentially. To determine the behavior of the function ${\widetilde}g_{\varepsilon}$ more precisely we need the following:
Conclusion of the proof: the Lipschitz property of solutions {#last step}
------------------------------------------------------------
With the above preparations, we are ready to prove our uniqueness theorem. Based on the results of the previous section we know that any solution with a small asymptotic angle can be written in the following way: $$u(\cdot;{\widetilde}g_{\varepsilon}, {\widetilde}\phi)={\widetilde}u_{\varepsilon}(\cdot;{\widetilde}g_{\varepsilon})+{\widetilde}\phi,$$ where ${\widetilde}u_{\varepsilon}$ is the approximate solution defined in (\[wtueps\]). Here and below we will indicate the dependence of this solution on the modulation function ${\widetilde}g_{\varepsilon}$ as well as on ${\widetilde}\phi$. Now, let us consider two solutions $u^{(j)}$, $j=1,2$ with the same asymptotic angle $\theta(u^{(j)})={\varepsilon}$. Sine the asymptotic angle is the same for both solutions, there is just one solution of the Toda system represented by the functions $q_{{\varepsilon},1}=-q_{{\varepsilon}, 2}$. On the other hand it may happen that ${\widetilde}g^{(1)}_{\varepsilon}\neq {\widetilde}g_{\varepsilon}^{(2)}$, and ${\widetilde}\phi^{(1)}\neq {\widetilde}\phi^{(2)}$. In the language of [@MR2557944] we have that ${\widetilde}g_{\varepsilon}^{(j)}\in \mathcal{C}^{2,\mu}_{{\varepsilon}\tau}({\mathbb R})\oplus D$ (see also section \[exists small eps\]) and in the previous section we have shown that $$\|{\widetilde}g^{(j)}_{\varepsilon}\|_{\mathcal{C}^{0,\mu}_{{\varepsilon}\tau}({\mathbb R})\oplus D}\leq C {\varepsilon}^\alpha,$$ with similar estimates for the higher order derivatives. In addition for the functions ${\widetilde}\phi^{(j)}$ we have (\[wtphi est\]). To prove the uniqueness of solutions with small angles is therefore enough to prove “local uniqueness” in the following sense: given two solutions associated to the same solution of the Toda system we have ${\widetilde}\phi^{(1)}={\widetilde}\phi^{(2)}$, and ${\widetilde}g_{\varepsilon}^{(1)}={\widetilde}g_{\varepsilon}^{(2)}$. Our strategy to prove this fact follows in some sense the strategy used to prove the existence of solutions with small angles employed in [@MR2557944]. Namely, we show the Lipschitz property of the map: ${\widetilde}g_{\varepsilon}\mapsto E({\widetilde}u_{\varepsilon}(\cdot; {\widetilde}g_{\varepsilon}))^\perp$ and then we use the linearized equation to show that ${\widetilde}\phi^{(1)}-{\widetilde}\phi^{(2)}$ can be controlled by a small constant times ${\widetilde}g_{\varepsilon}^{(1)}-{\widetilde}g_{\varepsilon}^{(2)}$. As a final step we show that the function ${\widetilde}g_{\varepsilon}^{(1)}-{\widetilde}g_{\varepsilon}^{(2)}$ satisfies the linearized Toda system with the right hand side again controlled by a small constant times ${\widetilde}g_{\varepsilon}^{(1)}-{\widetilde}g_{\varepsilon}^{(2)}$. This leads us to conclude that ${\widetilde}g_{\varepsilon}^{(1)}-{\widetilde}g_{\varepsilon}^{(2)}=0$, and a s a result we infer the uniqueness.
Now we will present some details of the argument outlined above. Many of the calculations are quite similar to the ones in [@MR2557944].
For future purpose it is convenient to introduce the following projection defined for any function $\psi\colon{\mathbb R}^2\to {\mathbb R}$: $$\psi^{\perp_{(j)}}=\psi-{\widetilde}{\tt x}_{{\varepsilon}, i}\circ \sum_{i=1,2}{\widetilde}c^{(j)}_i{\widetilde}\rho^{(j)}_{{\varepsilon}, i}{\widetilde}{H}_{{\varepsilon}, i}^{{(j)}\prime}\int_{\mathbb{R}}\left [{\widetilde}{\tt x}_{{\varepsilon}, i}^{*}\psi
{\widetilde}\rho_{{\varepsilon}, i}^{(j)}{\widetilde}{H}_{{\varepsilon}, i}^{{(j)}\prime}\right]({\widetilde}x_i, {\widetilde}y_i)\,d{\widetilde}y_{i}, \quad {\widetilde}c^{(j)}_i=\left(\int_{\mathbb R}[{\widetilde}{\tt x}_{i,{\varepsilon}}^{*}({\widetilde}\rho_{{\varepsilon}, i}^{(j)}{\widetilde}{H}_{{\varepsilon}, i}^{{(j)}\prime})]^2({\widetilde}x_i, {\widetilde}y_i)\,d{\widetilde}y_i\right)^{-1},$$ where $j=1,2$.
[10]{}
L. Ambrosio and X. Cabr[é]{}. Entire solutions of semilinear elliptic equations in [$\bold R^3$]{} and a conjecture of [D]{}e [G]{}iorgi. , 13(4):725–739 (electronic), 2000.
E. N. Dancer. Stable and finite [M]{}orse index solutions on [$\bold R^n$]{} or on bounded domains with small diffusion. , 357(3):1225–1243 (electronic), 2005.
M. del Pino, M. Kowalczyk, and F. Pacard. Moduli space theory for the [A]{}llen-[C]{}ahn equation in the plane. , 2010.
M. del Pino, M. Kowalczyk, F. Pacard, and J. Wei. Multiple-end solutions to the [A]{}llen-[C]{}ahn equation in [$\Bbb
R^2$]{}. , 258(2):458–503, 2010.
M. del Pino, M. Kowalczyk, and J. Wei. On [D]{}e [G]{}iorgi’s in dimension ${N}\geq 9$. , 2008.
N. Ghoussoub and C. Gui. On a conjecture of [D]{}e [G]{}iorgi and some related problems. , 311(3):481–491, 1998.
C. Gui. Hamiltonian identities for elliptic partial differential equations. , 254(4):904–933, 2008.
C. [Gui]{}. . , Feb. 2011.
H. Karcher. Embedded minimal surfaces derived from [S]{}cherk’s examples. , 62(1):83–114, 1988.
M. Kowalczyk, Y. Liu, and F. Pacard. he classification of [M]{}orse index [$1$]{} solutions of the [A]{}llen-[C]{}ahn equation in [$\Bbb R^2$]{}. .
M. Kowalczyk, Y. Liu, and F. Pacard. Four ended solutions to the [A]{}llen-[C]{}ahn equation on the plane. , 2011.
W. H. Meeks, III and M. Wolf. Minimal surfaces with the area growth of two planes: the case of infinite symmetry. , 20(2):441–465, 2007.
F. Pacard and J. Wei. Stable solutions of the allen-cahn equation in dimension 8 and minimal cones. , 2011.
J. P[é]{}rez and M. Traizet. The classification of singly periodic minimal surfaces with genus zero and [S]{}cherk-type ends. , 359(3):965–990 (electronic), 2007.
O. Savin. Regularity of flat level sets in phase transitions. , 169(1):41–78, 2009.
M. Schatzman. On the stability of the saddle solution of [A]{}llen-[C]{}ahn’s equation. , 125(6):1241–1275, 1995.
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---
abstract: 'Recently, numerical solutions to the field equations of Einstein-scalar-Gauss-Bonnet gravity that correspond to black-holes with non-trivial scalar hair have been reported. Here, we employ the method of the continued-fraction expansion in terms of a compact coordinate in order to obtain an analytical approximation for the aforementioned solutions. For a wide variety of coupling functionals to the Gauss-Bonnet term we were able to obtain analytical expressions for the metric functions and the scalar field. In addition we estimated the accuracy of these approximations by calculating the black-hole shadows for such black holes. Excellent agreement between the numerical solutions and analytical approximations has been found.'
author:
- 'Roman A. Konoplya'
- Thomas Pappas
- Alexander Zhidenko
bibliography:
- 'References.bib'
title: 'Einstein-scalar-Gauss-Bonnet black holes: Analytical approximation for the metric and applications to calculations of shadows'
---
=1
Introduction {#intro}
============
Nowadays black holes are the most important objects for understanding the regime of strong gravity. Observations in the gravitational [@abbott2016; @abbott2016a; @abbott2016b] and electromagnetic [@Akiyama:2019cqa; @Goddi:2017pfy] spectra support General Relativity, but, at the same time, leave ample room for alternative theories of gravity [@Yunes:2016jcc; @Konoplya:2016pmh]. One of the most interesting alternative approaches is related to adding higher-curvature corrections to the Einstein action. This kind of extension of the Einstein gravity is inspired by the low-energy limit of string theory [@Callan:1988hs; @Kanti1996] and, presumably, could describe quantum corrected black holes. The lowest-order correction is given by the (second order in curvature) Gauss-Bonnet term, which is pure divergence in four-dimensional spacetimes, but, when coupled to a scalar field, it leads to modifications of the Einstein equations.
All the known black-hole solutions in the four-dimensional Einstein-scalar-Gauss-Bonnet gravity are obtained either numerically [@Kanti1996; @Antoniou2018; @Kleihaus:2015aje; @Collodel:2019kkx; @Cunha:2016wzk; @Blazquez-Salcedo:2017txk], or perturbatively [@Ayzenberg:2014aka; @Ayzenberg:2014aka; @Maselli:2015tta], which makes the usage of a number of tools for analysis of behavior of such solutions either difficult or impossible. Analytical expressions for such numerical black-hole metrics, which are valid in the whole space outside the event horizon, would allow us to see the explicit dependence of the metric on physical parameters of the system and to work with the metric as, essentially, an exact solution. The approach to finding analytical approximations of numerical solutions was based on the general parametrization for spacetimes of static spherically symmetric black holes [@Rezzolla2014] and extended in [@Konoplya:2016jvv] to axial symmetry. For spherical symmetry the parametrization uses a continued-fraction expansion in terms of a compactified radial coordinate. This choice leads to superior convergence properties and allows one to approximate a black-hole metric with a much smaller set of coefficients. This approach was used to construct the analytical approximation of numerical black-hole solutions in the Einstein-Weyl [@Kokkotas2017a], Einstein-dilaton-Gauss-Bonnet [@Kokkotas2017] and Einstein-scalar-Maxwell [@Konoplya:2019goy] theories. Further studies of observables in these parametrized spacetimes [@Younsi:2016azx; @Konoplya:2018arm; @Nampalliwar:2018iru; @Konoplya:2019hml; @Konoplya:2019ppy; @Zinhailo:2018ska; @Zinhailo:2019rwd] showed that usually only 2 to 3 orders of the continued-fraction expansion are sufficient in order to achieve reasonable accuracy.
In [@Kokkotas2017] the analytical approximation was found for the particular choice of the scalar field coupling functional – the dilaton, exponential coupling, which was considered numerically in [@Kanti1996]. Recently this approach was extended in [@Antoniou2018] to various types of the scalar-field functional and, therefore, allowed one to look whether there are some common features for all the considered couplings of the scalar field. A similar problem was attacked numerically for the case of the Einstein-scalar-Maxwell theory [@Herdeiro:2018wub] and the study of its analytical approximation [@Konoplya:2019goy] showed that the radius of the black-hole shadow is increased for any of the considered couplings of the scalar field. *Scalarization*, that is, the phenomenon of spontaneous acquiring of a scalar hair by the black hole as a result of the nonminimal coupling of a scalar field to the system, has been actively studied in [@Doneva:2017bvd; @Silva:2017uqg; @Minamitsuji:2018xde; @Cunha:2019dwb; @Fernandes:2019rez].
Here we generalize the procedure for finding the analytical approximation in the Einstein-scalar-Gauss-Bonnet (EsGB) theory to the cases of various coupling functionals of the scalar field. Then, we apply the obtained parametrized black-hole metrics to the calculation of the radii of shadows in order to estimate the relative error due to the truncation of the continued-fraction expansion which we used. We also present the analytical expressions for both the radius of the photon sphere and the black-hole shadow.
The paper is organized as follows. In Sec. \[sec:ESGB\] we present the basics of the EsGB theory. Sec. \[sec:CF\] is devoted to the introduction of the continued-fraction expansion, while in Sec. \[sec:AAESGB\] we apply this procedure to the numerical solution of the EsGB black holes. Finally in Sec. \[sec:shadows\] we find black-hole shadows for the above numerical and parametrized black-hole metrics. In the Conclusions we summarize the obtained results and discuss the open questions.
Black holes in Einstein-scalar-Gauss-Bonnet gravity {#sec:ESGB}
===================================================
The Lagrangian for EsGB gravity reads = +f() R\^2\_[GB]{}- \_\^ , \[action\] where $\kappa^2 \equiv 16 \pi G c^{-4}=1$ is the Einstein’s constant. The Gauss-Bonnet (GB) term is defined as R\^2\_[GB]{} ( R\^2 - 4 R\_R\^+R\_ R\^ ), while $\alp$ is the GB constant and $f(\varphi)$ is an arbitrary smooth function of the scalar field $\vph$ corresponding to GB-coupling functional.
In four dimensions, if $f(\vph)$ is a constant, then the GB term is *topological* in the sense that it does not contribute to the field equations. In the case of an exponential coupling functional $f(\vph) = e^{\vph}$ black-hole solutions with scalar hair emerge for EsGB gravity and the first solutions were obtained numerically in [@Kanti1996]. More recently, the authors of [@Antoniou2018a] have reported that regular black-hole solutions with scalar hair appear as a generic feature of the theory .
Let us start by considering the following line element for a static and spherically symmetric spacetime: ds\^2=-g\_[tt]{}(r) dt\^2+g\_[rr]{}(r) dr\^2+r\^2 ( d \^2 +\^2d \^2 ) . \[metric ansatz\] We also assume that the scalar field shares the symmetries of the underlying spacetime and it thus depends solely on the radial coordinate $r$.
The Einstein equations that are derived from the theory are the following:
\[Tmn\]
&R\_- R g\_=-g\_\_\^+\_\_\
& -(g\_g\_+g\_g\_) \^\^\_ \_\_,
where \[g\] \^\_=\^ R\_= R\_, and the scalar-field equation of motion is \^2 +f’()R\^2\_[GB]{}=0, \[phi-eq\_0\]
where it is understood that throughout this article a prime indicates differentiation with respect to the argument of the function.
Numerical solutions to the field equations of EsGB gravity corresponding to black holes with scalar hair have been recently found in [@Antoniou2018] for a wide range of GB couplings. Here, by employing the method of [@Rezzolla2014] we obtain analytical approximations of these numerical solutions.
The continued-fraction approximation {#sec:CF}
====================================
In this section we outline the method of the continued-fraction approximation (CFA) [@Rezzolla2014] and introduce the notations we use in the rest of the article.
In the original coordinate system of , the radius of the event horizon of the black hole $r_0$ is determined by the vanishing of the norm of the timelike Killing vector associated with the invariance of the metric under time translations. This condition eventually translates to $g_{tt}(r_0)=0$. Then, we may perform a radial coordinate transformation and introduce the compact coordinate x 1-, that ranges from $x=0$ at the location of the horizon up to $x=1$ at spatial infinity.
In the CFA, we consider a new metric ansatz that is suitable for approximating any spherically symmetric metric to high accuracy with only a small number of parameters [@Kokkotas2017a; @Kokkotas2017]. The metric coefficients of are written in terms of the new set of functions $A(x)$ and $B(x)$ defined via the following relations:
& g\_[tt]{}(r)= x A(x),\
& g\_[tt]{}(r)g\_[rr]{}(r)= B(x)\^2,
\[A(x) and B(x) defs\] with
A(x)&1-(1-x)+(a\_0-)(1-x)\^2 +(x)(1-x)\^3\
B(x)&1+b\_0 (1-x) + (x)(1-x)\^2,
\[RZ ansatz\] where the parameter $\eps$ is determined by the value of the asymptotic mass $M$ of the black hole and the location of its event horizon $r_0$ as -(1- ). \[eps definition\] The parameter $\eps$ indicates the amount of the deviation of the EsGB black-hole geometry from the Schwarzschild black hole, for which $r_0 = 2\,M $. The parameters $a_0$ and $b_0$ are defined in terms of $\eps$ and the so-called parametrized post-Newtonian parameters $\beta$ and $\gamma$ as $$\begin{aligned}
a_0 &\equiv& \frac{(\beta -\gamma)(1+\epsilon)^2}{2},\\
b_0 &\equiv& \frac{(\gamma-1)(1+\epsilon)^2}{2}.\end{aligned}$$
The functions ${\tilde A}(x)$ and ${\tilde B}(x)$ have the delicate role of describing the metric near the horizon ($x=0$) and are defined in terms of continued-fraction expansions as follows:
$$\begin{aligned}
\nonumber
{\tilde A}(x)=\frac{a_1}{\displaystyle 1+\frac{\displaystyle
a_2x}{\displaystyle 1+\frac{\displaystyle a_3x}{\displaystyle
1+\frac{\displaystyle a_4x}{\displaystyle
1+\ldots}}}}\,,\\\label{contfrac}
%%%%%%
{\tilde B}(x)=\frac{b_1}{\displaystyle 1+\frac{\displaystyle
b_2x}{\displaystyle 1+\frac{\displaystyle b_3x}{\displaystyle
1+\frac{\displaystyle b_4x}{\displaystyle
1+\ldots}}}}\,.\end{aligned}$$
The values of the parameters $a_i$ and $b_i$ for $i\geqslant 1$ can be obtained numerically upon expanding both sides of Eqs. near the horizon and comparing coefficients of the same order in the expansion.
At this point let us mention that at spatial infinity the metric functions and the scalar field can be approximated as [@Antoniou2018a]
g\_[tt]{}(r)=&1-++\
& +(1/r\^5), \[gtt infinity\]
g\_[rr]{}(r)=&1++ +\
& +\
& +(1/r\^5), \[grr infinity\]
(r)=& \_ +++\
& ++(1/r\^5), \[phi infinity\]
where $\vph_{\infty}$ is the asymptotic value of the scalar field and $D$ is its charge. Notice that the exact form of $f(\vph)$ plays no role in the asymptotic expansions up to the third order. The form of Eqs. and implies that $\beta=\gamma=1$ and thus $a_0=b_0=0$ for any GB-coupling functional.
In the same spirit, an analytical approximation for the scalar field can also be obtained by means of the CFA [@Kokkotas2017]. For this purpose we define a new function of the compact coordinate that is related to the scalar field and its asymptotic value at spatial infinity via the following relation: \[Fx\] F(x) = e\^[(r)-\_]{}, where the left-hand side is expanded as F(x)=1+f\_0(1-x)+[F]{}(x)(1-x)\^2. \[F(x) def\] The coefficient $f_0=D/r_0$ is determined by the value of the charge of the scalar field and (x)=. \[F(x) CFA\] Again, by expanding near the event horizon one can obtain numerically the values of the coefficients $f_i$ for $i \geqslant 1$.
Analytical approximations for EsGB black holes {#sec:AAESGB}
==============================================
By employing the method described in the previous section we have derived analytical approximations for numerical black-hole solutions emerging in EsGB gravity. More precisely, for all the numerical solutions obtained for the different coupling functionals studied in [@Antoniou2018] we give here the approximate analytic metric coefficients.
Near the location of the event horizon we may expand the metric functions and the scalar field as follows:
g\_[tt]{}(r) &=& p\_1 , \[metric horizon expansion\]\
g\_[rr]{}(r) &=& \_[n=1]{}\^ q\_n (r-r\_0)\^n, and (r)=\_[n=0]{}\^ (r-r\_0)\^n , \[phi horizon expansion\] where $\vph_n \equiv \vph^{(n)}(r_0)$ is the $n$-th order derivative of the scalar field evaluated on the event horizon.
The value of the scalar field on the horizon $\vph_0$ is a free parameter, subject to the requirement $\vph_1 \in \mathbb{R}$ in order for a black-hole solution to exist. Upon specifying the form of the coupling functional $f(\vph)$, the first derivative of the scalar field $\vph _1$ on the horizon is uniquely determined for each value of $\vph_0$ through the constraint [@Antoniou2018] \_1 =( -1 ) . \[phi 1\] Once the values of the parameters $p_1,\vph_0,$ and $\vph_1$ have been specified, the rest of the parameters $p_i,q_i$ and $\vph_i$ in the expansions can be determined recursively up to an arbitrary order. This is achieved by substituting Eqs. and into the field equations and then solving the corresponding equations order by order in the expansion.
It is convenient to introduce a new dimensionless parameter $p$ instead of $\vph_0$ to parametrize the family of black-hole solutions for each GB coupling as follows: p . \[dimensionless param\] Notice that in order to have a regular black-hole solution with a well-defined horizon [@Kanti1996; @Antoniou2018], the following constraint must hold via Eq. : p , with the Schwarzschild limit corresponding to $p = 0$, while for $p \rightarrow 1$ the maximal-coupling regime is approached.
After fixing the units of length in such a way that $r_0=1$, $p$ depends on two parameters, $\alp$ and $\vhp_0$. In this paper we consider $\alp=1/4$ and collect numerical data for the family of black-hole solutions by varying $\vph_0$. Further, by comparing the numerical solutions for other values of $\alp$ we see that, for a fixed value of $p$ and varying $\alpha$, the variation of the black-hole geometry is negligibly small; i.e. for practical purposes we need to take into account only the value of $p$.
For each value of $p$ we numerically integrate the field equations to obtain the accurate numerical solutions for the metric functions and the scalar field[^1]. The parameter $p_1$ is then fine-tuned such that for $r \rightarrow \infty$ we have $g_{tt}(r) \rightarrow 1$ and $g_{rr}(r) \rightarrow 1$ and this way recover the asymptotically flat limit.
With these solutions at hand, the next step is to determine the values of the asymptotic parameters of the system. The asymptotic mass $M$ is computed by expanding the solution for $g_{tt}(r)$ at large values of the radial coordinate and isolating the numerical coefficient of the term $\sim 1/r$. Then, according to , $M$ simply corresponds to $-1/2 \times$(value of coefficient). This also determines the value of the parameter $\eps$ via . Similarly, the asymptotic value for $D$ of the scalar-field expansion is determined via the corresponding coefficient of the expansion of the numerical solution for $\vph(r)$.
The numerical values for the parameters $(p_i,q_i,\vph_i)$ are thus determined as described above for each value of $\vph_0$ and $p_1$ and in this way through Eqs. and one finally ends up with numerical values for the set $(a_i,b_i,f_i)$.
The above steps are repeated for different values of $p$ that span the allowed range $[0,1]$ and numerical data are assembled for $(a_i,b_i,f_i)$. Then, one is able to perform a fitting of these data in order to obtain analytical expressions for the CFA parameters as functions of $p$. It is then straightforward to write down approximate analytical expressions for the metric functions and the scalar field to the desired order in the CFA via and .
The even-polynomial coupling functional
---------------------------------------
The first case we study is the even-polynomial coupling functional f()= \^[2 n]{},n \^[+]{}. \[coupling: even polynomial\]
The form of the dimensionless parameter for this family of black-hole solutions ($\alp=1/4$) is p = (24 n\^2) \_0 \^[4 n-2]{}, \[p even pol\] and the allowed values for $\vph_0$ are thus |\_0 | (24 n\^2)\^. \[even allowed phi0s\] In order to be able to perform the analysis we need to further reduce the number of free parameters and so we must also choose a specific value for $n$ in .
As illustrative cases for this family of functionals we study $n=1$ and $n=2$ that correspond to the quadratic and quartic couplings, respectively.
### The quadratic GB-coupling functional
The obtained analytical expressions for the parameters of the CFA ,, and up to second order are given below
a\_1= , \[a1 even\]
a\_2= , \[a2 even\]
= . \[eps even\]
The profile of the $\eps$ parameter with respect to $p$ is depicted in Fig. \[fig: epsilons\_vs\_p\] for all the GB couplings we have studied in this article.
![The asymptotic parameter $\eps$ as a function of the dimensionless parameter p for different GB-coupling functionals. []{data-label="fig: epsilons_vs_p"}](epsilons_vs_p.pdf){width="\linewidth"}
The above parameters alone suffice for the determination of the analytical representation of $g_{tt}(r)$. We point out that a general feature of the approximate expressions for both the metric functions and the scalar field is that the relative error (RE) increases with $p$. For the GB coupling $f(\vph)=\vph^2$ when $p=0.8$ in Fig. \[fig: RE\_gtt\_Eve\_08\] we plot the RE between the fourth-order analytical approximation for the $g_{tt}(r)$ metric function and its accurate numerical solution. The maximum error occurs around the photon sphere radius at $r \approx 1.5 \,r_0$ and is less than $0.24\%$.
![The relative error of the fourth-order analytical approximation for $g_{tt}(r)_{(p)}$ from the accurate numerical solution $g_{tt}(r)$ for $p=0.8$. []{data-label="fig: RE_gtt_Eve_08"}](RE_gtt_Eve_08.pdf){width="\linewidth"}
In turn, the analytical approximation of $g_{rr}(r)$ emerges via and thus requires also the expressions for the parameters $b_i$ that are listed below
b\_1=, \[b1 even\]
b\_2=, \[b2 even\]
Finally for the scalar field the analytically-approximated parameters for the CFA are found to be
\_=, \[phiInf even\]
f\_0 = , \[f0 even\]
f\_1 = , \[f1 even\]
f\_2 = , \[f2 even\]
In Fig. \[fig:fDM Error for grr and phi\] we plot the corresponding REs for both the $g_{rr}(r)$ metric function and the scalar field $\vph(r)$, both at the fourth order in the CFA. The expressions for the second-order analytical approximations for the metric functions and the scalar field can be found in Appendix A for all the GB couplings studied in this article[^2].
{width="0.5\linewidth"}{width="0.5\linewidth"}
### The quartic GB-coupling functional
The analytic approximations for the parameters of the CFA expansion in this case are
a\_1= , \[a1 even phi4\]
a\_2= , \[a2 even phi4\]
= , \[eps even phi4\]
b\_1=, \[b1 even phi4\]
b\_2=, \[b2 even phi4\]
\_=, \[phiInf even phi4\]
f\_0 = , \[f0 even phi4\]
f\_1 = , \[f1 even phi4\]
f\_2 = , \[f2 even phi4\]
The odd-polynomial coupling functional
--------------------------------------
The odd-polynomial coupling functional is f()= \^[2 n+1]{},n . \[coupling: odd polynomial\] For $\alp=1/4$ the dimensionless parameter has the following form: p=6 (2 n+1)\^2 \_0 \^[4 n]{}, and the allowed values of $\vph_0$ are |\_0 | (6 (2 n+1)\^2)\^[-]{}.
For $n=1$, the approximate analytic expressions for the parameters are given below
a\_1=, \[a1 odd\]
a\_2=, \[a2 odd\]
= , \[eps odd\]
b\_1 = , \[b1 odd\]
b\_2 = , \[b2 odd\]
\_ = , \[phiInf odd\]
f\_0 = , \[f0 odd\]
f\_1 = , \[f1 odd\]
f\_2 = , \[f2 odd\]
The inverse-polynomial coupling functional
------------------------------------------
The inverse-polynomial coupling functional is f()= \^[-n]{},n \^[+]{}, \[coupling: inverse polynomial\] and the dimensionless parameter for $\alp=1/4$ turns out to be p =6 n\^2 ( \_0 )\^[-2 (n+1)]{} . The allowed range of values for $\vph_0$ in this case is |\_0| ( 6 n\^2 )\^ . Once again, we fix $n=1$ in order to perform the analysis.
The approximate analytic expressions for the parameters in this case are given below
a\_1 = , \[a1 inv\]
a\_2 = , \[a2 inv\]
= , \[eps inv\]
b\_1 = , \[b1 inv\]
b\_2 = , \[b2 inv\]
\_ = , \[phiInf inv\]
f\_0 = , \[f0 inv\]
f\_1 = , \[f1 inv\]
f\_2 = , \[f2 inv\]
The logarithmic coupling functional
-----------------------------------
Finally we turn to the logarithmic coupling functional f()= , \[coupling: log\] the dimensionless parameter for $\alp=1/4$ is p = , and the allowed values of $\vph_0$ are |\_0| .
The approximate analytic expressions for the parameters in this case are given below
a\_1 = , \[a1 log\]
a\_2 = , \[a2 log\]
= , \[eps log\]
b\_1 = , \[b1 log\]
b\_2 = , \[b2 log\]
\_ = , \[phiInf log\]
f\_0 = , \[f0 log\]
f\_1 = , \[f1 log\]
f\_2 = , \[f2 log\]
At this point, one must mention an important phenomenon, the *eikonal instability*, which takes place when the Gauss-Bonnet term is turned on. Once the Gauss-Bonnet coupling constant is not small enough, the black-hole solution suffers from a dynamical instability: if linearly perturbed, the perturbation grows unboundedly. The linear instability breaks down in the regime of small perturbations, indicating that the black hole cannot exist in this range of parameters.
The instability brought by the Gauss-Bonnet term is of special kind: it develops at high multipole numbers, so that the summation over the multipole numbers cannot be valid anymore. This kind of instability was first observed for the higher-dimensional Einstein-Gauss-Bonnet black holes [@Dotti:2005sq] and later observed for a number of other cases, including black branes [@Takahashi:2011du], asymptotically de Sitter and anti-de Sitter black holes [@Cuyubamba:2016cug; @Konoplya:2017ymp; @Konoplya:2017zwo], black holes and branes in theories with higher than the second order in curvature corrections [@Takahashi:2010gz; @Takahashi:2011qda; @Grozdanov:2016fkt; @Konoplya:2017lhs]. In some cases, the instability occurs not only for the gravitational perturbations, but also for the test scalar field [@Gonzalez:2017gwa].
As the eikonal instability is a very wide phenomenon which, it seems, does not depend on a particular form of the higher-curvature correction, we believe that it must be present also for the Einstein-scalar-Gauss-Bonnet theory at least once the scalar coupling is strong enough. Therefore, the regime of near extremal $p$, corresponding to the maximal coupling, most probably does not represent any realistic stable black hole. Exactly in this regime our continued-fraction expansion converges slowly. On the contrary, in the regime where one can expect stable configuration the second-order expansion is sufficient to constrain the relative error by a fraction of one percent. In other words, our analytical approximation is very accurate already at the second order, once one is limited by stable configurations.
Black-hole shadows and accuracy of the analytical approximation {#sec:shadows}
===============================================================
In the previous sections we have obtained approximate analytical expressions for the metric functions and the scalar field up to fourth order in the CFA. In all cases, we have found excellent agreement between the numerical and analytical solutions by computing the RE. Still, the metric itself is not gauge invariant and comparison of various metric functions does not allow us to determine the accuracy of the analytical approximation. For the latter one needs to consider some gauge-invariant, observable quantity.
Recently black-hole shadows have been intensively studied for various theories of gravity and astrophysical environment (an extensive, but not exhaustive list of works can be found in [@Contreras:2019cmf; @Tsukamoto:2017fxq; @Mizuno:2018lxz; @Zhu:2019ura; @Dokuchaev:2018kzk; @Xu:2018mkl; @Ovgun:2018tua; @Huang:2018rfn; @Hou:2018avu; @Wei:2018xks; @Davelaar:2018dfp; @Dokuchaev:2018ibr; @Mishra:2019trb; @Abdikamalov:2019ztb; @Held:2019xde; @Shaikh:2019fpu; @Wang:2019tjc; @Ali:2019khp; @Long:2019nox; @Ovgun:2019jdo; @Contreras:2019cmf; @Wang:2017hjl] and references therein). In this section we perform the computation of the shadows cast by the EsGB black holes numerically. For different orders in the continued-fraction approximation we compute the shadows and compare them with the numerical ones. This way we have a gauge-invariant measure of the accuracy of our approximation.
{width="0.5\linewidth"}{width="0.5\linewidth"}
The radius of the photon sphere $r_{ph}$ of a black hole in the coordinate system of is determined by means of the following function (see, for example, [@Perlick2015; @Konoplya2019] and references therein): h\^2(r) , \[h2 definition\] as the solution to the equation h\^2 (r)=0. Then, the radius of the black-hole shadow $R_{sh}$ as seen by a distant static observer located at $r_O$ is R\_[sh]{} = = , \[shadow def\] where in the last equation we have assumed that the observer is located sufficiently far away from the black hole so that she/he is deep in the asymptotically flat regime, i.e. $g_{tt}(r_O) \approx 1$.
In the case of the Schwarzschild black hole it is known that $r_{ph}= 1.5 \, r_0$ and so according to the shadow is $R_{sh} \approx 2.59808 \, r_0$. For the EsGB black holes, the deviations from these two limiting values are expected to increase with the parameter $p$ as we move further and further away from the Schwarzschild limit $(p=0)$. This is indeed the case as the plots for the numerical values of $r_{ph}$ and $R_{sh}$ reveal in Fig. \[fig: rph and shadows all cases\]. We point out that although $r_{ph}$ is a nonobservable auxiliary quantity, which is not gauge invariant, it is very useful in many applications beyond the computation of black-hole shadows. To this end, its profile with $p$ as depicted in Fig. \[fig: rph and shadows all cases\] provides useful information. Also, in Appendix C, the interested reader can find approximate analytical expressions for these two quantities.
Having obtained the accurate solutions for the shadows numerically we can now compare how each order in the CFA stands against the numerical solutions. By terminating the series of the expansion of $\tilde{A}(x)$ each time at $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ we obtain the first-, second-, third-, fourth-, and fifth-order analytical approximation for the $g_{tt}(r)$ metric function, respectively.
-------------- ------------ ------------ ------------ --------------- ---------------
$f=\phi^2$ $f=\phi^3$ $f=\phi^4$ $f=\ln(\phi)$ $f=\phi^{-1}$
$p=0.5$ $p=0.3$ $p=0.4$ $p=0.5$ $p=0.9$
$R_{sh}/r_0$ $2.62984 $ $2.62456 $ $2.63095$ $2.70299$ $2.74766 $
$RE_1$ $1.3759\%$ $0.5597\%$ $0.8669\%$ $0.1694\%$ $1.5947\%$
$RE_2$ $0.0782\%$ $0.0111\%$ $0.0138\%$ $0.0198\%$ $0.7590\%$
$RE_3$ $0.3380\%$ $0.0625\%$ $0.1307\%$ $0.4864\%$ $0.6678\%$
$RE_4$ $0.1224\%$ $0.0214\%$ $0.0188\%$ $0.0495\%$ $0.4288\%$
$RE_5$ $0.0413\%$ $0.0015\%$ $0.0104\%$ $0.0105\%$ $0.3065\%$
-------------- ------------ ------------ ------------ --------------- ---------------
: The accurate value of the black-hole shadow radius and the relative error for the approximations of the first five orders.[]{data-label="tabl:convergence"}
The absolute RE of the analytical approximation of the black-hole shadow from the numerical solution $R_{sh}$ for different couplings and values of the dimensionless parameter $p$ are given in Table \[tabl:convergence\]. Starting from the fourth-order approximation the relative error quickly decreases for small $p$. For larger values of $p$ the convergence becomes slower, so that for the near-extremal black holes we need more orders to achieve a reasonable approximation. However, for the nonextremal black holes, the second- and fourth-order approximations give RE of fractions of a percent. Notice that the third-order approximation usually leads to a slightly worse accuracy than those of the second and fourth order.
The analytic expressions for the metric functions in the second and fourth order in CFA deviate from the numerical ones by less than $1\%$ for almost the entirety of the GB couplings that we have studied in this work. Only for the inverse polynomial case, the deviation is slightly larger, but still smaller than $ 1.4\%$. It is noteworthy that these maximal values for the RE actually emerge in the large-$p$ regime where the black holes are presumably unstable. Thus for viable black-hole solutions the RE is quite small.
On the approximation for different values of the coupling constant
==================================================================
In the analysis performed in this article we considered $\alp=1/4$ in order to perform the fitting and obtain the approximate analytical expressions for the metric functions and the scalar field. In principle with the help of the analytical expressions obtained here one is able to consider a range of values for the GB constant $\alpha$.
{width="0.5\linewidth"}{width="0.5\linewidth"} {width="0.5\linewidth"}{width="0.5\linewidth"}
By considering other values of the coupling constant $\alp$ we have observed that for small $p$ the variation of the metric function is negligible. From Fig. \[fig:MRE\_metric\] we see that the relative difference between the accurate metric function and the function obtained by fixing $\alp=1/4$ is as small as fraction of a percent when $p\lesssim0.2$.
We are now in a position to find the appropriate approximation for the scalar field $\vph(r)$ when $\alp\neq1/4$. First, we can introduce |(r)=C\_(r), where $C_{\alp}$ is a constant, such that \[Cscale\] =. Assuming, that the metric depends on the parameter $p$ only, we notice that $\bar{\vph}(r)$ satisfies for $\alp=1/4$, i.e. can be well approximated by with some effective value of the parameter $p$, p = 6 f’(|\_0)\^2=96\^2C\_\^2f’(\_0)\^2=C\_\^2 p\_. Here $p_{\alp}$ is the actual value of the parameter given by , which should be used in the expression for the metric functions.
For any of the polynomial couplings $f(\vph)=\vph^r$ Eq. yields \[Cscalesol\] C\_=(4)\^, and, similarly, for $f(\vph)=\ln(\vph)$, we take $r=0$ in .
{width="0.5\linewidth"}{width="0.5\linewidth"} {width="0.5\linewidth"}{width="0.5\linewidth"}
On Fig. \[fig:MRE\_phi\] we show the relative error of the above approximation for the scalar field. Namely, we compare $C_{\alp}\vph(r)$ for various values of $\alp$ and $p_{\alp}$ and the function $\bar{\vph}(r)$ for $\alp=1/4$ and the corresponding effective value of $p$, $$p=C_{\alp}^2 p_{\alp}=\left(4\alpha\right)^{\frac{2}{r-2}}p_{\alp}.$$ We conclude that the scalar field can be approximated as (r)|(r), when $p_{\alp}$ is sufficiently small.
The only exception is the coupling functional $f(\phi) = \vph^2$, for which (\[Cscale\]) cannot be satisfied. Nevertheless, the obtained approximation for the metric functions can still be used in this case.
Conclusions
===========
In the context of Einstein-scalar-Gauss-Bonnet gravity, a plethora of black hole solutions with nontrivial scalar hair emerge for different coupling functionals to the Gauss-Bonnet term [@Antoniou2018a]. This has been recently demonstrated in [@Antoniou2018] where numerical solutions to the field equations have been obtained for four different GB couplings (even-, odd-, inverse-polynomial and logarithmic). In this work, we employed the powerful method of the continued-fraction approximation [@Rezzolla2014] in order to obtain analytic expressions for the metric functions and the scalar field for the aforementioned GB couplings.
For each coupling functional we parametrized the family of black-hole solutions that emerge in terms of a dimensionless compact parameter $p$ that ranges from $0$ (Schwarzschild limit) to $1$. The analytical representation is based on the continued-fraction expansion which converges quickly for all values of $p$ except the regime of near extremal coupling, when $p$ is close to unity. It is known that in this regime, Gauss-Bonnet black holes (as well as all the other known higher-curvature corrected black holes and branes whose gravitational perturbations were investigated) are unstable and, therefore, cannot exist. Although the (in)stability for the above considered couplings of the scalar field have not been studied in the literature so far, we assume that at least in the regime of the strong scalar field, the instability should remain. It would be interesting to check this supposition on the instability of EsGB black holes in the future and the obtained here analytical approximations for the black-hole metric and scalar field makes further investigation of stability easier.
We performed the computation up to the fourth order in the continued-fraction expansion and we have found that the deviation of the analytic expressions from the accurate numerical ones is at most of the order of $\mathcal{O}(1)\, \%$ for black-hole configurations which are expected to be gravitationally stable. This observation alone is not sufficient to guaranty the high accuracy of the approximation since the metric coefficients are not gauge-invariant quantities.
To this end, in order to make a concrete and gauge-invariant statement about the accuracy of the approximation we turned to the black-hole shadows cast by the EsGB black holes. We computed the shadows for five GB couplings numerically and compared them against the approximate results obtained via the analytical approximation to second, third and fourth order. We found that already in the second order, the largest relative error for the analytical approximations emerges in the maximal coupling limit $p \rightarrow 1$ and is less than $1\, \%$.
We noticed that all the considered coupling functionals lead to an increase of the radius of the black-hole shadow with respect to the Schwarzschild value $R_{sh}\approx2.59808 r_0$. In addition, we have obtained analytical expressions for the photon sphere which increases for all the couplings as well. The analytical representation obtained here for the black-hole metrics and scalar fields in the Einstein-scalar-Gauss-Bonnet theory allows one to explore various analytical, semianalytical, and numerical tools in order to study various effects in the background of these solutions, such as accretion of matter, quasinormal modes, scattering, Hawking radiation and others.
We thank G. Antoniou, A. Bakopoulos, and P. Kanti for sharing their numerical code with us. A. Z. was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). The authors acknowledge the support of the grant 19-03950S of Czech Science Foundation (GAČR). This publication has been prepared with partial support of the “RUDN University Program 5-100”.
Analytical expressions for the metric functions and the scalar field to second order in the CFA {#App A}
===============================================================================================
In this appendix we give the explicit expressions for the analytical approximations for the metric functions and the scalar field in second order in the CFA.
These functions are rational functions of $r$, which we give in the following form: $$\begin{aligned}
g_{tt}(r)&\approx & \mathcal{N}^{(1)}/\mathcal{D}^{(1)} \left(1-\frac{r_0}{r} \right) \,,
\\
\sqrt{g_{tt}(r) g_{rr}(r)}& \approx & \mathcal{N}^{(2)}/\mathcal{D}^{(2)}\,,
\\
e^{\vph (r)-\vph_\infty} &\approx & \mathcal{N}^{(3)}/\mathcal{D}^{(3)}\,,\end{aligned}$$
where the numerators, $\mathcal{N}^{(1)},\mathcal{N}^{(2)},\mathcal{N}^{(3)},$ and the denominators, $\mathcal{D}^{(1)},\mathcal{D}^{(2)},\mathcal{D}^{(3)},$ for each of the functions are given for each coupling separately.
Even-polynomial GB coupling: $f(\vph)=\vph^{2}$
-----------------------------------------------
$$\begin{aligned}
g_{tt}(r)&\approx & \mathcal{N}_{eve}^{(1)}/\mathcal{D}_{eve}^{(1)} \left(1-\frac{r_0}{r} \right) \,,
\\
\sqrt{g_{tt}(r) g_{rr}(r)}& \approx & \mathcal{N}_{eve}^{(2)}/\mathcal{D}_{eve}^{(2)}\,,
\\
e^{\vph (r)-\vph_\infty} &\approx & \mathcal{N}_{eve}^{(3)}/\mathcal{D}_{eve}^{(3)}\,,\end{aligned}$$
where
$$\begin{aligned}
\mathcal{N}_{eve}^{(1)}&=&p^6 (0.0273395\, r_0^3-0.0273395\, r^2 r_0)+p^5 ( r^3-0.737869 \,r^2 r_0-0.0114842\, r r_0^2-0.273615\, r_0^3)+p^4 \,(-10.4088 \,r^3
\nonumber \\
&+&10.1241\, r^2 r_0+0.121014\, r r_0^2+1.15224\, r_0^3)+p^3 \,(29.8132\, r^3-34.4461\, r^2 r_0-0.419751\, r r_0^2-3.0735 \,r_0^3)
\nonumber \\
&+&p^2 (-7.36175\, r^3+25.9689\, r^2 r_0+0.540161\, r r_0^2+3.67187\, r_0^3)+p \,(-51.7172\, r^3+27.879 \,r^2 r_0-0.230379\, r r_0^2
\nonumber \\
&-&1.48583 \,r_0^3)+39.8966\, r^3-29.9903 \,r^2 r_0 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{eve}^{(1)}&=& p^5 r^2 ( r- r_0)+p^4 r^2 (10.8289 \,r_0-10.4088\, r)+p^3 r^2 (29.8132 r-34.5843 r_0)+p^2 r^2 (24.7365 r_0-7.36175 r)
\nonumber\\
&+&p r^2 (28.8068 r_0-51.7172 r)+r^2 (39.8966 r-29.9903 r_0)\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{eve}^{(2)}&=& p^5 ( r^2- r r_0)+p^4 (5.97995\, r r_0-5.97995\, r^2)+p^3 (9.25097 r^2-9.03038\, r r_0+0.233666\, r_0^2)+p^2 (-13.5895 r^2
\nonumber\\
&+&12.1594 \,r r_0-0.516068 r_0^2)+p\, (22.3021 r^2-19.9922\, r r_0+0.283453 r_0^2)-13.2868\, r^2+12.1842\, r r_0 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{eve}^{(2)}&=& p^5 r ( r- r_0)+p^4 r\, (5.97995 r_0-5.97995 r)+p^3 r (9.25097 r-9.03038 r_0)+p^2 r (12.1594 r_0-13.5895 r)
\nonumber\\
&+&p r \,(22.3021 r-19.9922 r_0)+r\, (12.1842 r_0-13.2868 r) \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{eve}^{(3)}&=& p^8 (r^2-1.06383 r r_0-0.210319 r_0^2)+p^7 (2.75803 r^2-1.60013 r r_0+0.823148 r_0^2)+p^6 (-4.75602 r^2+4.56272 r r_0
\nonumber\\
&-&1.0166 r_0^2)+p^5 (-9.42419 r^2+5.36685 r r_0+0.353077 r_0^2)+p^4 (7.2356 r^2-5.19214 r r_0+0.000836036 r_0^2)
\nonumber\\
&+&p^3 (4.81869 r^2-3.44584 r r_0+0.0311472 r_0^2)+p^2 (0.132709 r^2-0.0952131 r r_0+0.0000162571 r_0^2)
\nonumber\\
&+&p(0.000867808 r^2-0.000623499 r r_0)\,,\end{aligned}$$
and
$$\begin{aligned}
\mathcal{D}_{eve}^{(3)}&=& p^8 r (r-1.14693 r_0)+p^7 r (2.75803\, r-1.80427\, r_0)+p^6 r (5.00688 r_0-4.75602 r)+p^5 r (5.97025 r_0-9.42419\, r)
\nonumber\\
&+&p^4 r (7.2356 r-5.96539 r_0)+p^3 r (4.81869\, r-3.52478\, r_0)+p^2 r (0.132709 r-0.0963952 r_0)+p\, r (0.000867808 \,r
\nonumber\\
&-&0.000623594 \,r_0) \,.\end{aligned}$$
Even-polynomial GB coupling: $f(\vph)=\vph^{4}$
-----------------------------------------------
$$\begin{aligned}
g_{tt}(r)&\approx & \mathcal{N}_{eve4}^{(1)}/\mathcal{D}_{eve4}^{(1)} \left(1-\frac{r_0}{r} \right) \,,
\\
\sqrt{g_{tt}(r) g_{rr}(r)}& \approx & \mathcal{N}_{eve4}^{(2)}/\mathcal{D}_{eve4}^{(2)}\,,
\\
e^{\vph (r)-\vph_\infty} &\approx & \mathcal{N}_{eve4}^{(3)}/\mathcal{D}_{eve4}^{(3)}\,,\end{aligned}$$
where
$$\begin{aligned}
\mathcal{N}_{eve4}^{(1)}&=&p^8 (r^3-1.089 r^2 r_0+0.0889959 r_0^3)+p^7 (-1.87642 r^3+2.42973 r^2 r_0-0.0140959 r r_0^2-0.324535 r_0^3)
\nonumber\\
&+&p^6 \,(-6.91475\, r^3+6.18264\, r^2 r_0+0.0641805 \,r r_0^2+0.536171\, r_0^3)+p^5\, (13.6824\, r^3-14.7421 \,r^2 r_0
\nonumber\\
&-&0.0782894\, r r_0^2-0.400357 \,r_0^3)+p^4\, (-1.21904 \,r^3+3.58592\, \,r^2 \,r_0+0.0175305\, r r_0^2+0.0804516\, r_0^3)
\nonumber\\
&+&p^3 (-4.24836 r^3+3.42978 r^2 r_0+0.0100198\, r r_0^2+0.0134892\, r_0^3)+p^2 \,(-0.643342 r^3+0.437795\, r^2 r_0
\nonumber\\
&+&0.000722858\, r \,r_0^2+0.000594032\, r_0^3)+p \,(-0.02586 \,r^3+0.0142478 \,r^2 r_0)\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{eve4}^{(1)}&=&p^8 r^2 (r- r_0)+p^7 r^2 (2.0348 r_0-1.87642 r)+p^6 r^2 (6.59925 r_0-6.91475 r)+p^5 r^2 (13.6824 r-14.6863 r_0)
\nonumber\\
&+&p^4 r^2 (3.44811 r_0-1.21904 r)+p^3 r^2 (3.3963 r_0-4.24836 r)+p^2 r^2 (0.436086 r_0-0.643342 r)
\nonumber\\
&+&p r^2 (0.0142432 r_0-0.02586 r)\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{eve4}^{\,(2)}&=&p^5 \,(r^2-\,r\,r_0)+p^4 \,(-6.02549\,r^2+6.109\,r\,r_0+0.0865879\,r_0^2)+p^3 \,(7.52143\,r^2-8.05501\,r\,r_0-0.189351\,r_0^2)
\nonumber\\
&+&p^2 \,(0.812262\,r^2+0.0379139\,r\,r_0+0.100843\,r_0^2)+p \,(-3.41709\,r^2+3.02529\,r\,r_0+0.00237323\,r_0^2)
\nonumber\\
&-&0.0430078\,r^2+0.0337265\,r\,r_0
\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{eve4}^{(2)}&=&p^5 r (r-r_0)+p^4 r (6.109 r_0-6.02549 r)+p^3 r (7.52143 r-8.05501 r_0)+p^2 r (0.812262 r+0.0379139 r_0)
\nonumber\\
&+&p r (3.02529 r_0-3.41709 r)+r (0.0337265 r_0-0.0430078 r)
\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{eve4}^{(3)}&=& p^9 (0.00714286 r_0^2-0.00714286 r r_0)+p^8 ( r^2-0.830984 r r_0-0.169016 r_0^2)+p^7 (-1.71501 r^2+1.43139 r r_0
\nonumber\\
&+&0.287647 r_0^2)+p^6 (-2.8681 r^2+1.90683 r r_0-0.0388565 r_0^2)+p^5 (2.68777 r^2-2.38079 r r_0-0.239279 r_0^2)
\nonumber\\
&+&p^4 (3.08141 r^2-1.93109 r r_0-0.0512424 r_0^2)+p^3 (0.69013 r^2-0.376209 r r_0+0.000923929 r_0^2)
\nonumber\\
&+&p^2 (0.0167125 r^2-0.0085256 r r_0-0.0000264241 r_0^2)+p (0.00011179 r^2-0.0000470767 r r_0)
\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{eve4}^{(3)}&=& p^8 r (r- r_0)+p^7 r (1.71501 r_0-1.71501 r)+p^6 r (2.30399 r_0-2.8681 r)+p^5 r (2.68777 r-2.85841 r_0)
\nonumber\\
&+&p^4 r (3.08141 r-2.28967 r_0)+p^3 r (0.69013 r-0.413742 r_0)+p^2 r (0.0167125 r-0.00910661 r_0)
\nonumber\\
&+&p\, r (0.00011179 r-0.000048073 r_0)
\,.\end{aligned}$$
Odd-polynomial GB coupling: $f(\vph)=\vph^{3}$
----------------------------------------------
$$\begin{aligned}
g_{tt}(r)&\approx & \mathcal{N}_{odd}^{(1)}/\mathcal{D}_{odd}^{(1)} \left(1-\frac{r_0}{r} \right) \,,
\\
\sqrt{g_{tt}(r) g_{rr}(r)}& \approx & \mathcal{N}_{odd}^{(2)}/\mathcal{D}_{odd}^{(2)}\,,
\\
e^{\vph (r)-\vph_\infty} &\approx & \mathcal{N}_{odd}^{(3)}/\mathcal{D}_{odd}^{(3)}\,,\end{aligned}$$
where
$$\begin{aligned}
\mathcal{N}_{odd}^{(1)}&=& p^8 (0.00537634 r^2 r_0-0.00537634 r_0^3)+p^7 (-0.0518836 r^2 r_0+0.00105719 r r_0^2+0.0529408 r_0^3)
\nonumber\\
&+&p^6 (r^3-0.86139 r^2 r_0-0.0104076 r r_0^2-0.149018 r_0^3)+p^5 (-3.96079 r^3+4.05559 r^2 r_0+0.029677 r r_0^2
\nonumber\\
&+&0.211035 r_0^3)+p^4 (3.89682 r^3-4.75407 r^2 r_0-0.0293161 r r_0^2-0.143895 r_0^3)+p^3 (0.572272 r^3+0.467441 r^2 r_0
\nonumber\\
&+&0.00542841 r r_0^2+0.0284754 r_0^3)+p^2 (-1.35561 r^3+1.05084 r^2 r_0+0.00352281 r r_0^2+0.00513373 r_0^3)
\nonumber\\
&+&p (-0.17693 r^3+0.113522 r^2 r_0+0.000048065 r r_0^2+0.000151556 r_0^3)-0.000822131 r^3 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{odd}^{(1)}&=& p^6 r^2 (r-r_0)+p^5 r^2 (4.15743 r_0-3.96079 r)+p^4 r^2 (3.89682 r-4.71385 r_0)+p^3 r^2 (0.572272 r+0.426992 r_0)
\nonumber\\
&+&p^2 r^2 (1.04068 r_0-1.35561 r)+p r^2 (0.113474 r_0-0.17693 r)-0.000822131 r^3 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{odd}^{(2)}&=& p^5 ( r^2- r r_0)+p^4 (-5.85214 r^2+5.95711 r r_0+0.108754 r_0^2)+p^3 (6.41291 r^2-7.08335 r r_0-0.238311 r_0^2)
\nonumber\\
&+&p^2 (2.69182 r^2-1.62168 r r_0+0.127734 r_0^2)+p\, (-4.37086 r^2+3.87431 r r_0+0.00238993 r_0^2)-0.0482146 r^2
\nonumber\\
&+&0.0388754 r r_0 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{odd}^{(2)}&=& p^5 r (r-r_0)+p^4 r (5.95711 r_0-5.85214 r)+p^3 r (6.41291 r-7.08335 r_0)+p^2 r (2.69182 r-1.62168 r_0)
\nonumber\\
&+&p r \,(3.87431 r_0-4.37086 r)+r (0.0388754 r_0-0.0482146 r) \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{odd}^{(3)}&=& p^8 (0.00790584 r_0^2-0.00704225 r r_0)+p^7 (r^2-0.974721 r r_0-0.261618 r_0^2)+p^6 (2.50004 r^2-1.58963 r r_0
\nonumber\\
&+&0.231062 r_0^2)+p^5 (-0.277785 r^2+0.826815 r r_0+0.0230046 r_0^2)+p^4 (-3.50087 r^2+2.17273 r r_0+0.111646 r_0^2)
\nonumber\\
&+&p^3 (-1.36372 r^2+0.827451 r r_0+0.00569516 r_0^2)+p^2 (-0.0768515 r^2+0.0400392 r r_0-0.000361993 r_0^2)
\nonumber\\
&+&p (-0.00130153\, r^2+0.000634807\, r\, r_0)
\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{odd}^{(3)}&=& p^7 r (r-1.12263 r_0)+p^6 r (2.50004 r-1.95942 r_0)+p^5 r (0.945795 r_0-0.277785 r)+p^4 r (2.69322 r_0-3.50087 r)
\nonumber\\
&+&p^3 r (0.910352 r_0-1.36372 r)+p^2 r (0.0433891 r_0-0.0768515 r)+p r\, (0.000670351 r_0-0.00130153 r)
\,.\end{aligned}$$
Inverse-polynomial GB coupling $f(\vph)=\vph^{-1}$
--------------------------------------------------
$$\begin{aligned}
g_{tt}(r)&\approx & \mathcal{N}_{inv}^{(1)}/\mathcal{D}_{inv}^{(1)} \left(1-\frac{r_0}{r} \right) \,,
\\
\sqrt{g_{tt}(r) g_{rr}(r)}& \approx & \mathcal{N}_{inv}^{(2)}/\mathcal{D}_{inv}^{(2)}\,,
\\
e^{\vph (r)-\vph_\infty} &\approx & \mathcal{N}_{inv}^{(3)}/\mathcal{D}_{inv}^{(3)}\,,\end{aligned}$$
where
$$\begin{aligned}
\mathcal{N}_{inv}^{(1)}&=& p^8 (0.00630915 r^2 r_0-0.00630915 r_0^3)+p^7 (-0.270873 r^2 r_0+0.00158747 r r_0^2+0.272461 r_0^3)+p^6 ( r^3+1.68132 r^2 r_0
\nonumber\\
&-&0.0676888 r r_0^2-1.87658 r_0^3)+p^5 (-30.771 r^3+27.3744 r^2 r_0+0.65521 r r_0^2+3.28027 r_0^3)+p^4 (52.7707 r^3
\nonumber\\
&-&61.4833 r^2 r_0-0.744326 r r_0^2-0.405224 r_0^3)+p^3 (-0.028509 r^3+13.4581 r^2 r_0-0.310375 r r_0^2-2.09652 r_0^3)
\nonumber\\
&+&p^2 (-24.1726 r^3+25.4044 r^2 r_0+0.435715 r r_0^2+0.619006 r_0^3)+p\, (-0.865964 r^3-3.69793 r^2 r_0+0.0327455 r r_0^2
\nonumber\\
&+&0.11195 r_0^3)+0.604044 r^3-0.93029 r^2 r_0 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{inv}^{(1)}&=& p^6 r^2 (r- r_0)+p^5 r^2 (31.0226 r_0-30.771 r)+p^4 r^2 (52.7707 r-60.4391 r_0)+p^3 r^2 (11.1237 r_0-0.028509 r)
\nonumber\\
&+&p^2 r^2 (25.2874 r_0-24.1726 r)+p r^2 (-0.865964 r-3.6373 r_0)+r^2 (0.604044 r-0.93029 r_0) \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{inv}^{(2)}&=& p^4 ( r^2-1.16356 r r_0-0.148553 r_0^2)+p^3 (-4.69714 r^2+6.11751 r r_0+0.733313 r_0^2)+p^2 (1.51773 r^2-5.7017 r r_0
\nonumber\\
&-&1.07164 r_0^2)+p\, (9.04644 r^2-4.16948 r r_0+0.487947 r_0^2)-7.0095\, r^2+5.05739\, r r_0 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{inv}^{(2)}&=& p^4 r ( r-1.16356 r_0)+p^3 r (6.11751 r_0-4.69714 r)+p^2 r (1.51773 r-5.7017 r_0)+p r\, (9.04644 r-4.16948 r_0)
\nonumber\\
&+&r (5.05739 r_0-7.0095 r) \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{inv}^{(3)}&=& p^8 (-0.0140845 r r_0-0.136912 r_0^2)+p^7 (r^2+0.455729 r r_0-0.327295 r_0^2)+p^6 (-3.28574 r^2+1.52376 r r_0
\nonumber\\
&+&0.538745 r_0^2)+p^5 (0.397664 r^2-1.30549 r r_0+0.0866968 r_0^2)+p^4 (1.67086 r^2-0.71786 r r_0-0.115423 r_0^2)
\nonumber\\
&+&p^3 (0.381375 r^2-0.127462 r r_0-0.0145457 r_0^2)+p^2 (0.0258233 r^2-0.00845295 r r_0-0.000366578 r_0^2)
\nonumber\\
&+&p (0.00034624 r^2-0.000100545 r r_0)
\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{inv}^{(3)}&=& p^7 r (r+0.79726 r_0)+p^6 r (0.109978 r_0-3.28574 r)+p^5 r (0.397664 r-0.585227 r_0)+p^4 r (1.67086 r-0.367609 r_0)
\nonumber\\
&+&p^3 r (0.381375 r-0.0841323 r_0)+p^2 r (0.0258233 r-0.00668003 r_0)+p \,r\, (0.00034624 r-0.0000924255 r_0)
\,.\end{aligned}$$
Logarithmic GB coupling: $f(\vph)=\ln{(\vph)}$
----------------------------------------------
$$\begin{aligned}
g_{tt}(r)&\approx & \mathcal{N}_{log}^{(1)}/\mathcal{D}_{log}^{(1)} \left(1-\frac{r_0}{r} \right) \,,
\\
\sqrt{g_{tt}(r) g_{rr}(r)}& \approx & \mathcal{N}_{log}^{(2)}/\mathcal{D}_{log}^{(2)}\,,
\\
e^{\vph (r)-\vph_\infty} &\approx & \mathcal{N}_{log}^{(3)}/\mathcal{D}_{log}^{(3)}\,,\end{aligned}$$
where
$$\begin{aligned}
\mathcal{N}_{log}^{(1)}&=& p^7 (0.0626084 r_0^3-0.0626084 r^2 r_0)+p^6 ( r^3-0.621986 r^2 r_0-0.378014 r_0^3)+p^5 (-4.81894 r^3+4.13961 r^2 r_0
\nonumber\\
&-&0.0186288 r r_0^2+0.65135 r_0^3)+p^4 (7.27136 r^3-6.76272 r^2 r_0+0.117437 r r_0^2-0.0407656 r_0^3)+p^3 (-1.24665 r^3
\nonumber\\
&+&0.285681 r^2 r_0-0.248593 r r_0^2-0.910056 r_0^3)+p^2 (-6.33774 r^3+8.6188 r^2 r_0+0.219636 r r_0^2+0.845321 r_0^3)
\nonumber\\
&+&p (5.38018 r^3-7.52226 r^2 r_0-0.0698525 r r_0^2-0.230411 r_0^3)-1.24797 r^3+1.92521 r^2 r_0 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{log}^{(1)}&=& p^6 r^2 (r-r_0)+p^5 r^2 (4.81894 r_0-4.81894 r)+p^4 r^2 (7.27136 r-6.97382 r_0)+p^3 r^2 (-1.24665 r-0.266442 r_0)
\nonumber\\
&+&p^2 r^2 (9.14686 r_0-6.33774 r)+p r^2 (5.38018 r-7.65098 r_0)+r^2 (1.92521 r_0-1.24797 r) \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{log}^{(2)}&=& p^4 (r^2-0.593401 r r_0+0.394399 r_0^2)+p^3 (-7.9072 r^2+4.88145 r r_0-1.33585 r_0^2)+p^2 (20.3089 r^2-13.2461 r r_0
\nonumber\\
&+&1.50407 r_0^2)+p\, (-21.2712 r^2+14.5616 r r_0-0.562773 r_0^2)+7.88315 r^2-5.61692 r r_0 \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{log}^{(2)}&=& p^4 r (r-0.593401 r_0)+p^3 r (4.88145 r_0-7.9072 r)+p^2 r (20.3089 r-13.2461 r_0)+p\, r\, (14.5616 r_0-21.2712 r)
\nonumber\\
&+&r (7.88315 r-5.61692 r_0) \,,\end{aligned}$$
$$\begin{aligned}
\mathcal{N}_{log}^{(3)}&=& p^8 (0.311804 r r_0-1.56376 r_0^2)+p^7 (r^2-0.552096 r r_0+5.09849 r_0^2)+p^6 (5.22019 r^2-1.79963 r r_0-5.43249 r_0^2)
\nonumber\\
&+&p^5 (-16.4094 r^2+4.36303 r r_0+1.52051 r_0^2)+p^4 (8.21674 r^2-1.77521 r r_0+0.254627 r_0^2)
\nonumber\\
&+&p^3 (2.36595 r^2-0.651287 r r_0+0.0795184 r_0^2)+p^2 (0.160339 r^2-0.0446282 r r_0+0.00490208 r_0^2)
\nonumber\\
&+&p (0.0026305 r^2-0.000740342 r r_0)
\,,\end{aligned}$$
$$\begin{aligned}
\mathcal{D}_{log}^{(3)}&=& p^7 r (r-2.35737 r_0)+p^6 r (5.22019 r+2.40632 r_0)+p^5 r (4.7997 r_0-16.4094 r)+p^4 r (8.21674 r-4.25309 r_0)
\nonumber\\
&+&p^3 r (2.36595 r-0.95373 r_0)+p^2 r (0.160339 r-0.0538552 r_0)+p r (0.0026305 r-0.000823782 r_0)
\,.\end{aligned}$$
Analytical expressions for the higher-order CFA coefficients up to fourth order {#App B}
===============================================================================
Even-polynomial GB coupling: $f(\vph)=\vph^2$
---------------------------------------------
a\_3= , \[a3 even\]
a\_4= , \[a4 even\]
b\_3=, \[b3 even\]
b\_4=, \[b4 even\]
f\_3 = , \[f3 even\]
f\_4 = . \[f4 even\]
Even-polynomial GB coupling: $f(\vph)=\vph^4$
---------------------------------------------
a\_3= , \[a3 even quartic\]
a\_4= , \[a4 even quartic\]
b\_3=, \[b3 even quartic\]
b\_4= , \[b4 even quartic\]
f\_3 = , \[f3 even quartic\]
f\_4 = . \[f4 even quartic\]
Odd-polynomial GB coupling: $f(\vph)=\vph^3$
--------------------------------------------
a\_3=, \[a3 odd\]
a\_4=, \[a4 odd\]
b\_3 = , \[b3 odd\]
b\_4 = , \[b4 odd\]
f\_3 = , \[f3 odd\]
f\_4 = . \[f4 odd\]
Inverse-polynomial GB coupling: $f(\vph)=\vph^{-1}$
---------------------------------------------------
a\_3 = , \[a3 inv\]
a\_4 = , \[a4 inv\]
b\_3 = , \[b3 inv\]
b\_4 = , \[b4 inv\]
f\_3 = , \[f3 inv\]
f\_4 = . \[f4 inv\]
Logarithmic GB coupling: $f(\vph)=\log{(\vph)}$
-----------------------------------------------
a\_3 = , \[a3 log\]
a\_4 = , \[a4 log\]
b\_3 = , \[b3 log\]
b\_4 = , \[b4 log\]
f\_3 = , \[f3 log\]
f\_4 = . \[f4 log\]
Analytical expressions for the photon-sphere radii and the black-hole shadows {#App C}
=============================================================================
{width="0.5\linewidth"}{width="0.5\linewidth"}
Here we present approximate analytical expressions for the radius of the photon sphere and the black-hole shadow for the four GB couplings that we have considered in this work.
Notice that in order to obtain these analytical expressions no approximate expression for the metric functions has been involved. Instead, we employed only the accurate numerical solution for $g_{tt}(r)$ aiming to get the most accurate results.
For various values of $p$ we computed the corresponding values of $r_{ph}$ and $R_{sh}$ and in turn we performed a fitting of the collected data. Eventually, as any fitting procedure unavoidably introduces some error we have also included Fig. \[fig: RE\_rph\_and\_shadow\_anal\_all\] to quantify the accuracy of the fitting of the numerical data at each value of the dimensionless parameter $p$.
Even-polynomial GB coupling: $f(\vph)=\vph^2$
---------------------------------------------
r\_[ph]{}= , \[AppC eq1\]
R\_[sh]{}= .
Even-polynomial GB coupling: $f(\vph)=\vph^4$
---------------------------------------------
r\_[ph]{}= ,
R\_[sh]{}= .
Odd-polynomial GB coupling: $f(\vph)=\vph^3$
--------------------------------------------
r\_[ph]{}= ,
R\_[sh]{}= .
Inverse-polynomial GB coupling: $f(\vph)=\vph^{-1}$
---------------------------------------------------
r\_[ph]{}= ,
R\_[sh]{}= .
Logarithmic GB coupling: $f(\vph)=\ln{(\vph)}$
----------------------------------------------
r\_[ph]{}= ,
R\_[sh]{}= . \[AppC eq8\]
[^1]: The interested reader can find more details about the numerical black-hole solutions emerging in EsGB gravity in [@Antoniou2018].
[^2]: We only give the second-order expressions in the appendix for reasons of compactness but in the accompanying Mathematica file one can obtain the analytical expressions up to fourth order. The file is available from <https://arxiv.org/src/1907.10112/anc/Approximation.nb>.
|
---
author:
- 'A. A. Vidotto'
- 'J.-F. Donati'
date: 'Received date / Accepted date '
title: 'Predicting radio emission from the newborn hot Jupiter [V830Tau]{} b and its host star'
---
Introduction {#sec.intro}
============
Detecting radio emission from exoplanets would revolutionise the area of exoplanetary studies. First, it would open up a new avenue for the direct detection of exoplanets [@1999JGR...10414025F]. Second, it would allow us to measure exoplanetary magnetic fields, which so far have only been elusively probed [@2008ApJ...676..628S; @2010ApJ...722L.168V; @2014Sci...346..981K]. Exoplanetary magnetic fields can reveal information about a planet’s interior structure and dynamics (e.g. whether it has a convecting, electrically conducting interior) and are also believed to play an important role in conditions for habitability, by protecting the planet’s surface from energetic cosmic particles, protecting its atmosphere from violent chemical changes, and potentially helping atmospheric retention .
The magnetised planets in the solar system generate radio emission through the electron cyclotron maser instability. The power of this emission is related to the kinetic and/or magnetic powers of the incident solar wind over several orders of magnitude . Although the physics of this relation are still not well understood, this “radiometric Bode’s relation” indicates that planetary radio emission is somehow powered by the interaction between the planetary magnetic field and the solar wind. In analogy to the solar system, it has been suggested that exoplanets may also produce radio emission due to their interaction with the winds of their host stars [@1999JGR...10414025F].[^1]
Calculations based on the solar-system analogy suggest that radio emission from close-in exoplanets should be several orders of magnitude larger than the largest radio emitter of the solar system, Jupiter . This is because, at the short orbital distances of close-in planets, the local power of the host star wind is significantly higher than, for example, the local power of the solar wind dissipated at the distances of the planets in our solar system. These and many other theoretical predictions have motivated a good number of observational searches of exoplanetary radio emission, yielding mostly negative results (e.g. , but see also , who found ambiguous hints of the existence of exoplanetary radio emissions). There are several reasons for the non-detections, such as a lack of exoplanetary magnetism, a mismatch of the frequency search, and/or a low sensitivity of the present-day instruments (@2000ApJ...545.1058B, @2011RaSc...46.0F09G).
One might overcome the issue of instrumental sensitivity by turning to exoplanets that, not only orbit closer to their stars, but that orbit stars that have winds that are more powerful than solar-type winds, and whose magnetic fields are also stronger. Stars that are more active than the Sun are indeed expected to host more powerful winds . In particular, stars in the pre-main sequence with close-in planets would be ideal targets to search for planetary radio emission . With the recent detection of the first hot giants orbiting stars younger than $10$–$20$ Myr ([V830Tau]{} b, @2016Natur.534..662D, and K2-33b, @2016Natur.534..658D, TAP26b, @2017MNRAS.tmp...26Y), now is an excellent timing for testing theoretical expectations of exoplanetary radio emission. In this paper, we compute the expected radio emission from the newborn hot Jupiter [V830Tau]{} b.
With the spectropolarimetric monitoring of [V830Tau]{}, @2016Natur.534..662D were able to extract the planetary signature of [V830Tau]{} b, a 2 Myr-old hot Jupiter orbiting at 0.057 au, while, at the same time, reconstructing the large-scale magnetic field topology of the host star. It is precisely the latter measurements that we use in this paper to derive the wind properties of the host star (Section \[sec.windmodel\]). With that, we are then able to derive the expected radio emission of [V830Tau]{} b, starting from the assumption that [V830Tau]{} b is magnetised (Section \[sec.radiomodel\]). Given the high densities of the wind of young active stars, these winds can become optically thick to free-free radiation at radio wavelengths. We thus investigate in which circumstances the planet would be embedded in the radio-emitting region of the stellar wind and whether planetary radio emission would be able to propagate through the stellar wind plasma, given that the planetary cyclotron frequency of emission should exceed the plasma frequency of the stellar wind (Section \[sec.radiowind\]). For physically-reasonable conditions, we show that planetary radio emission from [V830Tau]{} b is expected to peak at 12 mJy, presenting therefore a high potential of detection with LOFAR (the Low-Frequency Array) the upgraded UTR-2 (Ukrainian T-shaped Radio telescope) and GMRT (Giant Metrewave Radio Telescope ,see Section \[sec.conclusion\]). Interestingly, its host star has been detected using VLA (the Karl G. Jansky Very Large Array) at 4.5 and 7.5 GHz and with VLBA (the Very Long Baseline Array) at 8.4 GHz, becoming the first exoplanet host with detected radio emission [@bower].
Stellar wind modelling {#sec.windmodel}
======================
We model the wind of [V830Tau]{} by means of three-dimensional (3D) magnetohydrodynamics (MHD) simulations, similarly to @2012MNRAS.423.3285V [@2015MNRAS.449.4117V]. We refer the reader to these papers for more details of the model, which are summarised next. We use the 3D MHD numerical code BATS-R-US [@1999JCoPh.154..284P; @2012JCoPh.231..870T], modified as in @2012MNRAS.423.3285V. BATS-R-US solves the set of ideal MHD equations for the mass density, velocity, magnetic field, and gas pressure. We assume the wind is polytropic with a polytropic index $\gamma=1.15$ and a fully ionised hydrogen wind.
For the physical characteristics of the star, we use a rotation period of $2.741$ days, mass of $1 M_\odot$, and radius of $R_\star=2 R_\odot$. The radial part of the stellar magnetic field, anchored at the wind base, is constrained from observations collected in November-December 2015 [@2016Natur.534..662D]. Initially, the field is considered to be in its lowest (potential) state, but as the simulation evolves in time the wind particles self-consistently interact with the magnetic field lines (and vice-versa), removing the field from its initial potential state. At the wind base, we adopt a temperature of $10^6$ K and a number density of $n_0=10^{12}$cm$^{-3}$, which are free parameters of our model. The star rotates as a solid body with a rotation axis along the $z$-axis. Our grid is the same as described in @2014MNRAS.438.1162V. The resultant wind solution, obtained self-consistently, is found when the system reaches steady state in the reference frame corotating with the star. For the parameters we adopted, we obtain a wind mass-loss rate of ${\dot{M}}\sim 3{\times 10^{-9}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$. This mass-loss rate is an upper limit for the case [V830Tau]{}, as we will demonstrate in Section \[sec.radiowind\].
Figure \[fig.wind\] illustrates the output of the wind simulations of [V830Tau]{}. We overplot the (assumed equatorial and circular) orbital radius of [V830Tau]{} b (black circle) and a cut of the Alfvén surface $S_A$ at the equatorial plane ($xy$ plane). For the parameters we adopted, the planet’s orbit lies almost completely inside $S_A$. The Alfvén surface is expected to expand for lower values of $n_0$ [@2009ApJ...699..441V]. This indicates that [V830Tau]{} b is truly orbiting in the sub-Alfvénic regime, if not at all times, at least during most of its orbit. The Alfvén surface represents the boundary between a magnetically-dominated stellar wind (interior to $S_A$) and a region that is dominated by the wind inertia. The Alfvén surface is relevant for computing angular momentum losses [@1967ApJ...148..217W] and indicates the types of star-planet interactions. Planets orbiting in the sub-Alfvénic regime can be directly connected to the magnetic field of the star and perturbations can travel to/from the star . In this case, the planetary magnetosphere is embedded in that of the star and it is believed that there is continuous reconnection of the planet’s and stellar magnetic fields, similar to the Ganymede-Jupiter system. When the planet orbits outside $S_A$ (in the super-Alfvénic regime), as in the case of the Earth, the magnetospheric cavity surrounding the planet deflects the stellar wind particles. We estimate the size of the magnetosphere of [V830Tau]{} b in the next section.
![Simulated wind of [V830Tau]{}. Thin grey lines represent the magnetic field of [V830Tau]{} that is embedded in the wind. The circle depicts the orbital radius of [V830Tau]{} b, assumed to lie in the equatorial plane of the star ($xy$ plane). The white line represents a cut of the Alfvén surface at the equatorial plane, while the colour shows the total wind pressure.[]{data-label="fig.wind"}](fig1.png){width="\columnwidth"}
Exoplanetary radio emission {#sec.radiomodel}
===========================
To calculate the radio emission of [V830Tau]{} b, we use the radiometric Bode’s law for the solar system, in which the planetary radio power $P_{\rm radio}$ can be decomposed into a power released from the dissipation of kinetic energy of the stellar wind $P_k$ and/or a power released from the dissipation of magnetic energy of the wind $P_B$ $$\label{eq.pwrrec}
P_{\rm radio} = \eta_k P_k \,\,\,\, \,\,\,\, \,\,\,\, \textrm{or } \,\,\,\, \,\,\,\, \,\,\,\, P_{\rm radio} = \eta_B P_B\, ,$$ where $\eta_k $ and $\eta_B$ are efficiency ratios. In the solar system, $\eta_k = 1\times 10^{-5}$ and $\eta_B=2\times 10^{-3}$ ; we assume the same efficiency ratios here. In the past, it has been argued that it was not possible to decide which incident power actually drives the radio power observed from the magnetic planets of the solar system . Recently, however, @2010ASPC..430..175Z argued that the magnetic field is likely to be a determinant for extracting part of the flow power and converting it to energetic particles . Here, for completeness, we compute the radio powers coming from both kinetic and magnetic energies. As we will see, the latter is much larger than the former and, following the arguments in @2010ASPC..430..175Z, is the preferred method for estimating radio fluxes.
The dissipated kinetic and magnetic powers of the impacting wind on the planet are approximated as, respectively, (Appendix \[sec.apA\]) $$\begin{aligned}
P_k \simeq \rho (\Delta u)^3 \pi r_M^2 \, , && \label{eq.pK} \\
P_B \simeq \frac{B_{\perp}^2 }{4\pi} (\Delta u) \pi r_M^2\, , && \label{eq.pB}\end{aligned}$$ where $B_{\perp}$ is the magnetic field component perpendicular to $\Delta u$. Here, $|\Delta {\bf u}| = |{\bf u} - {\bf v}_{K}|$ is the relative velocity between the wind and the Keplerian velocity ($v_K$) of the planet, and $\rho$, $p$ , and $B$ are the local density, pressure, and magnetic field intensity of the stellar wind. Neglecting planet thermal pressure, the size of the planet’s magnetopause $r_M$ can be estimated by pressure balance between the stellar wind total pressure and the planet’s magnetic pressure: $p_{\rm tot}= {B_{p}^2(r_M)}/({8\pi})$, where $p_{\rm tot} = {\rho \Delta u^2} + \frac{B^2}{8\pi} + p$ is the sum of the ram, magnetic, and thermal pressures of the wind; $B_{p}(r_M)$ is the intensity of the planet’s magnetic field at the nose of the magnetopause. For a dipolar planetary magnetic field: $B_{p}(r_M) = \frac12 B_p (R_p/R)^3$, where $R_p$ is the planetary radius, $R$ is the radial coordinate centred at the planet, and $B_{p}$ is the polar magnetic field intensity (at the equator, the intensity is $\frac12 B_p $). Assuming the dipole is aligned with the planetary orbital spin axis, the magnetospheric radius (where $R=r_M$) is $$\label{eq.rM}
\frac{r_M}{R_p}= 2^{1/3}\left[ \frac{(B_p/2)^2/{8\pi} }{{\rho \Delta u^2} + p + B^2/{8\pi} } \right]^{1/6} .$$ Figure \[fig.rm\]a shows the stand-off distance of [V830Tau]{} b’s magnetopause calculated using Eq. (\[eq.rM\]) as the planet orbits around the star.[^2] We assume three different values of $B_{p}$: $10$, $50,$ and 100 G. For comparison, we note that the maximum intensity of Jupiter’s magnetic field is $14.3$ G [@1992AREPS..20..289B]. For $B_{p} = 10$ G, the average magnetospheric radius is $\sim 1.31R_p$ (i.e. very close to the planet surface), while in the second and third cases, $r_M \sim 2.2R_p$ and $2.8 R_p$, respectively. If the wind densities (and therefore mass-loss rates) were to decrease by a factor of $100$, the magnetospheric sizes would increase by about $20\%$ of the values presented in Figure \[fig.rm\]a (in such a case, the planet’s orbit would lie completely within the Alfvén surface). This shows that, even in the case of a “high” planetary magnetic field and low wind densities, the magnetosphere of [V830Tau]{} b is expected to be quite small, due to the harsh conditions of the external environment.
![(a) Estimated sizes of the planetary magnetospheres along a planetary year assuming three different values of the polar planetary magnetic field: $10$, $50,$ and $100$ G. (b) Predicted maximum frequency of the radio emission of [V830Tau]{} b. This frequency is calculated at the border of the polar-cap boundary, which is located at different colatitudes depending on the size of the planet magnetosphere (*cf*, Eq. \[eq.alpha0\]). The magnetic field intensity at this position in shown in the right axis.[]{data-label="fig.rm"}](fig2.png){width="0.98\columnwidth"}
The radio flux is related to the radio power as $$\phi_{\rm radio} = \frac{P_{\rm radio}}{d^2 \omega \Delta f} \, ,$$ where $\omega$ is the solid angle of the (hollow) cone of emission, $d$ is the distance to the system ($147$ pc in the case of [V830Tau]{}), and $\Delta f$ is the emission bandwidth, approximately the cyclotron frequency . In Appendix \[sec.apA\], we show that the radio flux density due to the dissipated wind kinetic and magnetic powers simplify to $$\begin{aligned}
\phi_{\rm radio, kin} = \eta_k^\prime \frac{R_p^2}{d^2 }\frac{\rho (\Delta u)^3}{{{ p_{\rm tot}}^{1/2}}} f(\alpha_0) , \label{eq.phiradio_maintext} \\
\phi_{\rm radio, mag} = \eta_B^\prime \frac{R_p^2}{d^2 }\frac{B_\perp^2 (\Delta u)}{{{ p_{\rm tot}}^{1/2}}} f(\alpha_0) , \label{eq.phiradio_B_2_maintext}\end{aligned}$$ respectively. Here $\eta_k^\prime$ and $\eta_B^\prime$ are constants and are related to $\eta_k$ and $\eta_B$ following Eqs. (\[eq.etakprime\]) and (\[eq.etaBprime\]). The planetary magnetic field dependence is hidden in the function $f(\alpha_0)$, which varies between $0$ and $3.3$ for any planetary magnetic field intensity (Figure \[fig.function\_variation\]). In particular, we assume that the beaming angle of the radio emission occurs at a ring with colatitude $\alpha_0$ and thickness $\delta\alpha = 17.5^{\rm o}$ [@2004JGRA..10909S15Z]. Using the values for $\eta_k$ and $\eta_B$ from the solar system, we have $ \eta_k^\prime \simeq 1.8 {\times 10^{-13}} \textrm{\, [cgs~units]}$ and $\eta_B^\prime \simeq 2.8{\times 10^{-12}} \textrm{\, [cgs~units]}$. From Eqs. (\[eq.phiradio\_maintext\]), (\[eq.phiradio\_B\_2\_maintext\]), (\[eq.etakprime\]), and (\[eq.etaBprime\]), we have that $$\label{eq.ratio}
\frac{\phi_{\rm radio, kin} }{\phi_{\rm radio, mag} } = \frac{\eta_k}{\eta_B (4\pi)^2} \frac{\rho (\Delta u)^2/2}{B_\perp^2/8\pi} = \frac{1}{3200 \pi^2} \frac{\rho (\Delta u)^2/2}{B_\perp^2/8\pi} .$$ Equation (\[eq.ratio\]) shows that the flux released in the dissipation of Poynting flux becomes dominant when the kinetic energy ($\rho (\Delta u)^2/2$) of the impacting wind is smaller than $3200 \pi^2$ times the magnetic energy ($B_\perp^2/8\pi$). This is the case of the [V830Tau]{} system, due to the intense stellar magnetism.
From Eqs. (\[eq.phiradio\_maintext\]) and (\[eq.phiradio\_B\_2\_maintext\]), we see that the radio fluxes are functions of the angular size of the planet ($R_p/d$) and the properties of the stellar wind surrounding the planet. We also note that radio fluxes depend relatively weakly on the magnetic intensity of the planetary dipolar field. This dependence is hidden in $f(\alpha_0)$ (*cf* Appendix \[sec.apA\]). For example, for polar magnetic fields of 10, 50, and 100 G, respectively, the (northern) polar-cap boundaries are located at colatitudes $\alpha_0$ of $61^{\rm o}$, $42^{\rm o}$ , and $37^{\rm o}$, respectively. For these values of $\alpha_0$, $f(\alpha_0)$ is $2.2$, $1.4,$ and $1.2$, respectively. Therefore, as radio fluxes are proportional to $f(\alpha_0)$, a change of field intensities from 10 to 100 G results in a rather small change (a factor of 1.8) in the computed radio fluxes. This is good news, given that exoplanetary magnetism remains observationally elusive (*cf* Sect. \[sec.intro\]).
Figure \[fig.radio\] shows the predicted radio flux densities along a planetary year assuming dissipation of kinetic (top) and magnetic (bottom) stellar wind powers. We found that radio emission arising from the dissipation of wind magnetic power is considerably larger ($ \sim 30$ times) than from kinetic power and this difference could become even larger should a lower stellar mass-loss rate be adopted. This is due to the kinetic-to-magnetic ratio being smaller than $3200\pi^2$ (*cf* Eq. \[eq.ratio\]). In Appendix \[sec.apB\], we show that our radio predictions for [V830Tau]{} b are robust against a reduction of stellar wind densities and mass-loss rates, given the high magnetism of [V830Tau]{}. We assume here that the planet has a radius of $1$ to $2~R_{\rm jup}$, but we note that, although the mass of [V830Tau]{} b has been constrained by observations to be about $0.7~M_{\rm jup}$ [@2016Natur.534..662D], no constraints on its radius currently exist. Since radio fluxes are proportional to $R_p^2$, the fluxes calculated for $R_p=2R_{\rm jup}$ are four times larger than for the calculations with $R_p=R_{\rm jup}$. For the case of $R_p=R_{\rm jup}$, the estimated magnetic radio fluxes are on average $\sim 6$ mJy, with peaks at $11$ mJy. This can be considered as a conservative estimate. Due to its youth and the close distance to the host star, it is likely that the radius of [V830Tau]{} b is larger than $1R_{\rm jup}$, since [V830Tau]{} b is likely to be still in the contracting phase [e.g. K2-29 b, which has a similar mass, but orbits an older, 450 Myr-old star, has a radius of $1.19~R_{\rm jup}$, @2016ApJ...824...55S]. For the case of $R_p=2R_{\rm jup}$, the estimated magnetic radio fluxes are on average $\sim 24$ mJy, with peaks at $44$ mJy.
![Predicted radio flux density along a planetary year assuming dissipation of kinetic (top) and magnetic (bottom) stellar wind powers onto the magnetospheric cross section of the exoplanet. At the low frequency range, values of a few mJy are potentially observable with LOFAR, the upgraded UTR-2 and GMRT.[]{data-label="fig.radio"}](fig3.png){width="0.98\columnwidth"}
We adopted in Figure \[fig.radio\] a dipolar magnetic field whose polar intensity is $10$ G, but again note that an increase in magnetic field intensity by a factor of 10 would result in a change on the predicted radio flux by a factor of 1.8. The planetary magnetic field has, however, a strong influence on the frequency of emission. Since this is cyclotron emission, by measuring its frequency one is able to derive the intensity of the planetary magnetic field (*cf* Eq. \[eq.fcyc\], see for a suggestion on how to conduct this estimation). Figure \[fig.rm\]b shows the predicted maximum emission frequency, considering the same polar planetary magnetic fields as in Figure \[fig.rm\]a. This frequency is calculated at the border of the polar-cap boundary, which is located at different colatitudes depending on the size of the planet magnetosphere (*cf* Eq. \[eq.alpha0\], @2011MNRAS.414.1573V). The frequencies are 18, 114, and 240 MHz for polar magnetic fields of 10, 50, and 100 G, respectively. The polar-cap boundaries extend down to latitudes of $\pm 29^{\rm o}$, $\pm 48^{\rm o}$ , and $\pm 53^{\rm o}$, respectively. The magnetic field intensity at these latitudes is shown in the right axis of Figure \[fig.rm\]b. For comparison, at the Earth, the size of the polar cap is about $17$ to $20^{\rm o}$ [@2009AnGeo..27.2913M], that is, extending to latitudes of $\pm 70$ to $\pm 73^{\rm o}$, showing once more the extreme case of the magnetosphere of [V830Tau]{} b.
Stellar versus exoplanetary radio emissions {#sec.radiowind}
===========================================
If [V830Tau]{} b is indeed a magnetised planet, it can emit at radio wavelengths due to a physical process called electron cyclotron maser instability (Section \[sec.radiomodel\]). However, [V830Tau]{} b may not be the only radio emitter in this exoplanetary system. The hot plasma from the stellar wind can also emit at radio wavelengths, although through a different physical process. Here, we estimate radio emission from the host star’s wind.
Radio emission from the stellar wind
------------------------------------
A thermal ionised plasma emits bremsstrahlung (free-free) radiation across the electromagnetic spectrum. In the case of stellar winds, free-free radio emission is more intense in the innermost regions of the stellar wind, where the densities are higher . For large enough densities, these regions can become optically thick to radio wavelengths and, if a planet is embedded in this region, then it is possible that most of the planetary radio emission gets absorbed and does not escape.
A simple way to calculate the radio emission of stellar winds was presented in , who assume that the wind is spherically symmetric and isothermal with temperature $T$. These hypotheses are, however, not true in our simulations: the asymmetric distribution of surface magnetic fields generates an asymmetric stellar wind [@2014MNRAS.438.1162V]. We also adopt a polytropic wind model, such that the wind temperature is not constant. To take into account the asymmetries of the stellar wind, a radiative transfer calculation performed on each cell of the numerical grid and integrated along the line-of-sight would provide more accurate predictions for the radio-emitting wind. This detailed study will be done in a future work. In the present paper, we compute the free-free emission of the inner regions of the wind of [V830Tau]{} using the model developed by coupled to our simulation results. We caution that these computations should be considered as estimates of the radio emission.
The stellar wind radio flux density at a frequency $\nu$ is given by $$\begin{aligned}
\label{eq.Snu}
S_{\nu} = 10^{-29} A(\alpha) R_\star^2 \left[ 5.624{\times 10^{-28 }}I(\alpha) n_0^2 R_\star \right]^{\frac{2}{2\alpha -1}} \nonumber \\
\left( \frac{\nu}{10 \mathrm{GHz}} \right)^{\frac{-4.2}{2\alpha -1}+2}\left( \frac{T}{10^4 \mathrm{K}} \right)^{\frac{-2.7}{2\alpha -1} +1} \left( \frac{d}{1 \textrm{kpc}} \right)^{-2} \mathrm{mJy} ,
\end{aligned}$$ where the functions $I(\alpha)$ and $A(\alpha)$ are given by $$I(\alpha) = \int_{0}^{\pi/2} (\sin \theta)^{2(\alpha -1)} \mathrm{d} \theta ,$$ $$A(\alpha)=1 + 2 \sum_{j=1}^{\infty} (-1)^{j+1} \frac{\tau_c^j}{j! j (2 \alpha -1) - 2} ,$$ and $\tau_c=3$. The wind density is assumed to decay as a power-law with exponent $\alpha$ $$n = n_0 \left(\frac{R_\star}{r}\right)^\alpha ,$$ with $n_0$ being the wind base density. The distance $ R_{\nu}$ within which half of the emission $S_\nu$ is produced is $$\label{eq.rnu}
\frac{R_{\nu}}{R_\star} = \left[ 4.23 \times10^{-27} I(\alpha) n_0^2 R_\star \right]^{\frac{1}{2\alpha -1}} \left( \frac{\nu}{10 \mathrm{GHz}} \right)^{\frac{-2.1}{2\alpha -1}}\left( \frac{T}{10^4 \mathrm{K}} \right)^{\frac{-1.35}{2\alpha -1}} .$$ For the wind temperature $T$ in Equations (\[eq.Snu\]) and (\[eq.rnu\]), we use the temperature adopted at the wind base ($10^6$ K) and we use the base density $n_0=10^{12}$cm$^{-3}$ adopted in the model described in Section \[sec.windmodel\]. To estimate $\alpha$, we compute a power-law fit for regions within $r<20R_\star$, which resulted in $\alpha\simeq 3.08$.[^3] With this value of $\alpha$, the temperature and frequency dependencies of $R_\nu$ become $$\label{eq.r1}
R_\nu \propto \nu^{-0.40} T^{-0.26} \, ,$$ which is flatter than an emitting wind that has reached asymptotic speed ($R_\nu \propto \nu^{-0.7} T^{-0.45}$, ).
Figure \[fig.RSnu\]a shows the size of the radio-emitting region, where the dotted line indicates the orbital radius of [V830Tau]{} b. We also explore the effects of the wind mass-loss rates on the radio emission of the wind. For this estimate, we scale the wind densities of the model presented in Section \[sec.windmodel\], such that the mass-loss rates of our parametric study vary between $10^{-12}$ and $3{\times 10^{-9}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$. We remind readers that the model presented in Section \[sec.windmodel\] (our fiducial model) has ${\dot{M}}\sim 3{\times 10^{-9}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$. We note that $R_\nu$ gets smaller for higher frequencies and/or lower mass-loss rates. At 10 MHz and for ${\dot{M}}$ in the range $[3{\times 10^{-12}}, 3{\times 10^{-9}}]~{{\rm M}_\odot ~{\rm yr}^{-1}}$, the planet’s orbit is always within $R_\nu$, that is, the planet is embedded in the radio-emitting wind. For 100 MHz and in the same range of ${\dot{M}}$, $R_\nu \sim 6$ to $90~R_\star$ and $r_{\rm orb}\gtrsim R_\nu$ for ${\dot{M}}\lesssim 3{\times 10^{-12}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$. For 1000 MHz, $r_{\rm orb}> R_\nu$ for ${\dot{M}}\lesssim 3{\times 10^{-11}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$.
![(a) Size of the stellar wind radio-emitting region as a function of frequency for a range of stellar mass-loss rates: $3{\times 10^{-12}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$ to $3{\times 10^{-9}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$ from bottom to top solid lines. The dotted line indicates the orbital radius of [V830Tau]{} b. (b) The same as in (a), but for the flux density of the emitting wind, $S_\nu$. Circles are the upper limits for the radio emission of [V830Tau]{} b as presented by @bower.[]{data-label="fig.RSnu"}](fig4.png){width="\columnwidth"}
Regarding the flux density of the emitting wind, $S_\nu$ increases for higher frequencies and/or higher mass-loss rates. For all the range of ${\dot{M}}$ we investigated and the range of frequencies ($[1,6000]$ MHz), the flux density of the emitting wind never reaches the mJy level and $S_\nu \lesssim 0.1$ mJy. Given the low values of $S_\nu$ at low frequencies ($S_\nu \lesssim 10^{-5}$ mJy for $\nu \lesssim 100$ MHz), detecting radio emission from [V830Tau]{}’s wind at these frequencies with present-day technology is not feasible (Section \[sec.conclusion\]).
By comparing our radio emission estimates with the observations presented in @bower, we can place constraints on the mass-loss rates of [V830Tau]{}. At certain observing epochs, @bower detected [V830Tau]{} in a flaring state, while at other observing epochs, only upper limits for radio emission of [V830Tau]{} could be derived, namely, $<0.066$ and $<0.147$ mJy at 6 GHz (VLA observations) and $<0.117$ mJy at 8.4 GHz (VLBA). These upper limits are shown as circles in Figure \[fig.RSnu\]b. From that, we infer an upper limit of ${\dot{M}}\lesssim 3{\times 10^{-9}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$ for the mass-loss rate of [V830Tau]{}, which is consistent to the value used in our fiducial model (Section \[sec.windmodel\]).
Can radio emission from [V830Tau]{} b propagate through the stellar wind?
-------------------------------------------------------------------------
The planetary radio emission can only propagate in the stellar wind plasma if the (maximum) frequency of emission $\Omega_c = e B(\alpha_0)/(2 \pi m_e c)$ is larger than the stellar wind plasma frequency $\omega_p = [n_e e^2/ (\pi m_e)]^{1/2}$ everywhere along the propagation path. In these expressions, $n_e$ is the local electron density of the stellar wind ($n_e=n/2$ for a fully ionised hydrogen wind), $e$ and $m_e$ are the electron charge and mass, respectively, and $B(\alpha_0)$ is the planetary magnetic field at colatitude $\alpha_0$ (half-aperture of the polar-cap boundary). In the scenario in which the planetary emission propagates towards wind lower densities, the condition $\Omega_c > \omega_p$ is met when $$\label{eq.bp}
{B(\alpha_0)} > \left( \frac{n_e}{10^5 ~\textrm{cm}^{-3}} \right)^{1/2} \textrm{G},$$ where $n_e$ is taken as the electron density at the planetary orbit. Figure \[fig.BpMin\] shows the minimum planetary magnetic field intensity of [V830Tau]{} b required for the propagation of planetary radio emission through the wind of the host star, as a function of the stellar wind mass-loss rates.
![Minimum planetary magnetic field intensity required for the propagation of planetary radio emission through the wind of the host star (Equation \[eq.bp\]) as a function of stellar wind mass-loss rate. The right hand side axis shows the corresponding frequency of planetary emission (cyclotron).[]{data-label="fig.BpMin"}](fig5.png){width="\columnwidth"}
Mass-loss rates of weak-lined T Tauri stars have not been observationally constrained. It is expected that these winds are intermediate between those of classical T Tauri stars with dense winds ($\dot{M}\sim 10^{-10} - 10^{-8}~{{\rm M}_\odot ~{\rm yr}^{-1}}$, @1995ApJ...452..736H [@2016RAA....16b..10I]) and zero-age main sequence stars with less dense winds. Very active, main sequence stars can have mass-loss rates of $\dot{M}\sim 2\times 10^{-12}~{{\rm M}_\odot ~{\rm yr}^{-1}}$ [@2005ApJ...628L.143W]. In our work, we estimate an upper limit for the wind of [V830Tau]{} to be ${\dot{M}}\lesssim 3{\times 10^{-9}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$ (Figure \[fig.RSnu\]b). Given these values, we speculate that the most probable ${\dot{M}}$ for the winds of weak-lined T Tauri stars lies in the range $\sim 10^{-12}$ – $10^{-10}~{{\rm M}_\odot ~{\rm yr}^{-1}}$. We use this range of values for the estimates we present next.[^4]
For ${\dot{M}}\sim 10^{-12}$ – $10^{-10}~{{\rm M}_\odot ~{\rm yr}^{-1}}$, the size of the wind-emitting region ranges between $R_\nu \sim 3$ – 15 $R_\star$ at $275$ MHz and between $R_\nu \sim 5$ – 30 $R_\star$ at $50$ MHz (cf. Figure \[fig.RSnu\]a). At these frequencies, $R_\nu$ is comparable to the planetary orbital radius ($6.1~ R_\star$), implying that [V830Tau]{} b might be embedded in the regions of the stellar wind that are optically thick to radio wavelengths, although not so deeply embedded. Given that the radio flux we computed for the planet ($\sim 6$ – $24$ mJy, Figure \[fig.radio\]b) is several orders of magnitude larger that the stellar radio flux ($\sim 10^{-6}$ – $10^{-3}$ mJy, Figure \[fig.RSnu\]b), it is possible that, even after attenuation, a significant fraction of the planetary radio flux can escape. An accurate radiative transfer calculation of the amount of planetary flux that reaches the observer is left for a more detailed future study.
For the same range of ${\dot{M}}$, planetary radio emission can propagate through the stellar wind if the planetary magnetic field strength is $\gtrsim1.3$ – $13$ G (Figure \[fig.BpMin\]). These minimum values encompass Jupiter’s magnetic field intensities and appear to be physically reasonable. The equivalent (minimum) frequency of planetary emission is about $\gtrsim 4$ – $40$ MHz. At a frequency of $\sim 50$ MHz (see previous paragraph) the wind is (radio) optically thick out to $\sim 5$ to $30~R_\star$ and the planet is not so deeply embedded in the radio-emitting wind. Altogether, these conditions seem very encouraging for planetary radio emission to escape the stellar wind of [V830Tau]{} and be detected at Earth.
Discussion and Conclusions {#sec.conclusion}
==========================
Based on 3D MHD simulations, we presented here estimates of the radio emission expected for [V830Tau]{} b, the youngest detected exoplanet to date. It orbits an active 2 Myr-old weak-lined T Tauri star at a distance of $0.057$ au, which is about $6.1$ stellar radii. At this distance, the environment surrounding this exoplanet is quite harsh. Using the observationally reconstructed stellar magnetic field, we simulated the wind of this star, allowing us to infer the conditions of the external ambient medium surrounding [V830Tau]{} b. With these conditions, we then computed the planetary radio emission, using, as analogy, the radiometric Bode’s law derived for the magnetised planets in the solar system. According to this empirical law, there exists a relation between the dissipated power of the stellar wind impacting on the magnetosphere of the planet and the power released in the planetary emission. Our model uses simple approximations and should be considered as an initial attempt at calculating the radio flux of [V830Tau]{} b, which is by far currently the best target for detection of exoplanetary radio emission. Detecting [V830Tau]{} b at radio wavelengths would ultimately help us to refine our models.
We showed that, although the frequency of the radio emission is intimately related to the assumed magnetic field of the planet, the radio fluxes are only weakly dependent on that. The estimated flux densities from dissipated magnetic wind energy are on the order of $6$ mJy, with peaks at $11$ mJy, for an assumed planetary radius $R_p=R_{\rm jup}$. Given the youth of [V830Tau]{} b, it is likely that this is a lower limit for the planetary radius. Alternatively, for $R_p=2 R_{\rm jup}$, the radio fluxes we estimated peak at $44$ mJy, with average emission of $24$ mJy. If [V830Tau]{} b were to have a polar magnetic field intensity of 14.3 G (the maximum value of Jupiter’s magnetic field), this means that the emission would occur at a frequency of about 28 MHz, originating at latitudes of about 34$^{\rm o}$, where the magnetic field intensity is $\sim 10 $ G. Given that LOFAR sensitivity for a 1-h integration time at 20–40 MHz is 3–30 mJy [@2011RaSc...46.0F09G], we conclude that [V830Tau]{} b is an excellent target to look for exoplanetary radio emission with LOFAR. Other present-day instruments that have the potential to detect such radio fluxes in the low-frequency range would be the upgraded UTR-2 and GMRT . In the near future, the Square Kilometre Array (SKA)-low array system is expected to outperform LOFAR in terms of sensitivity limits for low frequency ranges [@2015aska.confE.120Z] and it would be an ideal tool for detecting exoplanets at radio wavelengths.
In @2012MNRAS.423.3285V [@2015MNRAS.449.4117V], the radio flux densities of several hot Jupiters (namely: $\tau$ Boo b, HD 46375b, HD 73256b, HD 102195b, HD 130322b, HD 179949b, and HD 189733b) were computed using the same method as here, that is, one in which the data-driven 3D MHD simulations provide the stellar wind characteristics used in the radio flux computations. Compared to the radio flux densities derived in those past works ($0.02$ to $1$ mJy), [V830Tau]{} b presents the best prospect for detecting radio emission (in spite of it not being the closest among the studied systems). This is because [V830Tau]{} is an active star with a large-scale surface magnetic field intensity reaching $700$G [@2015MNRAS.453.3706D; @2016Natur.534..662D], up to two orders of magnitude larger than the magnetism of the stars studied in @2015MNRAS.449.4117V. The latter are main-sequence solar-type stars, which are significantly older than [V830Tau]{}. In their youth, stars are more active and their magnetism more intense [@2014MNRAS.441.2361V].
In addition to the variability of radio emission seen along one planetary year (Figs. \[fig.rm\] and \[fig.radio\], see also @2010MNRAS.406..409F [@2010ApJ...720.1262V; @2012MNRAS.423.3285V; @2015MNRAS.449.4117V]), variation within a few years timescale is also expected due to the evolution of the host star magnetism. This means that, within a few years, the conditions of the stellar wind surrounding the planet are likely to change, causing also changes in exoplanetary radio emission. This could be challenging for radio detection, which will likely require some monitoring of the system.
The star itself may also contribute at radio frequencies (and potentially through flares that could make the planet modulation harder to detect). We computed thermal radio emission generated in the high density regions of the stellar wind. Comparing our estimated values with the non-detections reported in @bower, we were able to place a constraint in the mass-loss rate of [V830Tau]{}: ${\dot{M}}\lesssim 3{\times 10^{-9}}~{{\rm M}_\odot ~{\rm yr}^{-1}}$. We argued that the most likely values of ${\dot{M}}$ lie between $\sim 10^{-12}$ and $10^{-10}~{{\rm M}_\odot ~{\rm yr}^{-1}}$. With these values, the radio-emitting wind extends to distances of $\sim 3$ to $30~R_\star$ at frequencies of 50 to 275 MHz. This implies that the orbit of [V830Tau]{} b (at $6.1~R_\star$) might sit inside the radio-emitting wind, but does not seem to be deeply embedded. Given that the radio fluxes of the planet are estimated to be $\sim 3$ to 7 orders of magnitude larger than the radio flux of the stellar wind, it is possible that even after undergoing absorption by the stellar wind, a significant fraction of the planetary radio emission can escape the optically (radio) thick wind and reach the observer. Altogether, the [V830Tau]{} system is a very encouraging system for conducting radio observations from both the perspective of radio emission from the planet as well as from the host star’s wind. Equally important, the lack of detection of emission at radio wavelengths can place important constraints on the characteristics of the yet unobserved winds from weak-lined T Tauri stars.
Finally, exoplanetary radio emission may offer a new avenue for detecting young hot giants. The recent discovery of three of them ([V830Tau]{} b, K2-33b, and TAP26b) may, in fact, indicate that these planets are more frequent than their equivalents around mature Sun-like stars. However, the extreme activity levels of their hosts prevents the use of traditional planet-search techniques. Radio searches can thus boost the findings of young hot Jupiters. It should also give us the chance to probe their magnetic fields from the detected radio frequency, and to simultaneously study stellar winds and star-planet interactions.
This work used the BATS-R-US tools developed at the University of Michigan Center for Space Environment Modeling and made available through the NASA Community Coordinated Modeling Center. This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s516. JFD thanks the IDEX initiative of Université Fédérale Toulouse Midi-Pyrénées for awarding a Chaire d’Attractivité in the framework of which the study of V830 Tau was carried out. The authors would like to thank Manuel Guedel and the anonymous referee for their constructive comments, which greatly improved this manuscript.
[66]{} natexlab\#1[\#1]{}
, F. 1992, Annual Review of Earth and Planetary Sciences, 20, 289
, T. S., [Dulk]{}, G. A., & [Leblanc]{}, Y. 2000, , 545, 1058
, G. C., [Loinard]{}, L., [Dzib]{}, S., [et al.]{} 2016, ArXiv e-prints \[\]
, T. E. 2004, [Physics of Solar System Plasmas]{}, 495
, T. J., [Hillenbrand]{}, L. A., [Petigura]{}, E. A., [et al.]{} 2016, , 534, 658
, Jr., J.-D., [Vidotto]{}, A. A., [Petit]{}, P., [et al.]{} 2016, , 820, L15
, J.-F., [H[é]{}brard]{}, E., [Hussain]{}, G. A. J., [et al.]{} 2015, , 453, 3706
, J. F., [Moutou]{}, C., [Malo]{}, L., [et al.]{} 2016, , 534, 662
, R., [Donati]{}, J., [Moutou]{}, C., [et al.]{} 2010, , 406, 409
, W. M., [Desch]{}, M. D., & [Zarka]{}, P. 1999, , 104, 14025
, B., [G[ü]{}del]{}, M., [Mutel]{}, R. L., [et al.]{} 2017, ArXiv e-prints \[\]
, E. J., [Guedel]{}, M., & [Blake]{}, G. A. 2000, , 27, 501
, J.-M., [Motschmann]{}, U., [Mann]{}, G., & [Rucker]{}, H. O. 2005, , 437, 717
, J.-M., [Preusse]{}, S., [Khodachenko]{}, M., [et al.]{} 2007, , 55, 618
, J.-M., [Tabataba-Vakili]{}, F., [Stadelmann]{}, A., [Grenfell]{}, J. L., & [Atri]{}, D. 2016, , 587, A159
, J.-M., [Zarka]{}, P., & [Girard]{}, J. N. 2011, Radio Science, 46, 0
, M. 2002, , 40, 217
, G., [Sirothia]{}, S. K., [Antonova]{}, A., [et al.]{} 2013, , 762, 34
, P., [Edwards]{}, S., & [Ghandour]{}, L. 1995, , 452, 736
, S. L. G. & [Zarka]{}, P. 2011, , 531, A29
, T. W. 2001, , 106, 8101
, N. & [Itoh]{}, Y. 2016, Research in Astronomy and Astrophysics, 16, 010
, M. & [Cameron]{}, A. C. 2008, , 490, 843
, C. P., [Guedel]{}, M., [Brott]{}, I., & [L[ü]{}ftinger]{}, T. 2015, , 577, A28
, K. G., [Holmstr[ö]{}m]{}, M., [Lammer]{}, H., [Odert]{}, P., & [Khodachenko]{}, M. L. 2014, Science, 346, 981
, T. J. W. & [Farrell]{}, W. M. 2007, , 668, 1182
, T. J. W., [Farrell]{}, W. M., [Dietrick]{}, J., [et al.]{} 2004, , 612, 511
, A., [Sirothia]{}, S. K., [Gopal-Krishna]{}, & [Zarka]{}, P. 2013, , 552, A65
, J. & [White]{}, S. M. 1996, , 462, L91+
, R., [Lammer]{}, H., & [Ribas]{}, I. 2007, , 129, 245
, S. E., [Hutchinson]{}, J., [Boakes]{}, P. D., & [Hubert]{}, B. 2009, Annales Geophysicae, 27, 2913
, J. D. & [Milan]{}, S. E. 2016, , 461, 2353
, N. & [Felli]{}, M. 1975, , 39, 1
, K. G., [Roe]{}, P. L., [Linde]{}, T. J., [Gombosi]{}, T. I., & [de Zeeuw]{}, D. L. 1999, , 154, 284
, S., [Kopp]{}, A., [B[ü]{}chner]{}, J., & [Motschmann]{}, U. 2005, , 434, 1191
, V. B., [Vavriv]{}, D. M., [Zarka]{}, P., [et al.]{} 2010, , 510, A16
, V. B., [Zarka]{}, P., & [Ryabov]{}, B. P. 2004, , 52, 1479
, A., [H[é]{}brard]{}, G., [Lillo-Box]{}, J., [et al.]{} 2016, , 824, 55
, J., [Grambusch]{}, T., [Duling]{}, S., [Neubauer]{}, F. M., & [Simon]{}, S. 2013, , 552, A119
, V., [Jardine]{}, M., [Vidotto]{}, A. A., [et al.]{} 2014, , 570, A99
, E., [Bohlender]{}, D. A., [Walker]{}, G. A. H., & [Collier Cameron]{}, A. 2008, , 676, 628
, S. K., [Lecavelier des Etangs]{}, A., [Gopal-Krishna]{}, [Kantharia]{}, N. G., & [Ishwar-Chandra]{}, C. H. 2014, , 562, A108
, G. L. & [Chen]{}, C.-K. 1975, , 80, 4675
, A., [Brun]{}, A. S., [Matt]{}, S. P., & [R[é]{}ville]{}, V. 2015, , 815, 111
, J. A., [Cottrell]{}, R. D., [Watkeys]{}, M. K., [et al.]{} 2010, Science, 327, 1238
, G., [van der Holst]{}, B., [Sokolov]{}, I. V., [et al.]{} 2012, Journal of Computational Physics, 231, 870
, A. A., [Fares]{}, R., [Jardine]{}, M., [et al.]{} 2012, , 423, 3285
, A. A., [Fares]{}, R., [Jardine]{}, M., [Moutou]{}, C., & [Donati]{}, J.-F. 2015, , 449, 4117
, A. A., [Gregory]{}, S. G., [Jardine]{}, M., [et al.]{} 2014, , 441, 2361
, A. A., [Jardine]{}, M., & [Helling]{}, C. 2010, , 722, L168
, A. A., [Jardine]{}, M., & [Helling]{}, C. 2011, , 414, 1573
, A. A., [Jardine]{}, M., [Morin]{}, J., [et al.]{} 2014, , 438, 1162
, A. A., [Opher]{}, M., [Jatenco-Pereira]{}, V., & [Gombosi]{}, T. I. 2009, , 699, 441
, A. A., [Opher]{}, M., [Jatenco-Pereira]{}, V., & [Gombosi]{}, T. I. 2010, , 720, 1262
, J., [Hallinan]{}, G., [Bourke]{}, S., [Guedel]{}, M., & [Rupen]{}, M. 2014, , 788, 112
, E. J. & [Davis]{}, L. J. 1967, , 148, 217
, B. E. 2004, Living Reviews in Solar Physics, 1, 2
, B. E., [M[ü]{}ller]{}, H.-R., [Zank]{}, G. P., [Linsky]{}, J. L., & [Redfield]{}, S. 2005, , 628, L143
, A. E. & [Barlow]{}, M. J. 1975, , 170, 41
, L., [Donati]{}, J.-F., [H[é]{}brard]{}, E. M., [et al.]{} 2017, \[\]
, P. 2007, , 55, 598
, P. 2010, in Astronomical Society of the Pacific Conference Series, Vol. 430, Pathways Towards Habitable Planets, ed. V. [Coud[é]{} du Foresto]{}, D. M. [Gelino]{}, & I. [Ribas]{}, 175
, P., [Cecconi]{}, B., & [Kurth]{}, W. S. 2004, (Space Physics), 109, 9
, P., [Lazio]{}, J., & [Hallinan]{}, G. 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), 120
, P., [Treumann]{}, R. A., [Ryabov]{}, B. P., & [Ryabov]{}, V. B. 2001, , 277, 293
, J. I., [Bustamante]{}, S., [Cuartas]{}, P. A., & [Hoyos]{}, J. H. 2013, , 770, 23
Detailed derivation of planetary radio flux {#sec.apA}
===========================================
In this work, we assume that radio emission arises near the colatitude $\alpha_0$ of the open-closed magnetic field line boundary; we refer to the region of open field lines as the polar cap.[^5] We note that the colatitudes of the polar cap boundary and that of the auroral ring might not match exactly [@1975JGR....80.4675S; @2001JGR...106.8101H], so this is to be considered as a first-order approximation. The half-aperture of the polar cap boundary $\alpha_0$ can be related to the size of the planet magnetosphere as [@1975JGR....80.4675S; @2010Sci...327.1238T; @2011MNRAS.414.1573V; @2013ApJ...770...23Z] $$\label{eq.alpha0}
\alpha_0 = \arcsin{[ (R_p/r_M)^{1/2}} ].$$ The (dipolar) planetary magnetic field at this colatitude $\alpha_0$ is $$\label{eq.balpha0}
B(\alpha_0) = \frac{B_p}{2} (1+3\cos^2 \alpha_0)^{1/2},$$ which corresponds to a maximum electron cyclotron frequency $$\label{eq.fcyc}
f_c = \frac{e B(\alpha_0) }{2 \pi m_e c},$$ where $m_e$ and $e$ are the electron mass and charge, and $c$ is the speed of light (Fig. \[fig.function\_variation\]). Here, we assume that the emission bandwidth $\Delta f$ is approximately the cyclotron frequency : $$\label{eq.deltafcyc}
\Delta f = \frac{e B(\alpha_0) }{2 \pi m_e c} = 2.8 \left( \frac{B(\alpha_0)}{1~{\rm G}}\right) ~{\rm MHz} .$$ The radio flux is related to the radio power as $$\label{eq.radio}
\phi_{\rm radio} = \frac{P_{\rm radio}}{d^2 \omega \Delta f} \, ,$$ where $d$ is the distance to the system and $$\label{eq.omega}
\omega= 2\times \int_{\alpha_0-\delta\alpha/2}^{\alpha_0+\delta\alpha/2} \sin \alpha {\rm d}\alpha {\rm d}\varphi = 2\times 2 \pi [\cos(\alpha_0-\delta\alpha/2) - \cos (\alpha_0+\delta\alpha/2)]$$ is the solid angle of the hollow cone where emission is arising (the factor of two was included in order to account for emission coming from both Northern and Southern hemispheres). The thickness of the hollow cone is assumed to be $\delta\alpha=17.5^{\rm o}$ [@2004JGRA..10909S15Z].
Flux released in the dissipation of the stellar wind kinetic power
-------------------------------------------------------------------
The dissipated kinetic power of the impacting wind on the planet is approximated as the ram pressure of the wind $\rho (\Delta u)^2$ impacting in the magnetospheric cross section $\pi r_M^2$, at a (relative) velocity $\Delta u$ $$\label{eq.pwrK}
P_k \simeq \rho (\Delta u)^3 \pi r_M^2 .$$ The flux that would be emitted when the kinetic power of the wind gets dissipated in the magnetospheric cross section of the planet - if the radio-magnetic scaling law is effective - can be written as $$\phi_{\rm radio,kin} = \frac{P_{\rm radio}}{d^2 \omega \Delta f} = \frac{\eta_k P_{\rm k}}{d^2 \omega \Delta f} =\frac{\eta_k \rho (\Delta u)^3 \pi r_M^2}{d^2 \omega \Delta f} ,$$ where we used Equations (\[eq.radio\]) and (\[eq.pwrK\]). Using Eqs. (\[eq.alpha0\]), (\[eq.balpha0\]), (\[eq.deltafcyc\]), and (\[eq.omega\]), the previous expression becomes $$\label{eq.phiradio}
\phi_{\rm radio,kin} = \frac{\eta_k \rho (\Delta u)^3 R_p^2 [\cos(\alpha_0-\delta\alpha/2) - \cos (\alpha_0+\delta\alpha/2)]^{-1} }{d^2 2 e/[2 \pi m_e c] {B_p} (1+3\cos^2 \alpha_0)^{1/2}\sin^4{\alpha_0}} \, .$$ We note that $\alpha_0$ correlates to $B_p$ through Equations (\[eq.alpha0\]) and (\[eq.rM\]) as $$\label{eq.sinalpha}
B_p = \frac{2 \sqrt{8\pi p_{\rm tot}}}{\sin^6{\alpha_0}} ,$$ where $p_{\rm tot}={{(\rho \Delta u^2} + p+ B^2/8\pi)}$ is the total pressure of the ambient medium external to the planet. Therefore, Eq. (\[eq.phiradio\]) becomes $$\begin{aligned}
\label{eq.phiradio2}
\phi_{\rm radio,kin} &=& \frac{\eta_k \rho (\Delta u)^3 R_p^2 \sin^2{\alpha_0}[\cos(\alpha_0-\delta\alpha/2) - \cos (\alpha_0+\delta\alpha/2)]^{-1}}{d^2 4 e/[2 \pi m_e c ] {\sqrt{8\pi p_{\rm tot}}} (1+3\cos^2 \alpha_0)^{1/2}} \nonumber \\
&=& \eta_k \frac{2\pi m_e c}{4 e} \frac{R_p^2}{d^2 } \frac{\rho (\Delta u)^3}{\sqrt{8\pi p_{\rm tot}}}f(\alpha_0) \, \end{aligned}$$ with $$\label{eq.falpha}
f(\alpha_0) = \frac{\sin^2\alpha_0}{ [\cos(\alpha_0-\delta\alpha/2) - \cos (\alpha_0+\delta\alpha/2)] (1+3\cos^2 \alpha_0)^{1/2}} \, .$$
We note that the planet magnetic field $B_p$ is no longer explicit in Eq. (\[eq.phiradio2\]); the dependence of the $\phi_{\rm radio,kin}$ with $B_p$ is hidden in $\alpha_0$ (Eq. \[eq.sinalpha\]) and, consequently, in $f(\alpha_0)$. Figure \[fig.function\_variation\] shows the variation of $f$ as a function of a given co-latitude of the polar-cap boundary $\alpha_0$. As can be seen, this function varies between $0$ and $3.3$ for any aperture angle of the polar-cap boundary.
![Black solid curve: variation of $f$ (Eq. \[eq.falpha\]) as a function of the planetary co-latitude of the polar-cap boundary $\alpha_0$. The top axis indicates the (non-linear) conversion from $\alpha_0$ to a normalised magnetospheric size following Eq. (\[eq.alpha0\]). Red dashed curve: the emission frequency at the polar-cap boundary (Eq. \[eq.fcyc\], right axis) for a magnetic field intensity at the pole of 14.3 G. \[fig.function\_variation\]](figA1.png){width="\columnwidth"}
If we then group all the constants of Equation (\[eq.phiradio2\]) into $$\label{eq.etakprime}
\eta_k^\prime = \eta_k \frac{2\pi m_e c}{4 e \sqrt{8\pi}} \simeq \eta_k 1.8{\times 10^{-8}} \textrm{\, [cgs~units]},$$ the radio flux due to the impacting wind (kinetic power) simplifies to $$\label{eq.phiradio3}
\phi_{\rm radio,kin} = \eta_k^\prime \frac{R_p^2}{d^2 }\frac{\rho (\Delta u)^3}{{{ p_{\rm tot}}^{1/2}}} f(\alpha_0),$$ which is a function of the angular size of the planet ($R_p/d$), the properties of the ambient medium surrounding the planet (i.e. the stellar wind), and $f(\alpha)$.
Flux released in the dissipation of the stellar wind Poynting flux
------------------------------------------------------------------
The magnetic power $P_B$ can be estimated as the Poynting flux of the stellar wind impacting on the planetary magnetospheric cross-section $S$ $$\label{eq.pwrB}
P_B = \int c \frac{{\bf E}\times {\bf B}}{4\pi} \cdot {\rm d}{\bf S} \simeq \frac{B_{\perp}^2 }{4\pi} (\Delta u) \pi r_M^2\, ,$$ where the electric field is ${\bf E} = - \Delta \mathbf{u} \times {\bf B}/c $ and $B_\perp$ is the perpendicular component of the stellar magnetic field to the velocity of the wind particles at the planet’s referencial frame. It can be written as $$B_\perp^2 = B^2 - B_\parallel^2 = B^2 - \left( \frac{\mathbf{B} \cdot \Delta \mathbf{u}}{|\Delta \mathbf{u}|}\right)^2 .$$ The power dissipated in interaction is often ascribed as a fraction $\epsilon$ of Eq. (\[eq.pwrB\]), with $\epsilon \sim 0.1$ to $0.2$ . However, this efficiency factor is often included in the radiometric Bode’s law by means of the $\eta_B$ parameter, such that $P_{\rm radio}=\eta_B P_{\rm B}$, with $\eta_B = 2{\times 10^{-3}} $ and $P_{\rm B}$ as in Eq. (\[eq.pwrB\]).
The flux that is emitted with the dissipation of wind magnetic power can be written as $$\phi_{\rm radio, mag} = \frac{P_{\rm radio}}{d^2 \omega \Delta f} = \frac{\eta_B P_{\rm B}}{d^2 \omega \Delta f} = \frac{\eta_B B_{\perp}^2 (\Delta u) \pi r_M^2}{ 4\pi d^2 \omega \Delta f} ,$$ where we used Eqs. (\[eq.pwrB\]) and (\[eq.radio\]). Using Eqs. (\[eq.alpha0\]), (\[eq.balpha0\]), (\[eq.deltafcyc\]), (\[eq.omega\]), and (\[eq.sinalpha\]), the previous expression becomes $$\begin{aligned}
\label{eq.phiradio_B}
\phi_{\rm radio, mag} &=& \frac{\eta_B B_\perp^2 (\Delta u) [\cos(\alpha_0-\delta\alpha/2) - \cos (\alpha_0+\delta\alpha/2)]^{-1} R_p^2 \sin^2{\alpha_0}}{4\pi d^2 4 e/[2\pi m_e c] {\sqrt{8\pi p_{\rm tot}}} (1+3\cos^2 \alpha_0)^{1/2}} \nonumber \\
&=& \eta_B \frac{2\pi m_e c}{4 e} \frac{R_p^2}{d^2 } \frac{B_\perp^2 (\Delta u) }{4\pi \sqrt{8\pi p_{\rm tot}}} f(\alpha_0).\end{aligned}$$ The magnitude of the planetary magnetic field is again embedded in $f(\alpha_0)$ (*cf* Fig. \[fig.function\_variation\]). If we then group all the constants of Eq. (\[eq.phiradio\_B\]) into $$\label{eq.etaBprime}
\eta_B^\prime = \eta_B \frac{2\pi m_e c}{4 e \sqrt{8\pi}} \frac{1}{4\pi} \simeq \eta_B 1.4{\times 10^{-9}} \textrm{\, [cgs~units]},$$ the radio flux due to the impacting wind (magnetic power) simplifies to $$\label{eq.phiradio_B_2}
\phi_{\rm radio, mag} = \eta_B^\prime \frac{R_p^2}{d^2 }\frac{B_\perp^2 (\Delta u)}{{{ p_{\rm tot}}^{1/2}}} f(\alpha_0),$$ which is a function of the angular size of the planet ($R_p/d$), the properties of the ambient medium surrounding the planet (i.e. the stellar wind), and $f(\alpha)$.
Effects of wind base density in our predicted radio emission {#sec.apB}
============================================================
In our simulations of [V830Tau]{} wind, we noted in Section \[sec.windmodel\] that the base density $n_0$ of the wind is a free parameter that affects the computed mass-loss rates and that, with our choice of $n_0$, our computed mass-loss rates were likely near an upper limit of the expected mass-loss rates for weak-lined T Tauri stars. Next, we verify how the radio fluxes estimated in this paper would have changed if the wind densities were to decrease by one to two orders of magnitude, bringing down the wind mass-loss rates of [V830Tau]{} to $\sim 10^{-11}$ – $10^{-10}~{{\rm M}_\odot ~{\rm yr}^{-1}}$.
With the wind densities assumed in this paper, the planet orbit changes from a magnetically-dominated region to a ram-pressure dominated region, as shown in Fig. \[fig.wind\]. A decrease in wind density by a factor of $10$ – $100$ would imply that the planet’s orbit would be completely inside the magnetically-dominated region (sub-Alfvénic). We remind readers also that, even if a given stellar wind is not dominated by the magnetic pressure, $\phi_{\rm radio, mag}$ can still dominate over $\phi_{\rm radio, kin}$ because of the $3200 \pi^2$ factor in Eq. (\[eq.ratio\]).
To simplify our following estimates, we study two limiting cases: (i) one in which the ram pressure dominates the stellar wind total pressure (similar to the conditions surrounding the Earth) and (ii) one in which the magnetic pressure is the dominating one (likely to be the condition surrounding [V830Tau]{} b). In case (i), we have that $p_{\rm tot} \to \rho (\Delta u)^2$, and Eqs. (\[eq.phiradio3\]) and (\[eq.phiradio\_B\_2\]) become $$\label{eq.phiradio4}
\phi_{\rm radio, kin} \to \eta_k^\prime \frac{R_p^2}{d^2 }{\rho^{1/2} (\Delta u)^2} f(\alpha_0)\,\,\,\, \textrm{[ram-pressure~dominated]}$$ and $$\label{eq.phiradio_B_3}
\phi_{\rm radio, mag} \to \eta_B^\prime \frac{R_p^2}{d^2 }\frac{B_\perp^2}{\rho^{1/2}} f(\alpha_0) \,\,\,\, \textrm{[ram-pressure~dominated]}.$$ That is, $\phi_{\rm radio, kin}$ becomes proportional to the square root of the stellar wind density, and $\phi_{\rm radio, mag}$ becomes inversely proportional to $\rho^{1/2}$. In case (ii), where the magnetic pressure of the stellar wind dominates the total pressure ($p_{\rm tot} \to B^2/(8\pi)$), Eqs. (\[eq.phiradio3\]) and (\[eq.phiradio\_B\_2\]) become $$\label{eq.phiradio5}
\phi_{\rm radio, kin} \to \eta_k^\prime \frac{R_p^2}{d^2 }\sqrt{8\pi}\frac{\rho (\Delta u)^3}{B} f(\alpha_0)\,\,\,\, \textrm{[magnetic-pressure~dominated]},$$ and $$\label{eq.phiradio_B_4}
\phi_{\rm radio, mag} \to \eta_B^\prime \frac{R_p^2}{d^2 } \sqrt{8\pi} B \Delta u f(\alpha_0)\,\,\,\, \textrm{[magnetic-pressure~dominated]} .$$ That is, $\phi_{\rm radio, kin}$ becomes linearly proportional to the stellar wind density, while $\phi_{\rm radio, mag}$ becomes independent of that.
Therefore, for planets orbiting farther out (in ram-pressure dominated regions), the stellar wind densities can either increase or decrease radio emission depending on whether it is the kinetic or magnetic flux of the impacting wind that is converted into radio power. On the other hand, for close-in planets orbiting around highly magnetised stars (in the magnetic-pressure dominated region), the radio flux becomes roughly independent of the density, being mainly affected by the local conditions of the stellar magnetic field and the relative velocity of the planet through the stellar wind (Eq. \[eq.phiradio\_B\_4\]). Since the latter condition is the most likely for [V830Tau]{} b, we conclude that the stellar wind density, an unknown in our numerical simulations, plays a minor role in our estimates of planetary radio fluxes.
[^1]: In particular, @2010ASPC..430..175Z [@2015aska.confE.120Z] found that the scaling between the magnetic power of the incident stellar wind on the magnetospheric cross-section and the radio power apply also to any plasma flow-obstacle interaction, including interactions between Io and Ganymede with Jupiter and the magnetised binary system V711 Tau. This generalised radio-magnetic Bode’s law covers about 13 orders of magnitude in stellar wind magnetic power and radio power.
[^2]: The right hand side of Eq. (\[eq.rM\]) is often multiplied by a correction factor $2^{1/3}$ used to account for the effects of currents [e.g. @2004pssp.book.....C].
[^3]: Note that, in the case of a stellar wind that has reached asymptotic wind speed, $\alpha=2$. This reduces, for example, to the case presented in .
[^4]: The non-detection of radio emission from stellar winds can be a useful way to observationally constrain mass-loss rates in solar-type stars, which have rarefied winds [e.g. @2000GeoRL..27..501G; @2014ApJ...788..112V; @2017arXiv170208393F].
[^5]: Although we refer to $\alpha_0$ as the “polar cap boundary”, we note that, due to the harsh conditions imposed by the stellar wind environment, the magnetospheric sizes might be small and, therefore, the polar cap might occupy a large area of the surface of the star (i.e. not only immediately around the poles, as happens for instance with the polar cap area of the Earth).
|
---
abstract: |
Coupled-channel three-body calculations of an $I=1/2$, $J^{\pi}=0^-$ $\bar{K}NN$ quasi-bound state in the $\bar{K}NN -
\pi \Sigma N$ system were performed and the dependence of the resulting three-body energy on the two-body $\bar{K}N - \pi
\Sigma$ interaction was investigated. Earlier results of binding energy $B_{K^-pp} \sim 50 -70$ MeV and width $\Gamma_{K^-pp} \sim
100$ MeV are confirmed \[N.V. Shevchenko [*et al.*]{}, Phys. Rev. Lett. [**98**]{}, 082301 (2007)\]. It is shown that a suitably constructed energy-independent complex $\bar{K}N$ potential gives a considerably shallower and narrower three-body quasi-bound state than the full coupled-channel calculation. Comparison with other calculations is made.
author:
- 'N.V. Shevchenko[^1]'
- 'A. Gal'
- 'J. Mareš'
- 'J. Révai'
title: '$\bar{K}NN$ quasi-bound state and the $\bar{K}N$ interaction: coupled-channel Faddeev calculations of the $\bar{K}NN - \pi \Sigma N$ system'
---
Introduction
============
Hadronic nuclei are useful tools for studying hadron-nucleon interactions and in-medium properties of hadrons. The recent interest in kaonic nuclei was motivated by the strongly attractive antikaon-nucleus density-dependent optical potentials obtained from $K^-$ atomic data fits [@Batty]. Akaishi and Yamazaki [@Akaishi1] using G-matrix one-channel $\bar{K}N$ interactions, predicted the existence of deep and narrow $K^-$ bound states in ${}^3$He, ${}^4$He, and ${}^8$Be. Of particular interest is the lightest possible antikaon-nucleus system, $K^- p p$, for which these authors calculated in Ref. [@Akaishi2] values of 48 MeV and 61 MeV for the total binding energy and the decay width, respectively. Deeply bound kaonic states were searched in $^4$He$(K^-, N)$ reactions at KEK, with negative results so far [@Iwasaki], and by the FINUDA spectrometer collaboration at DA$\Phi$NE [@FINUDA] in stopped $K^-$ reactions on nuclear targets such as lithium and carbon. The latter experiment suggested evidence for a bound state $K^- p p$ ‘observed’ through its decay into approximately back-to-back $\Lambda$-proton pairs. The deduced binding energy (115 MeV), but not the width (67 MeV), differs considerably from the theoretical prediction of Ref. [@Akaishi2]. However, this interpretation of the measured $\Lambda$-proton spectrum in the FINUDA experiment was challenged by Magas [*et al.*]{} [@Oset] who also criticized the Yamazaki-Akaishi calculations [@Akaishi2] for using an effective $\bar{K}$ optical potential in lieu of genuine $\bar{K} N$ interactions.
The near-threshold $\bar{K}N$ interaction is mainly affected by the sub-threshold $I=0$ resonance $\Lambda(1405)$, which is usually assumed a $\bar{K}N$ bound state and a resonance in the $\pi \Sigma$ channel. Numerous theoretical works were devoted to constructing $\bar{K}N$ interactions within K-matrix models, dispersion relations, meson-exchange models, quark models, cloudy bag-models, and more recently by applying SU(3) meson-baryon chiral perturbation theory (see e.g. the recent review papers [@Oller; @Weise]). Scattering experiments for $K^- p$ are rather old and the data are not too accurate. Kaonic hydrogen provides additional information. Namely, there are two experimental measurements of the $1s$ level shift and width caused by the strong interaction, performed at KEK [@KEK] and recently by the DEAR collaboration at DA$\Phi$NE, Frascati [@DEAR]. The measured upward shift appears as due to a repulsive strong interaction, but in fact it is caused by an attractive interaction in the $I=0$ $\bar{K}N - \pi \Sigma$ channel, which is strong enough to generate a quasi-bound strong-interaction state. The effect of such a strong attractive interaction is to push the purely Coulomb level upwards. Using the Deser formula [@Deser], it is possible to obtain the $K^- p$ scattering length from the value of the $1s$ level energy shift. Unfortunately, several recent theoretical models could not simultaneously reproduce the DEAR value of the $K^- p$ scattering length together with the bulk of $K^-p$ scattering data [@Borasoy].
As should be clear from this brief introduction, the fields of $\bar{K}N$ and $\bar{K}$–nucleus interaction are abundant with open questions and problems. The elucidation of $\bar{K}$– nuclear properties would help considerably to derive significant information on the in-medium $\bar{K}N$ interaction and on the possibility of kaon condensation in dense nuclear matter, see Refs. [@kondens1; @kondens2] and previous works cited therein. Among $\bar{K}$– nuclear systems, the study of three-body ‘exotic’ systems offers the advantage that Faddeev equations [@Faddeev], which exactly describe the dynamics of few particles, provide a proper theoretical and computational framework. In the present work, we have generalized the Faddeev equations in the Alt-Grassberger-Sandhas form [@AGS] in order to include additional ‘particle’ channels and thus performed the first genuinely three-body ${\bar K}NN - \pi \Sigma N$ coupled-channel Faddeev calculation in search for quasi-bound states in the $K^- p p$ system. A preliminary report of this work was given in Ref. [@ourPRL]. The present paper provides a more detailed and complete version of the previous one, especially concerning the dependence of the three-body results on the two-body input. The main result of Ref. [@ourPRL] is reconfirmed, namely that a single $K^-pp$ $I=1/2,~J^{\pi}=0^-$ quasi-bound state exists with binding energy $B \sim 50 - 70$ MeV and width $\Gamma \sim 100$ MeV. It is shown that ‘equivalent’ single-channel ${\bar K}NN$ calculations of the type reported by Yamazaki and Akaishi [@Akaishi2] underestimate considerably the binding energy, and particularly the width resulting within the full ${\bar K}NN - \pi \Sigma N$ coupled-channel calculations.
The paper is organized as follows: in Section II we describe the derivation of the coupled-channel Faddeev equations in the AGS form. The two-body potentials which enter these equations are described in Section III. Results are given in Section IV for the full coupled-channel calculations, along with suitably chosen single-channel calculations that could provide a testground for comparison with the single-channel calculation of Ref. [@Akaishi2]. Conclusions are given in Section V.
Formalism
=========
Three-body Faddeev equations [@Faddeev] in the Alt-Grassberger-Sandhas (AGS) form [@AGS] $$\nonumber
U_{ij} = (1-\delta_{ij}) G_0^{-1} + \sum_{k=1}^3 (1-\delta_{ik}) \,
T_k \, G_0 \, U_{kj} \\
\label{AGS}$$ define unknown operators $U_{ij}$, describing the elastic and re-arrangement processes $j + (ki) \to i + (jk)$. The inputs for the AGS system of equations (\[AGS\]) are two-body $T$-matrices, immersed into three-body space. The operator $G_0$ is the free three-body Green’s function. Faddeev partition indices $i,j = 1,2,3$ denote simultaneously an interacting pair and a spectator particle. When the initial state is known, as is usually assumed, the system (\[AGS\]) consists of three equations.
The AGS equations are quantum-mechanical ones, describing processes in which the number and composition of particles are fixed. However, the two-body $\bar{K}N$ interaction, which is essential for the $K^- pp$ quasi-bound state calculation, is strongly coupled to other channels, particulary to the $\pi
\Sigma$ channel via $\Lambda(1405)$ . To take the $\bar{K}N - \pi
\Sigma$ coupling directly into account (we neglect the weaker coupled $I=1$ $\pi \Lambda$ channel), it is necessary to extend the formalism of Faddeev equations. To this end it is assumed that in addition to the usual Faddeev channels, which represent different partitions of the same set of particles, there are also ‘particle’ channels. Each of the three ‘particle’ channels consists of three usual Faddeev partitions (here we treat the two nucleons as distinguishable particles, with proper antisymmetrization introduced at a later stage). Thus, all three-body operators will have ‘particle’ indices ($\alpha$) for each state in addition to the usual Faddeev indices ($i$), see Table \[channels.tab\].
-------------------------- ----------------------- ------------------------------------------- -------------------------------------------
$i$ $\setminus$ $\alpha$ $1$ ($\bar{K}NN$) $2$ ($\pi \Sigma N$) $3$ ($\pi N \Sigma$)
\[\] 1 $NN_{\, I=0,1}$ $\Sigma N_{\, I=\frac{1}{2},\frac{3}{2}}$ $\Sigma N_{\, I=\frac{1}{2},\frac{3}{2}}$
2 $\bar{K}N_{\, I=0,1}$ $\pi N_{\, I=\frac{1}{2},\frac{3}{2}}$ $\pi \Sigma_{\, I=0,1}$
3 $\bar{K}N_{\, I=0,1}$ $\pi \Sigma_{\, I=0,1}$ $\pi N_{\, I=\frac{1}{2},\frac{3}{2}}$
\[\]
-------------------------- ----------------------- ------------------------------------------- -------------------------------------------
: Interacting two-body subsystems for three partition ($i$) and three ‘particle’ channel ($\alpha$) indices. The interactions are further labelled by the two-body isospin values, entering the AGS equations with total three-body isospin $I=1/2$.[]{data-label="channels.tab"}
All operators in Eq. (\[AGS\]) now act in this additional ‘particle’ space: $T_i$ transform to $T_i^{\alpha \beta}$, $G_0
\to G_0^{\alpha \beta}$, and $U_{ij} \to U_{ij}^{\alpha \beta}$ ($\alpha, \beta = 1, 2, 3$). The two-body $T$-matrices have the following form: $$\label{T3x3}
T_1 \to \left(
\begin{tabular}{ccc}
$T_1^{NN}$ & 0 & 0 \\
0 & $T_1^{\Sigma N}$ & 0 \\
0 & 0 & $T_1^{\Sigma N}$
\end{tabular}
\right), \quad
T_2 \to \left(
\begin{tabular}{ccc}
$T_2^{KK}$ & 0 & $T_2^{K \pi}$ \\
0 & $T_2^{\pi N}$ & 0 \\
$T_2^{\pi K}$ & 0 & $T_2^{\pi \pi}$
\end{tabular}
\right), \quad
T_3 \to \left(
\begin{tabular}{ccc}
$T_3^{KK}$ & $T_3^{K \pi}$ & 0 \\
$T_3^{\pi K}$ & $T_3^{\pi \pi}$ & 0 \\
0 & 0 & $T_3^{\pi N}$
\end{tabular}
\right) \,,$$ where $T_i^{NN}$, $T_i^{\pi N}$ and $T_i^{\Sigma N}$ are the usual one-channel two-body $T$-matrices in three-body space, describing $NN$, $\pi N$, and $\Sigma N$ interactions, respectively. The elements of the coupled-channel $T$-matrix, $T_i^{KK}$, $T_i^{\pi \pi}$, $T_i^{\pi K}$, and $T_i^{K \pi}$, are labelled by two meson indices: $$\begin{aligned}
T_i^{KK}:& \qquad \bar{K} + N & \to \bar{K} + N \\
T_i^{\pi K}:& \qquad \bar{K} + N & \to \pi + \Sigma \\
T_i^{K \pi}:& \qquad \pi + \Sigma & \to \bar{K} + N \\
T_i^{\pi \pi}:& \qquad \pi + \Sigma & \to \pi + \Sigma ~.\end{aligned}$$ The free Green’s function is diagonal in channel indices: $G_0^{\alpha \beta} = \delta_{\alpha \beta} \, G_0^{\alpha}$, while the transition operators $U_{ij}^{\alpha \beta}$ have the most general form.
Searching for quasi-bound states assumes working at low energies. Low-energy interactions are satisfactorily described by $s$-waves, hence for all the relevant two-body interactions we use $L_i=0$. The total orbital angular momentum is then $L=0$. For the $K^- pp$ system, the total spin is $S=0$ due to the spin zero of the two protons and spin zero of the $K^-$ meson. All two-body baryon-baryon interactions are then spin-zero interactions. The remaining quantum number is isospin. It is possible to work in either particle or isospin basis, but since the Coulomb interaction is not included in the present calculation and charge independence is assumed for all two-body interactions, it is quite natural to choose the isospin basis. The total isospin $I$ is a conserved quantum number for charge-independent interactions, so a bound (or a quasi-bound) state must have a definite value of $I$. For $I=1/2$ there are two possible (unadmixed) states corresponding to the total spin $S$ of the system. In the $\bar{K}NN - \pi \Sigma N$ case $S$ coincides with the spin of the two baryons ($S_i=0,1$) and due to their indistinguishability the spin value also fixes the isospin of the two nucleons, $I_{NN}=1,0$, respectively. In these states – let us call them $pp$- and $d$-configuration – a more attractive combination of $\bar{K}N$ $I=0,1$ forces and a weaker $NN$ singlet force in the $pp$ is competing with a weaker $\bar{K}N$ attraction and a stronger $NN$ triplet force in $d$. Therefore it is not clear a priori, which of them has a lower energy. We have chosen to calculate the $I=1/2$, $S=0$ $pp$ configuration due to its connection to experiment. Moreover, simple isospin re-coupling arguments indicate, that it might have a lower energy. However, a similar calculation should be performed for the other, $I=1/2$, $S=1$ $d$-configuration, too. As for the $I=3/2$ state, it is governed by a weaker $\bar{K}N$ attraction than the one in the $I=1/2$ state under consideration in this work.
Separable potentials, and the corresponding $T$-matrices, are widely used in Faddeev calculations for reducing the dimension of integrals in the equations. The separable-potential approximation is justified by the fact that the kernels of two-particle equations are of the Hilbert-Schmidt type, at least under suitable conditions on the two-particle interactions [@Meetz]. Namely, the separable approximation is valid when each of the two-particle subsystems is dominated by a limited number of bound states or resonances [@Lovelace]. This condition is satisfied for the ‘main’ two-body interactions entering our system, $\bar{K}N-\pi \Sigma$ and $NN$. For the remaining $\Sigma N$ and $\pi N$ interactions we expect weaker contributions to the bound-state complex energy (as already demonstrated for $\Sigma N$ in Ref. [@ourPRL]). Hence we use for all two-body potentials a separable form: $$\label{Voperator}
V_{i,I}^{\alpha \beta} = \lambda_{i,I}^{\alpha \beta} \,
|g_{i,I}^{\alpha} \rangle \langle g_{i,I}^{\beta} | \,,$$ which leads to a separable form of $T$-matrices: $$\label{Toperator}
T_{i,I}^{\alpha \beta} = |g_{i,I}^{\alpha} \rangle
\tau_{i,I}^{\alpha \beta} \langle g_{i,I}^{\beta} | \,.$$ For $\alpha = \beta$ the corresponding $T$-matrix coincides with the usual one. With the relation (\[Toperator\]), the AGS system (\[AGS\]) can be expressed using new transition and kernel operators: $$\begin{aligned}
\label{X_definition}
X_{ij, I_i I_j}^{\alpha \beta} &=& \langle
g_{i,I_i}^{\alpha} | G_0^{\alpha} \, U_{ij, I_i I_j}^{\alpha
\beta} G_0^{\beta} | g_{j,I_j}^{\beta} \rangle \,, \\
\label{Z_definition}
Z_{ij, I_i I_j}^{\alpha \beta} &=&
\delta_{\alpha \beta} \, Z_{ij, I_i I_j}^{\alpha} =
\delta_{\alpha \beta} \, (1-\delta_{ij}) \,
\langle g_{i,I_i}^{\alpha} | G_0^{\alpha} | g_{j,I_j}^{\alpha}
\rangle \,.\end{aligned}$$ Substituting isospin-dependent $T_i^{\alpha \beta}$, $Z_{ij}^{\alpha}$, and $X_{ij}^{\alpha \beta}$ into the AGS system (\[AGS\]) we obtain the following system of operator equations: $$\label{full_oper_eq}
X_{ij, I_i I_j}^{\alpha \beta} = \delta_{\alpha \beta} \,
Z_{ij, I_i I_j}^{\alpha} +
\sum_{k=1}^3 \sum_{\gamma=1}^3 \sum_{I_k}
Z_{ik, I_i I_k}^{\alpha} \, \tau_{k, I_k}^{\alpha \gamma} \,
X_{kj, I_k I_j}^{\gamma \beta} \,.$$ The number of equations in the system is defined by the number of possible form-factors $g_{i, I_i}^{\alpha}$. As is seen from Table \[channels.tab\], before antisymmetrization our system consists of 18 equations.
Three sets of Jacobi momentum coordinates should be introduced for each ‘particle’ channel $\alpha$: $| \vec{k_{i}}^{\alpha},
\vec{p_i}^{\alpha} \rangle$, $i=1,2,3$, $\alpha=1,2,3$. Here, $\vec{k_{i}}^{\alpha}$ is the center-of-mass momentum of the $(jk)$ pair and $\vec{p_{i}}^{\alpha}$ is the momentum of spectator $i$ with respect to the pair $(jk)$, $i \neq j \neq k$. In these coordinates the three-body free Hamiltonian in the channel $\alpha$ is defined as $$H_0^{\alpha} = \frac{(k_{i}^{\alpha})^2}{2 \, m_{jk}^{\alpha}} +
\frac{(p_{i}^{\alpha})^2}{2 \, \mu_{i}^{\alpha}} \,,$$ where the reduced masses also have ‘particle’ channel indices: $$m_{jk}^{\alpha} = \frac{m_j^{\alpha} m_k^{\alpha}}
{m_j^{\alpha} + m_k^{\alpha}}, \quad
\mu_{i}^{\alpha} = \frac{m_i^{\alpha} (m_j^{\alpha} + m_k^{\alpha})}
{m_i^{\alpha} + m_j^{\alpha} + m_k^{\alpha}}, \quad i \neq j \neq k \,.$$ In contrast to the usual AGS formalism we have to use not the kinetic energy, but the total energy of the system, including rest masses. We introduce threshold energies: $z_{th}^{\alpha} = \sum_{i=1}^3 m_i^{\alpha}$, so that the total energy is $z_{tot} = z_{th}^{\alpha} + z_{kin}^{\alpha}$, where $z_{kin}^{\alpha}$ denotes the kinetic energy in channel $\alpha$. The integrations in Eqs. (\[X\_definition\]) and (\[Z\_definition\]) are performed over one of the Jacobi momenta, namely, over $\vec{k_{i}}^{\alpha}$, which describes the motion of an interacting pair of particles $j$ and $k$ ($i \neq j \neq k$). Thus, the operators $X$ and $Z$ act on the second momentum, $\vec{p_{i}^{\alpha}}$: $$\begin{aligned}
\label{X_mom}
&{}& X_{ij, I_i I_j}^{\alpha \beta} \to
\left\langle \vec{p_i}^{\alpha} | X_{ij, I_i I_j}^{\alpha \beta}
(z_{tot}) | \vec{p_j}'^{\beta} \right \rangle
= X_{ij, I_i I_j}^{\alpha \beta}
(\vec{p_i}^{\alpha}, \vec{p_j}'^{\beta}; z_{kin}^{\alpha} + z_{th}^{\alpha}) \,, \\
\label{Z_mom}
&{}& Z_{ij, I_i I_j}^{\alpha} \to
\left\langle \vec{p_i}^{\alpha} | Z_{ij, I_i I_j}^{\alpha}
(z_{tot}) |
\vec{p_j}'^{\alpha} \right \rangle
= Z_{ij, I_i I_j}^{\alpha}
(\vec{p_i}^{\alpha}, \vec{p_j}'^{\alpha}; z_{kin}^{\alpha} + z_{th}^{\alpha})
\,.\end{aligned}$$ The energy-dependent part of a two-body $T$-matrix, embedded in the three-body space is defined by the following relation: $$\label{tau_mom}
\tau_{i,I_i}^{\alpha \beta} \to
\left\langle \vec{p_i}^{\alpha} | \tau_{i, I_i}^{\alpha \beta}(z_{tot}) |
\vec{p_j}'^{\beta} \right \rangle \equiv
\delta_{ij} \, \delta(\vec{p_i}^{\alpha} - \vec{p_j}'^{\beta}) \,
\tau_{i,I_i}^{\alpha \beta}\left(
z_{tot} - z_{th}^{\alpha} - \frac{(p_i^{\alpha})^2}{2 \, \mu_i}
\right) \,.$$ It is worth noting that all elements of the two-channel two-body $\bar{K}N - \pi \Sigma$ $T$-matrix depend on the kinetic energies in both channels ($z_{kin}^1$ and $z_{kin}^2$) simultaneously. Here we define the argument of the corresponding $\tau^{\alpha \beta}$ using the left ‘particle’ index $\alpha$. The second kinetic energy can be simply found from the relation $z_{kin}^{\alpha} + z_{th}^{\alpha} = z_{kin}^{\beta} + z_{th}^{\beta}$.
The calculation of the kernels $Z$ involves transformation from one set of Jacobi coordinates to another one and isospin re-coupling, using the property of free Green’s function: $$\left\langle \vec{p_i}^{\alpha}, I_i^{\alpha} | G_0^{\alpha} |
\vec{p_j}'^{\alpha}, I_j^{\alpha} \right\rangle
= \left\langle \vec{p_i}^{\alpha} | G_0^{\alpha} | \vec{p_j}'^{\alpha}
\right\rangle_{I_i^{\alpha} I_j^{\alpha}} \,
\left\langle i_j^{\alpha} \, i_k^{\alpha} (I_i^{\alpha}) \, i_i^{\alpha},
I I_z | i_i^{\alpha} \, i_k^{\alpha} (I_j^{\alpha}) \, i_j^{\alpha},
I I_z \right\rangle \,,$$ where $i_j^{\alpha}$ and $I_j^{\alpha}$ denote one-particle and two-particle isospins, respectively, with partition subscripts $i \neq j \neq k$, the total three-body isospin and its projection being $I=1/2, I_z = 1/2$.
To search for a resonance or a bound state means to look for a solution of the homogeneous system corresponding to Eq. (\[full\_oper\_eq\]). But before solving the system $$\label{homog_oper_eq}
X_{i, I_i}^{\alpha} =
\sum_{k=1}^3 \sum_{\gamma=1}^3 \sum_{I_k}
Z_{ik, I_i I_k}^{\alpha} \, \tau_{k, I_k}^{\alpha \gamma} \,
X_{k, I_k}^{\gamma} \,,$$ we must antisymmetrize operators involving two identical baryons with antisymmetric spin components ($S_i=0$) and symmetric spatial components ($L_i=0$). Here, in Eq. (\[homog\_oper\_eq\]), and in the following we omit right-hand indices of $X$: $X_{ij,I_i I_j}^{\alpha \beta} \to X_{i,I_i}^{\alpha}$, which are unnecessary for a homogeneous system. The operator $X_{1,0}^1$ has antisymmetric $NN$ isospin components, so it drops out of the equations. In contrast, the operator $X_{1,1}^1$ has the correct symmetry properties. All the remaining operators form symmetric and antisymmetric pairs, the symmetric ones which are used in the calculation are: $$\begin{aligned}
\nonumber
&{}&X_{2,0}^{1,-} = X_{2,0}^{1} - X_{3,0}^{1}, \quad
X_{2,1}^{1,+} = X_{2,1}^{1} + X_{3,1}^{1}, \\
\nonumber
&{}&X_{2,0}^{3,-} = X_{2,0}^{3} - X_{3,0}^{2}, \quad
X_{2,1}^{3,+} = X_{2,1}^{3} + X_{3,1}^{2}, \\
\label{X_symmetrical}
&{}&X_{1,\frac{3}{2}}^{2,-} = X_{1,\frac{3}{2}}^{2} - X_{1,\frac{3}{2}}^{3},
\quad
X_{1,\frac{1}{2}}^{2,+} = X_{1,\frac{1}{2}}^{2} + X_{1,\frac{1}{2}}^{3}, \\
\nonumber
&{}&X_{2,\frac{3}{2}}^{2,-} = X_{2,\frac{3}{2}}^{2} - X_{3,\frac{3}{2}}^{3},
\quad
X_{2,\frac{1}{2}}^{2,+} = X_{2,\frac{1}{2}}^{2} + X_{3,\frac{1}{2}}^{3} \,.\end{aligned}$$ Taking into account equalities of some kernel functions, we end up with a system of nine coupled operator equations in the eight new operators (\[X\_symmetrical\]) and $X_{1,1}^1$, all of which have the required symmetry properties. Since the Faddeev equations are dynamical ones, their final number after antisymmetrization corresponds to the number of different form-factors entering the interactions. Similar antisymmetrization procedures have been implemented in several multi-channel Faddeev calculations, e.g. the fairly recent $K^- d$ work of Ref.[@Bahaoui].
To solve the homogeneous system we transform the integral equations into algebraic ones and then search for the complex energy at which the determinant of the kernel matrix becomes equal to zero. We are looking for a three-body pole, the real part of which is situated between the $\bar{K}NN$ and $\pi \Sigma N$ thresholds, corresponding to a resonance in the $\pi \Sigma N$ channel and a quasi-bound state (a bound state with non-zero width) in the $\bar{K}NN$ channel. Therefore, we must work on the physical energy sheet of channel one and on an unphysical sheet of the second channel.
Input
=====
The separable potential (\[Voperator\]), in momentum representation, has a form: $$\label{Vseprb}
V_{i,I_i}^{\alpha \beta}(k_i^{\alpha},k_i'^{\beta}) =
\lambda_{i,I_i}^{\alpha \beta} \;
g_{i,I_i}^{\alpha}(k_i^{\alpha}) \, g_{i,I_i}^{\beta}(k_i'^{\beta}).$$ For the $NN$, $\Sigma N$ and $\pi N$ interactions we have $\alpha = \beta$, whereas for the coupled-channel $\bar{K}N - \pi \Sigma$ interaction $\alpha, \beta = K$ ($\bar{K}N$-channel) or $\pi$ ($\pi \Sigma$-channel). We constructed our own coupled-channel $\bar{K}N - \pi \Sigma$ interactions, plus complex and real one-channel $\bar{K}N$ test potentials discussed below. We also constructed one-channel $\Sigma N$ interaction and used the PEST $NN$ potential [@NNpot]. Here we neglect the $\pi N$ interaction since its dominant part is in the (3,3) $p$-wave channel.
$\bar{K} N$ interaction
-----------------------
### Two-channel $\bar{K}N - \pi \Sigma$
There are many models of strangeness $-1$ meson-baryon scattering, constructed using different methods, see e.g. Refs. [@Borasoy; @Oller2] and references therein. These recent papers describe coupled-channel models of the $\bar{K}N$ interaction, constructed within the framework of Chiral perturbation theory. The exclusive use of on-shell amplitudes and the amount of coupled channels involved in such works renders them impractical for Faddeev calculations. We therefore constructed our own potentials for the coupled-channel $\bar{K}N - \pi \Sigma$ interaction in the form (\[Vseprb\]) with form-factors $$g_{I}^{\alpha}(k^{\alpha}) = \frac{1}{(k^{\alpha})^2 +
(\beta_{I}^{\alpha})^2} \, .
\label{formfactorKN}$$ To obtain the parameters $\lambda_{I}^{\alpha \beta}$ and $\beta_{I}^{\alpha}$ we used the following experimental data:
1. Mass $M_{\Lambda}$ and width $\Gamma_{\Lambda}$ of the $\Lambda(1405)$ resonance, assuming that it is a quasi-bound state in the $I=0$ $\bar{K}N$ channel and a resonance in the $I=0$ $\pi \Sigma$ channel. For the energy of $\Lambda(1405)$ $E_{\Lambda} = M_{\Lambda} - {\rm i}~\Gamma_{\Lambda}/2$, ($c=\hbar=1$), we adopted the PDG value [@PDG] $E_{\Lambda}^{\, \rm PDG} = 1406.5 - {\rm i}~25$ MeV. In some cases we used also other values of $M_{\Lambda}$ and $\Gamma_{\Lambda}$.
2. The $K^- p$ scattering length as derived from the atomic $1s$ level shift and width in the KEK experiment [@KEK] $$a_{K^-p} = (-0.78 \pm 0.15 \pm 0.03) + {\rm i}~(0.49 \pm 0.25 \pm 0.12)
\; {\rm fm}$$ and in the DEAR collaboration experiment [@DEAR] $$a_{K^-p} = (-0.468 \pm 0.090 \pm 0.015) + {\rm i}~(0.302 \pm 0.135 \pm
0.036) \; {\rm fm} \,.$$ In the following we denote the KEK value as $a_{K^-p}^{\rm KEK} = -0.78 + {\rm i}~0.49$ fm and the DEAR value as $a_{K^-p}^{\rm DEAR} = -0.468 + {\rm i}~0.302$ fm. Due to the fairly large experimental errors and also the large difference between the results of these two measurements, we fitted our parameters to a variety of values for the $K^- p$ scattering length. In Ref. [@ourPRL] we studied the sensitivity of the Faddeev calculations’ results to varying the KEK value within its error bars. The three-body pole energy was found to depend strongly on the input $K^- p$ scattering length. As for the DEAR value of the $K^- p$ scattering length, we note the controversy about its consistency with the bulk of the $K^- p$ scattering data [@Borasoy; @Oller2].
3. The very accurately measured threshold branching ratio [@gammaKp]: $$\gamma = \frac{\Gamma(K^- p \to \pi^+ \Sigma^-)}{\Gamma(K^- p \to
\pi^- \Sigma^+)} = 2.36 \pm 0.04 \, .$$ The value $2.36$ was used in our fits.
4. Elastic $K^- p \to K^- p$ and inelastic $K^- p \to \pi^+
\Sigma^-$ total cross sections. We chose these two reactions because among all available cross section data they have sufficient experimental data points with reasonable experimental errors.
We fitted the potential parameters to points (i)–(iii) of this list and then checked how well the resulting potential reproduces the cross sections (iv). The calculated cross-sections for four sets of parameters, in comparison with the experimental data, are shown in Figs. \[KpKpcross.fig\] and \[KppiSigcross.fig\]. These sets differ from each other by the value of the range parameter $\beta$; the remaining parameters were also changed in order to reproduce the same $\gamma$, $a_{K^- p}^{\rm KEK}$ and $E_{\Lambda}^{\, \rm PDG}$ data. We conclude from the figures that the best value of the $\bar{K}N$ range parameter is $\beta=3.5$ fm$^{-1}$. In the following we denote the set with $a_{K^- p}^{\rm KEK}$, $E_{\Lambda}^{\, \rm PDG}$, and $\beta=3.5$ fm$^{-1}$ as the ‘best set’.
Figure \[Sigmapi\_I0.fig\] shows the calculated $I=0$ elastic $\pi \Sigma$ cross section, demonstrating that $\Lambda(1405)$ is indeed a resonance in this channel.
We were unable to find a value for $\beta$, using the DEAR scattering length $a_{K^- p}^{\rm DEAR}$ and $E_{\Lambda}^{\, \rm PDG}$, such that the corresponding set of parameters provided a good description of both cross-sections. The elastic $K^- p \to K^- p$ cross-sections can be described with $1.5 \leq \beta \leq 2.5$ fm$^{-1}$, but the inelastic $K^- p \to
\pi^+ \Sigma^-$ cross sections for these values are situated much lower than the experimental data points. Given this situation, we did not perform three-body calculations with $\bar{K}N$ interaction parameters that reproduce the DEAR value of the $K^- p$ scattering length.
### One-channel complex and real $\bar{K}N$
In order to investigate all possible dependencies of our three-body results on two-body inputs we constructed additionally real and complex one-channel $\bar{K}N$ potentials. The imaginary part of the complex potential accounts for absorption to all other channels. Both potentials have the same form-factors as the coupled-channel potential \[Eq.(\[formfactorKN\])\], but for only one channel index $\alpha = \beta = K$. To fit the strength parameters $\lambda$ of the complex variant, we used experimental data (i) and (ii), i.e. the energy of $\Lambda(1405)$ and $a_{K^- p}$. For the complex $\bar{K}N$ potential we used ‘best set’ plus one more set of data, which is the same as was used in Refs. [@Akaishi1; @Akaishi2]: $E_{\Lambda}^{\,\rm AY} = 1405 - {\rm i}~20$ MeV, $a_{K^- p}^{\,\rm AY} = -0.70 + {\rm i}~ 0.53$ fm, and a range parameter $\beta=1.5$ fm$^{-1}$. We denote it as ‘AY set’.
A one-channel real $\bar{K}N$ potential was constructed by fitting its parameters to reproduce the real parts of $E_{\Lambda}^{\, \rm PDG}$ and $a_{K^- p}^{\rm KEK}$, with $\beta=3.5$ fm$^{-1}$. Here we assumed that $\Lambda(1405)$ is a real bound state of the $I=0$ $\bar{K}N$ subsystem.
$\Sigma N$ interaction
----------------------
Only few experimental data exist for this interaction. There are different models of it, for example several Nijmegen models, but due to the lack of data it is not possible to give preference to any of these over the other ones. A separable potential (\[Vseprb\]) with Yamaguchi form-factors $$g_{I}^{\Sigma N}(k) = \frac{1}{k^2 + (\beta_{I}^{\Sigma N})^2}$$ was used for the two isospin states. The parameters of the $I=3/2$ $\Sigma N$ interaction were fitted to:
1. the scattering length and effective radius $$a(I=3/2) = 3.8 \; {\rm fm}, \qquad r_{{\rm eff}}(I=3/2) = 4.0 \; {\rm fm}$$ from the Nijmegen potential model F [@SigmaN1] (we denote this set of $I=3/2$ $\Sigma N$ parameters as ’$\Sigma N$ set 1’).
2. the Nijmegen model NSC97 $YN$ phase shifts [@SigmaN2]. This ’$\Sigma N$ set 2’ gives the following scattering length and effective range $$a(I=3/2) = 4.15 \; {\rm fm}, \qquad r_{{\rm eff}}(I=3/2) = 2.4 \; {\rm fm.}$$
3. the scattering length and effective radius $$a(I=3/2) = 4.1 \; {\rm fm}, \qquad r_{{\rm eff}}(I=3/2) = 3.5 \; {\rm fm}$$ from the most recent Nijmegen potential ESC04a [@SigmaN3] (’$\Sigma N$ set 3’).
’$\Sigma N$ set 1’ ’$\Sigma N$ set 2’ ’$\Sigma N$ set 3’ no $\Sigma N$
---------------------- ---------------------- ---------------------- ----------------------
$-55.1-{\rm i}~50.9$ $-55.4-{\rm i}~51.9$ $-55.3-{\rm i}~51.1$ $-52.9-{\rm i}~50.9$
: Three-body pole energy $E_{K^- pp}$ (in MeV) of the $I=1/2,~J^{\pi}=0^-$ quasi-bound state of the $\bar{K} NN$ system with respect to the $K^-pp$ threshold calculated with the ’best set’ of $\bar{K}N - \pi \Sigma$ parameters using ’$\Sigma N$ set 1’, ’$\Sigma N$ set 2’, ’$\Sigma N$ set 3’, and with both $I=1/2$ and $I=3/2$ $\Sigma N$ interactions switched off.[]{data-label="SigmaN.tab"}
The dependence of the three-body pole position on the $\Sigma N$ parameters was investigated in Ref. [@ourPRL]. Table \[SigmaN.tab\] illustrates the sensitivity of the binding energies and widths of the $I=1/2,~J^{\pi}=0^-$ quasi-bound state of the $\bar{K} NN$ system to the $\Sigma N$ interaction parameters. Due to the weak dependence of the three-body pole position on the $\Sigma N$ interaction we used in the following only one (the first) set of $I=3/2$ $\Sigma N$ parameters.
For the $I=1/2$ $\Sigma N$ interaction only the scattering length was approximately determined: $a(I=1/2) = -0.5$ fm [@DalitzSigmaN]. We fitted the separable-potential parameters to this value, restricting the fit by imposing ‘natural’ values on the parameters and producing a reasonable value for the $I=1/2$ effective radius.
$NN$ interaction
----------------
We used the nucleon-nucleon PEST potential from Ref. [@NNpot], which is a separable approximation of the Paris potential. The strength parameter was set to $\lambda=-1$ and the form-factor is: $$g_{I}^{NN}(k) = \frac{1}{2 \sqrt{\pi}} \, \sum_{i=1}^6
\frac{c_{i,I}^{NN}}{k^2 + (\beta_{i,I}^{NN})^2} \,.$$ The constants $c_{i,I}^{NN}$ and $\beta_{i,I}^{NN}$ are listed in Ref. [@NNpot]. PEST is on- and off-shell equivalent to the Paris potential up to $E_{\,\rm lab} \sim 50$ MeV and is repulsive at distances shorter than 0.8 fm. It reproduces the deuteron binding energy $E_{\,\rm d} = -2.2249$ MeV, as well as the triplet and singlet $NN$ scattering lengths, $a(\,{}^3S_1) = -5.422$ fm and $a(\,{}^1S_0) = 17.534$ fm, respectively.
Results
=======
Results of full coupled-channel $\bar{K}NN - \pi \Sigma N$ calculation
----------------------------------------------------------------------
Full coupled-channel calculations were done systematically, studying various dependencies of the three-body pole position on different input parameters of the $\bar{K}N - \pi \Sigma$ potential. Here the three-body energy is defined as $E_{K^- pp} = -B_{K^- pp}
- {\rm i} \, \Gamma_{K^- pp} /2$, where $B_{K^- pp}$ is a binding energy with respect to the $K^- pp$ threshold, $\Gamma_{K^- pp}$ is a width of a quasi-bound state. The dependence of the real and imaginary parts of the three-body pole energy as function of the range parameter $\beta$ is shown in Figs. \[Re\_diffbetas.fig\] and \[Im\_diffbetas.fig\], respectively. It is seen that the dependence of the real part on $\beta$ is rather weak, whereas the imaginary part strongly depends on this parameter.
Other values which are varied are the mass $M_{\Lambda}$ and the width $\Gamma_{\Lambda}$ of the $\Lambda(1405)$ resonance. The results of such variations are shown in Table \[E\_Lambda.tab\]. All other input data used in this calculation are fixed at $\beta=3.5$ fm$^{-1}$ and $a_{K^- p}^{\rm KEK}$. As expected, the broadening of $\Lambda(1405)$ leads to a considerable increase of the three-body width, whereas the three-body binding energy depends on $\Gamma_{\Lambda}$ rather weakly. However, increasing the $\Lambda(1405)$–resonance mass strongly affects both real and imaginary parts of the three-body pole, leading to a fast decrease of both.
$\Gamma_{\Lambda}/2$ $\setminus$ $M_{\Lambda}$ 1400 1410 1420
------------------------------------------------ ------------------------- ---------------------- ------------------------------------
$20$ $-62.1-{\rm i}~46.9\, $ $-47.5-{\rm i}~37.6$ [*no $T_{\bar{K}N-\pi \Sigma}$*]{}
$25$ $-64.9-{\rm i}~58.4$ $-50.8-{\rm i}~47.4$ $-40.6-{\rm i}~39.4$
$30$ $-65.7-{\rm i}~72.2$ $-52.5-{\rm i}~59.8$ $-42.8-{\rm i}~50.8$
: Calculated three-body pole energy $E_{K^- pp}$ in MeV, of the $I=1/2,~J^{\pi}=0^-$ quasi-bound state of the $\bar{K} NN$ system with respect to the $K^-pp$ threshold, for different two-body input, mass $M_{\Lambda}$ and half-width $\Gamma_{\Lambda}/2$ of the $\Lambda(1405)$. For $E_{\Lambda}=1420-{\rm i}~20$ MeV no reasonable $T_{\bar{K}N-\pi \Sigma}$ parameters can be found. []{data-label="E_Lambda.tab"}
One-channel real and complex $\bar{K}NN$ calculations
-----------------------------------------------------
We also performed a test calculation for the one-channel $\bar{K}NN$ system using a one-channel real $\bar{K}N$ potential ($T$-matrix). For fitting we used the real part of $a_{K^- p}^{\rm KEK}$, the real part of $E_{\Lambda}^{\, \rm PDG}$, and assumed $\Lambda(1405)$ as a real bound state of the $I=0$ $\bar{K}N$ subsystem. For these data, and using $\beta=3.5$ fm$^{-1}$, we found a real bound state for $I=1/2$, $J^{\pi}=0^-$ $\bar{K}NN$ at $-43.8$ MeV below the $K^- pp$ threshold (the first column in Table \[E\_complex.tab\]).
$E_{1 \,\rm real}^{\, \rm best}$ $E_{1 \,\rm complex}^{\, \rm best}$ $E_{2 \,\rm coupled}^{\, \rm best}$ $E_{1 \,\rm complex}^{\,\rm AY}$ $E$ from Ref. [@Akaishi2]
---------------------------------- ------------------------------------- ------------------------------------- ---------------------------------- ---------------------------
$-43.8$ $-40.2-{\rm i}~38.7$ $-55.1-{\rm i}~50.9$ $-46.6-{\rm i}~29.6$ $-48.0-{\rm i}~30.5$
: Results of different calculations of the three-body pole energy $E_{K^- pp}$ in MeV, with respect to the $K^-pp$ threshold: real and complex $\bar{K}NN$ one-channel (first two columns), and full coupled-channel calculations (third column) using the ‘best set’ of $\bar{K}N - \pi \Sigma$ parameters. Fourth column: complex $\bar{K}NN$ one-channel calculation with ‘AY set’. Fifth column: AY’s result [@Akaishi2]. []{data-label="E_complex.tab"}
Another test calculation was performed with a one-channel complex $\bar{K}N$ potential. The strength parameters $\lambda$ of the potential were fitted to the $a_{K^- p}^{\rm KEK}$ and $E_{\Lambda}^{\, \rm PDG}$ data, and the dependence of the three-body pole on the range parameter $\beta$ was investigated. Results are presented in Fig. \[diff\_betas\_cmplxKN.fig\].
It is seen from the plot that increasing the range of the $\bar{K}N$ interaction, by decreasing the range parameter $\beta$, gives rise to a deeper and somewhat narrower three-body level. The dependence of the calculated $\bar{K}NN$ energy on the range parameter $\beta$, as displayed in Fig. \[diff\_betas\_cmplxKN.fig\], is rather strong. Therefore, using a too large or a too small range parameter for the complex $\bar{K}N$ interaction leads to substantial underestimate or overestimate, respectively, of the three-body energy. The ‘best set’ of $\bar{K}N$ parameters with a fixed value for the range parameter, $\beta = 3.5$ fm$^{-1}$, yields the three-body pole energy $E_{1 \,\rm complex}^{\,\rm best}$ shown in the second column of Table \[E\_complex.tab\]. The result of the full coupled-channel calculation $E_{2 \,\rm coupled}^{\,\rm best}$ is shown in the third column.
The transition within a three-body single-channel $\bar{K}NN$ calculation from using a real $\bar{K}N$ interaction to using the complex $\bar{K}N$ interaction, fitted to $E_{\Lambda}^{\, \rm PDG}$ and to $a_{K^- p}^{\rm KEK}$, is demonstrated in Fig. \[epsilonKN\_cmplxKN.fig\] by the trajectory of complex three-body energies starting with the real $E_{1 \,\rm real}^{\,\rm best}$ at the upper-left corner and ending with the complex $E_{1 \,\rm complex}^{\,\rm best}$ at the lower-right corner. This trajectory is generated by varying a real parameter $\varepsilon$ between 0 to 1, $\varepsilon = 0$ for $E_{1 \,\rm real}^{\,\rm best}$ and $\varepsilon = 1$ for $E_{1 \,\rm complex}^{\,\rm best}$, such that the imaginary parts of the fitted $E_{\Lambda}^{\, \rm PDG}$ and $a_{K^- p}^{\rm KEK}$ are scaled down by $\varepsilon$: $${\rm Im \,}{E_{\Lambda}^{\, \rm PDG}} \to \varepsilon \,{\rm Im \,}
{E_{\Lambda}^{\, \rm PDG}} \,, \quad
{\rm Im \,} a_{K^- p}^{\rm KEK} \to \varepsilon \,{\rm Im \,}{a_{K^- p}^{\rm KEK}} \,.$$
It is interesting to note that although the $I=0$ and $I=1$ strength parameters $\lambda_{\rm complex}$ provide stronger attraction in the $\bar{K}N$ systems than the attraction provided by $\lambda_{\rm real}$, yet $E_{1 \,\rm complex}$ signifies less binding than $E_{1 \,\rm real}$. This generalizes the well known property in two-body problems where including absorptivity leads effectively to adding repulsion. Here we find that absorption of flux from the $\bar{K}N$ channel into other unspecified channels represented by an imaginary part of a complex $\bar{K}N$ potential reduces also the three-body binding energy.
Comparing the result of the one-channel complex $\bar{K}NN$ calculation with the coupled-channel $\bar{K}NN$ (see Table \[E\_complex.tab\]) shows that $E_{2 \,\rm coupled}$ is much deeper and broader than $E_{1 \,\rm complex}$. This means that the $\pi \Sigma$ channel, within a genuinely three-body coupled-channel calculation plays an important dynamical role in forming the three-body resonance (quasi-bound state), over its obvious role of absorbing flux from the $\bar{K}N$ channel. The poor applicability of an optical potential approach (or some low-order perturbation calculation) in searching for a quasi-bound state was shown, for example, by Ueda [@Ueda], who studied the $\eta NN - \pi NN$ coupled-channel system using Faddeev equations, finding a large deviation of the calculated results from optical-model predictions.
In order to compare the present results with the results of calculations by Yamazaki and Akaishi [@Akaishi2], the one-channel $\bar{K}NN$ calculation was repeated using the complex $\bar{K}N$ potential corresponding to the ‘AY set’ of $\bar{K}N$ parameters. The result obtained by us ($E_{1 \,\rm complex}^{\,\rm AY}$) and $E$ from Ref. [@Akaishi2] are shown in the last two columns of Table \[E\_complex.tab\]. It is remarkable that in spite of different forms of the two-body potentials and different three-body formalisms, the calculated three-body energies in these single-channel $\bar{K}NN$ calculations come out very close to each other, provided the same set of $\bar{K}N$ parameters is fitted to. Nevertheless, both values of three-body energy are far away from the three-body energy of the complete coupled-channel calculation. One of the reasons is the use of a complex $\bar{K}N$ potential in the single-channel $\bar{K}NN$ calculations, another reason is the too small value, $\beta = 1.5$ fm$^{-1}$, for the range parameter used in these approximate calculations.
Conclusion
==========
We performed coupled-channel few-body calculations of the $I=1/2$, $J^{\pi}=0^-$ $\bar{K}NN$ system, finding a deeply bound and broad quasi-bound state, which is a resonance in the $\pi \Sigma N$ channel. The calculations yielded binding energy $B_{K^-pp} \sim 50 -70$ MeV and width $\Gamma_{K^-pp} \sim 100$ MeV, in agreement with our earlier results [@ourPRL]. It was shown that the explicit inclusion of the second channel is crucial for this system. The dependence of the three-body energy pole on different forms and parameters of the $\bar{K}N$ interaction, and on different ways of reproducing $\bar{K}N - \pi \Sigma$ observables, was studied. Most of these dependencies were found to be strong. In particular, it was shown that a complex $\bar{K}N$ potential gives much shallower and narrower three-body quasi-bound state than the full coupled-channel calculation, which has the same range parameter and reproduces the same $\bar{K}N - \pi \Sigma$ observables.
We compared our results with those of Yamazaki and Akaishi [@Akaishi2], demonstrating the shortcomings of these single-channel $\bar{K}NN$ calculations. Two more calculations of the same system appeared recently. Dote and Weise [@DoteWeise] have presented preliminary results of a variational Anitsymmetrized Molecular Dynamics calculation for the $K^- pp$ system within a single-channel $\bar{K}NN$ framework. Their calculation focuses attention to the dependence of the calculated real three-body binding energy on the range parameter of the Gaussian $\bar{K}N$ interaction used. It includes perturbatively also a $p$-wave $\bar{K}N$ interaction. Whereas a direct comparison between our coupled-channel calculations and these single-channel calculations cannot be made, the general criticism expressed above of the use of a single-channel formalism applies also to this work.
A coupled-channel $\bar{K}NN - \pi \Sigma N$ calculation of the same $K^- pp$ system was performed recently by Ikeda and Sato [@IkedaSato] with less emphasis on reproducing low-energy $\bar{K}N$ data. The obtained binding energies are in a similar range to those presented here, while the widths are consistently lower than those calculated in the present work.
It is worthwhile to note that all the theoretical calculations discussed above, including the present calculations, obtain binding energies which are considerably below the binding energy $\approx 115$ MeV deduced for the $K^-pp$ identification proposed in Ref. [@FINUDA]. This FINUDA $K^-_{\rm stop}$ experiment on lithium and heavier targets, as mentioned in the Introduction, leaves room for other interpretations as well. The use of a more restrictive $^3$He target in order to search for a $K^-pp$ quasibound state in a ($K^-,n$) reaction was approved as a ‘day-1’ experiment in J-PARC [@Nagae]. The spectrum calculated recently for this reaction [@Koike] demonstrates how the large width predicted for $K^-pp$ in the present work is expected to wipe out any clear peak structure in this reaction.
Additional calculations are necessary to study other features of the coupled $\bar{K}NN$ system. These include the secondary effect of the $\pi \Lambda$ channel beyond that of the primary inelastic $\pi \Sigma$ channel incorporated here, of $p$-wave $\bar{K}N$ and $\pi N$ interactions, and the use of relativistic kinematics. Finally, in order to understand better the $\bar{K}N$ interaction, it is desirable to perform coupled-channel Faddeev calculations of a quasi-bound state in the $S=1$ $\bar{K}NN$ system as well.
The work was supported by the Czech GA AVCR grant A100480617 and by the Israel Science Foundation grant 757/05.
[99]{}
E. Friedman, A. Gal, C.J. Batty, Phys. Lett. B 308, 6 (1993); Nucl. Phys. A 579, 518 (1994); C.J. Batty, E. Friedman, A. Gal, Phys. Rep. 287, 385 (1997); A. Cieplý, E. Friedman, A. Gal, J. Mareš, Nucl. Phys. A 696, 173 (2001); J. Mareš, E. Friedman, A. Gal, Nucl. Phys. A 770, 84 (2006).
Y. Akaishi, T. Yamazaki, in [*Proc. DA$\Phi$NE Workshop*]{}, Frascati Phys. Ser. 16, 59 (1999); Phys. Rev. C 65, 044005 (2002).
T. Yamazaki, Y. Akaishi, Phys. Lett. B 535, 70 (2002).
M. Iwasaki [*et al.*]{}, arXiv:0706.0297 \[nucl-ex\].
M. Agnello [*et al.*]{}, Phys. Rev. Lett. 94, 212303 (2005).
V.K. Magas, E. Oset, A. Ramos, H. Toki, Phys. Rev. C 74, 025206 (2006).
J.A. Oller, J. Prades, M. Verbeni, Eur. Phys. J. A 31, 527 (2007).
W. Weise, arXiv:nucl-th/0701035.
M. Iwasaki [*et al.*]{}, Phys. Rev. Lett. 78, 3067 (1997); T.M. Ito [*et al.*]{}, Phys. Rev. C 58, 2366 (1998).
G. Beer [*et al.*]{}, Phys. Rev. Lett. 94, 212302 (2005).
S. Deser [*et al.*]{}, Phys. Rev. 96, 774 (1954); T. L. Trueman, Nucl. Phys. 26, 57 (1961).
B. Borasoy. R. Ni[ß]{}ler, W. Weise, Phys. Rev. Lett. 94, 213401 (2005); [*ibid.*]{} 96, 199201 (2006); Eur. Phys. J. A 25, 79 (2005); B. Borasoy, U.-G. Mei[ß]{}ner, R. Ni[ß]{}ler, Phys. Rev. C 74, 055201 (2006).
G.E. Brown, C.-H. Lee, H.-J. Park, M. Rho, Phys. Rev. Lett. 96, 062303 (2006).
Y. Kim, K. Kubodera, D.-P. Min, F. Myhrer, M. Rho, Nucl. Phys. A 792, 249 (2007).
L.D. Faddeev, Soviet Phys. JETP 12, 1014 (1961); Mathematical aspects of the three-body problem in quantum scattering theory, Steklov Math. Institute 69 (1963).
E.O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B 2, 167 (1967).
N.V. Shevchenko, A. Gal, J. Mareš, Phys. Rev. Lett. 98, 082301 (2007).
K. Meetz, J. Math. Phys. 3, 690 (1962).
C. Lovelace, Phys. Rev. 135, B1225 (1964).
A. Bahaoui, C. Fayard, T. Mizutani, B. Saghai, Phys. Rev. C 68, 064001 (2003).
H. Zankel, W. Plessas, J. Haidenbauer, Phys. Rev. C 28, 538 (1983).
J.A. Oller, J. Prades, M. Verbeni, Phys. Rev. Lett. 95, 172502 (2005); [*ibid.*]{} 96, 199202 (2006); J.A. Oller, Eur. Phys. J. A 28, 63 (2006).
W.-M. Yao [*et al.*]{} (Particle Data Group), J. Phys. G 33, 1 (2006).
D.N. Tovee [*et al.*]{}, Nucl. Phys. B 33, 493 (1971); R.J. Nowak [*et al.*]{}, Nucl. Phys. B 139, 61 (1978).
W.E. Humphrey, R.R. Ross, Phys. Rev. 127, 1305 (1962).
M. Sakitt [*et al.*]{}, Phys. Rev. 139, B719 (1965).
J.K. Kim, Phys. Rev. Lett. 14, 29 (1965); Columbia University Report, Nevis 149 (1966); Phys. Rev. Lett. 19, 1074 (1967).
J. Ciborowski [*et al.*]{}, J. Phys. G 8, 13 (1982).
M.M. Nagels, Th.A. Rijken, J.J. de Swart, Phys. Rev. D 20, 1633 (1979).
http://nn-online.org by the Nijmegen group.
Th. A. Rijken, Y. Yamamoto, Phys. Rev. C 73, 044008 (2006).
R.H. Dalitz, Nucl. Phys. A 354, 101c (1981).
T. Ueda, Phys. Rev. Lett. 66, 297 (1991).
A. Dote, W. Weise, arXiv:nucl-th/0701050.
Y. Ikeda, T. Sato, arXiv:nucl-th/0701001; Phys. Rev. C 76, 035203 (2007).
T. Nagae, [*Spectroscopy of Hypernuclei with Meson Beams*]{}, in [*Topics in Strangeness Nuclear Physics*]{}, Lecture Notes in Physics 724, eds. P. Bydzovsky, A. Gal, J. Mares (Springer, Berlin Heidelberg, 2007) pp. 81–111.
T. Koike, T. Harada, Phys. Lett. B 652, 262 (2007).
[^1]: Corresponding author: shevchenko@ujf.cas.cz
|
---
abstract: 'This article concerns the expressive power of depth in deep feed-forward neural nets with $\operatorname{ReLU}$ activations. Specifically, we answer the following question: for a fixed $d_{in}\geq 1,$ what is the minimal width $w$ so that neural nets with $\operatorname{ReLU}$ activations, input dimension $d_{in}$, hidden layer widths at most $w,$ and arbitrary depth can approximate any continuous, real-valued function of $d_{in}$ variables arbitrarily well? It turns out that this minimal width is exactly equal to $d_{in}+1.$ That is, if all the hidden layer widths are bounded by $d_{in}$, then even in the infinite depth limit, $\operatorname{ReLU}$ nets can only express a very limited class of functions, and, on the other hand, any continuous function on the $d_{in}$-dimensional unit cube can be approximated to arbitrary precision by $\operatorname{ReLU}$ nets in which all hidden layers have width exactly $d_{in}+1.$ Our construction in fact shows that any continuous function $f:[0,1]^{d_{in}}\to\mathbb R^{d_{out}}$ can be approximated by a net of width $d_{in}+d_{out}$. We obtain quantitative depth estimates for such an approximation in terms of the modulus of continuity of $f$.'
address:
- 'Department of Mathematics, Texas A&M, College Station, United States'
- 'Trinity College, Cambridge, CB2 1TQ, UK'
author:
- Boris Hanin
- Mark Sellke
bibliography:
- 'bibliography.bib'
title: Approximating Continuous Functions by ReLU Nets of Minimal Width
---
Introduction
============
Over the past several years, artificial neural networks, especially deep networks, have become the state of the art in a wide variety of machine learning tasks. These tasks include important benchmark problems in machine vision ([@krizhevsky2012imagenet]) and machine translation ([@sutskever2014sequence; @wu2016google]) as well as superhuman performance at games such as Go [@silver2016mastering]. Despite these varied and striking successes, a theory of why neural nets provide such good approximations to interesting functions and can be effectively trained is only beginning to take shape.
While non-linear activations help neural nets express a wide variety of functions, repeated non-linearities can also “garble” the signal, leading to a loss of mutual information between the input and the activations at various hidden layers. Such an information theoretic point of view on neural nets has recently been systematically taken up in the work of Tishby with Shwartz-Ziv, Moshkovitz, and Zaslavsky [@shwartz2017opening; @moshkovitz2017mixing; @tishby2015deep]. In the present article, we answer a basic information theoretic question about neural nets. Namely, for each $d\geq 1,$ what is the minimal width $w_{\text{min}}(d)$ so that neural nets whose hidden layers have width at least $w_{\text{min}}(d)$ and arbitrary depth can approximate arbitrarily well any scalar continuous function of $d$ variables? We treat only neural nets with a popular and particularly simple activation function called rectified linear units, defined $$\operatorname{ReLU}(t):=\max{\ensuremath{\left(0,t \right)}}.$$
It have been known since the 1980’s (e.g. the work of Cybenko [@cybenko1989approximation] and Hornik-Stinchcombe-White [@hornik1989multilayer]) that feed-forward neural nets with a single hidden layer can approximate essentially any function if the hidden layer is allowed to be arbitrarily wide. Such results hold for a wide variety of activations, including $\operatorname{ReLU}.$ However, part of the recent renaissance in neural nets, is the empirical observation that deep neural nets tend to achieve greater expressivity per parameter than their shallow cousins. There are now a number of rigorous results about this so-called expressive power of depth [@arora2016understanding; @mhaskar2016learning; @lin2017does; @mhaskar2016deep; @poole2016exponential; @raghu2016expressive; @rolnick2017power; @telgarsky2015representation; @telgarsky2016benefits; @telgarsky2017neural; @yarotsky2016error]. We refer the reader to §3 in [@hanin2017universal] for a discussion of the relationships between some of these articles.
The main result of this article shows a sharp transition in the representational power of deep feed-forward neural nets with $\operatorname{ReLU}$ activations as a function of the widths of their hidden layers. To state it, we need some notation. We say that $\mathcal N$ is a feed-forward neural net with $\operatorname{ReLU}$ activations, input dimension $d_{in}$, output dimension $d_{out}$, and widths $d_{in}=d_1,d_2,\ldots,d_k,d_{k+1}=d_{out} $ (a $\operatorname{ReLU}$ net for short) if it computes a function $f_{\mathcal N}$ of the form $$\label{E:relunet-def}
A_k\circ \operatorname{ReLU}\circ A_{k-1}\circ \cdots \circ \operatorname{ReLU}\circ
A_1,$$ where $A_i:{{\mathbb R}}^{d_i}{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{i+1}}$ are affine transformations and for any $m\geq 1$ $$\operatorname{ReLU}{\ensuremath{\left(x_1,\ldots, x_m \right)}}={\ensuremath{\left(\max{\ensuremath{\left(0,x_1 \right)}},\ldots,
\max{\ensuremath{\left(0,x_m \right)}} \right)}}.$$ The integers $d_2,\ldots, d_k$ are said to be the widths of the hidden layers of $\mathcal N,$ and the integer $k$ is the depth of $\mathcal
N.$ Notice that for fixed $d_1,\ldots d_{k+1},$ the family of neural nets is a finite dimensional family of non-linear functions parameterized by the affine transformations $A_i.$ Our main result concerns the numbers $w_{\text{min}}(d_{in},d_{out}),$ defined to be the minimal value of $w$ such that for every continuous function $f:[0,1]^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}$ and every ${\varepsilon}>0$ there is a $\operatorname{ReLU}$ net $\mathcal N$ with input dimension $d_{in},$ hidden layer widths at most $w$, and output dimension $d_{out}$ that ${\varepsilon}-$approximates $f:$ $$\sup_{x\in [0,1]^{d_{_{in}}}}{\left\lVertf(x)-f_{\mathcal N}(x)\right\rVert}\leq {\varepsilon}.$$ The main result of this article is the following estimate for $w_{\text{min}}(d_{in}, d_{out}).$
\[T:main\] For every $d_{in},d_{out}\geq 1,$ $$d_{in}+1\leq w_{\text{min}}(d_{in},d_{out})\leq d_{in}+d_{out}.$$
Proving the upper bound $w_{\text{min}}(d_{in},d_{out})\leq d_{in}+d_{out}$ in Theorem \[T:main\] requires a novel construction by which any continuous function with $d_{in}$ input variables and $d_{out}$ output variables can be approximated to arbitrary precision by a $\operatorname{ReLU}$ net with width $d_{in}+d_{out}$ and depth depending on its modulus of continuity ${\omega}_f$. Recall that ${\omega}_f(\delta)\leq\varepsilon$ when $|x-y|\leq \delta$ implies that $|f(x)-f(y)|\leq \varepsilon$ uniformly over all inputs $x,y.$ Since ${\omega}_f$ need not be continuous or bijective, define $${\omega}^{-1}_f(\varepsilon)=\sup\{\delta:{\omega}_f(\delta)\leq\varepsilon\}.$$ We will show that if $K\subseteq {{\mathbb R}}^{d_{in}}$ is any compact set and $f:K{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}$ is continuous, then there exists a $\operatorname{ReLU}$ net $\mathcal N$ with input dimension $d_{in}$, all hidden layers of width $d_{in}+d_{out}$, output dimenion $d_{out},$ and depth $O(\operatorname{diam}(K)/{\omega}^{-1}_f({\varepsilon}))^{d_{in}+1}$ that ${\varepsilon}$-approximates $f$ on $K:$ $$\sup_{x\in K} {\left\lVertf(x)-f_{\mathcal N}(x)\right\rVert}\leq {\varepsilon}.$$ We refer the reader to Proposition \[P:density-maxmin\] for the precise statement. The construction is carried out in §\[S:UB\]. In contrast, obtaining the lower bound $$w_{\text{min}}(d_{in},d_{out})\geq w_{\text{min}}(d_{in},1)\geq d_{in}+1,$$ requires constructing, for every $d_{in}\geq 1,$ a continuous function $f:[0,1]^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}$ and a constant $\eta>0$ so that any width $d$ $\operatorname{ReLU}$ net $\mathcal N$ must satisfy $$\label{E:LB}
\sup_{x\in[0,1]^{d_{in}}}{\ensuremath{\left| f(x)-f_{\mathcal N}(x) \right|}}> \eta.$$ Our construction in §\[S:LB\] only requires that the function have some compact level set (connected component of a fiber $f^{-1}(a)$) and be non-constant inside that level set.
Before proceeding to the proof of Theorem \[T:main\], we make two remarks. First, the neural nets we consider here are not allowed to have skip (e.g. residual) connections, popularized in the ResNets introduced by He-Zhang-Ren-Sun in [@he2016deep] and in the Highway Nets introduced by Srivastava-Greff-Schidhuber in [@srivastava2015highway]. A skip connection allows the input to a given hidden layer to be an affine function of the all the outputs of all the previous hidden layers, instead of just the one preceeding it. If one allows skip connections, then a $\operatorname{ReLU}$ net whose hidden layers have width $1$ can already approximate any continuous function if the net is allowed to be arbitrarily deep. The reason is that any feed-forward neural net with one hidden layer of width $k$ can be converted into a neural net with $k$ hidden layers, each of width $1,$ that computes the same function. The construction is simply to “turn the hidden layer on its’ side.” That is, each neuron in the single hidden layer in the original shallow net becomes its own hidden layer. The input to the net is connected to the single neural in every new hidden layer, which is in turn connected to the output. In this construction, each hidden layer is connected only to the input and output. In the language of Veit-Wilber-Belongie [@veit2016residual], the resulting ResNet implements an ensemble of paths of length $1$. Second, it is tempting to generalize Theorem \[T:main\] to arbitrary piecewise linear activations. However, it seems that such a generalization is not straightforward, even for activations of the form $\sigma(t)=\max{\ensuremath{\left(\ell_1(t),\ell_2(t) \right)}}$, where $\ell_1,\ell_2$ are two affine functions with different slopes.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The first author would like to thank Zhangyang Wang for several stimulating discussions about extending the results in this article to allowing residual connections and to more general activations. We are also grateful to Dmitry Yarotsky for pointing out several inaccuracies and a mistake (now corrected) in the proof of Lemma \[L:LB-key\] in a previous version.
Proof of the Upper Bound in Theorem \[T:main\] {#S:UB}
==============================================
Fix ${\varepsilon}>0,$ $d_{in}, d_{out}\geq 1$, a compact set $K\subseteq {{\mathbb R}}^{d_{in}},$ and a continuous function $f:K{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}.$ In this section, we prove that there exists a ReLU net $\mathcal N$ with input dimension $d_{in},$ hidden layer widths $d_{in}+d_{out},$ and output dimension $d_{out}$ such that $$\label{E:goal1}
{\left\lVertf-f_{\mathcal N}\right\rVert}_{C^0(K)}=\sup_{x\in K}{\left\lVertf(x)-f_{\mathcal N}(x)\right\rVert}\leq {\varepsilon}.$$ We will use the following definition.
A function $g:{{\mathbb R}}^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}$ is a max-min string of length $L\geq 1$ on $d_{in}$ input variables and $d_{out}$ output variables if there exist affine functions $\ell_1,\ldots,
\ell_{L}:{{\mathbb R}}^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}$ such that $$g=\sigma_{L-1}(\ell_{L},\sigma_{L-2}(\ell_{L-1},\ldots,
\sigma_2(\ell_3, \sigma_1(\ell_1,\ell_2))\cdots),$$ where each $\sigma_i$ is either a coordinate-wise max or a min.
The statement follows immediately from the following two propositions.
\[P:relu-maxmin\] For every max-min string $g$ on $d_{in}$ input variables and $d_{out}$ ouput variables with length $L$ and every compact $K\subseteq {{\mathbb R}}^{d_{in}}$, there exists a $\operatorname{ReLU}$ net with input dimension $d,$ hidden layer width $d_{in}+d_{out},$ output dimension $d_{dout}$, and depth $L$ that computes $x\mapsto g(x)$ for every $x\in K.$
\[P:density-maxmin\] For every compact $K\subseteq {{\mathbb R}}^{d_{in}},$ any continuous $f:K{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}$ and each ${\varepsilon}>0$ there exists a max-min string $g$ on $d_{in}$ input variables and $d_{out}$ output variables with length $$\left(\frac{O(\operatorname{diam}(K))}{{\omega}^{-1}_f({\varepsilon})}\right)^{d_{in}+1}$$ for which $${\left\lVertf-g\right\rVert}_{C^0(K)}\leq {\varepsilon}.$$
Proposition \[P:relu-maxmin\] is essentially Lemma 4 in [@hanin2017universal]. We include a short proof for the reader’s convenience in §\[S:relu-pf\]. Proposition \[P:density-maxmin\] appears to be new, however, and is the main technical result in the present article. It is proved in §\[S:density-pf\]. It is related in spirit to results in the literature (e.g. [@scholtes2012introduction Prop. 2.2.2.]) that express a continuous piecewise affine $h:K{\ensuremath{\rightarrow}}{{\mathbb R}}$ on a convex domain as $$\max_{1\leq i \leq N}\min_{1\leq j\leq M(i)}
{\ensuremath{\{\ell_{1,i},\ldots,\ell_{M(i),i}\}}},\qquad \ell_{j,i}:K{\ensuremath{\rightarrow}}{{\mathbb R}}\quad \text{affine}.$$ Nonetheless, Proposition \[P:density-maxmin\] is of a rather different nature since we are allowed to take only max and min of two affine functions at a time.
Proof of Proposition \[P:relu-maxmin\] {#S:relu-pf}
--------------------------------------
We may assume without loss of generality that $K$ is contained in the positive orthant: $$K\subseteq {{\mathbb R}}_+^{d_{in}}={\ensuremath{ \left\{ {\ensuremath{\left(x_1,\ldots, x_{d_{in}} \right)}}\in {{\mathbb R}}^{d_{in}}\,\right|\left.\,x_i\geq
0,\qquad 1\leq i \leq d_{in} \right\}}}$$ since we can always shift the input to a neural net by a fixed vector. Let us fix a max-min string $$g=\sigma_{L-1}(\ell_{L},\sigma_{L-2}(\ell_{L-1},\ldots,
\sigma_2(\ell_3, \sigma_1(\ell_1,\ell_2))\cdots).$$ We can assume $g$ is non-negative since we can subtract a constant in the final linear transformation. Note that for any constant $C,$ the function $g+C$ is also a max-min string whose affine tranformations are $\ell_i+C.$ Since we may subtract an arbitrary constant in the output of the last layer in a $\operatorname{ReLU}$ net, we may additionally assume that each $\ell_i$ is non-negative on $K.$ With these reductions, we construct the neural net that computes $g(x)$ for every $x\in K.$ For all $j=2,\ldots, L$ define invertible affine tranformations $A_j:{{\mathbb R}}^{d_{in}+d_{out}}{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{in}+d_{out}}$ $$A_j(x,y)=
\begin{cases}
(x, y-\ell_j(x)), &\text{if }\sigma_{j-1}=\text{max}\\
(x, -y+\ell_j(x)), &\text{if }\sigma_{j-1}=\text{min}.
\end{cases},$$ where $x\in {{\mathbb R}}^{d_{in}}$ and $y\in {{\mathbb R}}^{d_{out}}.$ Their inverses are given by $$A_j^{-1}(x,y) =
\begin{cases}
(x, y+\ell_j(x)), &\text{if }\sigma_{j-1}=\text{max}\\
(x, -y+\ell_j(x)), &\text{if }\sigma_{j-1}=\text{min}.
\end{cases}.$$ Further, set $$A_1(x)=(x,\ell_1(x)),\qquad x\in {{\mathbb R}}^{d_{in}}.$$ Write $H_1:=A_1$ and $$H_j:=A_j\circ \operatorname{ReLU}\circ A_j^{-1},\qquad j=2,\ldots, L.$$ The image of $K$ under $H_0$ is the graph of $\ell_1,$ and the image of the graph of any function $g:K{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}$ under $H_j$ is the graph of $\sigma_{j-1}{\ensuremath{\left(\ell_j,g \right)}}.$ Hence, the image of $K$ under the $\operatorname{ReLU}$ net $$\operatorname{ReLU}\circ H_L\circ\cdots \circ H_1$$ is the graph of $g.$ Note that the final $\operatorname{ReLU}$ is trivial since $g$ is non-negative. Appending a final layer $(x_1,\ldots,x_{d_{in}}, y_1,\ldots, y_{d_{out}}
)\mapsto {\ensuremath{\left(y_1,\ldots, y_{d_{out}} \right)}}$ yields the desired net.
Proof of Proposition \[P:density-maxmin\] {#S:density-pf}
-----------------------------------------
Note that if $g$ is a max-min string on $d_{in}$ input variables and $d_{out}$ output variables, then so is $g(x-x_0)$ for any $x_0\in {{\mathbb R}}^{d_{in}}.$ Using also that every compact set is contained in a ball shows that we may assume without loss of generality that $K$ is a ball $B_r$ of radius $r$ centered at the origin.
Fix a continuous function $f:B_r{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}.$ We first explain how to uniformly approximate $f$ by max-min strings in the model case when we seek to approximate $f$ on an arbitrary finite subset of $\mathbb R^{d_{in}}$.
\[P:discrete\] Let $S\subseteq\mathbb R^{d_{in}}$ be a finite set. Then any function $f:S\to
{{\mathbb R}}^{d_{out}}$ can be computed exactly by a max-min string.
We prove the proposition by induction on $|S|$. If $S={\ensuremath{\{s\}}},$ then the constant max-min string $f(s)$ suffices. Suppose now that ${\ensuremath{\left| S \right|}}\geq 2.$ The idea is to consider the convex hull $\widehat{S}$ of the points in $S$ and “repeatedly cut off a corner.” Let $s_0\in S$ be an extreme point of $\widehat S$, a vertex of $\widehat{S}$ that is not contained in any proper face. By the inductive hypothesis, there is a max-min string $g$ on $d_{in}$ input variables and $d_{out}$ output variables that agrees with $f$ on $S\backslash\{s_0\}$. Moreover, for every $t>0,$ we can find an affine function $\ell:{{\mathbb R}}^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{out}}$ with $\ell(s_0)=0$ and $\ell(s)\geq t$ for $s\in S\backslash
{\ensuremath{\{s_0\}}}$ (the inequality holds for each of the $k$ components of $\ell$). Taking $t$ large, define the max-min string $$\widehat{g}=\max(\min(g, f(s_0)+\ell),f(s_0)-\ell) ,$$ where the max and min are componentwise. By construction, $\widehat{g}(s_0)=f(s_0).$ Further, because $t$ is large, $\widehat{g}(s)=f(s)$ for $s\in S\backslash {\ensuremath{\{ s_0\}}}$. Hence $\widehat{g}$ and $f$ agree on $S$, completing the proof.
We carry out the same proof idea for continuous functions on ${{\mathbb R}}^{d_{in}}.$ We focus for simplicity on the construction for $d_{in}=2$ and $d_{out}=1$. The extension to general $d_{out}$ is immediate and requires only that various inequalities below hold for every component of vectors in ${{\mathbb R}}^{d_{out}}.$ The extension to $d_{in}\geq 3$ requires a minor modification, which we present after the $d_{in}=2$ proof. Before getting into the details, we emphasize the main difference between the discrete case treated in Proposition \[P:discrete\] above and the continuous case below. The issue is that now when we cut off a corner from the convex hull of the set where we have ${\varepsilon}$-approximated the function $f$, we have to approximate $f$ correctly on the entire piece we cut off, not just at a single vertex. To get an $\varepsilon$-approximation, we need our corner piece to have diameter $O\left({\omega}^{-1}_f({\varepsilon})\right)$ so that the variation of $f$ on the piece is $O(\varepsilon)$ (recall that ${\omega}^{-1}_f({\varepsilon})$ is the inverse modulus of continuity). That is, we can only cut off small-diameter pieces at a time. Thus, to build an approximation to $f$ on ball of radius $R$ from an approximation to $f$ on a ball of radius $r<R$, we have to slowly add small pieces to $B_r$ in all directions until the resulting set grows to contain $B_R.$ Our precise construction repeatedly uses the following observation. We state the observation for $d_{in}=2$ and explain below its extension to $d_{in}\geq 3.$
\[L:extend\] Fix ${\varepsilon}>0$ and a continuous function $f:\mathbb R^2\to\mathbb
R$. Suppose $K\subseteq {{\mathbb R}}^2$ and $\triangle ABC$ is an triangle with $$\operatorname{diam}(\triangle ABC)\leq{\omega}^{-1}_f({\varepsilon})$$ such that $K$ is contained in the infinite planar sector $\angle BAC$. Then if there exists a max-min string $g$ with $$\sup_{x\in K}{\ensuremath{\left| f(x)-g(x) \right|}}\leq {\varepsilon},$$ then there also exists a max-min string $\widehat{g}$ with $$\sup_{x\in K\cup ABC}{\ensuremath{\left| f(x)-\widehat{g}(x) \right|}}\leq {\varepsilon}.$$
Let $g$ be a max-min string that $\varepsilon$-approximates $f$ on $K$. Let $\ell$ be the affine function with $\ell(A)=0$, $\ell(B)=\ell(C)=\varepsilon$. As in Proposition \[P:discrete\], define $$\ell_-(x):=f(A)-\ell(x),\qquad \ell_+(x):=f(A)+\ell(x)$$ and consider the max-min string $$\widehat{g}=\max(\ell_-, \min(\ell_+, g)).$$ Next, by the definition of ${\omega}^{-1}_f,$ we have $${\ensuremath{\left| f(x)-\widehat{g}(x)-f(x) \right|}}\leq {\varepsilon},\qquad \qquad x\in ABC.$$ We now show that this estimate continues to hold for $x\in K$ as well. We claim that on $K\cup\triangle ABC$ we have $$\ell_--{\varepsilon}\leq f\leq \ell_++{\varepsilon}\label{eq:linbound}$$ These inequalities follow essentially from the fact that the absolute values of the slopes of $\ell_\pm$ on the rays $Ap$ for any point $p$ on segment $BC$ are bounded below by $\frac{{\omega}^{-1}_f({\varepsilon})}{{\varepsilon}}$. This is true since for any such $p$ we have $\ell(p)=\varepsilon$ and $|A-p|\leq\omega_f^{-1}(\varepsilon)$. Now to prove Inequality \[eq:linbound\] we note that for any $x\in K$ we have $$|f(x)-f(A)|\leq {\varepsilon}+\frac{{\varepsilon}}{{\omega}^{-1}_f({\varepsilon})}|x-A|.$$ Indeed, suppose that ray $Ax$ has length $n\omega^{-1}_f({\varepsilon})+r$ for integer $n$ and real remainder $r<\omega^{-1}_f({\varepsilon})$. Then by the triangle inequality we have $$|f(x)-f(A)|\leq n{\varepsilon}+{\varepsilon}\leq {\varepsilon}+\frac{{\varepsilon}}{{\omega}^{-1}_f({\varepsilon})}|x-A|$$ as desired.
These estimates imply that on $K$ $$f-{\varepsilon}=\min(f, f-{\varepsilon})\leq \min(\ell_+,g)\leq
\max{\ensuremath{\left(\ell_-,\min(\ell_+,g) \right)}}=\widehat{g}$$ and $$\widehat{g}=\max{\ensuremath{\left(\ell_-,\min(\ell_+,g) \right)}}\leq \max(\ell_-, g)\leq \max(f, f+{\varepsilon})=f+{\varepsilon}.$$ Therefore, $\widehat{g}-{\varepsilon}\leq f \leq \widehat{g}+{\varepsilon},$ as desired.
![To extend an $\varepsilon$-approximation of $f$ on the inner disk of radius $r$ to the outer disk of radius $r'=r+\frac{\omega_f^{-1}(\varepsilon)^2}{r}$, we proceed in steps. Each step, we draw triangle $X'ZY'$ as shown and apply Lemma \[L:extend\] to extend our approximation to a larger region. Because the outer circle $B_{r'}(P)$ is contained in sector $X'ZY'$, we do not lose any area contained in $B_{r'}(P)$ when applying Lemma \[L:extend\].[]{data-label="fig:extension"}](NeuralNetExtension1.png){width="10cm"}
![In Figure \[fig:extension\], after applying Lemma \[L:extend\], the region on which we approximated $f$ has grown to include the shaded circular sector $X_0PY_0$. (This is just because it is contained in the union of the two shaded regions in Figure \[fig:extension\].) Since $d(X,Y)\asymp \varepsilon$, this means that applying Lemma \[L:extend\] to $O\left(\frac{r}{\varepsilon}\right)$ rotated configurations of this form extends the region of $\varepsilon$-approximation from $B_r(P)$ to $B_{r'}(P)$.[]{data-label="fig:extension2"}](NeuralNetExtension2.png){width="10cm"}
We now turn to the details of the proof of Proposition \[P:density-maxmin\]. We will explain how to approximate our fixed continuous function $f$ by a max-min string on a ball of radius $R>0$ centered at the origin. We will use Lemma \[L:extend\] to show that we can approximate $f$ on successively larger and larger balls. Observe that if $r\leq {\omega}^{-1}_f({\varepsilon})$ then $${\left\lVertf-f(0)\right\rVert}_{C^0(B_r)}\leq {\varepsilon},$$ so that the constant max-min string $f(0)$ is an $\varepsilon$-approximation to $f$ on the small ball $B_{{\omega}^{-1}_f({\varepsilon})}(0)$. To prove that we can approximate $f$ on larger balls, suppose $g$ is a max-min string on ${d_{in}}$ variables that approximates $f$ to within $\varepsilon$ on the ball $B_r(0)$ with $r\geq w_f^{-1}({\varepsilon}).$ We use Lemma \[L:extend\] to construct a new max-min string $\widehat{g}$ which uniformly $\varepsilon$-approximates $f$ on a ball of slightly larger radius $$R_{r,{\varepsilon}}:=r+\frac{{\omega}^{-1}_f({\varepsilon})^2}{10r}.$$ Since for every ${\varepsilon}>0,$ the function $R_{r,{\varepsilon}}$ is strictly increasing in $r$ it cannot have a fixed point and the $k-$fold composition $R_{r,{\varepsilon}}^{(k)}$ sends any $r>0$ to infinity with $k.$ Using this procedure repeatedly therefore allows us to increase $r$ without bound and will complete the proof. Our approach is illustrated in Figures \[fig:extension\] and \[fig:extension2\].
We begin with the construction when ${d_{in}}=2$ and will explain the simple modification for ${d_{in}}\geq 3$ below. For each $r'>r$ and any two sufficiently close points $X,Y$ on the boundary of $B_r$, let $X',Y'$ be the intersections of line $XY$ with the boundary circle of $B_{r'}(P)$. Also, denote by $Z$ be the intersection of the tangents to $B_{r'}$ through $X',Y'$ (see Figure 1). Then $B_r$ is contained in the planar sector $\angle X'ZY'$, and the diameter of $\triangle X'ZY'$ can be made arbitrarily small by taking $r'$ close to $r$ and $X$ close to $Y.$ In particular, for every $r\geq
{\omega}^{-1}_f({\varepsilon}),$ we take $$r'=R_{r,{\varepsilon}}=r+\frac{{\omega}^{-1}_f({\varepsilon})^2}{10 r},\qquad
{\ensuremath{\left| XY \right|}}={\omega}^{-1}_f({\varepsilon}){\ensuremath{\left(1-\frac{{\omega}^{-1}_f({\varepsilon})^2}{100 r^2} \right)}}^{1/2}.$$ This choice for $|XY|$ is valid because $|XY|\leq \omega^{-1}_f({\varepsilon})\leq r$, so we can indeed find points $X,Y$ at this distance with no problem. Then $$|X'Y'|=\sqrt{|XY|^2+4(r'-r)^2}={\omega}^{-1}_f({\varepsilon}).$$ We also know that $|X'Z|=|Y'Z|\leq |X'Y'|={\omega}^{-1}_f({\varepsilon})$ because $|X'Y'|={\omega}^{-1}_f({\varepsilon})\leq r\leq r'$, implying obtuseness of $\triangle X'ZY'$ at $Z$. Thus, $$\operatorname{diam}(\triangle X'ZY')=|X'Y'|={\omega}^{-1}_f({\varepsilon}).$$ Lemma \[L:extend\] therefore shows that there exists a max-min string $g'$ that uniformly ${\varepsilon}$ approximates $f$ on $K'=\triangle X'Y'Z\cup B_r.$ Notice that $K'$ contains the circular sector of $B_{R_{r,{\varepsilon}}}$ cut out by the rays $OX$ and $OY.$ Finally, consider a $\delta-$net ${\ensuremath{\{p_i\}}}$ on the circumference of $B_r$ with $$\delta =\frac{{\omega}^{-1}_f({\varepsilon})}{2}{\ensuremath{\left(1-\frac{{\omega}^{-1}_f({\varepsilon})^2}{100 r^2} \right)}}^{1/2.}$$ The size of this net is $O(r/{\omega}^{-1}_f({\varepsilon})).$ Applying Lemma \[L:extend\] $O(r/{\omega}^{-1}_f({\varepsilon}))$ times and repeating the above argument with $(X,Y)=(p_i, p_{i+1})$ completes the proof of the upper bound in Theorem \[T:main\] when ${d_{in}}=2.$
The argument when ${d_{in}}\geq 3$ is essentially the same. The idea is to take the diagrams depicted and rotate them around the axis $PZ.$ Lemma \[L:extend\] extends to higher dimensions with the triangle $\triangle ABC$ replaced by the tip of a cone with the same diameter requirement. Such a cone is obtained by rotating $X'ZY'$ in Figures 1 and 2. The rest of the argument then carries over verbatim.
Now we analyze the efficiency of this procedure. First, to complete a single radius increment requires covering the boundary of $B_r$ with balls of radius $O{\ensuremath{\left({\omega}^{-1}_f({\varepsilon}) \right)}}$. It is standard that in $\mathbb R^{d_{in}}$, this requires $$\left(\frac{O(r)}{{\omega}^{-1}_f({\varepsilon})}\right)^{{d_{in}}-1}$$ balls. We get one extra max and min in the max-min string we build to approximate $f$ for each such ball. Thus, at a cost of $\left(\frac{O(r)}{{\omega}^{-1}_f({\varepsilon})}\right)^{{d_{in}}-1}$ many maxes and mins, the radius on which we approximate $f$ increases $$r\mapsto R_{r,{\varepsilon}}=r+\frac{{\omega}^{-1}_f({\varepsilon})^2}{10r}.$$ Hence, if we fix $R>{\omega}^{-1}_f({\varepsilon}),$ then for every ${\omega}^{-1}_f({\varepsilon})\leq r \leq
R,$ we have $$R_{r,{\varepsilon}}-r\geq\frac{{\omega}^{-1}_f({\varepsilon})^2}{10R}$$ and to obtain an approximation of $f$ on $B_R,$ by a max-min we need to extend the approximation of $f$ from a small ball to a larger ball at most $10R^2/{\omega}^{-1}_f({\varepsilon})^2$ times. The number of maxes and mins required for each extension is $(O(R)/{\omega}^{-1}_f({\varepsilon}))^{{d_{in}}-1}$. Hence, the length of the max-min string we construct to approximate $f$ on $B_R$ is $$\left(\frac{O(R)}{{\omega}^{-1}_f({\varepsilon})}\right)^{d_{in}+1},$$ as claimed.
We have tacitly neglected the case $d_{in}=1$. This case is the same as $d_{in}=2$ but easier. In fact here we require only $$\left(\frac{O(R)}{{\omega}^{-1}_f({\varepsilon})}\right)$$ layers, which would naively correspond to $d_{in}=0$. The reason is that a $1$-dimensional ball can be increased in radius by $\omega_f^{-1}(\varepsilon)$ by adding only a single external line segment of length $\omega_f^{-1}(\varepsilon)$. In the higher dimensional cases, we need to add the external pieces mostly tangentially which requires more layers.
Proof of the Lower Bound in Theorem \[T:main\] {#S:LB}
==============================================
The purpose of this section is to prove that for every ${d_{in}}\geq 1,$ there exists $f\in C([0,1]^{d_{in}},\mathbb R)$ and $\eta=\eta({d_{in}},f)>0$ so $f$ satisfies the following property. For any $\operatorname{ReLU}$ net $\mathcal N$ with input dimension ${d_{in}}$, hidden layer width ${d_{in}}$, and output dimension $1$, we have $${\left\lVertf-f_{\mathcal N}\right\rVert}_{C^0}\geq \eta.$$
In fact, we will show that if there is $a$ such that a compact connected component of the pre-image $f^{-1}(a)$ disconnects a bounded region from the infinite component of $\mathbb R^{d_{in}}$, then $f$ is not approximable by depth-${d_{in}}$ $\operatorname{ReLU}$ nets.
Fix ${d_{in}}\geq 1,$ and consider a width ${d_{in}}$ $\operatorname{ReLU}$ net $$f_{\mathcal N} := A_n\circ \operatorname{ReLU}\circ A_{n-1}\cdots \circ\operatorname{ReLU}\circ A_1,$$ where the $A_i$’s are affine and $A_n$ maps ${{\mathbb R}}^{d_{in}}$ to ${{\mathbb R}},$ while for $1\leq i \leq n-1,$ the transformations $A_i$ map ${{\mathbb R}}^{d_{in}}$ to ${{\mathbb R}}^{d_{in}}.$ We may assume without loss of generality that $A_i$ have full rank for all $i$ since $f_{\mathcal N}$ is continuous with respect to the $A_i$’s and affine maps with full rank are dense among all affine maps. Define a *level set* of a function $f$ to be a connected component of a pre-image $f^{-1}(a)$ for some $a$. The following Lemma shows the level sets of any function $$f_j(x)=\operatorname{ReLU}\circ A_j\circ \cdots \operatorname{ReLU}\circ A_1(x)$$ computed by the first $j$ hidden layers of $\mathcal N$ are of a rather special form.
\[L:LB-key\]
For each $j\geq 1$, set $S_j$ to be the set of points on which all ReLU evaluations throughout the evaluation of $f_j$ are (strictly) positive. Then $S_j$ is open and convex, $f_j$ is affine on $S_j$, and every level set of $f_j$ that is bounded is contained in $S_j.$
For every $j,$ $S_j$ is open and convex since it is cut out by a collection of inequalities of the form ${\ensuremath{\{\ell_k(x)>0\}}},$ where $\ell_k:{{\mathbb R}}^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}$ are affine. Note that the level sets of $f_n:{{\mathbb R}}^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}$ are the union of level sets of $f_{n-1}:{{\mathbb R}}^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}^{d_{in}}.$ Hence, it is enough to show that for every $j\leq n-1$ if a level set of $f_j$ intersects the complement of $S_j,$ then it is unbounded. We prove this by induction on $j\geq 1.$ The base case $j=1$ is immediate if $S_1={{\mathbb R}}^{d_{in}}.$ Otherwise, consider $y\not \in S_1.$ Then, at least one, say the $k^{th}$, component of $f_1(y)$ is zero. The inverse image under $\operatorname{ReLU}$ of the point $f_1(y)$ therefore contains a ray (e.g. the ray $r$ starting at $f_1(y)$ and going to $-\infty$ parallel to the $k^{th}$ coordinate axis). The inverse image of $r$ under the affine map $A_1$ also contains a ray and hence is unbounded, proving the base case.
For the inductive step, fix some $j\geq 2$. There is nothing to prove if $S_j={{\mathbb R}}^{d_{in}}$. Otherwise, consider $y \not \in S_j.$ If $y\not \in S_{j-1}$, then we are done by induction since $f_{j-1}^{-1}(f_{j-1}(y))\subseteq f_{j}^{-1}(f_{j}(y)).$ If $y\in S_{j-1},$ then we argue as before. Namely, the $k^{th}$ component of $f_{j}(y)$ vanishes for some $k$ and the inverse image under $\operatorname{ReLU}\circ A_{j}$ of the point $f_{j}(y)$ therefore contains a ray. If this ray is contained in $f_{j-1}(S_{j-1})$, then its pre-image under $f_{j-1}$ also contains a ray since $f_{j-1}$ is affine when restricted to $S_{n-1}.$ Otherwise, this ray intersects the boundary of $f_{j-1}(S_{j-1})$ at some point $p.$ By induction, the pre-image $f_{j-1}^{-1}(p)$ is unbounded and hence so is $f_{j}^{-1}(f_{j}(y))$ since it contains $f_{j-1}^{-1}(p).$ This completes the proof.
We now complete the proof of the lower bound in Theorem \[T:main\]. Suppose that $f:{{\mathbb R}}^{d_{in}}{\ensuremath{\rightarrow}}{{\mathbb R}}$ is a continuous function such that for some $a$, the pre-image $f^{-1}(a)$ contains a compact connected component $A$ and that $\mathbb R^{d_{in}}\backslash A$ contains a bounded connected component $B.$ For example, we could take $$f(x_1,\ldots, x_{d_{in}}):=\sum_{j=1}^{d_{in}} \left(x_j-\frac{1}{2}\right)^2,\qquad a=\frac{1}{4}.$$ In this case $A$ is a sphere and $B$ is a ball. Suppose $y\in B$ and $f(y)=b\neq a$. Then for $\eta<\frac{|a-b|}{4}$, we show that $f$ is not $\eta$-approximable on $A\cup B$.
Set $c=\frac{a+b}{2}$, and let $C$ be the intersection of $f^{-1}(c)$ with $B$. Since $f$ is continuous, the intermediate value theorem implies that $C$ separates $y$ and $A$. Let $C'\subseteq C$ be the boundary of any connected component of $C$ that contains $y$. Informally, $A$ surrounds $C'$ which surrounds $y$. Now, suppose some $f_{\mathcal N}$ computed by a neural net satisfies $${\left\lVertf-f_{\mathcal N}\right\rVert}_{C^0(A\cup B)}\leq \eta.$$ Denote by $S_{\mathcal N}$ the set of points in ${{\mathbb R}}^{d_{in}}$ where all the $\operatorname{ReLU}$s in $\mathcal N$ are positive. Suppose first that $S_{\mathcal N}$ contains $C'.$ By Lemma \[L:LB-key\], $S_{\mathcal N}$ is convex. Since the intermediate value theorem implies any path from $y$ to $\infty$ intersects $C'$, we know that $y$ is in the convex hull of $C'$. Hence we have $y\in S_{\mathcal N}$ as well. Since $f_{\mathcal N}$ is affine on $S_{\mathcal N},$ this means that $f_{\mathcal N}(y)$ is between the minimum and maximum values of $f_{\mathcal N}$ on $C'$. As $f(y)=b$ and $f(C')=c$ we get a contradiction since $\eta<\frac{|b-c|}{2}=\frac{|a-b|}{4}.$
Suppose in the second case that $S_{\mathcal N}$ does not contain $C'$, so there is $x\in C'\backslash S_{\mathcal N}$. Then, by Lemma \[L:LB-key\], the level set of $f_{\mathcal N}$ containing $x$ must be unbounded, and hence must intersect $A$ (as $A$ separates $y$ from $\infty$ and $x\in C'$ is reachable from $y$ without intersecting A). This is again a contradiction for $\eta<\frac{|a-c|}{2}=\frac{|a-b|}{4}.$
In both cases, we showed that $f$ and $f_{\mathcal N}$ differed significantly; the first case used the affineness of $f_{\mathcal N}$ on $S_{\mathcal N}$ while the second used the unboundedness of level sets away from $S_{\mathcal N}$. We conclude that a width-$d$ net cannot uniformly approximate $f$. $\square$
|
---
address: 'Oleg Yaremko,Ekaterina Zhuravleva Penza State University,str. Lermontov, 37, 440038, Penza, Russia'
author:
- 'O. Yaremko, E.Zhuravleva'
title: Matrix Fourier transform with discontinuous coefficients
---
*[Mathematics Subject Classification 2010]{}:[35N30 Overdetermined initial-boundary value problems; 35Cxx Representations of solutions; 65R10 Integral transforms]{}.*\
[Penza state university, PO box 440026, Penza, Lermontov’s street, 37, Russia]{}
Introduction
============
Different representations of the solutions of the equilibrium equation through functions of tension are used when solving problems by the variable separation method. The required problem is taken to the solution of differential equations of a more simple structure with the help of such representations. Each functions of tension in these equations “is not fastened” with others, but it enters into boundary conditions together with the others. A.F.Ulitko \[7\] has offered rather effective method of research of problems of mathematical physics - a method Eigen vector-valued functions. This method is the vector analogue of the Fourier method.
The method of integral transformations is also an analytical method of the decision of solution of problems theory of elasticity. The method of integral transformations we consider and develop in this article. we come to the most simple problem in space of images with the help of the integral transformations (Fourier, Laplace, Hankel, etc.). The finding of the formula of direct is the main difficulty in solving of problems of this approach. Extensive enough bibliography of works on use of this method in problems of the theory of elasticity is resulted in J.S.Ufljand’s monography \[2\]. Method of the vector integral transforms of Fourier is equivalent the method Eigen vector-valued functions, however, unlike the last it can to be applied successfully be used, applied to the solution of problems of the theory of elasticity in a piece-wise homogeneous medium. The theory of integral transforms of Fourier with piece-wise constant coefficients in a scalar case was studied by Ufljand J.S. \[16\], \[17\], Najda L.S. \[11\], Protsenko V. S \[12\], \[13\], Lenjuk M. P \[8\], \[9\], \[10\]. The vector variant of a method adapted for the solution of problems in piece-wise homogeneous medium is developed by the author in \[2\], \[19\]. Unknown tension in the boundary conditions and in the internal conditions of conjugation don’t commit splitting in a considered dynamic problem, so the application of the scalar integral transforms of Fourier with piece-wise constant coefficients does not lead to success. Method of the vector integral transforms of Fourier with discontinuous coefficients is used for its solution in the present work. Conformable theoretical bases of a method are presented in item 4 for granted. The necessary proofs by the method of contour under the scheme developed in \[2\] and \[19\]. The closed form solution of the dynamic problem found in the use of this method in item 4. Integral transforms arise in a natural way through the principle of linear superposition in constructing integral representations of solutions of linear differential equations. First note that the structure of integral transforms with the relevant variables are determined by the type of differential equation and the kind of media in which the problem is considered. Therefore decision of integral transforms are the problem for mathematical physics piecewise-homogeneous (heterogeneous) media. It is clear this method is an effective for obtaining the exact solution of boundary-value problems for piecewise-homogeneous structures mathematical physics.
The author together with I.I.Bavrin has proposed integral transforms with non-separate variables for solving multidimensional problems in the work [@yar].
Let $V$ from $R^{n+1}$ be the half-space $$V=\left\{ {\left( {y_1 ,...,y_n ,x} \right)\in R^{n+1}:x>0}
\right\},$$ then solution of the Dirichlet’s problem for the half-space is expressed by Poisson formula takes the form: [@bes]
$$u(x,y)=\Gamma \left( {\frac{n+1}{2}} \right)\pi
^{-\frac{n+1}{2}}\int\limits_{y=0} {\frac{x}{\left[ (y-\eta )^2+x^2 \right]^{\frac{n+1}{2}}}f(\eta )d\eta } .$$ Obviously Poisson’s kernel is the form of integral Laplace transform and therefore expansion of the function $f(y)$ for the eigenfunctions of the Laplace operator $\Delta$ is obtained from the reproduce properties of the Poisson kernel: $$f(y)=\mathop {\lim }\limits_{\tau \to 0} \int\limits_0^\infty {\lambda
^{\frac{n}{2}}e^{-\lambda \tau }} \left(\frac{1}{\left( {\sqrt {2\pi }
} \right)^n}\int\limits_{R^n} {\frac{J_{\frac{n-2}{2}} \left( {\lambda
\left| {y-\eta } \right|} \right)}{\left| {y-\eta }
\right|^{^{\frac{n-2}{2}}}}} f\left( \eta \right)d\eta\right)d\lambda ,$$ here $J_{\nu}$ is Bessel’s function of order $\nu$ [@bes]. We may assume that integral transforms with non- separate variables are defined as follows [@yar] on the basis of this expansion:\
direct integral Fourier transform has the form
$$F\left[ f \right]\left( {y,\lambda } \right)=\frac{1}{\left( {\sqrt {2\pi }
} \right)^n}\int\limits_{R^n} {\frac{J_{\frac{n-2}{2}} \left( {\lambda
\left| {y-\eta } \right|} \right)}{\left| {y-\eta }
\right|^{^{\frac{n-2}{2}}}}} f\left( \eta \right)d\eta \equiv \hat {f}\left(
{y,\lambda } \right),$$
inverse Fourier integral transform has the form $$F^{-1} [\hat {f}](y)=
\mathop {\lim }\limits_{\tau \to 0} \int\limits_0^\infty {\lambda
^{\frac{n}{2}}e^{-\lambda \tau }} \hat {f}(y;\lambda )d\lambda \equiv f(y).$$
In our case the construction of multi-dimensional analogues for integral transforms (1)-(2) with discontinuous coefficients is the purpose of this research.
One-dimensional integral transforms with discontinuous coefficients
===================================================================
In this paper integral transforms with discontinuous coefficients are constructed in accordance with author’s work [@10]. Let $\varphi \left( {x,\lambda } \right)$ and $\varphi ^\ast \left(
{x,\lambda } \right)$ be eigenfunctions of primal and dual problems Sturm-Liouville for Fourier operator on sectionally homogeneous axis $ I_n $, $$I_n =\left\{ {x:\;x\in \mathop
U\limits_{j=1}^{n+1} \left( {l_{j-1} ,l_j } \right),\;\,l_0 =-\infty
,\;\,l_{n+1} =\infty ,\;\,l_j <l_{j+1} ,\;\,j=\overline {1,n} } \right\}.$$ Let us remark that eigenfunction $\varphi \left( {x,\lambda } \right)$, $$\varphi \left( {x,\lambda } \right)=\sum\nolimits_{k=2}^n {\theta \left(
{x-l_{k-1} } \right)\,\theta \left( {l_k -x} \right)\,\varphi _k \left(
{x,\lambda } \right)+}$$ $$+\,\theta \left( {l_1 -x} \right)\,\varphi _1 \left( {x,\lambda }
\right)+\theta \left( {x-l_n } \right)\,\varphi _{n+1} \left( {x,\lambda }
\right)$$ is the solution of separated differential equations system $$\left( {a_m^2 \frac{d^2}{dx^2}+{\kern 1pt}\lambda ^2} \right)\,\varphi _m
\left( {x,\lambda } \right)=0,\;\;x\in \left( {l_m ,l_{m+1} } \right);\quad
m=1,...,n+1,$$ by the coupling conditions $$\left[ {\alpha _{m1}^k \frac{d}{dx}+\beta _{m1}^k } \right]\varphi _k
=\left[ {\alpha _{m2}^k \frac{d}{dx}+\beta _{m2}^k } \right]\varphi _{k+1}
,$$ $$x=l_k ,\;\;k=1,...,n;\;\;m=1,2,$$ on the boundary conditions $$\left. {\varphi _1 } \right|_{x=-\infty } =0\,,\;\,\left. {\;\varphi _{n+1}
} \right|_{x=\infty } =0.$$ Similarly, the eigenfunction $\varphi ^\ast \left(
{x,\lambda } \right)$, $$\varphi ^\ast \left( {\xi ,\lambda } \right)=\sum\nolimits_{k=2}^n {\theta
\left( {\xi -l_{k-1} } \right)\,\theta \left( {l_k -\xi } \right)\,\varphi
_k^\ast \left( {\xi ,\lambda } \right)\,+}$$ $$+\theta \left( {l_1 -\xi } \right)\,\varphi _1^\ast \left( {\xi ,\lambda }
\right)+\theta \left( {\xi -l_n } \right)\,\varphi _{n+1}^\ast \left( {\xi
,\lambda } \right)$$ is the solution of separate differential equations system $$\left( {a_m^2 \frac{d^2}{dx^2}+{\kern 1pt}\lambda ^2} \right)\,\varphi
_m^\ast \left( {x,\lambda } \right)=0,\;\;x\in \left( {l_m ,l_{m+1} }
\right);\quad m=1,...,n+1,$$ by the coupling conditions $$\frac{1}{\Delta _{1,k} }\left[ {\alpha _{m1}^k \frac{d}{dx}+\beta _{m1}^k }
\right]\varphi _k^\ast =\frac{1}{\Delta _{2,k} }\left[ {\alpha _{m2}^k
\frac{d}{dx}+\beta _{m2}^k } \right]\varphi _{k+1}^\ast ,
\quad
x=l_k ,\;\;$$ where $$\Delta _{i,k} =\det \left( {{\begin{array}{*{20}c}
{\alpha _{1i}^k } \hfill & {\beta _{1i}^k } \hfill \\
{\alpha _{2i}^k } \hfill & {\beta _{2i}^k } \hfill \\
\end{array} }} \right)
k=1,...,n;\;\;\quad i,m=1,2,$$ on the boundary conditions $$\left. {\varphi _1 } \right|_{x=-\infty } =0\,,\;\,\left. {\;\varphi _{n+1}
} \right|_{x=\infty } =0.$$ Further normalization eigenfunctions is accepted by the following: $$\varphi _{n+1} \left( {x,\lambda }
\right)=e^{ia_{n+1}^{-1} x\lambda }. \quad \varphi _{n+\mbox{1}}^\ast \left(
{x,\lambda } \right)=e^{-ia_{n+1}^{-1} x\lambda }.$$ Let direct $F_{n} $ and inverse $F_{n}^{-1} $ Fourier transforms on the Cartesian axis with $ n $ division points be defined by the rules in \[10\] : $$F_{n} \left[ f \right]\,\left( \lambda \right)=\sum\limits_{m=0}^{n+1}
{\int\limits_{l_m-1 }^{l_{m} } \; } u_{m}^\ast \left( {\xi ,\lambda }
\right)\,f_{m} \left( \xi \right)d\xi \equiv \hat {f}\left( \lambda
\right),$$ $$f_k \left( x \right)=\frac{1}{\pi i}\int\limits_0^\infty {u_k \left(
{x,\lambda } \right)\hat {f}\left( \lambda \right)\lambda d\lambda .}$$
Vector Fourier transform with discontinuous coefficients
========================================================
Let’s develop the method of vector Fourier transform for the solution this problem. Let’s consider Sturm–Liouville vector theory \[1\] about a design bounded on the set of non-trivial solution of separate simultaneous ordinary differential equations with constant matrix coefficients $$\label{eq4}
\left( {A_m^2 \frac{d^2}{dx^2}+\lambda ^2{\rm E}+\Gamma _m^2 } \right)y_m
=0,\;\,q_m^2 =\lambda ^2{\rm E}+\Gamma _m^2 ,\;\,m=\overline {1,n+1}$$ on the boundary conditions. $$\label{eq5}
\left. {\left( {\left( {\alpha _{11}^0 +\lambda ^2\delta _{11}^0 }
\right)\frac{d}{dx}+\left( {\beta _{11}^0 +\lambda ^2\gamma _{11}^0 }
\right)} \right)y_1 } \right|_{x=l_0 } =0,\quad \left. {\left\| {y_{n+1} }
\right\|{\kern 1pt}{\kern 1pt}} \right|_{x=\infty } \,<\,\infty$$ and conditions of the contact in the points of conjugation of intervals $$\label{eq6}
\left( {\left( {\alpha _{j1}^k +\lambda ^2\delta _{j1}^k }
\right)\frac{d}{dx}+\left( {\beta _{j1}^k +\lambda ^2\gamma _{j1}^k }
\right)} \right)y_k =\left( {\left( {\alpha _{j2}^k +\lambda ^2\delta
_{j2}^k } \right)\frac{d}{dx}+} \right.\left. {\left( {\beta _{j2}^k
+\lambda ^2\gamma _{j2}^k } \right) } \right)y_{k+1} ,$$ $x=l_k ,\;\,k=\overline {1,n} ,\;\,j=1,2.,$ where $$y_m \left( {x,\lambda } \right)=\left( {{\begin{array}{*{20}c}
{y_{1m} \left( {x,\lambda } \right)} \hfill \\
\vdots \hfill \\
{y_{rm} \left( {x,\lambda } \right)} \hfill \\
\end{array} }} \right),
\left\| {y_m } \right\|=\sqrt {y_{1m}^2 +...+y_{rm}^2 } ,m=\overline {1,n+1}
.$$ Let for some $\lambda $ the considered the boundary problem has a non-trivial solution $$y\left( {x,\lambda } \right)=\sum\limits_{k=1}^n {\theta \left( {x-l_{k-1} }
\right)\,\theta \left( {l_k -x} \right)\,y_k \left( {x,\lambda }
\right)\,+\,\theta \left( {x-l_n } \right)\,y_{n+1} \left( {x,\lambda }
\right)} .$$ The number $\lambda $ is called an Eigen value in this case, and the corresponding decision $y\left( {x,\lambda } \right)$ is called Eigen vector-valued function.
$$\alpha _{11}^0 ,\beta _{11}^0 ,\gamma _{11}^0 ,\delta _{11}^0 ,\alpha
_{j1}^k ,\beta _{j1}^k ,\gamma _{j1}^k ,\delta _{j1}^k ,\alpha _{j2}^k
,\beta _{j2}^k ,\gamma _{j2}^k ,\delta _{j2}^k ,A_j -\;\left(
{j=1,2;\;\,m=1,n+1;\;\,k=1,n} \right)$$ are matrixes of the size $r\times r$. We will required invertible $$\label{eq7}
\det \;\;M_{mk} \ne 0,\;\;\lambda \in \left. {\left[ {0,\infty } \right.}
\right)$$ for matrixes $$M_{mk} \equiv \left( {{\begin{array}{*{20}c}
{\beta _{1m}^k +\lambda ^2\gamma _{1m}^k } \hfill & {\alpha _{1m}^k
+\lambda ^2\delta _{1m}^k } \hfill \\
{\beta _{2m}^k +\lambda ^2\gamma _{2m}^k } \hfill & {\alpha _{2m}^k
+\lambda ^2\delta _{2m}^k } \hfill \\
\end{array} }} \right),\;\,m=1,2;\;\,k=\overline {1,n} .$$ Matrixes $A_m^2 $ and $\Gamma _m^2 $ , are is $m=\overline {1,n+1} $ -positive-defined \[6\]. We denote $$\Phi _{n+1} \left( x \right)=e^{q_{n+1} xi};\;\,\Psi _{n+1} \left( x
\right)=e^{-q_{n+1} xi};\;\,q_{n+1}^2 =A_{n+1}^{-2} \left( {\lambda ^2{\rm
E}+\Gamma ^2} \right).$$ Define the induction relations the others n-pairs a matrix-importance functions $\left( {\Phi _k ,\Psi _k } \right),\;\;k=1,n:$ $$\left[ {\left( {\alpha _{j1}^k +\lambda ^2\delta _{j1}^k }
\right)\frac{d}{dx}+\left( {\beta _{j1}^k +\lambda ^2\gamma _{j1}^k }
\right)} \right]\,\left( {\Phi _k ,\Psi _k } \right)=$$ $$\label{eq8}
=\left[ {\left( {\alpha _{j2}^k +\lambda ^2\delta _{j2}^k }
\right)\frac{d}{dx}+\left( {\beta _{j2}^k +\lambda ^2\gamma _{j2}^k }
\right)} \right]\,\left( {\Phi _{k+1} ,\Psi _{k+1} } \right),\quad
k=\overline {1,n} ,\quad j=\overline {1,2} .$$ Let us introduce the following notation $$\left. {\mathop \Phi \limits^0 _1 \left( \lambda \right)=\left[ {\left(
{\alpha _{11}^0 +\lambda ^2\delta _{11}^0 } \right)\frac{d}{dx}+\left(
{\beta _{11}^0 +\lambda ^2\gamma _{11}^0 } \right)} \right]\Phi _1 \left(
{x,\lambda } \right)\,} \right|_{x=l_0 } ,$$ $$\left. {\mathop \Psi \limits^0 _1 \left( \lambda \right)=\left[ {\left(
{\alpha _{11}^0 +\lambda ^2\delta _{11}^0 } \right)\frac{d}{dx}+\left(
{\beta _{11}^0 +\lambda ^2\gamma _{11}^0 } \right)} \right]\Psi _1 \left(
{x,\lambda } \right)\,} \right|_{x=l_0 } ,$$ $$\Omega _k =\left( {{\begin{array}{*{20}c}
{\Phi _k } \hfill & {\Psi _k } \hfill \\
{\Phi _k^/ } \hfill & {\Psi _k^/ } \hfill \\
\end{array} }} \right),\quad i=\overline {1,n+1} .$$
The spectrum of the problem (\[eq4\]),(\[eq5\]),(\[eq6\]) is a continuous and fills all semi axis $\left( {0,\infty } \right)$. Sturm–Liouville theory r time is degenerate. To each Eigen value $\lambda $ corresponds to exactly $r$ linearly independent vector-valued functions. As the last it is possible to take $r$ columns matrix-importance functions. $$u\left( {x,\lambda } \right)=\sum\limits_{k=1}^n {\theta \left( {x-l_{k-1} }
\right)\,\theta \left( {l_k -x} \right)\,u_k \left( {x,\lambda }
\right)\,+\,\theta \left( {x-l_n } \right)\,u_{n+1} \left( {x,\lambda }
\right)} ,$$ $$\label{eq9}
u_j \left( {x,\lambda } \right)=\Phi _j \left( {x,\lambda } \right)\mathop
{\Phi _1^{-1} }\limits^{0\;\;\;} \left( \lambda \right)-\Psi _j \left(
{x,\lambda } \right)\mathop {\Psi _1^{-1} }\limits^{0\;\;\;} \left( \lambda
\right).$$ That is $$y^m\left( {x,\lambda } \right)=\left( {{\begin{array}{*{20}c}
{u_{1m} \left( {x,\lambda } \right)} \hfill \\
\vdots \hfill \\
{u_{rm} \left( {x,\lambda } \right)} \hfill \\
\end{array} }} \right).$$
Dual Sturm–Liouville theory consists in a finding of the non-trivial solution of separate simultaneous ordinary differential equations with constant matrix coefficients. $$\label{eq10}
\left( {A_m^2 \frac{d^2}{dx^2}+\lambda ^2{\rm E}+\Gamma _m^2 } \right)y_m
=0,\;\,q_m^2 =\lambda ^2{\rm E}+\Gamma _m^2 ,\;\,m=\overline {1,n+1}$$ on the boundary conditions $$\label{eq11}
\left. {\left( {\frac{d}{dx}y_1^\ast \left( {\beta _{11}^0 +\lambda ^2\gamma
_{11}^0 } \right)^{-1}+y_1^\ast \left( {\alpha _{11}^0 +\lambda ^2\delta
_{11}^0 } \right)^{-1}} \right){\kern 1pt}} \right|_{x=l_0 } =0,\quad
\;\;\left\| {y_{n+1}^\ast } \right\|\,<\,\infty ,$$ and conditions of the contact in the points of conjugation of intervals $$\left( {-\frac{d}{dx}y_k^\ast ,y_k^\ast } \right)\left(
{{\begin{array}{*{20}c}
{\beta _{11}^k +\lambda ^2\gamma _{11}^k } \hfill & {\alpha _{11}^k
+\lambda ^2\delta _{11}^k } \hfill \\
{\beta _{21}^k +\lambda ^2\gamma _{21}^k } \hfill & {\alpha _{21}^k
+\lambda ^2\delta _{21}^k } \hfill \\
\end{array} }} \right)^{-1}=$$ $$\label{eq12}
=\left( {-\frac{d}{dx}y_{k+1}^\ast ,y_{k+1}^\ast } \right)\left(
{{\begin{array}{*{20}c}
{\beta _{12}^k +\lambda ^2\gamma _{12}^k } \hfill & {\alpha _{12}^k
+\lambda ^2\delta _{12}^k } \hfill \\
{\beta _{22}^k +\lambda ^2\gamma _{22}^k } \hfill & {\alpha _{22}^k
+\lambda ^2\delta _{22}^k } \hfill \\
\end{array} }} \right)^{-1},\quad \quad x=l_k ,\quad k=\overline {1,n} .$$ The solution of the boundary value problem we write in the form of $$y^\ast \left( {\xi ,\lambda } \right)=\sum\limits_{k=2}^n {\theta \left(
{\xi -l_{k-1} } \right)\,\theta \left( {l_k -\xi } \right)\,y_k^\ast \left(
{\xi ,\lambda } \right)\,+\theta \left( {l_1 -\xi } \right)\,y_1^\ast \left(
{\xi ,\lambda } \right)+\theta \left( {\xi -l_n } \right)\,y_{n+1}^\ast
\left( {\xi ,\lambda } \right)} ,$$ $$y_m^\ast \left( {\xi ,\lambda } \right)=\left( {{\begin{array}{*{20}c}
{y_{m1}^\ast \left( {\xi ,\lambda } \right)} \hfill & \cdots \hfill &
{y_{mr}^\ast \left( {\xi ,\lambda } \right)} \hfill \\
\end{array} }} \right),$$ $$\left\| {y_m^\ast } \right\|=\sqrt {\left( {y_{1m}^\ast }
\right)^2+...+\left( {y_{rm}^\ast } \right)^2} ,m=\overline {1,n+1} .$$
The spectrum of the problem (\[eq4\]),(\[eq5\]),(\[eq6\]) is a continuous and fills semi axis $\left( {0,\infty } \right)$. Sturm–Liouville theory r time is degenerate. To each Eigen value $\lambda $ corresponds to exactly $r$ linearly independent vector-valued functions. As the last it is possible to take $r$ rows matrix-importance functions. $$u^\ast \left( {x,\lambda } \right)=\sum\limits_{k=1}^n {\theta \left(
{x-l_{k-1} } \right)\,\theta \left( {l_k -x} \right)\,u_k^\ast \left(
{x,\lambda } \right)\,+\,\theta \left( {x-l_n } \right)\,u_{n+1}^\ast \left(
{x,\lambda } \right)} ,$$ $$u_j^\ast \left( {x,\beta } \right)=\left( {\mathop \Phi \limits^0 _1 \left(
\beta \right),\mathop \Psi \limits^0 _1 \left( \beta \right)}
\right)\,\Omega _j^{-1} \left( {x,\beta } \right)\left(
{{\begin{array}{*{20}c}
0 \hfill \\
{\rm E} \hfill \\
\end{array} }} \right)A_j^{-2} ,$$ That is $$\label{eq13}
y^{\ast j}\left( {\xi ,\lambda } \right)=\left( {{\begin{array}{*{20}c}
{u_{j1}^\ast \left( {\xi ,\lambda } \right)} \hfill & \cdots \hfill &
{u_{jr}^\ast \left( {\xi ,\lambda } \right)} \hfill \\
\end{array} }} \right),
j=\overline {1,r} .$$
The existence of spectral functions $u\left( {x,\lambda } \right)$ and the conjugate spectral function $u^\ast \left( {x,\lambda } \right)$ allows to write the a vector decomposition theorem on the set of $I_n^+ $.
Let the vector-valued function $f (x)$ is defined on $I_n^+ $ continuous, absolutely integrated and has the bounded total variation. Then for any $x\in I_n^+ $ true formula of decomposition $$f\left( x \right)=-\frac{1}{\pi j}\int\limits_0^\infty {u\left( {x,\lambda }
\right)} \left( \right.\int\limits_{l_0 }^\infty {u^\ast
\left( {\xi ,\lambda } \right)f\left( \xi \right)d\xi +}$$ $$+\left( {\gamma _{11}^0 f_1 \left( {l_0 } \right)+\delta _{11}^0 {f}'_1
\left( {l_0 } \right)} \right)+\sum\limits_{k=1}^n {\left( {\phi _1^0 \left(
\lambda \right),\psi _1^0 \left( \lambda \right)} \right)\,\Omega _k^{-1}
\left( {l_k ,\lambda } \right)\,M_{k1}^{-1} \left( \lambda \right)\cdot }$$ $$\label{eq14}
\cdot \left. {\left\{ {\left( {{\begin{array}{*{20}c}
{\gamma _{21}^k } \hfill & {\delta _{21}^k } \hfill \\
{\gamma _{22}^k } \hfill & {\delta _{22}^k } \hfill \\
\end{array} }} \right)\,\left( {{\begin{array}{*{20}c}
{f_{k+1} \left( {l_k } \right)} \hfill \\
{{f}'_{k+1} \left( {l_k } \right)} \hfill \\
\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
{\gamma _{11}^k } \hfill & {\delta _{11}^k } \hfill \\
{\gamma _{12}^k } \hfill & {\delta _{12}^k } \hfill \\
\end{array} }} \right)\,\left( {{\begin{array}{*{20}c}
{f_k \left( {l_k } \right)} \hfill \\
{{f}'_k \left( {l_k } \right)} \hfill \\
\end{array} }} \right)} \right\}} \right)\lambda d\lambda .$$ The decomposition theorem allows to enter the direct and inverse matrix integral Fourier transform on the real semi axis with conjugation points: $$F_{n+} \left[ f \right]\left( \lambda \right)=\int\limits_{l_0 }^\infty
{u^\ast \left( {\xi ,\lambda } \right)f\left( \xi \right)d\xi +}$$ $$+\left( {\gamma _{11}^0 f_1 \left( {l_0 } \right)+\delta _{11}^0 f_1^/
\left( {l_0 } \right)} \right)+\sum\limits_{k=1}^n {\left( {\phi _1^0 \left(
\lambda \right),\psi _1^0 \left( \lambda \right)} \right)\,\Omega _k^{-1}
\left( {l_k ,\lambda } \right)\,M_{k1}^{-1} \left( \lambda \right)\cdot }$$ $$\label{eq15}
\cdot \left\{ {\left( {{\begin{array}{*{20}c}
{\gamma _{21}^k } \hfill & {\delta _{21}^k } \hfill \\
{\gamma _{22}^k } \hfill & {\delta _{22}^k } \hfill \\
\end{array} }} \right)\,\left( {{\begin{array}{*{20}c}
{f_{k+1} \left( {l_k } \right)} \hfill \\
{f_{k+1}^/ \left( {l_k } \right)} \hfill \\
\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
{\gamma _{11}^k } \hfill & {\delta _{11}^k } \hfill \\
{\gamma _{12}^k } \hfill & {\delta _{12}^k } \hfill \\
\end{array} }} \right)\,\left( {{\begin{array}{*{20}c}
{f_k \left( {l_k } \right)} \hfill \\
{f_k^/ \left( {l_k } \right)} \hfill \\
\end{array} }} \right)} \right\}\equiv \tilde {f}\left( \lambda \right),$$ $$\label{eq16}
F_{n+}^{-1} \left[ {\tilde {f}} \right]\,\left( x \right)=-\frac{1}{\pi
i}\int\limits_0^\infty {\lambda u\left( {x,\lambda } \right)\,\tilde
{f}\left( \lambda \right)d\lambda } \equiv f\left( x \right),$$ when $$f\left( x \right)=\sum\limits_{k=1}^n {\theta \left( {l_k -x}
\right)\,\theta \left( {x-l_{k-1} } \right)\,f_k \left( x \right)\,+\theta
\left( {x-l_n } \right)\,f_{n+1} \left( x \right)} .$$
Let’s result the basic identity of integral transform of the differential operator $$B=\sum\limits_{j=1}^n {\theta \left( {x-l_{j-1} } \right)\,\theta \left(
{l_j -x} \right)\left( {A_j^2 \frac{d^2}{dx^2}+\Gamma _j^2 }
\right)\,+\theta \left( {x-l_n } \right)\left( {A_{n+1}^2
\frac{d^2}{dx^2}+\Gamma _{n+1}^2 } \right)} .$$
If vector-valued function $$f\left( x \right)=\sum\limits_{k=1}^n {\theta \left( {x-l_{k-1} }
\right)\,\theta \left( {l_k -x} \right)f_k \left( x \right)\,+\theta \left(
{x-l_n } \right)\,f_{n+1} \left( x \right)} ,$$ is continuously differentiated on set three times, has the limit values together with its derivatives up to the third order inclusive $$f_k^{(m)} \left( {l_{k-1} } \right)=f_k^{(m)} \left( {l_{k-1} +0}
\right),\;\,m=0,1,2,3;\quad k=\overline {1,n+1}$$ Satisfies to the boundary condition on infinity $$\mathop {\lim }\limits_{x\to \infty } \;\left( {u^\ast \left( {x,\lambda }
\right)\frac{d}{dx}f\left( x \right)-\frac{d}{dx}u^\ast \left( {x,\lambda }
\right)\,f\left( x \right)} \right)=0$$ Satisfies to homogeneous conditions of conjugation (\[eq6\]), that basic identity of integral transform of the differential operator $B$ hold $$F_{n+} \left[ {B\left( f \right)} \right]\,\left( \lambda \right)=-\lambda
^2\tilde {f}\left( \lambda \right)-\left\{ {\left( {\beta _{11}^0 f_1 \left(
{l_0 } \right)+\alpha _{11}^0 f_1^/ \left( {l_0 } \right)} \right)-}
\right.$$ $$\label{eq17}
\left. {-\left( {\gamma _{11}^0 A_1^2 f_1^{//} \left( {l_0 } \right)+\delta
_{11}^0 A_1^2 f_1^{///} \left( {l_0 } \right)} \right)} \right\} .$$
The proof of theorems 1,2,3,4 is spent by a method of the method of contour integration. Similarly presented to work of the author \[19\].
[99]{} Ahtjamov A.M., Sadovnichy V. A, Sultanaev J.T.(2009) Inverse Sturm–Liouville theory with non disintegration boundary conditions. - Moscow: Publishing house of the Moscow university.
Bavrin I.I., Matrosov V.L., Jaremko O. E.(2006) Operators of transformation in the analysis, mathematical physics and Pattern recognition. Moscow, Prometheus, p 292.
Bejtmen G., Erdeji A., (1966) High transcendental function, Bessel function, Parabolic cylinder function, Orthogonal polynomials. Reference mathematical library, Moscow, p 296.
Brejsuell R., (1990) Hartley transform, Moscow, World, p 584.
Vladimirov V. S., Zharinov V.V., (2004) The equations of mathematical physics, Moscow, Phys mat lit, p 400.
Gantmaxer F.R., (2010) Theory matrix. Moscow, Phys mat lit, p 560.
Grinchenko V. T., Ulitko A.F., Shulga N.A., (1989) Dynamics related fields in elements of designs. Electro elasticity. Kiev. Naukova Dumka,p 279.
Lenyuk M.P., (1991) Hybrid Integral transform (Bessel, Lagrange, Bessel), the Ukrainian mathematical magazine. p. 770-779.
Lenyuk M.P., (1989) Hybrid Integral transform (Bessel, Fourier, Bessel), Mathematical physics and non-linear mechanics, p. 68-74
Lenyuk M.P.(1989) Integral Fourier transform on piece-wise homogeneous semi-axis, Mathematica, p. 14-18.
Najda L. S. (1984) Hybrid integral transform type Hankel- Legendary, Mathematical methods of dynamic systems analysis . Kharkov, p. 132-135.
Protsenko V. S., Solovev A.I.. (1982) Some hybrid integral transform and their applications in the theory of elasticity of heterogeneous medium. Applied mechanics, p 62-67.
Rvachyov V. L., , Protsenko V. S., (1977) Contact problems of the theory of elasticity for anon classical areas, Kiev. Naukova Dumka.
Sneddon I.. (1955) Fourier Transform, Moscow.
Sneddon I., Beri D. S., (2008) The classical theory of elasticity. University book, p. 215.
Uflyand I. S. (1967) Integral transforms in the problem of the theory of elasticity. Leningrad. Science, p. 402
Uflyand I. S..(1967) On some new integral transformations and their applications to problems of mathematical physics. Problems of mathematical physics. Leningrad, p. 93-106
Physical encyclopedia. The editor-in-chief A. M. Prokhorov, D.M. Alekseev, Moscow,(1988-1998).
Jaremko O. E., (2007) Matrix integral Fourier transform for problems with discontinuous coefficients and conversion operators. Proceedings of the USSR Academy of Sciences. p. 323-325.
|
---
abstract: 'We analyse the effects of atom–atom collisions on collective laser cooling scheme. We derive a quantum Master equation which describes the laser cooling in presence of atom–atom collisions in the weak–condensation regime. Using such equation, we perform Monte Carlo simulations of the population dynamics in one and three dimensions. We observe that the ground–state laser–induced condensation is maintained in the presence of collisions. Laser cooling causes a transition from a Bose–Einstein distribution describing collisionally induced equilibrium,to a distribution with an effective zero temperature. We analyse also the effects of atom–atom collisions on the cooling into an excited state of the trap.'
address: 'Institut für Theoretische Physik, Universität Hannover, Appelstr. 2, D–30167 Hannover, Germany'
author:
- Luis Santos and Maciej Lewenstein
title: Collisional effects on the collective laser cooling of trapped bosonic gases
---
Introduction {#sec:Intro}
============
In the recent years, laser cooling has constituted one of the most active research fields in atomic physics [@Nobel]. However, the laser cooling techniques by themselves have not allowed to reach temperatures for which the quantum statistical effects become evident. In particular, only the combination of laser cooling, and evaporative [@BEC] or sympathetic cooling [@Symp] has permited in the last years to observe experimentally, the Bose–Einstein condensation (BEC) in alkali gases, seventy years after its theoretical prediction [@Bose24]. The question whether it is or it is not possible to achieve the BEC only with laser cooling techniques remain, at least as an intellectual challenge. The laser–induced BEC is, however, not only an academic problem, but has several advantages with respect to the nowadays widely–employed collisional mechanisms (as evaporative cooling). These advantages are: (i) the number of atoms does not decrease during the cooling process; (ii) it is possible to design a non–destructive BEC detection by fluorescence measurements; (iii) reacher effects can appear, since now the system is open, i.e. it is not in thermal equilibrium; (iv) laser–induced condensation can be used to design techniques to pump atoms into the condensate. The latter is specially important in the contex of future atom–laser devices [@Spreeuw95; @Janicke96; @Janicke99; @Bar].
The main problem which prevents experimentalists from obtaining BEC by optical means is the reabsorption of spontaneously emitted photons. The most effective laser–cooling techniques (such as VSCPT [@VSCPT], or Raman cooling [@Raman]), are based on the crucial concept of dark states [@Orriols], i.e. states which cannot absorb the laser light, but can receive population via incoherent pumping, i.e. via spontaneous emission. Unfortunately, the atoms occupying the dark states are not unaffected by the photons spontaneously emitted by other atoms. This problem turns to be very important at high densities, as those required for the BEC [@Reabproblem]; in such conditions dark–state cooling techniques cease to work adequately. Several remedies to the reabsorption problem have been proposed, as the reduction of the dimensionality of the trap from three to two or one dimensions [@Reabproblem], or the use of traps with frequencies, $\omega$, of the order of the recoil frequency ($\omega_R=\hbar k_L^2/2M$, where $k_L$ is the laser wavevector and M is the atomic mass) [@Janicke96]. Other, perhaps more promising, idea consists in exploiting the dependence of the reabsorption probability on the fluorescence rate $\gamma$. In particular, in the so–called [*Festina Lente*]{} limit [@Festina], when $\gamma <\omega$ with $\omega$ the trap frequency, the heating effects of the reabsorption can be neglected. Another proposal consists in working in the so–called Bosonic Accumulation Regime [@Bar], in which the reabsorption can, under certain conditions, even help to build up the condensate. In the following we shall assume that the considered system fulfills the Festina Lente limit.
In a series of papers [@1Atom; @Manyatoms], we have proposed a cooling mechanism (which we have called Dynamical cooling) which permits the cooling of an atomic sample into an arbitrary single state of an harmonic trap, beyond the Lamb–Dicke limit (i.e. when the Lamb–Dicke parameter $\eta>1$, with $\eta^2=\omega_R/\omega$). The cooling mechanism employs laser pulses of different frequencies (and eventually different directions, phases and intensities), in such a way that a particular state of the trap remains dark during the cooling process, acting as a trapping state. Therefore, the population is finally transferred to this particular state. We have first analysed the particular situation of a single atom in the trap [@1Atom], and extended the analysis to a collection of trapped bosons [@Manyatoms]. We have shown that the bosonic statistics helps to achieve more robust and rapid condensation, as well as to produce non–linear effects, such as hysteresis and multistability phenomena.
However, all the calculations performed so far in the analysis of the dynamical cooling scheme do not take into account the atom–atom collisions, i.e. are considered in the so–called [*ideal gas*]{} limit. The ideal gas limit imposes important restrictions to the physical system, in particular the atomic density cannot be very large. An interesting possibility in order to achieve quasi–ideal gases consists in the “switching–off” of the $s$–wave scattering length $a$ (which is the main contribution to the atom–atom collisions for sufficiently low energies), either by employing magnetic fields (tuning the so–called Feshbach resonances [@Ketterle]) or by using a red–detuned laser tuned between molecular resonances as proposed by Fedichev [*et al*]{} [@Fedichev]. However, without special precautions, the effects of the atom–atom collisions play a substantial role. It is the aim of this paper to analyse such effects in the context of our dynamical laser cooling scheme.
In recent years, C. W. Gardiner, P. Zoller and collaborators have devoted a series of papers [@QK1; @QK2; @QK3; @QK4; @QK5] to the decription of interacting Bose gases with and without trapping potentials. These authors have developed a quantum kinetic theory of Bose gases. In particular, for the case of a weakly interacting gas, a so–called Quantum Kinetic Master Equation (QKME) has been formulated [@QK1], which is a quantum stochastic equation for the kinetics of the dilute Bose gas, that describes the behavior and formation of the condensate. This equation is very difficult to simulate, and therefore various simplifications have been proposed. Particularly interesting results are obtaining by using the so–called Quantum Boltzmann Master Equation (QBME) which neglects all spatial inhomogeinity of the trapped states [@QK2]. Although this is an extreme simplification, the (very much easier) simulation of the QBME give a good idea of the solutions that the QKME could produce. In the following, we shall show that the master equation (ME) which describes the laser cooling problem in the presence of atom–atom collisions can be, in the case of the weak–condensation regime, splitted into two independent parts, one accounting for the collisional effects (which has the form of the QBME proposed in Ref. [@QK1]), and another which describes the laser cooling process, and has the form of the ME already developed for the case without collisions [@Manyatoms].
The structure of the paper is as follows. In Sec. \[sec:Model\], we derive the quantum ME which describes the laser cooling plus collisions in the weak–condensation regime. In Sec. \[sec:1D\], we present the results for one–dimensional excited–state cooling. Sec. \[sec:3D\] is devoted to the three–dimensional results for the case of ground–state cooling. Here, we use additional ergodic approximation which assures fast redistribution of atoms within an energy shell. Finally, in Sec. \[sec:conclu\] we summarize some conclusions.
Model. Master Equation. {#sec:Model}
=======================
We assume in this paper the same atomic model as that presented in Refs. [@1Atom; @Manyatoms], i.e. a three–level $\Lambda$–system, composed of a ground–state level $|g\rangle$, a metastable state $|e\rangle$ and an auxiliary third fast–decaying state $|r\rangle$. Two lasers excite coherently the resonant Raman transition $|g\rangle\rightarrow|e\rangle$ (with an associated effective Rabi frequency $\Omega$), while the repumping laser in or off–resonance with the transition $|g\rangle\rightarrow|r\rangle$ pumps optically the atom into $|g\rangle$. With this three level scheme, one obtains an effective two–level system with an effective spontaneous emission rate $\gamma$, which can be easily controlled by varying the intensity or the detuning of the repumping laser [@Marzoli94]. In the following we follow the same notation as in Refs. [@Manyatoms]. Let us introduce the annihilation and creation operators of atoms in the ground (excited) state and in the trap level $m$ $(l)$, which we will call $g_{m}$, $g_{m}^{\dag}$ ($e_{l}$, $e_{l}^{\dag}$). These operators fulfill the bosonic commutation relations $[g_{m},g_{m'}^{\dag}]=\delta_{mm'}$ and $[e_{l},e_{l'}^{\dag}]=\delta_{ll'}$. Using standard methods of the theory of quantum stochastic processes [@Gardinerbook1; @Gardinerbook2; @Carmichaelbook; @Carmichaelbooknew] one can develop the quantum ME which describes the atom dynamics [@Manyatoms] $$\dot\rho(t)={\cal L}_0\rho+{\cal L}_1\rho+{\cal L}_2\rho,
\label{ME}$$ where
$$\begin{aligned}
{\cal L}_0\rho&=&-i\hat H_{eff}\rho(t)+i\rho(t)\hat H_{eff}^{\dag}+{\cal J}\rho(t), \\
{\cal L}_1\rho&=&-i[\hat H_{las},\rho(t)], \\
{\cal L}_2\rho&=&-i[\hat H_{col},\rho(t)], \end{aligned}$$
with
$$\begin{aligned}
\hat H_{eff}&=&\sum_{m}\hbar\omega_{m}^{g}g_{m}^{\dag}g_{m}+\sum_{l}\hbar(\omega_{l}^{e}
-\delta)e_{l}^{\dag}e_{l} \nonumber \\
&-&i\hbar\gamma\int d\phi d\theta sin\theta {\cal W}(\theta,\phi) \nonumber \\
&\times& \sum_{l,m}|\eta_{lm}(\vec k)|^{2}e_{l}^{\dag}g_{m}g_{m}^{\dag}e_{l}, \\
\hat H_{las}&=&\frac{\hbar\Omega}{2}\sum_{l,m}\eta_{lm}(k_{L})e_{l}^{\dag}g_{m}+H.c.,\\
\hat H_{coll}&=&\sum_{m_1,m_2,m_3,m_4}\frac{1}{2}U_{m_1,m_2,m_3,m_4}
g_{m_4}^{\dag}g_{m_3}^{\dag}g_{m_2}g_{m_1},\\
{\cal J}\rho(t)&=&2\hbar\gamma\int d\phi d\theta \sin\theta{\cal W}(\theta,\phi) \nonumber \\
&\times& \sum_{l,m}[\eta_{lm}^{\ast}(\vec k)g_{m}^{\dag}e_{l}]\rho(t)[\eta_{lm}(\vec
k)e_{l}^{\dag}g_{m}].\end{aligned}$$
where $2\gamma$ is the single–atom spontaneous emission rate, $\Omega$ is the Rabi frequency associated with the atom transition and the laser field, $\eta_{lm}(k_{L})=\langle e,l|e^{i\vec k_{L}\cdot\vec r}|g,m\rangle$ are the Franck–Condon factors, ${\cal W}(\theta,\phi)$ is the fluorescence dipole pattern, $\omega_m^g$ ($\omega_l^e$) are the energies of the ground (excited) harmonic trap level $m$ ($l$), and $\delta$ is the laser detuning from the atomic transition. The new term respect to what is considered in Refs. [@Manyatoms], is that of $H_{coll}$, which describes the two–body interactions in the Bose gas. Only ground–ground collisions are considered because we assume that the laser interaction is sufficiently weak to guarantee that only few atoms are excited (formally we consider only one). In the regime we want to study, only $s$–wave scattering is important, and then: $$U_{m_1,m_2,m_3,m_4}=\frac{4\pi\hbar^2a}{m}\int_{R^3}d^3x
\psi_{m_4}^{\ast}\psi_{m_3}^{\ast}\psi_{m_2}\psi_{m_1},$$ where $\psi_{m_j}$ denotes the harmonic oscillator wavefunctions and $a$ denotes the scattering length. Following the simplifications of the QBME [@QK1; @QK2], we exclude the spatial dependence and therefore no transport or wave–packet spreading terms appear in (\[ME\]).
In the following we are going to work in the so–called weak–condensation regime, where no mean–field efects are considered. This means that we consider that the typical energy provided by the collisions is smaller than the oscillator energy. As shown in [@QK2], in typical experiments this condition requires that the condensate cannot contain more than $1000$ particles. We shall work thus below such limit. We shall also consider that $\Omega\ll\omega$. Also, due to the Festina–Lente requirements, $\Omega$ is in general smaller than the typical collisional energy. Therefore we can formally establish the hierarchy ${\cal L}_0\gg{\cal L}_2>{\cal L}_1$.
Let us define a projector operator ${\cal P}$: $${\cal P}X=\sum_{\vec n}|\vec n\rangle\langle\vec n|\langle\vec n|X|\vec n\rangle,
\label{Proy}$$ and its complement ${\cal Q}=1-{\cal P}$, and $|\vec n\rangle\equiv |N_{0},N_{1},\dots;g\rangle$$\otimes
|0,0,\dots;e\rangle$ are the ground state configurations with $N_{j}$ atoms in the $j$–th level, and no excited atoms. It is easy to prove that: $${\cal L}_0{\cal P}={\cal PL}_1{\cal P}={\cal PL}_2{\cal P}=0.
\label{prop}$$ Projecting the ME (\[ME\]), one obtains:
$$\begin{aligned}
&& \dot v={\cal P}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_{2})w, \label {vdot} \\
&& \dot w={\cal Q}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_{2})v+
{\cal Q}({\cal L}_{1}+{\cal L}_{2})w, \label{wdot}\end{aligned}$$
where $v={\cal P}\rho$ and $w={\cal Q}\rho$. Laplace transforming ($v(t)\rightarrow\tilde v(s)$) and solving the system of equations (assuming for simplicity $w(0)=0$), one obtains: $$\begin{aligned}
s\tilde v(s)-v(0)&=&{\cal P}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_2) \nonumber \\
&\times&[s-{\cal Q}({\cal L}_{0}+{\cal L}_{1}+{\cal L}_{2})]^{-1}
({\cal L}_{1}+{\cal L}_2)\tilde v(s).
\label{lapl}\end{aligned}$$ Performing the inverse Laplace transform, using (\[prop\]) and ${\cal P}\exp[-{\cal L}_0\tau]{\cal L}_1{\cal P}=
{\cal P}\exp[-{\cal L}_0\tau]{\cal L}_2{\cal P}=0$, and applying the Markov approximation following Ref. [@QK1], the ME becomes up to order ${\cal O}({\cal L}_1^2)$: $$\dot v(t)={\cal L}_{coll}v(t) + {\cal L}_{cool}v(t)
\label{ME2}$$ where $${\cal L}_{coll}=-{\cal PL}_2{\cal L}_0{\cal L}_2
\label{Lcoll}$$ describes the collisional part. In principle in ${\cal L}_{coll}$ appears a second term coming from the term between brackets in Eq. (\[lapl\]), but since we consider the weak–condensation regime, we can employ the Born approximation as in Ref. [@QK1] to neglect these terms in the collisional part. Therefore the collisional part is described by a QBME as that of Refs. [@QK1; @QK2]. The laser–cooling dynamics is described by $$\begin{aligned}
{\cal L}_{cool}&=&-{\cal PL}_{1}[{\cal L}_{0}]^{-1}{\cal L}_{1} \nonumber \\
&+&\frac{1}{2}{\cal PL}_0\int_{0}^{\infty}d\tau\int_{0}^{\infty}d\tau' e^{-{\cal
L}_{0}(\tau-\tau')}{\cal L}_{1}e^{-{\cal L}_{0}\tau'}{\cal L}_{1},
\label{Lcool}\end{aligned}$$ which has the same form of the ME calculated for the case of the laser cooling without collisions [@Manyatoms].
Summaryzing, the dynamics of the system splits into two parts, (i) collisional part, described by a QBME, and (ii) laser–cooling part, described by the same ME as without collisions. The first correction to such splitting between both dynamics is of the order ${\cal L}_2{\cal L}_1^2$; therefore the independence between the collisions and laser cooling dynamics is only valid in the weak–interaction regime.
The independence of both dynamics, allows for an easy simulation of the laser cooling in presence of collisions. In particular, we simulate both dynamics using Monte Carlo methods, combining the numerical method of Ref. [@QK2], with the simulations already presented in Refs. [@Manyatoms]. In the following two sections we shall present the results for one–dimensional and three–dimensional simulations respectively.
One–dimensional results {#sec:1D}
=======================
This section analyzes the case of a one dimensional harmonic trap, and it is mainly devoted to the analysis of the laser cooling into states different that the ground state of the trap [@1Atom; @Manyatoms] (the ground–state case in analysed in the next section for the more interesting case of three dimensions). The laser–cooling into an excited state of the trap is only calculated in the one–dimensional case, because its analysis becomes very complicated in higher dimensions. The reason for that is that the ergodic approximation, which we employ in Sec. \[sec:3D\], is incompatible with the analysis of the cooling in excited states of the trap.
In this section we shall show that, as expected, the excited–state cooling is strongly affected by the collisions, even for modified low scattering lengths, in particular because the collisions act as a mechanism to empty the desired excited state, and therefore compete with the cooling mechanism which tends to populate such state. For usual scattering legths and atom densities, the collisional processes are much faster than the typical cooling time, and therefore the excited state cooling is completely suppressed. However, the use of Feshbach resonances [@Ketterle], or lasers [@Fedichev], can modify the scattering length, in such a way that the typical time between collisions can be comparable with the typical cooling time. In this section, we study the population dynamics for different modified scattering lengths, analysing the transition from an ideal–gas regime (with $a=0$ as that studied in Refs. [@Manyatoms]), to a regime in which atom–atom collisions constitute the dominant process.
Collisional probabilities
-------------------------
Following Ref. [@QK2] (Eq. (14)), the probability of a collision between two atoms respectivelly in the states $n_1$ and $n_2$ of an harmonic trap (of frequency $\omega$), to produce two atoms in the states $n_3$ and $n_4$ respectively, is given by: $$\begin{aligned}
&&P_{coll}(n_1,n_2\rightarrow n_3,n_4)= \nonumber \\
&&\frac{4\pi}{\hbar^2\omega}|U_{n_1,n_2,n_3,n_4}|^2 \nonumber \\
&&\times N_{n_1}(N_{n_2}-\delta_{n_1,n_2})(N_{n_3}+1)(N_{n_4}+1+\delta_{n_3,n_4}),
\label{P1234}\end{aligned}$$ where $N_n$ denotes the occupation number of the level $n$ of the trap. The Kronecker deltas in the previous expression account for the bosonic factors appearing when two atoms in a particular level are created, or destructed. Considering a highly anisotropic trap of frequencies $\omega_x=\omega_y=\lambda\omega$, $\omega_z=\omega$, the problem becomes one–dimensional, and the expression (\[P1234\]) takes the form $$\begin{aligned}
&&P_{coll}(n_1,n_2\rightarrow n_3,n_4)= \nonumber \\
&&\xi
\omega
\Delta_c(n_1n,n_2,n_3)\delta(n_1+n_2-n_3-n_4) \nonumber \\
&&\times N_{n_1}(N_{n_2}-\delta_{n_1,n_2})(N_{n_3}+1)(N_{n_4}+1+\delta_{n_3,n_4}),
\label{pcoll}\end{aligned}$$ where the Dirac $\delta$ accounts for the energy conservation. In the previous expression: $$\begin{aligned}
&&\Delta_c(n_1,n_2,n_3)=\left [2^{2(n_1+n_2)}n_1!n_2!n_3!(n_1+n_2-n_3)!\right ]^{-1} \nonumber \\
&& \left [\int_{-\infty}^{\infty}dzH_{n_1}(z)H_{n_2}(z)H_{n_3}(z)H_{n_1+n_2-n_3}(z)
e^{-2z^2} \right ] ^2,\end{aligned}$$ where $H_j(z)$ is the Hermite polynomial of order $j$, and $$\xi=\frac{16\lambda^2}{\pi} \left ( \frac{a}{a_{HO}} \right )^2,$$ with $a_{HO}=\sqrt{\hbar/m\omega}$. For typical experimental situations $a/a_{HO}\sim 10^{-3}$, so $\xi=5\times 10^-6 r^2$, where $r=\lambda a/a_0$, where $a_0$ is the scattering length without any external modification. Therefore $r=0$ accounts for an ideal gas, whereas $r=\lambda$ accounts for an unmodified scattering length.
Laser–cooling transition probabilities
--------------------------------------
The probability of laser–induced transition of an atom from the state $n_1$ to the state $n_2$ of the trap is calculated in Refs. [@Manyatoms], and it is given by $$P_{cool}(n_1\rightarrow n_2)=\frac{\Omega^2}{2\gamma}\Delta_l(n_1,n_2)N_{n_1}(N_{n_2}+1),
\label{pcool}$$ where $$\begin{aligned}
\Delta_l(n_1,n_2)&=&\frac{\Omega^{2}}{2\gamma}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}
d\theta\sin\theta{\cal W}(\theta,\phi) \nonumber \\
&\times&
\left
|\sum_{l}\frac{\gamma\eta_{ln_2}^{\ast}(\vec
k)\eta_{ln_1}(k_{L})}{[\delta-\omega(l-n_1)]+i\gamma R_{n_1l}}\right |^{2}.
\label{Gnm} \end{aligned}$$ with $$\begin{aligned}
R_{n_1l}&=&\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta{\cal W}(\theta,\phi) \nonumber \\
&\times& \sum_{n'}|\eta_{ln'}(\vec k)|^{2}(N_{n'}+1-\delta_{n',n_1})
\label{Rml}\end{aligned}$$
Dynamical cooling
-----------------
Let us briefly review at this point the dynamical cooling scheme proposed in refs. [@1Atom; @Manyatoms]. It is easy to observe from the form of the rates (\[Gnm\]), and from the same arguments as those used in Ref. [@1Atom], that as in the single–atom case two different dark–state mechanisms can be employed:
- “Franck–Condon”–dark–states. Let us assume a laser pulse with detuning $\delta=s\omega$ respect to the atomic transition, where $s$ is an integer number. It can be easily proved [@1Atom] that a particular level of the trap $|m\rangle$ remains unemptied (dark) if the Franck–Condon factor $\langle m+s|\exp(ikx)|m\rangle$ vanishes. In particular, the dark–state condition for $n=1$ is $$\eta^{2}=s+1; \label{dcond1}$$
- “Interference”–dark–states. These dark–state mechanism is characteristic for dimensions higher than one. Let us assume the two–dimensional problem, in which we have two orthogonal lasers characterised by two different Rabi frequencies: $\Omega$ in direction $x$, and $A\Omega$ in direction $y$. The factor $A$ indicates a possible difference between the intensities or phases of both lasers, and can be used to create a dark–state. If the laser detuning is zero and if we choose a value $A=-\langle m_{x}^{0}|e^{ikx}|m_{x}^{0}\rangle/\langle m_{y}^{0}|e^{iky}|m_{y}^{0}\rangle$, for a particular two–dimensional state $|m_{x}^{0},m_{y}^{0}\rangle$, then the selected level remains dark respect to the laser pulse [@1Atom].
In absence of collisions, the above mentioned dark–state mechanisms [@1Atom; @Manyatoms] allow to cool the atoms not only into the ground state, but also into an arbitrary excited state of the trap. In order to achieve that one has to use different dynamical cooling schemes. Each dynamical cooling cycle must contain sequences of pulses of appropriate frequencies. The following types of pulses are employed: i) [*confinement pulses*]{}: spontaneous emission may increase each of the quantum numbers $m_{x,y,z}$ by $O(\eta^2)$. In $D$-dimensions pulses with detuning $\delta = -D\hat \eta^{2} \omega$, where $\hat\eta^2$ is the closest integer to $\eta^2$, have thus an overall cooling effect, and confine the atoms in the energy band of $D$ recoils; ii) [*dark-state cooling pulses*]{}: these pulses should fulfill dark state condition for a selected state to which the cooling should occur; iii) [*sideband and auxiliary cooling pulses*]{}: in general, dark state cooling pulses might lead to unexpected trapping in other levels. In order to avoid it, auxiliary pulses that empty undesired dark states and do not empty the desired dark state are needed; iv) [*pseudo-confining pulses*]{}: with the use of pulses i)–iii) cooling is typically very slow. In order to shorten cooling time we use pulses with $\delta = -3
\eta^{2}\omega/2$ and $\delta = -\eta ^{2}\omega$, which pseudo-confine the atoms below $n=3\eta ^{2}/2$ and $n= \eta ^{2}$.
Numerical results
-----------------
In the following we simulate the dynamics of the atomic population in the different trap levels using standard Monte Carlo methods. We consider the case of an harmonic trap with Lamb–Dicke parameter $\eta=3$. We assume $\gamma=0.04\omega$, $\Omega=0.03\omega$ (consequent with the Festina Lente limit), and a number of atoms $N=133$ (well in the weak–condensation regime). The calculations have been performed taking into account $40$ trap levels. As an initial condition, we assume in all the following graphics a thermal distribution with mean $\langle n \rangle =6$. In order to compare with the calculations without collisions, we analyse the same cooling scheme into the level $n=1$ of the trap, as that studied in Refs. [@Manyatoms], for the case without collisions. We consider cycles of four laser pulses with detunings $\delta=s\omega$, where $s_{1,2,3,4}=-9,8,-10,-3$, and time duration $T=2\gamma/\Omega^2$. Pulses $1$ and $3$ are confining pulses, pulse $2$ is a dark–state pulse for $n=1$, and pulse $4$ is an auxiliary pulse. The one–atom emptying rates [@1Atom], $|\langle n+s|\exp(ikx)|n\rangle|^2$, for the first $10$ levels of the trap are presented in Fig. \[fig:1\]. As one can observe, the effect of the pulses is to empty all the states except $n=1$, which acts consequently as a trapping state. Observe, that due to the characteristics of the Franck–Condon factors, some levels of the trap are barely emptied, in particular for this case, $n=7$ is also a quasi–dark state for pulse $2$, and is poorly emptied by the auxiliary pulse $4$. This is not important in the case without collisions, because $n=1$ remains the darkest level throughout all the dynamics. We shall show in the following that this is no more true when the collions are accounted for.
=7.0cm\
=5.0cm\
\
\
\
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Fig. \[fig:2\] shows the evolution of the averaged population of the level $n=1$ when $r$ grows. We have evolved the system under 5000 cooling cycles (to avoid the effects of the initial conditions), and performed the average from the cycle $5000$ until the cycle $15000$. In Fig. \[fig:2\] we have depicted the averaged population distribution for the cases of (a) $r=0$ (ideal gas), (b) $r=0.4$, (c) $r=0.8$, (d) $r=1.2$, and (e) $r=5.0$. For the ideal–gas case $r=0$ one obtains that the population is completely condensed into the level $n=1$ [@Manyatoms]. When $r$ is increased the laser–induced condensation into the excited state $n=1$ is destroyed, but in a non–trivial way. We observe in Fig. \[fig:2\] (b) that for $r=0.4$ the population is basically distributed in two well defined peaks, one in $n=1$ and the other in $n=7$. The reason for this behavior can be understood very well using Fig. \[fig:1\]. In absence of collisions the level $n=1$ is not emptied at all, while, as pointed out previously, $n=7$ is emptied, but slowly, and therefore at the end the population is finally transferred to the level $n=1$. However in the presence of collisions, level $n=1$ is still a dark–state for the laser, but it is emptied by the collisions with a frequency proportional to $N_1^2$. This means that the population is pumped into $n=1$ due to the laser cooling, but the more the population we pump into $n=1$ the more the level is emptied via collisions. This effect can be well illustrated by Fig. \[fig:3\], where one can observe periods of filling of $n=1$ followed by abrupt decays of the level population. The emptying of level $n=1$ is mainly produced via collisions between two atoms in the level $n=1$ to produce two atoms in $n=0$ and $n=2$ respectively. The laser cooling provides a mechanism to repump such expelled population from the level $n=0$ and $n=2$ back to the level $n=1$. Such control is already maintained for large occupations of $n=1$, but in an unstable way, due to the highly non–linear character of the dynamics. A slight excess of population into the level $n=0$ and $n=2$ provoques a speed–up of the emptying process of $n=1$. This situation is reflected, for example, in the behavior of the system between cycles $4500$ and $5500$ in Fig. \[fig:3\]. In particular, level $n=1$ can become more emptied than $n=7$, and the latter turns to be the effective darkest level, i.e. the level less emptied. Therefore the population tends to be transferred into $n=7$. But, when $N_7$ increases so does the empty rate of the level $n=7$, which can become larger than that of $n=1$, and so on. Therefore, as consequence of this process a non–linear pseudo–oscillatory motion between the populations of $n=1$ and $n=7$ is produced, as observed in Fig. \[fig:3\]. This oscillatory motion leads to the two–peaked distribution of Figs. \[fig:2\]. Finally, when $r$ becomes very large the collision dynamics is much faster than the cooling time, and the peaked structure dissapears, as observed in Figs. \[fig:2\] (c), (d) and (e). Observe that nevertheless the effects of the laser cooling mechanism are nevertheless present in Fig. \[fig:2\] (f). In absence of laser cooling, it can be demonstrated that for $r=5$, the population has a maximum in $n=0$. On the contrary, in the presence of the laser, the population of $n=0$ is very efficiently and rapidly emptied by the pulse with detuning $s=8$, which is the most rapid cooling process. For larger $r$ even this process is eventually overcome, and one recovers the same distribution as that obtained only considering the collisions without laser cooling.
Three-dimensional results {#sec:3D}
=========================
In this section we analyze the case of the laser–cooling into the ground state of an isotropic three–dimensional harmonic trap, of frequency $\omega$. The numerical calculation of the system dynamics for the three–dimensional case is quite complicated, due to both the degeneracy of the levels, and the difficulties to obtain reliable values for the integrals $U(n_1,n_2,n_3,n_4)$. Therefore, we shall limit ourselves to the use of the ergodic approximation, i.e. we shall assume that states with the same energy are equally populated. The populations of the degenerate energy levels equalize on a time scale much faster than the collisions between levels of different energies, and than the laser–cooling typical time. This approximation leads to the correct steady–state distribution, although the dynamics can be slightly different than in the non–ergodic calculation [@QK2].
=6.0cm\
Following ref. [@Holland] the probability of a collision of two atoms in energy shells $n_1$ and $n_2$, to give two atoms in shells $n_3$ and $n_4$ (where this collision is assumed to change the energy distribution function), is of the form: $$\begin{aligned}
&&P(n_1,n_2\rightarrow n_3,n_4)=\Delta (n_j+1)(n_j+2) \nonumber \\
&&\times\frac{N_{n1}(N_{n2}-\delta_{n_2,n_1})(N_{n_3}+g_{n_3})(N_{n_4}+g_{n_4}+\delta_{n_3,n_4})}
{g_{n_1}g_{n_2}g_{n_3}g_{n_4}},\end{aligned}$$ where $g_{n_k}=(n_k+1)(n_k+2)/2$ is the degeneracy of the energy shell $n_k$, $n_j=min \{ n_1,n_2,n_3,n_4 \}$, and $\Delta=(4a^2\omega^2m)/(\pi\hbar)\sim 1.5\times 10^{-5}$. Concerning the laser–cooling probabilities we shall use the same expressions as those already developed in Refs. [@Manyatoms].
In the following we simulate the evolution of the system by using again Monte Carlo simulations. Due to numerical limitations we consider a Lamb–Dicke parameter $\eta=2$. We assume as previously $\gamma=0.04\omega$ and $\Omega=0.03\omega$, and a number of atoms $N=133$. As a first step, we begin with a thermal distribution of mean $\langle n\rangle=6$, and evolve the system just with collisions, until obtaining a Bose–Eintein distribution (BED) (which does not coincide exactly with the thermodynamical one, due to finite–size effects), see Fig. \[fig:4\]. The distribution obtained in this initial step serves as the initial state for laser cooling. As we see, it already contains quite subtantial amount of atoms condensed in the ground state, but also a lot of uncondensed ones. Laser cooling will transfer the latter ones into the ground state. We apply our laser cooling cycles, each one of them composed by two laser pulses of detuning $\delta=s\omega$, with $s_{1,2}=-4,0$, and time duration $T=(2\gamma)/\Omega^2$. The laser pulses are emitted in three orthogonal directions $x$, $y$ and $z$, and are characterized by their respective rabi frequencies $\Omega_x=\Omega_y=\Omega$, $\Omega_z=A_z\Omega$. For the first pulse we assume $A_z=1$, while for the second one $A_z=-2$ is considered. With this choice, the second pulse is an “interference”–dark–state pulse for the ground–state of the trap. Fig. \[fig:5\](a) shows (dashed line) that these two pulses are able to condense the population into the ground state of the trap, in absence of collisions; in particular no confinement pulses (of detunings $\delta=-12\omega$ in this case) are needed. This is due to the bosonic enhancement and the fact that initially the system is already partially condensed. The dark–state pulse is neccesary to repump the population in those states of the energy shells $1$, $2$ and $3$, which are dark respect to the pulses with detuning $\delta=-4\omega$. Fig. \[fig:5\] shows (solid line) the dynamics of the population of the ground-state in presence of collisions. After 600 cycles, all the population is transferred to the ground state of the trap. This means that applying the laser cooling scheme brigs the system into an effective BED of $T=0$. It is easy to undertand why the effect is maintained in presence of collisions, even considering that the collisional dynamics is much faster than the laser–cooling one. The laser–cooling mechanism tends to decrease the energy per particle (i.e. the chemical potential of the system), in the same way as evaporative cooling does, but without the losses of particles in the trap during the process. Thermalization via collisions brings the system to a lower temperature. Repeating the laser cooling sufficient times the system ends with an effective zero temperature. Finally, let us point out that some auxiliary pulses which are needed in the ideal gas, are not in presence of collisions. In particular for the previous example, the pulse of zero detuning (required for the ideal gas case, Fig. \[fig:5\](b) dashed line) is no more needed, as shown in Fig. \[fig:5\](b) (solid line). Thus, the laser–cooling scheme is not only possible in presence of collisions, but can be even significanly simplified.
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Conclusions {#sec:conclu}
===========
In this paper, we have analysed the effects of the atom–atom collisions on the colective laser cooling of bosonic gases trapped in an harmonic trap, under the Festina–Lente condition. In particular, we have studied the case in which the mean–field energy provided by the atom–atom collisions is much smaller than the typical energy of the harmonic trap. Under such conditions, we have derived the ME which describes the system, and observed that such ME splits into two parts:(i) a purely collisional part which has the form of a QBME, and (ii) a purely laser–cooling part, which has the same form as the ME which describes the laser–cooling in absence of collisions. By using this ME, we have simulated the dynamics of the trapped gas for different situations. First, we have analysed the cooling into an excited state of a one–dimensional trap. We have observed that the transition from the ideal–gas limit (in which the atoms are completely condensed into the chosen excited state) to the case in which the collisions dominate the dynamics, is not trivial, specially when the collisional and cooling time scale are comparable. In such a case, cooling and collisional processes enter in competition, and new phenomena can appear, as for example unstable population of an excited state followed by abrupt population decays, and non–linear pseudo–oscillations between different trap levels. We have finally analyzed the laser–cooling into the ground state of an isotropic three–dimensional harmonic trap, by using the ergodic approximation. We have shown that, although the collisional time is typically much faster than the cooling time scale, the laser cooling allows to transform a BED with a finite temperature into an effective BED with zero temperature. The laser cooling reduces the chemical potential of the trapped atoms, while the collisions provide the thermalization.
Let us finally present important remarks concerning the scaling of our theory, the situation beyond the weak–condensation regime, and the problem of the two- and three–body losses in the trap. First, we stress that we have presented here the results obtained for $\eta=2,3$ only for the reasons of numerical complexity which grows rapidly with $\eta$. Qualitatively, the same results can be obtained for larger $\eta$’s, and therefore, for lower densities. In fact, we have observed similar results for $\eta=5$ in one dimensional simulations. If we increase $\eta$ by factor $F$, the corresponding density (for fixed $N$) decreases as $F^{-3}$, the three body loss rates as $F^{-6}$, whereas the trap frequency decreases as $F^{-2}$, which means that the corresponding cooling time (to fulfill the [*Festina Lente*]{} conditions) will increase as $F^2$, i.e. much less than the lifetime due to three-body collisions.
Beyond the weak condensation regime, the mean–field energy cannot be neglected, and therefore the trap levels are no more the harmonic ones. This has a two–fold consequence: (i) The levels of the trap are non–harmonic, i.e. they are not equally separated, because their energies become dependent on the occupation numbers; (ii) the wavefunctions are different, and in particular the condensate wavefunction becomes broader (we consider here only the case of repulsive interactions, $a>0$). The fact that the energy levels are not harmonic any more, complicates the laser cooling, but the use of pulses with a variable frequency and band–width should produce the same results as those presented here. The point (ii) implies that the central density of the interacting gas is much lower than the one predicted for noninteracting particles. In fact, the ratio between the interacting–gas central density (in Thomas–Fermi (TF) approximation) and the ideal–gas central density, goes as [@Stringari] $$\frac{n_{TF}}{n_{ideal}}=\frac{15^{2/5}\pi^{1/2}}{8}\left ( \frac{Na}{a_{HO}} \right )^{-3/5},$$ where the central density for the ideal case is given by $n_{ideal}=N/(\pi^{3/2}a_{HO}^3)$.
The above result has important consequences, when one considers the problem of three–body collisions, which usually begin to play a role at densities of the order of $10^{15}$ atoms/cm$^3$. For example, let us analyse the case of Sodium, for which $a_R=\sqrt{\hbar/m\omega_R}=0.132 \mu {\rm m}$, and $a=2.75$[nm]{}. From the definition of $\eta$, $a_{HO}=\eta a_R$. For the ideal gas case, the regime in which three–body losses are important is reached for $N\simeq 12.8 \eta^3$. For $\eta=8$ this means $N=6.5\times 10^3$. Amazingly, for the interacting gas, the same is true for $N \simeq 5.1 \eta^6$, and therefore for $\eta=8$, the regime in which three–body losses are important is reached for $N=1.3\times 10^6$. Below this number, the interaction between the particles is dominated by the ellastic two–body collisions considered in this paper. As point out above, our laser cooling scheme could be extended beyond the weak–condensation regime, and therefore laser–induced condensations of more than $10^6$ atoms are feasible.
Concerning other loss mechanisms, we have to mention here the hyperfine changing two-body collisions, or generally speaking any inelastic two-body processes. These can be supressed completely if we cool atoms into the absolute ground internal state, which is possible in the dipole traps. For alkalis this is typically done by cooling to the lowest energy state in the lower hyperfine manifold (external static magnetic fields are used to split the levels within the hyperfine manifold).
Finally, yet another loss mechanism disregarded here is due to photoassociation, i.e. excitation of molecular resonances. This kind of loss rates are typically of the order $\gamma n(\lambda/2\pi)^3$ (where $\gamma$ is the linewidth of the auxiliary level $|r\rangle$, $\lambda$ is the laser wavelength, and $n$ is the atomic density), i.e. allow for achieving about $1000$ cooling cycles of duration $\simeq 1/\gamma$ provided the density remains smaller than $10^{12}-10^{13}$ atoms/cm$^2$. However, the photoassociation losses can be reduced by several orders of magnitude if the laser is red detuned, and tuned exactly in the middle of the molecular resonances [@shlypm] (note that this is the detuning respect to the one–photon transitions from the ground states to the state $|r\rangle$, and not the two–photon detuning). Other, possibility, is of course to use a more intense laser tuned below the Condon point, i.e. the minimum of the molecular potential.
We acknowledge support from Deutsche Forschungsgemeinschaft (SFB 407) and the EU through the TMR network ERBXTCT96-0002. We thank J. I. Cirac, Y. Castin, G. Birkl, K. Sengstock, W. Ertmer, T. Pfau and T. Esslinger for fruithful discussions.
S. Chu, Nobel Lecture, Rev. Mod. Phys. [**70**]{}, 685 (1998), C. Cohen–Tannoudji, Nobel Lecture, [*ibid*]{}., 707; W. D. Phillips, Nobel Lecture, [*ibid*]{}., 721. M. H. Anderson J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science [**269**]{}, 198 (1995); K.B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Drutten, D. S. Durfee, D. M. Kurn and W. Ketterle, Phys. Rev. Lett. [**75**]{}, 3969 (1995); C. C. Bradley, C. A. Sackett and R. G. Hulet , [*ibid.*]{} [**78**]{}, 985 (1997). C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell and C. E. Wieman, Phys. Rev. Lett. [**78**]{}, 586 (1997). S. N. Bose, Z. Phys. [**26**]{}, 178 (1924); A. Einstein, Sitzber. Kgl. Preuss. Akad. Wiss., p.261 (1924).
R. J. C. Spreeuw, T. Pfau, U. Janicke and M. Wilkens, Europhys. Lett. [**32**]{}, 469 (1995). U. Janicke and M. Wilkens, Europhys. Lett. [**35**]{}, 561 (1996). U. Janicke and M. Wilkens, Adv. At. Mol. Opt. Phys. [**41**]{}, 261 (1999). J. I. Cirac and M. Lewenstein, Phys. Rev. A [**53**]{}, 2466 (1996). A. Aspect, E. Arimondo, R. Kaiser, N. Vanteenkiste and C. Cohen-Tannoudji, Phys. Rev. Lett. [**61**]{}, 826 (1988). M. Kasevich and S. Chu, Phys. Rev. Lett. [**69**]{}, 1741 (1992). E. Arimondo and G. Orriols, Lett. Nuovo Cimento [**17**]{}, 333 (1976). D. W. Sesko, T. G. Walker and C. E. Wieman, J. Opt. Soc. Am B [**8**]{}, 946 (1991); M. Olshan’ii, Y. Castin and J. Dalibard, Proc. 12$^{\rm th}$ Int. Conf. on Laser Spectroscopy, M. Inguscio, M. Allegrini and A. Lasso, Eds. (World Scientific, Singapour, 1996); A. M. Smith and K. Burnett, J. Opt. Soc. Am. B [**9**]{}, 1256 (1992); K. Ellinger, J. Cooper and P. Zoller, Phys. Rev. A [**49**]{}, 3909 (1994).
Y. Castin, J. I. Cirac and M. Lewenstein, Phys. Rev. Lett. [**80**]{}, 5305 (1998). J. I. Cirac, M. Lewenstein and P. Zoller, Europhys. Lett. [**35**]{}, 647 (1996).
L. Santos and M. Lewenstein, Phys. Rev. A (in press); see also G. Morigi, J. I. Cirac, M. Lewenstein and P. Zoller, Europhys. Lett [**39**]{}, 13 (1997).
L. Santos and M. Lewenstein, Europhys. J. D, in press (1999); L. Santos and M. Lewenstein, Phys. Rev. A, in press (1999).
J. Stenger, S. Inouye, M. R. Andrews, H. J. Miesner, D. M. Stamper–Kurn and W. Ketterle, Phys. Rev. Lett. [**82**]{}, 2422 (1999).
P. O. Fedichev, Yu. Kagan, G. V. Shlyapnikov and J. T. M. Walraven, Phys. Rev. Lett. [**77**]{}, 2913 (1996).
C. W. Gardiner and P. Zoller, Phys. Rev. A, [**55**]{}, 2902 (1997). D. Jaksch, C. W. Gardiner and P. Zoller, Phys. Rev. A, [**56**]{}, 575 (1997). C. W. Gardiner and P. Zoller, Phys. Rev. A, [**58**]{}, 536 (1998). D. Jaksch,C. W. Gardiner, K. M. Gheri and P. Zoller, Phys. Rev. A, [**58**]{}, 1450 (1998). C. W. Gardiner and P. Zoller, cond-mat/9905087.
I. Marzoli, J. I. Cirac, R. Blatt and P. Zoller, Phys. Rev. A [**49**]{}, 2771 (1994). C. Gardiner, Handbook of Stochastical Methods, Springer Verlag,1985. C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991). H. Carmichael, An Open Systems Approach to Quantum Optics, Springer-Verlag, 1993. H. Carmichael, Statistical Methods in Quantum Optics 1, Springer-Verlag, 1999. M. Holland, J. Williams and J. Cooper, Phys. Rev. A [**55**]{}, 3670 (1997). F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. [**77**]{}, 463 (1999). T. W. Hijmans, G. V. Shlyapnikov and A. L. Burin, Phys. Rev. A [**54**]{} 4332 (1996); K. Burnett, P. S. Julienne, and K.-A. Suominen, Phys. Rev. Lett. [**77**]{}, 1416 (1996).
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abstract: 'The formation and field-induced switching of defect dipoles in acceptor doped lead titanate is described by a kinetic model representing an extension of the well established Arlt-Neumann model \[Ferroelectrics [**76**]{}, 303 (1987)\]. Energy barriers for defect association and reorientation of oxygen vacancy-dopant (Cu and Fe) complexes are obtained from first-principles calculations and serve as input data for the kinetic coefficients in the rate equation model. The numerical solution of the model describes the time evolution of the oxygen vacancy distribution at different temperatures and dopant concentrations in the presence or absence of an alternating external field. We predict the characteristic time scale for the alignment of all defect dipoles with the spontanenous polarization of the surrounding matrix. In this state the defect dipoles act as obstacles for domain wall motion and contribute to the experimentally observed aging. Under cycling conditions the fully aligned configuration is perturbed and a dynamic equilibrium is established with defect dipoles in parallel and anti-parallel orientation relative to the spontaneous polarization. This process can be related to the deaging behavior of piezoelectric ceramics.'
author:
- Paul Erhart
- Petra Träskelin
- Karsten Albe
title: ' Formation and switching of defect dipoles in acceptor doped lead titanate: A kinetic model based on first-principles calculations '
---
Introduction {#sect:intro}
============
Aging phenomena, namely the gradual change of physical properties with time, are observed in almost all ferroelectrics. [@Ple56; @IkeUed67; @CarHar78; @Tak82; @ArlNeu88; @LohNeuArl90; @WarDimPik95; @AfaPetPro01; @ZhaRen05; @ZhaErdRen08; @MorDam08; @GenGlaHir09; @ZhaRen10] In some acceptor doped barium titanate ($\rm BaTiO_3$) and lead zirconate titanate (PZT) ceramics aging goes along with an increasing shift of the hysteresis along the axis of the electrical field giving rise to an internal bias field. [@TagStoCol01] In the past, several plausible models have been developed to intepret the occurence of bias fields and aging phenomena in ferrocelectrics in terms of domain splitting [@IkeUed67], space-charge formation, [@Tak82] electronic charge trapping, [@WarDimPik95; @AfaPetPro01], ionic drift [@MorDam08] and reorientation of defect dipoles. [@ArlNeu88; @Ren04; @ShiGriChe07; @JiaMiUrb08]
In acceptor (“hard”) doped ferroelectrics transition metals usually substitute the $B$-site (Ti or Zr in PZT) and tend to bind strongly to oxygen vacancies. These acceptor center-oxygen vacancy associates form electric and elastic defect dipoles such as charged $({\text{Fe}}'_{\text{Zr},{\text{Ti}}}$-$V^{\bullet \bullet}_{\text{O}})^{\bullet}$ or $({\text{Cu}}''_{\text{Zr},{\text{Ti}}}$-$V_{\text{O}}^{\bullet\bullet})^\times$, [@MesEicKlo05; @ErhEicTra07] which contribute to the overall polarization in a ferroelectric compound [@WarDimPik95; @PoyCha99a; @MesEicDin04; @MesEicKlo05; @ErhEicTra07; @BooSmiChe07; @EicErhTra08; @MarEls11] and can be aligned either parallel, anti-parallel, or perpendicular to the polarization of the surrounding material as shown schematically in [Fig. \[fig:confs\]]{}.
In the parelectric state, defect dipoles of different orientation are energetically equivalent, whereas they have a preferred orientation in a polar matrix. Arlt and Neumann [@NeuArl87; @ArlNeu88] have attributed the occurence of internal bias fields to the switching of defect dipoles and described the transient orientation of dipoles by a kinetic model. As quantitative data on the energy landscape for these defect dipoles was unavailable at the time they relied on a very simple electrostatic estimate of the energy difference. [^1] The energetic asymmetry between the parallel and anti-parallel dipoles obtained in this fashion for BaTiO$_3$ was about 30meV and thus much smaller than the energy differences calculated more recently by first-principles methods for PbTiO$_3$, [@PoyCha99a; @MesEicDin04; @ErhEicTra07] which revealed that the energetic asymmetry is actually as large as the barriers for oxygen migration. Only recently, Marton and Elsässer [@MarEls11] showed that in Fe-doped lead titanate the barrier for reorientation sensitively depends on the position of the migrating oxygen vacancy with respect to the iron atom and the surrounding spontaneous ferroelectric polarization. During fast field cycling, the defect-dipoles are expected not to change orientation, because the characteristic rate for oxygen jumps around the acceptor center should be lower than the domain switching process. Experimentally, Zhang [*et al.*]{} [@ZhaErdRen08] followed the dynamics of $(\text{Mn}_{{\text{Ti}}}$-$V_{\text{O}})^\times$ dipoles in barium titanate by electron paramagnetic resonance studies and found support for the so-called “defect symmetry principle”, which assumes that non-switching defect dipoles impose a restoring force for reversible domain switching. [@Ren04] Jakes [*et al.*]{} could show that in $\rm Fe^{3+}$ doped PZT defect dipoles are not preferentially located at domain walls but within the domains. [@JakErdEic11] Morozov [*et. al.*]{} studied aging-deaging process in hard PZT ceramics using the harmonic analysis of polarization response under switching conditions and concluded that two or more mechanisms are responsible for domain stabilization. [@MorDam08] Activation energies of about 0.6eV were attributed to short-range charge hopping, which could be due to local reorientation of microdipoles.
Since the switching dynamics of defect dipoles depends on the electric and thermal energy provided to change polarization direction, the contribution of dipole reorientation can only be reliably assessed if realistic numbers for the migration and association energies are available, which allow to quantitatively model the switching dynamics of defect dipoles in a comprehensive way.
The objective of the present work is to develop a kinetic model that captures the formation of defect dipoles as well as their reorientation both in the absence and presence of electric fields. [^2] Cu and Fe-doped lead titanate are considered as representative examples and the energy landscape for oxygen vacancy migration in these materials is obtained using first-principles calculations. We consider both free oxygen vacancies and oxygen vacancies associated with Fe or Cu. Starting from a statistical distribution the majority of oxygen vacancies is initially unbound. Over time vacancies are captured by impurity atoms and subsequently converted into the lowest energy configuration, which corresponds to a defect dipole that is aligned parallel to the macroscopic polarization ($M_B-V_{c1}$ in [Fig. \[fig:confs\]]{}). While the exact time scales for these processes are dependent on dopant type, concentration, and temperature our results demonstrate that the ground state is reached within seconds at temperature slightly above room temperature and thus that already the pistine material can be considered as “aged”. In the presence of an oscillating external field our model predicts that a gradual reorientation of defect dipoles leads to a dynamic equilibrium, in which the parallel and anti-parallel configurations occur with equal probability. This is in accord with the experimental observation of deaging by the application of AC fields. [@GraSuvKun06; @GlaGenKun12]
This paper is organized as follows. First, we describe the kinetic model and discuss its features. This is followed in [Sect. \[sect:dft\]]{} by a description of the first-principles calculations that were carried to determine the model parameters. In [Sect. \[sect:results\]]{} we apply the kinetic model to study vacancy redistribution as a function of temperature and impurity concentration both in the absence and presence of an oscillating external electric field. The implications of the present findings for aging and fatigue are discussed in [Sect. \[sect:discussion\]]{} and conclusions are summarized in [Sect. \[sect:conclusions\]]{}.
Kinetic model {#sect:kinmodel}
=============
In this section we formulate a kinetic model that describes the redistribution of oxygen vacancies between different types of sites as a function of time. It captures the temperature, impurity concentration and frequency dependence of this process within a mean-field approximation. Figure \[fig:confs\] provides an overview of the different types of oxygen vacancies that are taken into account by this model.
{width="0.98\linewidth"}
\[fig:confs\]
In general, the temporal variation of the concentration of vacancies of type $i$ can be described by a rate equation $$\begin{aligned}
\frac{\partial c_i}{\partial t}
&=
- \sum_j \Phi_{ij} \mathrm{K}_{ij} c_i
+ \sum_j \Phi_{ji} \mathrm{K}_{ji} c_j
\label{eq:crate},\end{aligned}$$ where the first term on the right hand side accounts for the “loss” of vacancies of type $i$ while the second term describes the “gain” due to vacancy jumps from sites of type $j$ to sites of type $i$. At typical device operation temperatures near 300K the creation or annihilation of vacancies at surfaces or interfaces is negligible and the total concentration of vacancies can be assumed as constant, $$\begin{aligned}
\sum_i c_i = c_{tot}.
\label{eq:ctot}\end{aligned}$$ The rate at which vacancies of type $i$ jump onto sites of type $j$ is given by $$\begin{aligned}
{\mathrm K}_{ij} &=
\nu^i_0 \exp\left(-\frac{\Delta G_m^{i-j}}{k_B T}\right)\end{aligned}$$ where $\nu_0^i$ is the attempt frequency and $\Delta G_m^{i-j}$ is the free energy of migration encountered by a vacancy jumping from a site of type $i$ to a site of type $j$. The attempt frequency is for all jumps approximated by the frequency of the lowest optical mode at $\Gamma$, $\nu_0\approx 2\,{\text{THz}}$. [@GhoCocWag99]
The probability $\Phi_{ij}$ for a vacancy to jump from a site of type $i$ to a site of type $j$ is given by the fraction of sites of type $j$ in the first nearest neighbor shell of sites of type $i$. Using a simple mapping to index different defect configurations, [-]{}$\rightarrow $ (1), [-]{}$\rightarrow$ (2), [-]{}$\rightarrow$ (3), $\rightarrow$ (4), and $\rightarrow$ (5) \[see [Fig. \[fig:migration\_paths\]]{}(g) for examples\] and taking into account the geometry of the lattice (see [Fig. \[fig:migration\_paths\]]{}) the following probability matrix is obtained $$\begin{aligned}
{\ensuremath\boldsymbol{\mathrm{\Phi}}}
&=
\frac{1}{8}
\left(\begin{matrix}
0 & 0 & 4 & 0 & 4 \\
0 & 0 & 4 & 0 & 4 \\
1 & 1 & 2 & 2 & 2 \\
0 & 0 & 8\alpha & 0 & 8(1-\alpha) \\
4\alpha & 4\alpha & 8\alpha & 8(1-\alpha) & 8(1-\alpha)
\end{matrix}\right)
\label{eq:probmatrix},\end{aligned}$$ where $\alpha=6{\text{$f_{M}$}}$. Here ${\text{$f_{M}$}}$ is the fraction of $B$ sites which have been replaced by impurity atoms. The recurrence of the factor eight in [Eq. (\[eq:probmatrix\])]{} results from the number of oxygen sites in the second neighbor shell of any given oxygen site, while the factor six stems from the number of oxygen sites in the first neighbor shell of a $B$ site. Introducing $\mathrm{W}_{ij}=\Phi_{ij} \mathrm{K}_{ij}$ and $\mathrm{V}_{ij}=\delta_{ij}\sum_k\mathrm{W}_{ik}$, [Eq. (\[eq:crate\])]{} can be rewritten in a convenient matrix form $$\begin{aligned}
{\ensuremath\boldsymbol{\dot{c}}}
&= \left({{\ensuremath\boldsymbol{\mathrm{W}}}^{\mathrm{T}}}-{\ensuremath\boldsymbol{\mathrm{V}}}\right) {\ensuremath\boldsymbol{c}},
\label{eq:model}\end{aligned}$$ which in this work has been numerically [^3] using an adaptive time step algorithm for stiff differential equations. [^4]
It should be noted that the model does not take into account the possibility of two or more oxygen vacancies associating with the same impurity atom. Both experiments and calculations indicate, however, that this is unlikely to occur for the dopants considered in the present work. Similarly the possibility that two or more impurity atoms form an aggregate can be ruled out based on experimental evidence. [@MesEicKlo05]
In the present form the model does not include any constraints to allow for the number of oxygen vacancies to be larger than the number of impurity atoms or vice versa. This situation can, however, be implemented rather easily by solving the kinetic model in steps. For instance, consider a case in which the vacancy concentration is $[V_{{\text{O}}}]=0.01$ and the dopant/impurity concentration is $[M]=0.005$. The sum of the [*relative*]{} concentrations of complexed vacancies $[M]/[V_{\text{O}}]$ can, therefore, not exceed $c_{max}=0.5 > c_1 + c_2 + c_3$. Starting from some initial distribution, one solves the kinetic model until $c_1+c_2+c_3$ equals $c_{max}$. At this point all impurities are complexed with vacancies, and one can “remove” the free vacancy concentrations $c_4$ and $c_5$ from the model. This is achieved by reducing the $5\times 5$ matrices in Eqs. (\[eq:crate\]-\[eq:probmatrix\]) and [Eq. (\[eq:model\])]{} to $3\times 3$ matrices, only keeping elements $(i,j)\in \{1,2,3\}$. The opposite scenario, in which the number of impurity/dopant atoms exceeds the number of free vacancies, can be implemented in a similar fashion. For the sake of clarity and because the key conclusions of this work are unaffected by these conditions, we do not consider any of these cases in the remainder part of this paper.
Applying the model to a specific material requires knowledge of the energy differences between various vacancy configurations as well as migration energies. To provide these parameters, we have carried out first-principles calculations that are described in the following section. At this level we neglected the vibrational entropy contribution to the migration barriers and approximated $\Delta G_m \approx \Delta E_m$.
First-principles calculations {#sect:dft}
=============================
Computational parameters
------------------------
The barriers for oxygen vacancy migration in pure as well Cu and Fe-doped lead titanate were calculated within density functional theory (DFT) using the Vienna ab-initio simulation package. [@KreHaf93; @*KreHaf94; @*KreFur96a; @*KreFur96b] The potentials due to the ions and the core electrons were represented by the projector-augmented wave method. [@Blo94; @*KreJou99] The $5d$ electrons of Pb, the $3s$ and $3p$ electrons of Ti as well as the $3p$ electrons of Fe and Cu were treated as part of the valence. The exchange-correlation potential was represented using the local spin density approximation, [@CepAld80; @*PerZun81]. Supercells containing $2\times 2\times 4$ unit cells equivalent to 80 atoms were employed and the Brillouin zone was sampled using a $2\times 2\times 2$ Monkhorst-Pack mesh. Similar computational parameters were successfully used in previous studies of Cu and Fe-doped lead titanate. [@MesEicKlo05; @ErhEicTra07; @EicErhTra08] For several configurations we also carried out calculations using a $4\times 4\times 4$ mesh and found negligible differences on the order of $0.05\,{\text{eV}}$ and below. The computations were performed at the theoretical lattice constant of $a_0=3.866\,\text{\AA}$ and the theoretical value for the axial ratio of $c/a=1.05$, both of which are in reasonable agreement with experiment ($a_0=3.905\,\text{\AA}$, $c/a=1.064$ at room temperature, Refs. ). The calculated band gap of 1.47eV is considerably smaller than the experimental value, but consistent with the well known band gap error of DFT. As argued in Ref. the band gap error is, however, expected to have a minor effect in the context of migration barrier calculations. Migration paths and barriers were determined using the climbing image nudged elastic band method [@HenJohJon00; @HenUbeJon00] and configurations were optimized until the maximum force was less than 30meV/Å. For charged defects a homogeneous background charge was added.
Free oxygen vacancies {#sect:free_VO}
---------------------
![ Schematic representation of the barriers for the migration of free (a) and complexed (b-d) oxygen vacancies in units of eV. Each figure shows the projection of a $B$O$_6$ octahedron onto the (100) plane. The numbers in circles indicate the indices used to distinguish the different processes. Oxygen sites (and thus possible vacancy sites) are shown as red circles while the position of the $B$-site cation (Ti, Cu, or Fe), which is situated at the center of the oxygen octahedron, is indicated by blue (Ti), green (Cu) and yellow (Fe) circles. []{data-label="fig:barriers"}](fig2.eps){width="0.92\linewidth"}
As indicated in [Fig. \[fig:confs\]]{} there are two crystallographically distinct oxygen vacancy sites in the tetragonal perovskite lattice (space group P$4mm$). Vacancies can be situated along the $c$-axis (${\text{$V_c$}}$, Wyckoff site $1b$) or within the $ab$-plane (${\text{$V_{ab}$}}$, Wyckoff site $2c$). [@ParCha98; @ErhEicTra07] [^5] Thus, three different migration paths are possible between nearest neighbor sites: pure in-plane migration (${\text{$V_{ab}$}}\rightarrow{\text{$V_{ab}$}}$), out-of-plane migration along the positive direction of the tetragonal axis (${\text{$V_{ab}$}}\rightarrow{\text{$V_c$}}~[001]$), and out-of-plane migration along the negative direction of the tetragonal axis (${\text{$V_{ab}$}}\rightarrow{\text{$V_c$}}~[00\bar{1}]$).
[l\*[3]{}[ll]{}]{} & & & +2\
$V_{ab}\rightarrow V_{ab}$ & 0.98 & & 0.62 & & 0.53 &\
$V_{c}\rightarrow V_{ab}~[00\bar{1}]$ & 0.83 & (0.91) & 0.94 & (0.58) & 1.10 & (0.54)\
$V_{c}\rightarrow V_{ab}~[001]$ & 0.50 & (0.59) & 0.58 & (0.22) & 0.70 & (0.13)\
The calculated barriers are compiled in [Fig. \[fig:barriers\]]{}(a) and [Table \[tab:migration\_VO\]]{}. In general, the barriers found to be charge state dependent, which is in line with calculations for cubic PbTiO$_3$ [@Par03] and cubic BaTiO$_3$. [@ErhAlb07] For migration within the $ab$-plane the barriers decrease as electrons are removed from the defect, which is in accord with the calculations on cubic perovskite structures.[@Par03; @ErhAlb07] The barriers for migration via $c$-type vacancies, in constrast, increase. Since the charge state $q=+2$ prevails vor the oxygen vacancy almost over the entire band gap, [@ErhEicTra07] its respective barriers were used in the construction of the energy landscapes used in the kinetic model.
As detailed in the appendix, we can obtain the oxygen diffusivity (excluding the formation energy contribution) from our calculated barriers and compare it with experimental data. We find that the calculated activation energy of 0.7eV is in good agreement with the experimental value of 0.87eV. [@GotHahFle08]
Complexed oxygen vacancies {#sect:complexed_VO}
--------------------------
The incorporation of an impurity on the $B$ site breaks translational symmetry along the tetragonal axis and lifts the degeneracy of the oxygen sites in this direction. One therefore obtains three distinct types of first-neighbor impurity atom–oxygen vacancy associates (compare Figs. \[fig:migration\_paths\] and \[fig:barriers\]), which leads to three distinct migration paths. Migration within the second neighbor shell of impurity atoms was not considered, since it has been previously shown that the binding energy between oxygen vacancies and Cu and Fe impurities is the largest in the first neighbor shell. [@ErhEicTra07] Thus, once an oxygen vacancy arrives in the vicinity of an impurity via diffusion, it will be attracted to the impurity and eventually reside in its first neighbor shell. The barriers for different paths for the migration of oxygen vacancies in the first nearest neighbor shell of copper and iron impurities are shown in [Fig. \[fig:barriers\]]{}(b-d).
Construction of energy landscape {#sect:landscape}
--------------------------------
{width="0.82\linewidth"} {width="0.82\linewidth"}
So far, we have calculated the migration barriers for free oxygen vacancies, which determine the elements of the rate matrix $\mathrm{K}_{ij}$ for $(i,j)=\{4,5\}$, and the migration barriers for oxygen vacancies in the first neighbor shell of an impurity atoms, which provide the elements with $(i,j)=\{1,2,3\}$. Using these data some parts of the energy landscape can already be constructed as indicated by the migration barriers shown in black in [Fig. \[fig:energy\_surface\_Cu\]]{}.
To determine the barriers for the remaining combinations of $i$ and $j$, e.g., $1-5$ or $3-4$, one would require noticeably larger supercells than the ones employed in the present work. This results from the long ranged Coulombic attraction between the impurity atom and the oxygen vacancy which leads to a gradual transition from a free oxygen vacancy to a vacancy in the first neighbor shell of an impurity ion over several lattice spacings. This complexity can in principle be captured by increasing the dimensionality of the probability and rate matrices, $\Phi$ and $\mathrm{K}$. In the present more simple description, we, however, consider already oxygen vacancies in the second impurity neighbor shell as “free”.
For completing the rate matrix, we then assume the migration barriers for jumps [*between free*]{} oxygen vacancies to hold for jumps [*to complexed*]{} vacancies as well (values marked in green in [Fig. \[fig:energy\_surface\_Cu\]]{}). To determine the barriers for the reverse jumps, we resort to the binding energies calculated for ${\text{Cu}}_{{\text{Ti}}}-V_{{\text{O}}}$ and ${\text{Fe}}_{{\text{Ti}}}-V_{{\text{O}}}$ complexes calculated in Ref. (values marked in blue in [Fig. \[fig:energy\_surface\_Cu\]]{}). For Cu and Fe the binding energy amounts to $-2.38\,{\text{eV}}$ and $-1.32\,{\text{eV}}$ for Fermi levels near mid gap, which leads to the energy surfaces shown in [Fig. \[fig:energy\_surface\_Cu\]]{}. We have tested the sensitivity of the results of the kinetic model to the barriers for jumps between free and complexed vacancies, which showed the assumptions made in determining their values to be of little consequence.
Results {#sect:results}
=======
In this section we present results obtained using the kinetic model for Cu and Fe-doped lead titanate described in [Sect. \[sect:kinmodel\]]{}. To illustrate the general features of vacancy redistribution, we first discuss the results for Cu-doped PbTiO$_3$ in the absence and presence of electric fields in Sects. \[sect:results\_dc\] and \[sect:results\_ac\], respectively. The results for Fe-doped material are qualitatively very similar and will be summarized in [Sect. \[sect:Fe\_results\]]{}.
Vacancy redistribution in the absence of electric fields {#sect:results_dc}
--------------------------------------------------------
![ Vacancy redistribution in the absence of electric fields for Cu-doped lead titanate: (a) Temporal evolution of the relative concentrations of different vacancy types at a temperature 300K and for a Cu concentration of 5%. (b) Temperature dependence of the characteristic transition times marked by white circles in (a). The transition time for the conversion from $V_c$ to ${\text{Cu}}_{\text{Ti}}-V_{c2}$ depends on the Cu concentration as exemplified by the three green dashed lines of varying thickness. []{data-label="fig:Cu_dc"}](fig4a.eps "fig:"){width="0.85\linewidth"} ![ Vacancy redistribution in the absence of electric fields for Cu-doped lead titanate: (a) Temporal evolution of the relative concentrations of different vacancy types at a temperature 300K and for a Cu concentration of 5%. (b) Temperature dependence of the characteristic transition times marked by white circles in (a). The transition time for the conversion from $V_c$ to ${\text{Cu}}_{\text{Ti}}-V_{c2}$ depends on the Cu concentration as exemplified by the three green dashed lines of varying thickness. []{data-label="fig:Cu_dc"}](fig4b.eps "fig:"){width="0.85\linewidth"}
\[fig:chartimes\]
Using the energy landscape for Cu-doped PbTiO$_3$ shown in [Fig. \[fig:energy\_surface\_Cu\]]{} as input data for the kinetic model one can obtain the temporal evolution of the relative concentrations of different types of vacancies as exemplarily shown in [Fig. \[fig:Cu\_dc\]]{}(a) for a temperature of 300K and a dopant concentration of ${\text{$f_{M}$}}=5\%$. In this example, the vacancies are initially statistically distributed over all available sites. In [Fig. \[fig:Cu\_dc\]]{}(a) four distinct time regimes with characteristically different dynamic balance can be identified that are separated by the transitions marked A, B, and C:
*(i)* The first regime (up to $t\lesssim 10^{-10}\,{\text{s}}$ at $300\,{\text{K}}$) is associated with the redistribution of unbound vacancies. As can be seen from [Fig. \[fig:energy\_surface\_Cu\]]{}, is energetically preferred over . Figure \[fig:Cu\_dc\](a) shows that even for a temperature as low as 300K the redistribution between these two types of vacancies, [*i.e.*]{} the installation of the (partial) equilibrium over the subset of unbound vacancies, takes place within fractions of a second.
*(ii)* During the second stage ($10^{-10}\,{\text{s}}- 10^0\,{\text{s}}$ at $300\,{\text{K}}$) unbound $c$-type vacancies dominate. Concurrently, dopants begin to capture vacancies. The dynamic equilibrium at this point is such that ${\text{Cu}}_{{\text{Ti}}}-V_{c2}$ complexes dominate over ${\text{Cu}}_{{\text{Ti}}}-V_{c1}$ and ${\text{Cu}}_{{\text{Ti}}}-V_{ab}$ dipoles. The former are energetically less favorable but are much more easily accessible since $\Delta G_m({\text{$V_{ab}$}}\rightarrow{{\text{$M_B$}}-\text{$V_{c2}$}})$ is only 0.13eV compared to $\Delta G_m({\text{$V_{ab}$}}\rightarrow{{\text{$M_B$}}-{\text{$V_{ab}$}}})=0.53\,{\text{eV}}$, $\Delta G_m({\text{$V_c$}}\rightarrow{{\text{$M_B$}}-{\text{$V_{ab}$}}})=0.70\,{\text{eV}}$, and $\Delta G_m({\text{$V_{ab}$}}\rightarrow{{\text{$M_B$}}-\text{$V_{c1}$}})=0.54\,{\text{eV}}$.
*(iii)* In the third stage ($10^{0}\,{\text{s}}- 10^5\,{\text{s}}$ at $300\,{\text{K}}$) vacancy–dopant complexes take over with ${\text{Cu}}_{{\text{Ti}}}-V_{c2}$ being the dominant defect. The prevalence of ${\text{Cu}}_{{\text{Ti}}}-V_{c2}$ is inherited from the second stage which determines the initial concentrations for the third stage.
*(iv)* Eventually, the system reaches equilibrium ($t\gtrsim 10^5\,{\text{s}}$ at $300\,{\text{K}}$), [*i.e.*]{} virtually all vacancies occupy the lowest energy site. [^6]
We can now discuss the ependence of vacancy migration on temperature and dopant concentration. As indicated by the letters A, B, and C in [Fig. \[fig:Cu\_dc\]]{}(a), characteristic times can readily identified, at which the majority defect type changes. As shown in [Fig. \[fig:chartimes\]]{}(b) the temperature dependence of these characteristic times can be fit to an Arrhenius equation, , using the migration barrier between the states involved as the activation energy. This analysis also demonstrates that the effect of changing the dopant concentration is small and is only visible for the transition between vacancy types $V_c$ and ${\text{Cu}}_{\text{Ti}}-V_{c2}$.
Already at temperatures $\gtrsim\!450\,{\text{K}}$ the full equilibrium is established within less than a second. Since during growth these temperatures are easily reached, the vacancy distribution in tetragonal lead titanate should be in thermal equilibrium, [*i.e.*]{} virtually all dopants are complexed with vacancies in the ground state configuration, in which the defect dipoles are aligned with the domain polarization.
Vacancy redistribution in the presence of electric fields {#sect:results_ac}
---------------------------------------------------------
In the present model the perturbation introduced by an external electric field enters in two ways. First the barriers for vacancy jumps with components along the direction of the electric field are distorted (“direct effect”), $\Delta G_m\rightarrow \Delta G_m-\delta E$ where $\delta E=E_{loc} \Delta r_{[001]} q e$. Here, $\Delta r_{[001]}$ denotes the displacement along the direction of the electric field which is positive (negative) if the displacement is parallel (anti-parallel) to the electric field; $q$ is the charge state of the defect, $E_{loc}$ is the local electric field, and $e$ denotes the unit charge. Typical electric fields used for poling ferroelectric ceramics are on the order of 2–5kV/cm; the local electric field can, however, be larger than this value due to inhomogeneities. [@GlaGenKun12] Assuming a value of $E_{loc}=100\,\text{kV/cm}$ and choosing $\Delta r_{[001]}=1\,\text{\AA}$, we obtain an upper limit for $\delta E$ of 0.01eV.
Obviously the “direct” effect of the electric field is rather small compared to the energy difference between different vacancy types and pertains to charged vacancies only. This implies that with regard to vacancy redistribution a material in a constant external field will behave almost identical to the situation without electric fields.
The situation does, however, change if we consider an oscillating external field. The field induced reversal of the polarization has a much larger effect on the energetics of the system than the direct contribution since reorientation of the polarization implies that the (average) displacement of $B$-site atoms is reversed, thus transforming [-]{} into [-]{} complexes. In the present model, this is equivalent to exchanging rows 1 and 2 of the migration rate matrix, $\mathrm{K}_{ij}$. One can therefore include the effect of an oscillating electric field by (*i*) periodically modifying the barriers for out-of-plane jumps by $\delta E$ and (*ii*) simultaneously exchanging the barriers for jumps involving [-]{} or [-]{}.
![ Vacancy redistribution in the presence of an oscillating external field: Equilibration over vacancy types [-]{} (solid lines) and [-]{} (dashed lines) in the presence of an external oscillating field with a cycling frequency of 1Hz for temperatures between 300 and 360K. []{data-label="fig:efield_equilibration"}](fig5.eps){width="0.85\linewidth"}
Figure \[fig:efield\_equilibration\] illustrates the temporal evolution of the concentrations of [-]{} and [-]{} vacancies in the presence of an oscillating electric field. The plot shows that under prolonged cycling a dynamic balance between the two configurations [-]{} and [-]{} is established. The time after which this balance is obtained depends sensitively on temperature e.g., at room temperature it is reached after about $10^6\,{\text{s}}$ (approximately two weeks) while at 340K it requires only about one hour.
The dynamic balance between [-]{} and [-]{} occurs because the characteristic time required to reach full equilibrium in the absence of external electric fields \[see [Fig. \[fig:chartimes\]]{}(b)\] exceeds the cycling period. In a fast switching field the restoring force of the spontaneous polarization is changing signs on a short time scale. As a result, a mean-field composed of parallel and anti-parallel polarization of the matrix is acting on the dipoles, which evantually populate both orientations along the c-axis. This dynamic equilibrium is sensitive to temperature and the assumption of a static distribution of defect dipoles in a fast switching field should be taken with care. The results show that by applying a bipolar electric field defect dipoles are redistributed and on average the clamping effect on domain walls is reduced. This is in line with recent results on deaging of doped PZT.[@GraSuvKun06; @GlaGenKun12]
Results for Fe-doped lead titanate {#sect:Fe_results}
----------------------------------
![ (a) Temporal evolution of the relative concentrations of different vacancy types for three different temperatures in Fe-doped lead titante. (b) Dependence of characteristic time scales for the transitions indicated by white circles in (a) on temperature and dopant concentration. []{data-label="fig:Fe_dc"}](fig6a.eps "fig:"){width="0.85\linewidth"} ![ (a) Temporal evolution of the relative concentrations of different vacancy types for three different temperatures in Fe-doped lead titante. (b) Dependence of characteristic time scales for the transitions indicated by white circles in (a) on temperature and dopant concentration. []{data-label="fig:Fe_dc"}](fig6b.eps "fig:"){width="0.85\linewidth"}
\[fig:chartimes\_Fe\]
Complexes of Fe with oxygen vacancies act as donors leading to electron chemical potentials in the upper half of the band gap. The most stable charge state is $q=+1$. [@ErhEicTra07] Under such conditions the binding energy is $-1.32\,{\text{eV}}$ from which one can construct an energy surface shown in [Fig. \[fig:energy\_surface\_Cu\]]{}(b).
The temporal evolution of different vacancy configurations in the absence of an external electric field is shown in [Fig. \[fig:Fe\_dc\]]{}(a) which allows us to infer the temperature dependence of the characteristic time scales summarized in [Fig. \[fig:chartimes\_Fe\]]{}(b). Comparing [Fig. \[fig:Cu\_dc\]]{} and [Fig. \[fig:Fe\_dc\]]{} we find that the results for Fe and Cu-doped lead titanate are very similar. This is expected since the first two transitions ($V_{ab}\rightarrow V_{c}$ and $V_c\rightarrow {\text{Fe}}_{\text{Ti}}-V_{c2}$, compare [Sect. \[sect:landscape\]]{}) are determined by the migration barriers in the pure host. With regard to the third transition between $M_{\text{Ti}}-V_{c2}$ and $M_{\text{Ti}}-V_{c1}$ the situation is different as the effective barrier in Fe-doped material is 1.00eV and thus slightly smaller than in Cu-doped lead titanate (1.06eV, compare [Fig. \[fig:barriers\]]{}), which speeds up the transition. We can thus expect that in the presence of an oscillating external field the dynamic equilibrium between ${\text{Fe}}_{\text{Ti}}-V_{c2}$ and ${\text{Fe}}_{\text{Ti}}-V_{c1}$ is established faster as well, which is confirmed by explicit calculation. Whereas for Cu-doped lead titanate our model calculations predict the equilibrium to be installed over two weeks at room temperature, in Fe-doped material the same process should occur on the order of a day. Similarly at 340K equilibration should take only on the order of tens of minutes.
Summary and Discussion {#sect:discussion}
======================
We have parametrized a kinetic model for defect dipole formation and switching by taking data from first-principles calculations for Fe and Cu-doped $\rm PbTiO_3$. We find that at temperatures $\gtrsim\!450\,{\text{K}}$, which is well below the Curie temperature of 720K, [@NohCerIgl95] the formation and alignment of defect dipoles in doped PbTiO$_3$ should occur within less than a second. [^7]
![ Schematic representation of defect dipole arrangements under different conditions as deduced from the kinetic model. The large value arrows indicates the matrix polarization in different domains while the small arrows represent defect dipoles. The thick solid line illustrates the position of a 90[$^\circ$]{} domain wall. []{data-label="fig:compound"}](fig7.eps){width="0.99\columnwidth"}
Bipolar poling leads to a dynamic equilibrium between defect dipoles that are aligned parallel and anti-parallel to the lattice polarization, respectively, and thus can be seen as one major contribution to deaging of PZT ceramics. This is in accord with the experimental observation of deaging by the application of AC fields. [@GraSuvKun06; @GlaGenKun12] In Cu-doped lead titanate the dynamic equilibration takes about two weeks at 300 K, but is massively accelerated if temperature is slightly increased. This points to the importance of closely monitoring the sample temperature during testing and studying aging and deaging processes.
For the [-]{} complex, which for both Cu and Fe is the ground state configuration, the local polarization is parallel to the polarization of the surrounding matrix. [@ErhEicTra07] Since in contrast [-]{} defects are aligned anti-parallel to the lattice polarization, an increase in their concentration causes an overall loss of switchable polarization. This direct contribution should scale linearly with the number of impurity atoms in the sample but due to the small magnitude of the defect dipole moment will amount to a rather small contribution on the macroscopic scale. Defect dipoles, however, also interact with domain walls and can affect their mobility. In lead titanate and tetragonal PZT one typically observes 90[$^\circ$]{} domain wall configurations, which is schematically indicated in Fig. \[fig:compound\]. It has been shown by first-principles calculations that the head-to-tail domain wall configuration shown e.g., in [Fig. \[fig:compound\]]{}, is energetically more stable than head-to-head or tail-to-tail configurations. [@MeyVan02] In the pristine material after cooling (middle panel of [Fig. \[fig:compound\]]{}) all defect dipoles are aligned with the lattice polarization and thus follow the head-to-tail pattern. This situation changes significantly after cyclic loading (right panel of [Fig. \[fig:compound\]]{}) since now half of the defect dipoles oppose the lattice polarization and thus create local high-energy head-to-head and tail-to-tail configurations.
Recent simulations of domain wall motion using an empirical force field [@ShiGriChe07] have provided impressive evidence that domain wall motion proceeds via a nucleation-and-growth process. It will be subject of future work to determine the role of defect dipoles quantitatively but already at the present stage one can imagine that defect dipoles that are aligned anti-parallel to the lattice polarization in the growing domain will locally impede both nucleation and growth while the opposite can be said for defect dipoles that are aligned parallel to the lattice polarization. One can expect that even though the direct contribution of defect dipoles to the macroscopic polarization is small they can have a significant indirect impact by pinning domain walls and reducing their mobility. It should be stressed that the fact that domain motion occurs via nucleation and growth is crucial in this context since it implies that domain wall motion occurs locally and can thus be strongly influenced by localized defect dipoles.
Conclusions {#sect:conclusions}
===========
In the present work we have derived a kinetic model that allows us to study the temporal evolution of the concentration of different types of vacancies both in the absence and presence of electric fields. The most important input parameter is the energy landscape for oxygen vacancy migration. Using parameters for Cu and Fe-doped obtained from density functional theory calculations, we found that the equilibration of the vacancy distribution occurs readily at temperatures considerably below the Curie temperature. As a result in the as-synthesized material virtually all impurity atoms are associated with vacancies forming [-]{} complexes. The complete realignment of vacancy-metal impurity dipoles parallel to the spontaneous polarization occurs on time scales of hours to days at room temperature, but is massively accelerated if temperature is slightly increased. This provides evidence for the fact that aging due to defect dipoles occurs instantaneously in PbTiO$_3$-based ferroelectrics.
In the presence of an oscillating electric field a dynamic balance between [-]{} and [-]{} is established. Prolonged cycling therefore leads to the accumulation of defect dipoles that oppose the polarization of the encompassing domain. While these defect dipoles directly reduce the switchable polarization, more importantly they can impede domain wall motion, which has been recently shown to proceed via nucleation and growth. [@ShiGriChe07]
The present results provide valuable insights into the switching kinetics of defect dipoles in ferroelectrics, which is relevant for understanding aging and deaging mechanisms. The rate equation approach can be adapated straightforwardly to describe more complex geometries and systems. This could be used to model the lattice geometries of e.g., BiFeO$_3$ or LaMnO$_3$.
Kinetic models similar to the one discussed in the present paper can also be used to interpret experimental measurements. To this end, one could employ probes which are sensitive to the orientation of the defect dipoles (e.g., electron spin resonance [@MesEicKlo05; @EicErhTra08]) and measure the intensity of the signal before, during and after cycling or heat treatments.
This project was partially funded by the *Sonderforschungsbereich 595* “Fatigue in functional materials” of the *Deutsche Forschungsgemeinschaft*. P.E. acknowledges funding from the “Areas of Advance – Materials Science” at Chalmers. Computer time allocations by the Swedish National Infrastructure for Computing are gratefully acknowledged.
Oxygen Diffusivity {#sect:diffusivity}
==================
We can employ the calculated migration barriers to derive the diffusivity of oxygen vacancies (see e.g., Ref. ). The rate at which a vacancy jumps along a given path $i$ is $$\begin{aligned}
\nu &= \nu_0 \exp\left(-\beta \Delta G_i\right)\end{aligned}$$ where $\nu_0$ is the attempt frequency, $\Delta G_i$ is the barrier which has to be surpassed along path $i$, and $\beta=1/k_BT$. Summing over all paths and including the jump lengths $\lambda_i$ as well as the path multiplicities $\zeta_i$ the defect diffusivity is then given by $$\begin{aligned}
D_d &= \frac{1}{2} \sum_i \nu_i \lambda_i^2 \zeta_i.
\label{eq:diffusivity}\end{aligned}$$ There are four symmetrically equivalent migration processes ($\zeta=4$) within the $ab$-plane (${\text{$V_{ab}$}}\rightarrow{\text{$V_{ab}$}}$), for which the jump lengths projected onto the $ab$-plane and the $c$-axis are $\lambda_{\perp}=a_0$ and $\lambda_{\parallel}=0$, respectively. With regard to of out-of-plane migration (${\text{$V_{ab}$}}\rightarrow{\text{$V_c$}}$) there are again four possibilities each for jumps with components along $[001]$ and $[00\bar{1}]$, respectively, associated with displacements $\lambda_{\perp}=a_0/\sqrt{2}$ and $\lambda_{\parallel}=\pm \frac{1}{2}c_0$. Inserting these parameters into [Eq. (\[eq:diffusivity\])]{} yields $$\begin{aligned}
D_{\perp}
&= a_0^2 \nu_0
\bigg[
\exp\left(-\beta\Delta G_{ab-c}^{[001]}\right)
+ \exp\left(-\beta\Delta G_{ab-c}^{[00\bar{1}]}\right)
\\
&\quad\quad\qquad
+ 2 \exp\left(-\beta\Delta G_{ab-ab}\right)
\exp\left(-\beta\Delta G_{ab-c}^{form}\right)
\bigg]
\\
D_{\parallel}
&= \frac{1}{2} c_0^2 \nu_0
\left[
\exp\left(-\beta\Delta G_{ab-c}^{[001]}\right)
+ \exp\left(-\beta\Delta G_{ab-c}^{[00\bar{1}]}\right)
\right]\end{aligned}$$ where the very last term takes into account the equilibrium occupancy of $ab$-sites with respect to $c$-site vacancies and $\Delta G_{ab-c}^{form}$ is the difference between the formation free energies of $c$ and $ab$-site vacancies. Finally, the isotropic diffusivity is given as the trace of the diffusivity tensor which for tetragonal symmetry yields $D = 2 D_{\perp} + D_{\parallel}$.
According to [Table \[tab:migration\_VO\]]{} the process ${\text{$V_c$}}\rightarrow{\text{$V_{ab}$}}[001]$ has the lowest barrier for all charge states (also compare the barriers between the four leftmost minima in [Fig. \[fig:energy\_surface\_Cu\]]{}). Therefore, the isotropic defect diffusivity is approximately $$\begin{aligned}
D &\approx
\left( 2 a_0^2 + \frac{1}{2} c_0^2 \right)
\nu_0 \exp\left(-\beta\Delta G_{ab-c}^{[001]}\right).\end{aligned}$$ Using the migration barriers for charge state $q=+2$, approximating the attempt frequency by the lowest optical mode at $\Gamma$, $\nu_0\approx 2\,{\text{THz}}$, [@GhoCocWag99] and using the experimental lattice constants one obtains $$\begin{aligned}
D &\approx 7.8 \times 10^{-3}
\exp\left(-0.7\,{\text{eV}}/k_B T\right) {\text{cm}}^2/{\text{s}}.\end{aligned}$$ The activation barrier in this expression is in reasonable agreement with recent diffusion measurements on lead titanate-zirconate (PZT) alloys [@GotHahFle08], in which the migration barrier for oxygen vacancy migration was found to be $0.87\,{\text{eV}}$.
It should be noted that the experiments in Ref. were carried out at low temperatures on samples that contained an extrinsic concentration of oxygen vacancies. As a result, the calculated and measured pre-factors cannot be directly compared since the latter contains the (unknown) concentration of extrinsic vacancies in the samples.
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[^1]: In Ref. an alternative estimate for the energy differences between different defect dipole alignments is given based on dipolar interaction, which leads larger energy difference that are closer to the ones obtained by first-principles calculations. These values were, however, not employed in said reference to actually model aging.
[^2]: Since the direct contribution of the electric field to the energy landscape is very small (see [Sect. \[sect:results\_ac\]]{}), within our model the results for non-oscillating (DC) fields are virtually identical to the situation without any external field.
[^3]: In principle, the solution of [Eq. (\[eq:model\])]{} can be written in terms of the eigenvalues and vectors of ${{\ensuremath\boldsymbol{\mathrm{W}}}^{\mathrm{T}}}-{\ensuremath\boldsymbol{\mathrm{V}}}$. Since the eigenvalues appear in an exponential function, the stability of the solution, which can be tested via [Eq. (\[eq:ctot\])]{} is highly sensitive to their numerical accuracy. In practice, we have therefore resorted to numerical solvers that approach [Eq. (\[eq:model\])]{} directly.
[^4]: Specifically, we used the `ode15s` solver of <span style="font-variant:small-caps;">matlab</span>. [@matlab]
[^5]: Park and Chady [@ParCha98] discuss two different configurations for the $ab$-site vacancy, $V_{ab}^{sw}$ and $V_{ab}^{ud}$, but the former one seems to be always lower in energy and therefore prevails. We are thus left with only two different types vacancies, $V_c$ and $V_{ab}$. Also compare the discussion in Ref. .
[^6]: At finite temperatures a fraction of vacancies will also occupy higher energy configurations. Their number, however, is very small (and indistinguishable from zero on the scale of [Fig. \[fig:Cu\_dc\]]{}(a)), since it is determined by the Boltzmann factor $\exp(-\Delta E/k_B T)$ where $\Delta E$ is the energy difference between the ground state configuration, $M_{{\text{Ti}}}-V_{c1}$, and the configuration in question.
[^7]: The Curie temperatures of PbTiO$_3$ and PbZrO$_3$ are 720 and 460K, respectively, [@NohCerIgl95] and the concentration dependent Curie temperature of PZT is bounded by these values.
|
---
abstract: 'Les deux résultats principaux de cette note sont d’une part que si $V$ est une représentation de $G_{{\mathbf{Q}_p}}$ de dimension $2$ qui est potentiellement trianguline, alors $V$ vérifie au moins une des propriétés suivantes (1) $V$ est trianguline déployée (2) $V$ est une somme de caractères ou une induite (3) $V$ est une représentation de de Rham tordue par un caractère, et d’autre part qu’il existe des représentations de $G_{{\mathbf{Q}_p}}$ de dimension $2$ qui ne sont pas potentiellement triangulines.'
address:
- |
Université de Lyon\
UMPA, ENS de Lyon\
46 allée d’Italie\
69007 Lyon\
France
- |
CMLS\
Ecole Polytechnique\
91128 Palaiseau Cedex\
France
author:
- Laurent Berger
- Gaëtan Chenevier
bibliography:
- 'potrig.bib'
date: Décembre 2009
title: Représentations potentiellement triangulines de dimension $2$
---
The two main results of this note are on the one hand that if $V$ is a $2$-dimensional potentially trianguline representation of $G_{{\mathbf{Q}_p}}$ then $V$ satisfies at least one of the following properties (1) $V$ is split trianguline (2) $V$ is a direct sum of characters or an induced representation (3) $V$ is a twist of a de Rham representation, and on the other hand that there exists some $2$-dimensional representations of $G_{{\mathbf{Q}_p}}$ which are not potentially trianguline.
Introduction {#introduction .unnumbered}
============
Dans le cadre de la correspondance de Langlands $p$-adique, Colmez a introduit dans [@CTR] la notion de représentation $p$-adique trianguline et a démontré plusieurs propriétés importantes de ces objets. Ses constructions ont été reprises et généralisées par Nakamura dans [@KN]. La définition de Colmez peut se faire en termes des [$({\varphi},\Gamma)$-modules sur l’anneau de Robba ]{} de Fontaine et Kedlaya ou bien en termes de [$B$-paires ]{}. Dans cette note qui est un complément à [@CTR] et [@KN], nous avons choisi de travailler avec les $B$-paires, qui sont plus commodes par certains aspects. A partir de maintenant, $K$ est une extension finie de ${\mathbf{Q}_p}$ et $G_K = {\operatorname{Gal}}({\overline{\mathbf{Q}}_p}/K)$.
Les $B$-paires sont des objets introduits dans [@LB8] qui généralisent les représentations $p$-adiques, et la catégorie des $B$-paires est alors équivalente à celle des $({\varphi},\Gamma)$-modules sur l’anneau de Robba. Pour définir les $B$-paires, on utilise les anneaux ${\mathbf{B}_{\mathrm{dR}}}^+$, ${\mathbf{B}_{\mathrm{dR}}}$ et ${\mathbf{B}_{\mathrm{e}}}={\mathbf{B}_{\mathrm{cris}}}^{{\varphi}=1}$ de Fontaine. Notons que ces trois anneaux sont filtrés et que leurs gradués respectifs sont ${\mathbf{C}_p}[t]$, ${\mathbf{C}_p}[t,t^{-1}]$ et $\{ P \in {\mathbf{C}_p}[t^{-1}]$ tels que $P(0) \in {\mathbf{Q}_p}\}$.
Si $D$ est un ${\varphi}$-module filtré qui provient de la cohomologie d’un schéma $X$ propre et lisse sur ${\mathcal{O}}_K$, alors le ${\varphi}$-module sous-jacent ne dépend que de la fibre spéciale de $X$ (c’en est la cohomologie cristalline) alors que la filtration ne dépend que de la fibre générique (c’est la filtration de Hodge de la cohomologie de de Rham, dans laquelle se plonge la cohomologie cristalline). Si $V=V_{\mathrm{cris}}(D)$, alors on voit que ${\mathbf{B}_{\mathrm{e}}}\otimes_{{\mathbf{Q}_p}} V = ({\mathbf{B}_{\mathrm{cris}}}\otimes_{K_0} D)^{{\varphi}=1}$ ne dépend que de la structure de ${\varphi}$-module de $D$ et de plus, les ${\varphi}$-modules $D_1$ et $D_2$ sont isomorphes si et seulement si les ${\mathbf{B}_{\mathrm{e}}}$-représentations ${\mathbf{B}_{\mathrm{e}}}\otimes_{{\mathbf{Q}_p}} V_1$ et ${\mathbf{B}_{\mathrm{e}}}\otimes_{{\mathbf{Q}_p}} V_2$ le sont. Par ailleurs, ${\mathbf{B}_{\mathrm{dR}}}^+ \otimes_{{\mathbf{Q}_p}} V = \mathrm{Fil}^0 ({\mathbf{B}_{\mathrm{dR}}}\otimes_{K_0} D)$ et les modules filtrés $K \otimes_{K_0} D_1$ et $K \otimes_{K_0} D_2$ sont isomorphes si et seulement si les ${\mathbf{B}_{\mathrm{dR}}}^+$-représentations ${\mathbf{B}_{\mathrm{dR}}}^+ \otimes_{{\mathbf{Q}_p}} V_1$ et ${\mathbf{B}_{\mathrm{dR}}}^+ \otimes_{{\mathbf{Q}_p}} V_2$ le sont.
L’idée sous-jacente à la construction des $B$-paires est d’isoler les phénomènes liés à la [fibre spéciale ]{} et à la [fibre générique ]{} en considérant non pas des représentations $p$-adiques $V$, mais des [$B$-paires ]{} $W=(W_e,W_{dR}^+)$ où $W_e$ est un ${\mathbf{B}_{\mathrm{e}}}$-module libre muni d’une action semi-linéaire et continue de $G_K$ et où $W_{dR}^+$ est un ${\mathbf{B}_{\mathrm{dR}}}^+$-réseau de $W_{dR} = {\mathbf{B}_{\mathrm{dR}}}\otimes_{{\mathbf{B}_{\mathrm{e}}}} W_e$ stable par $G_K$. Si $V$ est une représentation $p$-adique, alors $W(V)= ({\mathbf{B}_{\mathrm{e}}}\otimes_{{\mathbf{Q}_p}} V, {\mathbf{B}_{\mathrm{dR}}}^+ \otimes_{{\mathbf{Q}_p}} V)$ est une $B$-paire et cette construction permet de plonger la catégorie des représentations $p$-adiques dans celle strictement plus grande des $B$-paires.
La définition de Colmez est alors la suivante : si $V$ est une représentation $E$-linéaire de $G_K$ (ici $E$ est une extension finie de ${\mathbf{Q}_p}$ qui est un corps de coefficients pour les représentations que l’on considère) alors on peut lui associer comme ci-dessus une $B$-paire $E$-linéaire $W(V)$, et on dit que $V$ est trianguline déployée si $W(V)$ est une extension successive de $B$-paires de rang $1$. On dit que $V$ est trianguline s’il existe une extension finie $F$ de $E$ telle que $F \otimes_E V$ est trianguline déployée (notons cependant que les [triangulines ]{} de Colmez correspondent à nos [triangulines déployées ]{}). On dit que $V$ est potentiellement trianguline s’il existe une extension finie $L$ de $K$ telle que $V \vert_{G_L}$ est trianguline. Le premier résultat de cette note (le théorème \[main\]) est le suivant.
[Théorème A]{} Si $V$ est une représentation $E$-linéaire de $G_{{\mathbf{Q}_p}}$ de dimension $2$ qui est potentiellement trianguline, alors $V$ vérifie au moins une des propriétés suivantes :
1. $V$ est trianguline déployée;
2. $V$ est une somme de caractères ou une induite;
3. $V$ est une représentation de de Rham tordue par un caractère.
La démonstration du théorème A se fait en utilisant de la descente galoisienne. Les $B$-paires ont des pentes (les pentes de frobenius des $({\varphi},\Gamma)$-modules correspondants) et des poids (qui généralisent les poids de Hodge-Tate des représentations $p$-adiques). La combinatoire des pentes et des poids est assez rigide et si $V$ est une représentation potentiellement trianguline, soit sa triangulation descend et on est dans le cas (1), soit il y a des symétries supplémentaires suffisantes pour montrer qu’on est dans le cas (2) ou (3).
Les conditions (1), (2) et (3) du théorème A ne sont pas du tout mutuellement exclusives, et en fait pour tout $S \subset \{ 1, 2, 3\}$ non vide, il existe une représentation $V$ qui vérifie exactement $S$. Le cas $S=\emptyset$ revient à la construction d’une représentation $p$-adique qui n’est pas potentiellement trianguline, et dans la suite de cet article, nous montrons que de telles représentations existent.
[Théorème B]{} Il existe des représentations $p$-adiques de dimension $2$ de $G_{{\mathbf{Q}_p}}$ qui ne sont pas potentiellement triangulines.
Nous prouvons en fait un résultat plus fort, dont voici une conséquence.
[Théorème C]{} Soit $R : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2(E)$ une représentation résiduellement absolument irréductible. Il existe une extension finie $F/E$ et une representation continue $$\rho : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2(F \langle t \rangle)$$ telle que $\rho_0 \otimes_E F=R$ et telle que pour tout $t \in \overline{{\mathbf{Z}}}_p$, sauf peut-être pour un ensemble dénombrable d’entre eux, $\rho_t$ n’est pas potentiellement trianguline.
Ce résultat entraîne évidemment le précédent : considérer par exemple la représentation sur le module de Tate d’une courbe elliptique sur ${\mathbf{Q}_p}$ ayant bonne réduction supersingulière, ou encore une représentation induite convenable. La représentation $\rho$ que l’on construit est de polynôme de Sen et déterminant constants. Dans de nombreux cas, notamment dans l’exemple précédent, on peut prendre $F=E$ dans l’énoncé ci-dessus.
Remarquons pour terminer que la démonstration du théorème B n’est pas constructive, et le lecteur pourra chercher avec profit à construire explicitement une représentation qui n’est pas potentiellement trianguline.
La catégorie des $B$-paires {#defs}
===========================
Nous commençons par faire des rappels très succints sur les définitions (données dans [@FPP] par exemple) des divers anneaux que nous utilisons dans cette note. Rappelons que ${\widetilde{\mathbf{E}}^+}= \varprojlim_{x \mapsto x^p} {\mathcal{O}}_{{\mathbf{C}_p}}$ est un anneau de caractéristique $p$, complet pour la valuation ${{\operatorname{val}}_{\mathrm{E}}}$ définie par ${{\operatorname{val}}_{\mathrm{E}}}(x)= {{\operatorname{val}}_p}(x^{(0)})$ et qui contient un élément ${\varepsilon}$ tel que ${\varepsilon}^{(n)}$ est une racine primitive $p^n$-ième de l’unité. On fixe un tel ${\varepsilon}$ dans toute cette note. L’anneau ${\widetilde{\mathbf{E}}}= {\widetilde{\mathbf{E}}^+}[1/({\varepsilon}-1)]$ est alors un corps qui contient comme sous-corps dense la clôture algébrique de ${\mathbf{F}_p}(\!({\varepsilon}-1)\!)$. On pose ${\widetilde{\mathbf{A}}^+}=W({\widetilde{\mathbf{E}}^+})$ et ${\widetilde{\mathbf{B}}^+}={\widetilde{\mathbf{A}}^+}[1/p]$. L’application $\theta : {\widetilde{\mathbf{B}}^+}\to {\mathbf{C}_p}$ qui à $\sum_{k \gg -\infty} p^k [x_k]$ associe $\sum_{k \gg -\infty} p^k x_k^{(0)}$ est un morphisme d’anneaux surjectif et ${\mathbf{B}_{\mathrm{dR}}}^+$ est le complété de ${\widetilde{\mathbf{B}}^+}$ pour la topologie $\ker(\theta)$-adique, ce qui en fait un espace topologique de Fréchet. On pose $X=[{\varepsilon}]-1 \in {\widetilde{\mathbf{A}}^+}$ et $t=\log(1+X) \in {\mathbf{B}_{\mathrm{dR}}}^+$ et on définit ${\mathbf{B}_{\mathrm{dR}}}$ par ${\mathbf{B}_{\mathrm{dR}}}={\mathbf{B}_{\mathrm{dR}}}^+[1/t]$. Soit ${\widetilde}{p} \in {\widetilde{\mathbf{E}}^+}$ un élément tel que ${\widetilde}{p}^{(0)}=p$. L’anneau ${\mathbf{B}_{\mathrm{max}}}^+$ est l’ensemble des éléments de ${\mathbf{B}_{\mathrm{dR}}}^+$ qui peuvent s’écrire sous la forme $\sum_{n {\geqslant}0} b_n ([{\widetilde}{p}]/p)^n$ où $b_n \in {\widetilde{\mathbf{B}}^+}$ et $b_n \to 0$ quand $n \to \infty$ ce qui en fait un sous-anneau de ${\mathbf{B}_{\mathrm{dR}}}^+$ muni en plus d’un frobenius ${\varphi}$ qui est injectif, mais pas surjectif. On pose ${\mathbf{B}_{\mathrm{max}}}={\mathbf{B}_{\mathrm{max}}}^+[1/t]$ et ${\mathbf{B}_{\mathrm{e}}}={\mathbf{B}_{\mathrm{max}}}^{{\varphi}=1}$. Rappelons que les anneaux ${\mathbf{B}_{\mathrm{max}}}$ et ${\mathbf{B}_{\mathrm{dR}}}$ sont reliés, en plus de l’inclusion ${\mathbf{B}_{\mathrm{max}}}\subset {\mathbf{B}_{\mathrm{dR}}}$, par la suite exacte fondamentale : $0 \to {\mathbf{Q}_p}\to {\mathbf{B}_{\mathrm{e}}}\to {\mathbf{B}_{\mathrm{dR}}}/{\mathbf{B}_{\mathrm{dR}}}^+ \to 0$.
L’anneau ${\mathbf{B}_{\mathrm{dR}}}^+$ contient ${\overline{\mathbf{Q}}_p}$ ce qui fait que si $E$ est une extension finie de ${\mathbf{Q}_p}$, alors $E \otimes {\mathbf{B}_{\mathrm{dR}}}^+ \simeq ({\mathbf{B}_{\mathrm{dR}}}^+)^{[E:{\mathbf{Q}_p}]}$ (dans toute cette note, on écrit $E\otimes -$ plutôt que $E \otimes_{{\mathbf{Q}_p}} -$ pour alléger les notations). En ce qui concerne $E \otimes {\mathbf{B}_{\mathrm{e}}}$, on a le résultat suivant (rappelons qu’un anneau de Bézout est un anneau intègre tel que tout idéal de type fini est principal).
\[bebez\] Si $E$ est une extension finie de ${\mathbf{Q}_p}$, alors l’anneau $E \otimes {\mathbf{B}_{\mathrm{e}}}$ est un anneau de Bézout.
Si $E={\mathbf{Q}_p}$, alors c’est la proposition 1.1.9 de [@LB8] et le cas général fait l’objet du lemme 1.6 de [@KN].
Tous les anneaux construits ci-dessus admettent une action naturelle de $G_{{\mathbf{Q}_p}}$ et donc de $G_K$ si $K \subset{\overline{\mathbf{Q}}_p}$. On fait agir $G_{{\mathbf{Q}_p}}$ trivialement sur $E$ de sorte que $E \otimes {\mathbf{B}_{\mathrm{e}}}$ et $E \otimes {\mathbf{B}_{\mathrm{dR}}}$ sont munis d’une action $E$-linéaire de $G_{{\mathbf{Q}_p}}$.
Une $E \otimes {\mathbf{B}_{\mathrm{e}}}$-représentation de $G_K$ est un $E \otimes {\mathbf{B}_{\mathrm{e}}}$-module libre de rang fini muni d’une action semi-linéaire de $G_K$. Si $W_e$ est une $E \otimes {\mathbf{B}_{\mathrm{e}}}$-représentation de $G_K$, alors on pose $W_{dR} = (E \otimes {\mathbf{B}_{\mathrm{dR}}}) \otimes_{E \otimes {\mathbf{B}_{\mathrm{e}}}} W_e$.
\[defbp\] Une $B^{\otimes E}_{\vert K}$-paire est une paire $W=(W_e,W_{dR}^+)$ où $W_e$ est une $E \otimes {\mathbf{B}_{\mathrm{e}}}$-représentation de $G_K$ et où $W_{dR}^+$ est un $E \otimes {\mathbf{B}_{\mathrm{dR}}}^+$-réseau de $W_{dR}$ stable par $G_K$.
Si $E={\mathbf{Q}_p}$, alors on retrouve la définition du §2 de [@LB8] et dans le cas général, on retrouve les [$E$-$B$-paires de $G_K$ ]{} de [@KN]. Rappelons que l’on note $X \subset W$ si $X_e \subset W_e$ et $X_{dR}^+ \subset W_{dR}^+$ mais que l’on dit que $X$ est un sous-objet strict de $W$ seulement si en plus $X_{dR}^+$ est saturé dans $W_{dR}^+$. Dans ce cas, le quotient $W/X$ est aussi une $B^{\otimes E}_{\vert K}$-paire.
Si $W$ est une $B^{\otimes E}_{\vert K}$-paire, et si $F$ est une extension finie galoisienne de $E$ et $L$ est une extension finie de $K$, alors $F \otimes_E W \vert_{G_L}$ est une $B^{\otimes F}_{\vert L}$-paire, munie en plus d’une action de ${\operatorname{Gal}}(F/E)$ et d’une action de $G_K$ qui étend celle de $G_L$. On a alors le résultat suivant de [descente galoisienne ]{}.
\[galdesc\] Si $W$ est une $B^{\otimes E}_{\vert K}$-paire et si $X \subset F \otimes_E W \vert_{G_L}$ est stable sous les actions de ${\operatorname{Gal}}(F/E)$ et de $G_K$, alors $X^{{\operatorname{Gal}}(F/E)}$ avec l’action induite de $G_K$ est une $B^{\otimes E}_{\vert K}$-paire et $X = F \otimes_E X^{{\operatorname{Gal}}(F/E)} \vert_{G_L}$.
La proposition 2.2.1 de [@LB11] appliquée à $B=F$, $M=X$ et $S=E \otimes {\mathbf{B}_{\mathrm{e}}}$ puis $S=E \otimes {\mathbf{B}_{\mathrm{dR}}}^+$ (si l’on remplace le produit tensoriel complété $\widehat{\otimes}$ par un produit tensoriel simple $\otimes$, ce qui ne change pas la démonstration) implique que $X_e^{{\operatorname{Gal}}(F/E)}$ et $(X_{dR}^+)^{{\operatorname{Gal}}(F/E)}$ sont localement libres de rang fini sur $E \otimes {\mathbf{B}_{\mathrm{e}}}$ et $E \otimes {\mathbf{B}_{\mathrm{dR}}}^+$ et vérifient $X = F \otimes_E X^{{\operatorname{Gal}}(F/E)}$. Ils sont libres de rang fini (par la proposition \[bebez\] pour $X_e^{{\operatorname{Gal}}(F/E)}$ et car ${\mathbf{B}_{\mathrm{dR}}}^+$ est principal et le rang est constant pour $(X_{dR}^+)^{{\operatorname{Gal}}(F/E)}$) et comme $X^{{\operatorname{Gal}}(F/E)}$ est stable sous l’action induite de $G_K$, c’est bien une $B^{\otimes E}_{\vert K}$-paire.
Pentes et poids des $B$-paires {#ppbp}
==============================
Rappelons que par le théorème A de [@LB8] (si $E={\mathbf{Q}_p}$) et par le théorème 1.36 de [@KN] en général, on a une équivalence de catégories entre la catégorie des $B^{\otimes E}_{\vert K}$-paires et celle des $({\varphi},\Gamma_K)$-modules sur l’anneau $E \otimes {\mathbf{B}^{\dagger }_{\mathrm{rig} ,K}}$ où ${\mathbf{B}^{\dagger }_{\mathrm{rig} ,K}}$ est [l’anneau de Robba sur $K$ ]{}.
On sait associer, par exemple selon la méthode de [@KLMT], des pentes aux ${\varphi}$-modules sur l’anneau de Robba; en particulier, on dispose de la notion de ${\varphi}$-module isocline de pente $s$ où $s \in {\mathbf{Q}}$ et on peut donc définir la notion de $B^{\otimes E}_{\vert K}$-paire isocline de pente $s$ via l’équivalence de catégories entre $B^{\otimes E}_{\vert K}$-paires et $({\varphi},\Gamma_K)$-modules sur l’anneau $E \otimes {\mathbf{B}^{\dagger }_{\mathrm{rig} ,K}}$. On a alors le théorème suivant, qui suit de cette équivalence de catégories et du théorème 6.10 de [@KLMT], le théorèmeÊ de filtration par les pentes pour les ${\varphi}$-modules sur l’anneau de Robba.
\[slopefil\] Si $W$ est une $B^{\otimes E}_{\vert K}$-paire, alors il existe une filtration canonique $$0 = W_0 \subset W_1 \subset \cdots \subset W_\ell = W$$ par des sous-$B^{\otimes E}_{\vert K}$-paires, telle que :
1. pour tout $1 {\leqslant}i {\leqslant}\ell$, le quotient $W_i / W_{i-1}$ est isocline;
2. si l’on appelle $s_i$ la pente de $W_i / W_{i-1}$, alors $s_1 < s_2 < \cdots < s_\ell$.
Notons tout de même que $E \otimes {\mathbf{B}^{\dagger }_{\mathrm{rig} ,K}}$ n’est pas nécessairement intègre, et donc qu’il faut un petit argument supplémentaire (on oublie $E$ puis on utilise le fait que la filtration est canonique pour le faire réapparaître) pour obtenir le théorème ci-dessus, qui est alors le théorème 1.32 de [@KN]. Dans cette note, nous n’avons pas besoin de savoir comment calculer les pentes d’une $B$-paire. En plus du théorème \[slopefil\], nous n’utilisons que les deux résultats ci-dessous.
\[etalrep\] Si $V$ est une représentation $E$-linéaire de $G_K$ alors $$W(V)=((E \otimes {\mathbf{B}_{\mathrm{e}}}) \otimes_E V, (E \otimes {\mathbf{B}_{\mathrm{dR}}}^+) \otimes_E V)$$ est une $B^{\otimes E}_{\vert K}$-paire, et le foncteur $V \mapsto W(V)$ donne une équivalence de catégories entre la catégorie des représentations $E$-linéaires de $G_K$ et la catégorie des $B^{\otimes E}_{\vert K}$-paires isoclines de pente nulle.
Si $E={\mathbf{Q}_p}$, alors c’est le théorème 3.2.3 de [@LB8] appliqué à $a/h=0/1$ et le cas $E$-linéaire s’en déduit immédiatement.
Si $W$ est une $B^{\otimes E}_{\vert K}$-paire de rang $1$, alors elle n’a qu’une seule pente, que l’on appelle aussi le degré de $W$, noté $\deg(W)$. Si $W$ est de rang ${\geqslant}1$, alors on pose $\deg(W) = \deg \det(W)$ ce qui fait de $\deg(\cdot)$ une fonction additive sur les suites exactes.
\[degrgun\] Si $X \subset W$ sont deux $B^{\otimes E}_{\vert K}$-paires de rang $1$, alors $\deg(X) {\geqslant}\deg(W)$ et $\deg(X)=\deg(W)$ si et seulement si $X=W$.
Si $E={\mathbf{Q}_p}$, cela suit du corollaire 1.2.8 de [@LB8] et le cas $E$-linéaire est tout à fait semblable.
Les poids d’une $B$-paire $W$ sont une généralisation des poids de Hodge-Tate des représentations $p$-adiques. Rappelons que si $U$ est une ${\mathbf{C}_p}$-représentation de $G_K$, et que si l’on note $H_K = {\operatorname{Gal}}({\overline{\mathbf{Q}}_p}/K(\mu_{p^\infty}))$ et $\Gamma_K = G_K / H_K$, alors la réunion $U^{H_K}_{\mathrm{fini}}$ des sous-$K_\infty$-espaces vectoriels de dimension finie stables par $\Gamma_K$ de $U^{H_K}$ a la propriété que l’application ${\mathbf{C}_p}\otimes_{K_\infty} U^{H_K}_{\mathrm{fini}} \to U$ est un isomorphisme (cf. [@SN80]). L’espace $U^{H_K}_{\mathrm{fini}}$ est muni de l’application $K_\infty$-linéaire $\nabla_U = \log(\gamma) / \log_p(\chi(\gamma))$ avec $\gamma \in \Gamma_K \setminus \{1\}$ suffisamment proche de $1$. Le polynôme caractéristique de $\nabla_U$ est alors à coeffcients dans $K$, et même dans $E \otimes K$ si $U$ est de plus $E$-linéaire. Les racines de ce polynôme sont les poids de Sen de $U$ (le nombre de racines dépend de la décomposition de $E \otimes K$, ce qui explique l’inclusion éventuellement stricte dans le (2) de la proposition \[incsen\] ci-dessous).
Si $W$ est une $B^{\otimes E}_{\vert K}$-paire, alors $W_{dR}^+ / t W_{dR}^+$ est une $E \otimes {\mathbf{C}_p}$-représentation de $G_K$ et on pose ${\mathrm{D}_{\mathrm{Sen}}}(W) = (W_{dR}^+ / t W_{dR}^+)^{H_K}_{\mathrm{fini}}$. Les poids de $W$ sont alors les poids de Sen de $\nabla_W$ agissant sur ${\mathrm{D}_{\mathrm{Sen}}}(W)$. Notons $\operatorname{poids}(W)$ l’ensemble des poids de Sen de $W$ comptés avec multiplicité. La proposition ci-dessous est inspirée des calculs du §3 de [@FIHP]. On écrit “$\operatorname{poids}(X) \subset \operatorname{poids}(W)+{\mathbf{Z}}_{{\geqslant}0}$” pour exprimer le fait que tout poids de $X$ est de la forme $w+a$ où $w$ est un poids de $W$ et $a \in {\mathbf{Z}}_{{\geqslant}0}$.
\[incsen\] Si $X \subset W$ sont deux $B^{\otimes E}_{\vert K}$-paires, alors
1. $\operatorname{poids}(X) \subset \operatorname{poids}(W)+{\mathbf{Z}}_{{\geqslant}0}$;
2. si $X$ est un sous-objet strict de $W$, alors $\operatorname{poids}(W) \supset \operatorname{poids}(W/X) \cup \operatorname{poids}(X)$;
3. si $X$ et $W$ sont de même rang et $\operatorname{poids}(X) = \operatorname{poids}(W)$, alors $X=W$.
Commençons par montrer le (2). Si $X$ est un sous-objet strict de $W$, alors on a une suite exacte $0 \to X_{dR}^+ \to W_{dR}^+ \to (W/X)_{dR}^+ \to 0$ et donc $0 \to {\mathrm{D}_{\mathrm{Sen}}}(X) \to {\mathrm{D}_{\mathrm{Sen}}}(W) \to {\mathrm{D}_{\mathrm{Sen}}}(W/X) \to 0$ ce qui permet de conclure.
Le (2) implique que pour montrer le (1), on peut se ramener au cas où $X$ et $W$ sont de même rang. Notons $t W$ la $B$-paire $tW=(W_e,tW_{dR}^+)$ de sorte que $\operatorname{poids}(tW) = \operatorname{poids}(W) + 1$. Comme deux ${\mathbf{B}_{\mathrm{dR}}}^+$-réseaux sont commensurables, il existe $h {\geqslant}0$ tel que $t^h W \subset X$ et en considérant la suite d’inclusions $$X = X + t^h W \subset X + t^{h-1} W \subset \cdots \subset X + t W \subset X + W = W,$$ on voit que pour montrer le (1), on peut en plus supposer que $tW \subset X \subset W$. Dans ce cas, on a deux suites exactes $$\begin{gathered}
0 \to W_{dR}^+ / t W_{dR}^+ \to W_{dR}^+/X_{dR}^+ \\Ê0 \to t (W_{dR}^+ / X_{dR}^+) \to X_{dR}^+ / t X_{dR}^+ \to W_{dR}^+ / t W_{dR}^+\end{gathered}$$ qui montrent que les poids de $X$ sont dans $\operatorname{poids}(W) \cup \operatorname{poids}(W)+1$.
Enfin pour montrer le (3), on voit que si $\operatorname{poids}(X) = \operatorname{poids}(W)$, alors on a égalité à chaque étape ci-dessus et que cela implique $W=X$. On peut aussi se ramener au cas de rang $1$ en prenant le déterminant.
Si $W$ est une $B^{\otimes E}_{\vert K}$-paire, alors $W_{dR}$ est une ${\mathbf{B}_{\mathrm{dR}}}$-représentation de $G_K$ et ces objets sont étudiés dans le §3 de [@FIHP]. Si $W$ est une $B^{\otimes E}_{\vert K}$-paire, alors on dit qu’elle est de de Rham si la ${\mathbf{B}_{\mathrm{dR}}}$-représentation $W_{dR}$ est triviale. Remarquons que si $V$ est une représentation $E$-linéaire de $G_K$ alors $V$ est de de Rham si et seulement si $W(V)$ est de de Rham.
\[isdr\] Si $W$ est une $B^{\otimes E}_{\vert K}$-paire à poids entiers et si $X \subset W$ est une $B^{\otimes E}_{\vert K}$-paire de rang $1$, alors $X$ est de de Rham.
Comme ${\overline{\mathbf{Q}}_p}\subset {\mathbf{B}_{\mathrm{dR}}}$, on a une décomposition $E \otimes {\mathbf{B}_{\mathrm{dR}}}= {\mathbf{B}_{\mathrm{dR}}}^{[E:{\mathbf{Q}_p}]}$ qui est compatible à l’action naturelle de $G_L$ sur les deux membres si $L$ est une extension de ${\mathbf{Q}_p}$ qui contient $E$. Si l’on choisit une extension finie $L$ de ${\mathbf{Q}_p}$ qui contient $K$ et $E$, alors on en déduit une décomposition $(X \vert_{G_L})_{dR} = \oplus_{i=1}^{[E:{\mathbf{Q}_p}]} X_{dR}^{(i)}$. Chaque $X_{dR}^{(i)}$ est une ${\mathbf{B}_{\mathrm{dR}}}$-représentation de dimension $1$ dont le poids appartient à ${\mathbf{Z}}$ et qui est donc triviale par les résultats du §3.7 de [@FIHP]. C’est donc que $X \vert_{G_L}$ est de de Rham et donc $X$ aussi.
Représentations potentiellement triangulines {#potrig}
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Si $V$ est une représentation $E$-linéaire de $G_K$, alors on peut lui associer par la proposition \[etalrep\] une $B^{\otimes E}_{\vert K}$-paire $W(V)$, et on dit que $V$ est trianguline déployée si $W(V)$ est une extension successive de $B^{\otimes E}_{\vert K}$-paires de rang $1$. On dit que $V$ est trianguline s’il existe une extension finie $F$ de $E$ telle que $F \otimes_E V$ est trianguline déployée. Etant donné l’équivalence de catégories entre $B^{\otimes E}_{\vert K}$-paires et $({\varphi},\Gamma_K)$-modules sur $E \otimes {\mathbf{B}^{\dagger }_{\mathrm{rig} ,K}}$ cette définition est compatible avec les définitions 4.1 et 3.4 de [@CTR] (à ceci près que les [triangulines ]{} de Colmez correspondent à nos [triangulines déployées ]{}). On dit que $V$ est potentiellement trianguline s’il existe une extension finie $L$ de $K$ telle que $V \vert_{G_L}$ est trianguline.
\[main\] Si $V$ est une représentation $E$-linéaire de $G_{{\mathbf{Q}_p}}$ de dimension $2$ qui est potentiellement trianguline, alors $V$ vérifie au moins une des propriétés suivantes :
1. $V$ est trianguline déployée;
2. $V$ est une somme de caractères ou une induite;
3. $V$ est une représentation de de Rham tordue par un caractère.
Soit $W=W(V)$ la $B^{\otimes E}_{\vert {\mathbf{Q}_p}}$-paire isocline de pente nulle associée à $V$ comme dans la proposition \[etalrep\]. Si $V$ est potentiellement trianguline, alors il existe une extension finie $F$ de $E$, et une extension finie $K$ de ${\mathbf{Q}_p}$, que l’on peut supposer toutes les deux galoisiennes, telles que l’on puisse écrire $$0 \to X \to F \otimes_E W \vert_{G_K} \to Y \to 0$$ avec $X$ et $Y$ deux $B^{\otimes F}_{\vert K}$-paires de rang $1$. Si $g \in {\operatorname{Gal}}(F/E)$ ou bien si $g \in G_{{\mathbf{Q}_p}}$, alors $g(X)$ et $g(Y)$ sont aussi deux $B^{\otimes F}_{\vert K}$-paires de rang $1$ et on a $$0 \to g(X) \to F \otimes_E W \vert_{G_K} \to g(Y) \to 0.$$ Si $g(X)=X$ quel que soit $g \in {\operatorname{Gal}}(F/E)$ et quel que soit $g \in G_{{\mathbf{Q}_p}}$, alors la proposition \[galdesc\] montre que $X$ provient d’une $B^{\otimes E}_{\vert {\mathbf{Q}_p}}$-paire ce qui fait qu’on est dans le cas (1).
Le reste de la démonstration est donc consacré à examiner le cas où il existerait un $g \in {\operatorname{Gal}}(F/E)$ ou bien un $g \in G_{{\mathbf{Q}_p}}$ tel que $g(X) \neq X$. Dans ce cas, on a $g(X) \cap X = \{0\}$ et donc $g(X) \hookrightarrow Y$. Comme $W$ et donc $F \otimes_E W \vert_{G_K}$ est de pente nulle, le théorème \[slopefil\] implique que $\deg(X) {\geqslant}0$. Si $\deg(X)=0$, alors $\deg (X \oplus g(X)) = 0$ et la proposition \[degrgun\] appliquée à l’inclusion $\det(X \oplus g(X)) \subset \det(F \otimes_E W \vert_{G_K})$ montre que $F \otimes_E W \vert_{G_K}$ est somme directe de $X$ et $g(X)$. Par l’équivalence de catégories de la proposition \[etalrep\], la représentation $F \otimes_E V \vert_{G_K}$ est somme directe de deux caractères de $G_K$ ce qui fait par le lemme \[supersol\] ci-dessous que l’on est dans le cas (2).
Supposons donc que $\deg(X)>0$. Comme $V$ est une représentation $E$-linéaire de $G_{{\mathbf{Q}_p}}$ les poids de Sen de $V$ et donc de $W$ sont les deux racines $\lambda$ et $\mu \in {\overline{\mathbf{Q}}_p}$ du polynôme caractéristique de $\nabla_W$ qui est à coefficients dans $E$. Les poids de $F \otimes_E W \vert_{G_K}$ sont donc des uplets de $\lambda$ et de $\mu$. Par le (2) de la proposition \[incsen\], les poids de Sen de $X$ et de $Y$ sont aussi des uplets de $\lambda$ et de $\mu$.
Supposons tout d’abord que $\lambda-\mu \notin {\mathbf{Z}}$. Toujours par le (2) de la proposition \[incsen\], le poids de $g(X)$ est aussi un uplet de $\lambda$ et de $\mu$ et comme $g(X) \hookrightarrow Y$ c’est par ailleurs un uplet d’éléments de $\lambda+{\mathbf{Z}}$ et de $\mu+{\mathbf{Z}}$ par le (1) de la proposition \[incsen\]. Si $\lambda-\mu \notin {\mathbf{Z}}$, le poids de $g(X)$ est donc nécessairement égal à celui de $Y$ et par le (3) de la proposition \[incsen\], on a $g(X)=Y$. Comme $\deg g(X) = \deg(X) > 0$ et $\deg(Y)<0$, on a une contradiction et le cas $\lambda-\mu \notin {\mathbf{Z}}$ ne peut pas se produire s’il existe $g$ tel que $g(X) \neq X$ avec $\deg(X)>0$.
Nous sommes donc ramenés à examiner la situation où $\lambda-\mu = a \in {\mathbf{Z}}$ et $\deg(X)>0$. Quitte à tordre $V$ par un caractère de poids $-\mu$, on peut supposer que les poids de Sen de $V$ sont $0$ et $a \in {\mathbf{Z}}$. Nous allons montrer que $V$ est alors de de Rham. Pour cela, remarquons que $X \oplus g(X) \subset F \otimes_E W \vert_{G_K}$ et que bien que cette inclusion ne soit pas une égalité, on a $X_{dR} \oplus g(X_{dR}) = F \otimes_E W_{dR} \vert_{G_K}$ puisque les deux ${\mathbf{B}_{\mathrm{dR}}}$-espaces vectoriels sont de même dimension. Afin de terminer la démonstration, on applique la proposition \[isdr\] qui montre que $X$ et $g(X)$ sont de de Rham et donc $W$ aussi.
\[supersol\] Si $V$ est une représentation $E$-linéaire de $G_{{\mathbf{Q}_p}}$ de dimension $2$ telle qu’il existe une extension finie $K$ de ${\mathbf{Q}_p}$ vérifiant $V \vert_{G_K} = V_1 \oplus V_2$ alors soit $V$ est une somme de caractères, soit $V$ est une induite.
On peut supposer que $K$ est une extension galoisienne de ${\mathbf{Q}_p}$. Si $V_1 \neq V_2$ alors le lemme ne pose pas de difficulté. Si $V_1 = V_2$ alors posons $H=G_K$ et $G=G_{{\mathbf{Q}_p}}$. La théorie de la ramification montre qu’il existe une suite de groupes $$H = H_0 \subset H_1 \subset \cdots \subset H_n = G$$ telle que $H_i$ est distingué dans $G$ et $H_{i+1}/H_i$ est cyclique (en d’autres termes, $G/H$ est hyper-résoluble). Il existe alors $g_1,\hdots,g_n \in G$ tels que $H_i = \langle H_{i-1},g_i \rangle$. Si $V$ est une somme de caractères, alors on a terminé et sinon il existe $1 {\leqslant}m {\leqslant}n$ tel que $V \vert_{H_{i-1}}$ est somme de deux caractères égaux mais pas $V \vert_{H_i}$. La représentation $V \vert_{H_i}$ n’est pas irréductible car $H_i = \langle H_{i-1},g_i \rangle$ et $H_{i-1}$ agit par des homothéties, ce qui fait que quitte à remplacer $K$ par ${\overline{\mathbf{Q}}_p}^{H_i}$, on est ramené au cas $V_1 \neq V_2$.
Parties fines d’un espace analytique {#pfea}
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Afin de montrer les théorèmes B et C de l’introduction, nous avons besoin de faire quelques rappels et compléments sur les espaces rigides analytiques $p$-adiques. Si $\mathcal{X}$ est un tel espace et $x \in \mathcal{X}$ en est un point fermé, nous désignons par $K(x)={\mathcal{O}}_{\mathcal{X},x}/{\mathfrak{M}}_x$ le corps résiduel de $x$, qui est une extension finie de ${\mathbf{Q}_p}$. On rappelle que $\mathcal{X}$ est dit de dimension finie si l’ensemble $\{\dim {\mathcal{O}}_{\mathcal{X},x}\}_{x \in \mathcal{X}}$ est borné, auquel cas $\dim(\mathcal{X})$ est le maximum de cet ensemble. Tous les espaces ci-dessous sont supposés de dimension finie. Nous notons $\mathcal{B}$ la boule unité fermée de dimension $1$ sur ${\mathbf{Q}_p}$ (d’algèbre affinoïde ${\mathbf{Q}_p}\langle t \rangle $).
Un espace rigide est dit *de type dénombrable* s’il admet un recouvrement (non nécessairement admissible) par un nombre dénombrable d’ouverts affinoïdes. Cette propriété est bien entendue stable par réunions disjointes dénombrables quelconques.
\[gcun\] Si $\mathcal{X}$ est un affinoïde, alors il existe une famille dénombrable d’ouverts affinoïdes $\mathcal{U}=(\mathcal{U}_i)_{i {\geqslant}0}$ de $\mathcal{X}$ telle que pour tout $x \in \mathcal{X}$ et tout voisinage ouvert affinoïde $\mathcal{V}$ de $x$, il existe un entier $i$ tel que $x \in \mathcal{U}_i \subset
\mathcal{V}$.
Si $\mathcal{X}$ est la boule unité $\mathcal{B}^n$, de paramètres $t_1,\dots,t_n$, alors nous pouvons prendre pour $\mathcal{U}$ la collection de toutes les sous-boules affinoïdes dont le centre $x$ est tel que les $t_i(x)$ sont algébriques sur ${\mathbf{Q}}$ (rappelons que les rayons possibles sont dénombrables). En général, nous pouvons trouver par définition une immersion fermée $\mathcal{X} \to \mathcal{B}^n$ pour $n$ assez grand, et l’image inverse dans $\mathcal{X}$ de la collection précédente d’ouverts affinoïdes de $\mathcal{B}^n$ a les propriétés requises.
\[gcde\] Si $\mathcal{X}$ est de type dénombrable, alors de tout recouvrement de $\mathcal{X}$ par des ouverts admissibles on peut extraire un recouvrement dénombrable. De plus, tout fermé et tout ouvert de $\mathcal{X}$ est encore de type dénombrable.
Pour le premier point, $\mathcal{X}$ étant de type dénombrable on peut le supposer affinoïde, auquel cas cela découle du lemme précédent. La seconde assertion est évidente pour un fermé, et dans le cas d’un ouvert elle se ramène à voir qu’un ouvert d’un affinoïde est de type dénombrable, ce qui découle encore du lemme \[gcun\] ci-dessus.
\[gctr\] Si $\mathcal{X}$ est un espace rigide, une partie $\mathcal{A} \subset \mathcal{X}$ sera dite *fine* s’il existe un espace rigide $\mathcal{Y}$ de type dénombrable, ainsi qu’un morphisme analytique $f : \mathcal{Y} \to \mathcal{X}$, tels que $\dim(\mathcal{Y}) < \dim(\mathcal{X})$ et $\mathcal{A} \subset f(\mathcal{Y})$.
Par exemple, si $\dim(\mathcal{X})=1$, ses parties fines sont ses parties dénombrables. Le corps ${\mathbf{Q}_p}$ étant indénombrable il est bien connu qu’une telle partie est propre. Nous allons maintenant vérifier que ce résultat s’étend en toute dimension.
\[gcqu\] Soient $\mathcal{X}$ un espace analytique et $\mathcal{A} \subset \mathcal{X}$ une partie fine.
1. Si $\mathcal{U} \subset \mathcal{X}$ est un ouvert admissible de dimension $\dim(\mathcal{X})$, alors $\mathcal{A} \cap \mathcal{U}$ est une partie fine de $\mathcal{U}$.
2. Si $\nu :\mathcal{X}' \rightarrow \mathcal{X}$ est un morphisme fini tel que $\dim (\mathcal{X}') =\dim (\mathcal{X})$, alors $\nu^{-1}(A)$ est une partie fine de $\mathcal{X}'$. Cela vaut en pariculier si $\nu$ est la normalisation de $\mathcal{X}$.
3. Si $\mathcal{X}$ est irréductible et si $\{\mathcal{X}_\lambda \}_{\lambda \in \Lambda}$ est un ensemble indénombrable de fermés de $\mathcal{X}$ irréductibles, deux à deux distincts, et de dimension $\dim(\mathcal{X})-1$, alors hors d’un ensemble dénombrable de $\lambda \in \Lambda$, la partie $\mathcal{A} \cap \mathcal{X}_\lambda$ est une partie fine de $\mathcal{X}_\lambda$.
Le (1) découle de la définition et de ce qu’un ouvert d’un espace de type dénombrable l’est encore. Pour le (2), écrivons $\mathcal{A} \subset f(\mathcal{Y})$ avec $\mathcal{Y}$ de type dénombrable et de dimension $< \dim (\mathcal{X})$ ; l’ensemble $\nu^{-1}(\mathcal{A})$ est inclus dans l’image du morphisme naturel $\mathcal{Y} \times_{\mathcal{X}}{\mathcal{X}}' \rightarrow \mathcal{X}'$. Cela permet de conclure car l’espace $\mathcal{Y} \times_{\mathcal{X}}\mathcal{X}'$ est de type dénombrable, étant fini sur $\mathcal{Y}$ qui a cette propriété, et de dimension ${\leqslant}\dim (\mathcal{Y} )< \dim (\mathcal{X}')$ pour la même raison.
Vérifions à présent le (3). On peut supposer que $\mathcal{A}=f(\mathcal{Y})$ avec $\dim(\mathcal{Y}){\leqslant}n-1$ où $n=\dim (\mathcal{X})$. Soit $\Lambda' \subset \Lambda$ le sous-ensemble des $\lambda$ tels que $\mathcal{Y}$ ait une composante irréductible $\mathcal{T}$ avec $f(\mathcal{T})$ Zariski-dense dans $\mathcal{X}_\lambda$. Comme $\mathcal{Y}$ est de type dénombrable, il en va de même de sa normalisation, de sorte que $\mathcal{Y}$ n’a qu’un nombre dénombrable de composantes irréductibles, et donc $\Lambda'$ est dénombrable.
Posons $\mathcal{A}_\lambda = \mathcal{A} \cap \mathcal{X}_\lambda$ et considérons $\lambda \in \Lambda$ tel que $\mathcal{A}_\lambda$ n’est pas une partie fine de $\mathcal{X}_\lambda$. Nous allons montrer que $\lambda \in \Lambda'$. L’espace $\mathcal{Y}_\lambda=f^{-1}(\mathcal{X}_\lambda)$ est un fermé de $\mathcal{Y}$ et en particulier il est de type dénombrable. Comme $\mathcal{A}_\lambda \subset f(\mathcal{Y}_\lambda)$ n’est pas une partie fine de $\mathcal{X}_\lambda$, l’espace $\mathcal{Y}_\lambda$ est de dimension ${\geqslant}n-1$. Il vient que $\dim(\mathcal{Y}_\lambda)= \dim (\mathcal{Y})=n-1$ car $\dim (\mathcal{Y}){\leqslant}n-1$. La décomposition en composantes irréductibles de la nilréduction de $\mathcal{Y}_\lambda$ est donc de la forme $\mathcal{T}\cup \mathcal{T}'$ où $\mathcal{T}$ est une réunion non vide de composantes irréductibles $\mathcal{T}_i$ de $\mathcal{Y}$ et $\dim(\mathcal{T}') < n-1$. Si pour chaque $i$, l’adhérence Zariski $\mathcal{Z}_i$ de $f(\mathcal{T}_i)$ dans $\mathcal{X}_\lambda$ est stricte, donc de dimension $< \dim \mathcal{X}_\lambda$ par irréductibilité de $\mathcal{X}_\lambda$, alors $\mathcal{A}_\lambda$ est inclus dans la partie fine $f(\mathcal{T}') \cup (\cup_i \mathcal{Z}_i)$ de $\mathcal{X}_\lambda$. On en déduit que l’un des $f(\mathcal{T}_i)$ est Zariski-dense dans $\mathcal{X}_\lambda$, et donc que $\lambda \in \Lambda'$.
Si $K$ est une extension finie de ${\mathbf{Q}_p}$, nous entendons par *$K$-boule* de dimension $r$ l’affinoïde $\mathcal{B}^r_K$ sur ${\mathbf{Q}_p}$ d’algèbre $K \langle t_1,t_2,\hdots,t_r\rangle$. Si $\mathcal{X}$ est un affinoïde et si $x \in \mathcal{X}$ en est un point régulier, rappelons qu’un résultat classique dû à Kiehl [@K67 Thm. 1.18] assure l’existence d’un voisinage ouvert affinoïde $\mathcal{U}$ de $x$ dans $\mathcal{X}$ qui est isomorphe à une $K(x)$-boule[^1].
\[gcci\] Si $\mathcal{X}$ est un espace analytique de dimension $>0$, alors une partie fine de $\mathcal{X}$ en est une partie stricte.
Plus précisément, soit $\mathcal{A} \subset \mathcal{X}$ une partie fine et soit $x \in \mathcal{X}$ tel que $\dim {\mathcal{O}}_{\mathcal{X},x}=\dim (\mathcal{X})>0$. Si $x$ est régulier, alors on peut trouver un morphisme analytique $$\iota : \mathcal{B}^1_{K(x)} \longrightarrow \mathcal{X}$$ tel que $\iota(0)=x$, tel que $\iota^{-1}(\mathcal{A})$ est dénombrable, et qui est une immersion fermée vers un voisinage affinoïde de $x$. Si $x$ n’est pas régulier, on peut encore trouver un $\iota$ comme ci-dessus satisfaisant les deux première conditions, si l’on s’autorise à remplacer $K(x)$ par une extension finie.
Soit $\mathcal{A}=f(\mathcal{Y}) \subset \mathcal{X}$ une partie fine et $x \in \mathcal{X}$ de dimension $\dim X >0$. Démontrons tout d’abord l’assertion concernant le cas où $x$ est régulier. D’après le (1) du lemme \[gcqu\] et le résultat de Kiehl, on peut supposer que $\mathcal{X}$ est une $K(x)$-boule de dimension $\dim (\mathcal{X})=n>0$. Si $n=1$ alors $\mathcal{A}$ est dénombrable et le résultat est évident. Sinon on procède par récurrence sur $n$. On choisit une famille indénombrable de sous-$K(x)$-boules fermées centrées en $x$ et de dimension $n-1$ (par exemple $t_1=\lambda t_2$ pour $\lambda \in \mathbf{Z}_p^\times$), et on conclut par le (3) du lemme \[gcqu\].
La première assertion de la proposition s’en déduit car on peut supposer que $\mathcal{X}$ est réduit, auquel cas son lieu régulier est un ouvert Zariski et Zariski-dense, donc contient un point de dimension $\dim \mathcal{X}$.
Vérifions le dernier point. Quitte à remplacer $\mathcal{X}$ par un ouvert affinoïde de dimension $\dim(\mathcal{X})$, et d’après le (1) du lemme \[gcqu\], on peut supposer que $\mathcal{X}$ est affinoïde contenant $x$, puis que $\mathcal{X}$ est normal et connexe d’après le (2) du même lemme. Par normalisation de Noether-Tate, on peut donc trouver un morphisme fini et surjectif $\pi: \mathcal{X}\rightarrow \mathcal{B}^n$ avec $n=\dim(\mathcal{X})$. Par conséquent, $\pi(\mathcal{A})$ est une partie fine de $\mathcal{B}^n$. Par l’argument précédent, il existe une immersion fermée $\mathcal{B}^1_{K(\pi(x))} \rightarrow \mathcal{B}^n_{K(\pi(x))}$ ne rencontrant $\pi(\mathcal{A})$ qu’en un sous-ensemble dénombrable. L’image inverse $\mathcal{C}$ de cette boule dans $\mathcal{X} \times K(\pi(x))$ est un fermé d’équi-dimension $1$ contenant $x$ et ne recontrant $\mathcal{A}$ qu’en un sous-ensemble dénombrable. Quitte à normaliser $\mathcal{C}$, on peut finalement supposer que $\mathcal{X}$ est régulier de dimension $1$, et on conclut encore par le résultat de Kiehl.
Représentations non potentiellement triangulines
================================================
Rappel sur les espaces de déformations
--------------------------------------
Soit $q$ une puissance de $p$, ${\mathbf{F}_q}$ le corps fini à $q$ éléments, ${\mathbf{Q}_q}$ l’extension non ramifiée de ${\mathbf{Q}_p}$ de corps résiduel ${\mathbf{F}_q}$ et ${\mathbf{Z}_q}$ l’anneau des entiers de ${\mathbf{Q}_q}$. Soit $$r : G_{{\mathbf{Q}_p}}
\to {\mathrm{GL}}_2({\mathbf{F}_q})$$ une représentation continue et absolument irréductible. D’après un résultat classique de Mazur (cf. [@MDG]), le foncteur des déformations de $r$ aux ${\mathbf{Z}_q}$-algèbres locales finies de corps résiduel ${\mathbf{F}_q}$ est pro-représentable par une ${\mathbf{Z}_q}$-algèbre locale noethérienne complète $R(r)$ de corps résiduel ${\mathbf{F}_q}$. On désigne par $\mathcal{X}(r)$ l’espace analytique $p$-adique associé par Berthelot à $R(r)[1/p]$. D’après un théorème de Tate, si $\delta=\dim_{{\mathbf{F}_q}} {\rm Hom}_{G_{{\mathbf{Q}_p}}}(r,r(1))$, alors $\dim_{{\mathbf{F}_q}} H^2(G_{{\mathbf{Q}_p}},{\rm ad }(r))=\delta$ et $\dim_{{\mathbf{F}_q}}H^1(G_{{\mathbf{Q}_p}},{\rm ad}(r))=5+\delta$. Lorsque $r \not \simeq r(1)$, ce qui est par exemple toujours satisfait si $p>3$, il vient que $R(r) \simeq {\mathbf{Z}_q}{[\![ t_0,t_1,\hdots,t_4 ]\!]}$, de sorte que $\mathcal{X}(r)$ est isomorphe à la boule unité ouverte de dimension $5$ sur ${\mathbf{Q}_q}$. Dans tous les cas, comme on le verra ci-dessous, $\mathcal{X}(r)$ est régulier de dimension $5$.[^2]
Le ${\mathbf{Q}_q}$-espace analytique $\mathcal{X}(r)$ jouit d’une propriété universelle que nous rappelons à présent. Soit $\mathcal{Y}$ un ${\mathbf{Q}_q}$-affinoïde et $\rho : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2({\mathcal{O}}(\mathcal{Y}))$ une représentation continue. Pour $y \in \mathcal{Y}$, on note $\rho_y : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2(K(y))$ l’évaluation de $\rho$ en $y$. On note aussi $k_y$ le corps résiduel de $K(y)$, qui est alors muni d’un morphisme naturel ${\mathbf{F}_q}\to k_y$, ainsi que ${\overline{\rho}}_y : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2(k_y)$ la représentation résiduelle semi-simplifiée de $\rho_y$. On dit que $\rho$ est [*résiduellement constante et égale à $r$*]{} si ${\overline{\rho}}_y \simeq r \otimes_{{\mathbf{F}_q}} k_y$ pour tout $y \in \mathcal{Y}$. Si $\mathcal{Y}$ est connexe, il suffit pour cela que cela soit vrai pour un $y \in \mathcal{Y}$. [*Les points de $\mathcal{X}(r)$ dans un ${\mathbf{Q}_q}$-affinoïde $\mathcal{Y}$ sont en bijection canonique avec les classes d’isomorphisme de ${\mathcal{O}}(\mathcal{Y})$-représentations continues $\rho : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2({\mathcal{O}}(\mathcal{Y}))$ qui sont résiduellement constantes et égales à $r$*]{}. Cela vaut en particulier pour les points fermés $x \in \mathcal{X}(r)$, qui sont en bijection avec les classes d’isomorphisme de relèvements $r_x : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2(K(x))$ de $r$. Enfin, cette propriété universelle appliquée aux ${\mathbf{Q}_q}$-algèbres locales artiniennes assure que pour tout $x$ dans $\mathcal{X}(r)$, $\widehat{{\mathcal{O}}}_{\mathcal{X},x}$ est canoniquement isomorphe à la déformation (pro-)universelle de $r_x$ au sens de Mazur. Comme $r_x \not\simeq r_x(1)$ pout tout $x \in \mathcal{X}(r)$, les théorèmes de Tate montrent bien que $\mathcal{X}(r)$ est régulier de dimension $5$.
Points potentiellement triangulins de $\mathcal{X}(r)$
------------------------------------------------------
Etant donnée une propriété de représentations, on dira que $x \in \mathcal{X}(r)$ a cette propriété si la représentation asscoiée $r_x$ a cette propriété.
\[gcse\] L’ensemble des points potentiellement triangulins est une partie fine de $\mathcal{X}(r)$.
Un point technique nous empêche de démontrer cette conjecture, mais nous en montrons ci-dessous une variante à poids de Hodge-Tate-Sen et déterminant fixés. La théorie de Tate-Sen nous fournit un polynôme $P(T) = T^2+aT+b \in {\mathcal{O}}(\mathcal{X}(r))[T]$ tel que pour tout $x \in \mathcal{X}(r)$ l’évaluation (des coefficients) de $P(T)$ en $x$ est le polynôme de Sen de $r_x$. On dispose de plus d’une fonction $\lambda \in R(r)^\times$ qui est l’évaluation en $p \in {\mathbf{Q}_p}^\times=G_{{\mathbf{Q}_p}}^{\rm ab}$ du déterminant de la représentation universelle $G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2(R(r))$.
Fixons une extension finie $E$ de ${\mathbf{Q}_q}$ ainsi que $P_0 \in E[T]$ unitaire de degré $2$ et $\lambda_0 \in {\mathcal{O}}_E^\times$, et considérons $\mathcal{X}_0 \subset \mathcal{X}(r) \times_{{\mathbf{Q}_q}} E$ le fermé défini par les équations $P=P_0$ et $\lambda=\lambda_0$. C’est un espace rigide sur $E$ dont chaque composante irréductible est de dimension ${\geqslant}2$ par le *hauptidealsatz* de Krull.[^3]
\[gchu\] Pour tout $(P_0,\lambda_0)$, l’ensemble des points potentiellement triangulins de chacune des composantes irréductibles de $\mathcal{X}_0$ en est une partie fine.
Ce théorème entraîne le Théorème $C$ de l’introduction par la proposition \[gcci\], ainsi donc que le théorème $B$.
Soit $V$ une $E$-représentation trianguline de dimension $2$ qui est absolument irréductible et $D_{\rm rig}(V)$ le $({\varphi},\Gamma)$-module étale sur l’anneau de Robba associé à $V$. Soient $F$ une extension finie de $E$, ainsi que des caractères $\delta_i : {\mathbf{Q}_p}^\times \to F^\times$ pour $i=1,2$, tels que $D_{\rm rig}(V)\otimes_E F$ soit une extension de $({\mathbf{B}^{\dagger }_{\mathrm{rig} ,{\mathbf{Q}_p}}} \otimes F)(\delta_2)$ par $({\mathbf{B}^{\dagger }_{\mathrm{rig} ,{\mathbf{Q}_p}}}
\otimes
F)(\delta_1)$. On note $x : {\mathbf{Q}_p}^\times \rightarrow F^\times$ l’inclusion et on pose $\chi=x|x|$ (c’est le caractère cyclotomique). Rappelons que si $\delta_1\delta_2^{-1}\in \chi
x^{\mathbf{N}}$ alors quitte à tordre la représentation $V$ par un caractère, soit $V$ est semistable non cristalline, soit $V$ est cristalline telle que le quotient des deux valeurs propres de son frobenius cristallin est $p^{\pm 1}$.
Soit ${\mathcal{O}}$ la déformation pro-universelle de $V$ aux $E$-algèbres artiniennes de corps résiduel $E$, paramétrant les déformations de polynôme de Sen et déterminant constants. Si $\delta_1\delta_2^{-1} \notin \chi
x^{\mathbf{N}}$, alors ${\mathcal{O}}\simeq E{[\![ X,Y ]\!]}$.
Si ${\mathcal{O}}'$ est la déformation pro-universelle de $V$ aux $E$-algèbres artiniennes de corps résiduel $E$, alors on a déjà vu que ${\mathcal{O}}' \simeq E{[\![ X_1,\dots,X_5 ]\!]}$. Soient $T^2+aT+b \in {\mathcal{O}}[T]$ le polynôme de Sen universel, $\lambda \in {\mathcal{O}}^\times$ la valeur en $p$ du déterminant de la déformation universelle, et $(T^2+a_0T+b_0, \lambda_0)\in E[T] \times E^\times$ leurs évaluations en $0$. Par définition, $${\mathcal{O}}={\mathcal{O}}'/(a-a_0,b-b_0,\lambda-\lambda_0).$$ D’après un résultat classique sur les anneaux locaux réguliers, il suffit de voir que les images de $a-a_0$, $b-b_0$ et $\lambda-\lambda_0$ sont linéairement indépendantes sur $E$ dans ${\mathfrak{M}}/{\mathfrak{M}}^2$ où ${\mathfrak{M}}$ est l’idéal maximal de ${\mathcal{O}}'$. Il suffit de le vérifier après extension des scalaires à $F$, et donc de voir qu’il existe des déformations $\widetilde{V}$ de $V
\otimes_E F$ à $F[\varepsilon]/(\varepsilon)^2$ telles que $(a(\widetilde{V})-a_0,b(\widetilde{V})-b_0,
\lambda(\widetilde{V})-\lambda_0)$ soit quelconque dans $(\varepsilon F)^3$. Par l’hypothèse sur $\delta_1\delta_2^{-1}$, cela résulte de [@BCH Prop. 2.3.10 (ii)], qui montre que l’on peut même choisir $\widetilde{V}$ trianguline sur $F[\varepsilon]/(\varepsilon^2)$ au sens de [*loc.cit.*]{}
Ainsi, si la représentation $R$ de l’énoncé du théorème $C$ satisfait les hypothèses du lemme ci-dessus, alors le point correspondant à $R$ dans l’espace $\mathcal{X}_0 \subset \mathcal{X}(\overline{R})$ approprié est un point régulier. Dans ce cas, on peut donc prendre $F=E$ dans l’énoncé de ce théorème, d’après la proposition \[gcci\]. La précision suivant l’énoncé du théorème $C$ de l’introduction s’en déduit. Cela démontre en particulier qu’il existe des représentations non potentiellement triangulines qui sont à coefficients dans ${\mathbf{Q}_p}$.
Preuve du théorème \[gchu\]
---------------------------
Le reste du chapitre est consacré à la démonstration du théorème \[gchu\]. D’après le théorème A de l’introduction, il y a trois types (non exclusifs) de représentations potentiellement triangulines :
- les représentations de de Rham tordues par un caractère;
- les induites d’un caractère d’une extension quadratique de ${\mathbf{Q}_p}$;
- les représentations triangulines (déployées).
Ces représentations vivent dans des familles analytiques naturelles que nous décrivons à présent. Considérons tout d’abord le cas (b), qui est le plus simple. Il ne serait pas difficile d’expliciter la famille universelle (de dimension $3$) formée de toutes les représentations de type (b). C’est cependant inutile pour l’application au théorème \[gchu\] car à déterminant et polynôme de Sen fixés, il n’y a qu’un nombre dénombrable de telles représentations.
En effet, il n’y a d’une part qu’un nombre fini d’extensions quadratiques $K$ de ${\mathbf{Q}_p}$ ($3$ si $p {\geqslant}3$ et $7$ si $p=2$). D’autre part, fixons $K$ une extension finie de ${\mathbf{Q}_p}$ et notons $\Sigma$ l’ensemble des $[K:{\mathbf{Q}_p}]$ plongements de $K$ dans ${\overline{\mathbf{Q}}_p}$. Si $\eta : G^{\rm ab}_K=\widehat{K^\times} \to
{\overline{\mathbf{Q}}_p}^\times$ est un caractère continu, il existe des éléments uniques $a_\sigma \in
{\overline{\mathbf{Q}}_p}$, $\sigma \in \Sigma$, tels que $\eta(x)=\prod_{\sigma \in
\Sigma}\sigma(x)^{a_\sigma}$ pour tout $x$ dans un sous-groupe ouvert assez petit de ${\mathcal{O}}_K^\times$. En particulier, le caractère $\eta$ est d’ordre fini si, et seulement si, $a_\sigma=0$ pour tout $\sigma \in
\Sigma$ ; la donnée des $a_\sigma$ détermine donc $\eta$ à multiplication près par un caractère d’ordre fini (et en particulier, un ensemble dénombrable de caractères). On conclut car le polynôme de Sen de ${\rm Ind}_{G_K}^{G_{{\mathbf{Q}_p}}} \eta$ est exactement $\prod_{\sigma \in
\Sigma}(T-a_\sigma)$.
Intéressons nous maintenant au cas (c). Colmez a défini dans [@CTR] l’espace $\mathcal{S}$ des représentations triangulines (nous nous limitons ici aux représentations irréductibles). Par construction, c’est un espace analytique sur ${\mathbf{Q}_p}$ de type dénombrable, équi-dimensionnel de dimension $4$, et muni d’un morphisme naturel vers l’espace $\mathcal{D}$ des caractères $p$-adiques de $({\mathbf{Q}_p}^\times)^2$ (une réunion disjointe finie de copies de $\mathbb{G}_m^2 \times \mathcal{W}^2$ où $\mathcal{W}$ est la boule unité ouverte de dimension $1$ sur ${\mathbf{Q}_p}$). L’espace défini par Colmez est construit de manière ad-hoc de sorte que ses points fermés paramètrent les représentations triangulines. Contrairement à ce que l’on pourrait penser, il n’existe pas de famille analytique de représentations galoisiennes sur $\mathcal{S}$ qui se spécialise en tout point sur la représentation paramétrée par ce point ; une première obstruction vient de ce que la représentation résiduelle associée n’est pas constante sur $\mathcal{S}$. Cependant, Colmez démontre une forme faible de ce type d’énoncé qui est suffisante pour notre application (mais pas tout à fait pour la conjecture \[gcse\]). Si $x \in \mathcal{S}$ est un point fermé, il lui est associé un point $\delta(x) \in \mathcal{D}$, c’est à dire une paire de caractères continus $\delta_{i,x} : {\mathbf{Q}_p}^\times \to K(x)^\times$ pour $i=1$, $2$. Par construction, le $D_{\rm rig}$ de la $K(x)$-représentation $V_x$ associée à $x$ est une extension non triviale de $({\mathbf{B}^{\dagger }_{\mathrm{rig} ,{\mathbf{Q}_p}}} \otimes K(x))(\delta_{2,x})$ par $({\mathbf{B}^{\dagger }_{\mathrm{rig} ,{\mathbf{Q}_p}}} \otimes K(x))(\delta_{1,x})$. On note $$\mathcal{S}' \subset \mathcal{S}$$ l’ouvert admissible de $\mathcal{S}$ défini par la condition $(\delta_{1,x}/\delta_{2,x})(p) \notin p^{\mathbf{Z}}$. L’application naturelle $\mathcal{S}' \to \mathcal{D}$ est alors une immersion ouverte, et d’après [@CTR Prop. 5.2], pour tout $x \in \mathcal{S}'$ il existe un voisinage affinoïde $\mathcal{B}_x$ de $x$ dans $\mathcal{S}'$ qui est une $K(x)$-boule, ainsi qu’une représentation continue $\rho^{\mathcal{B}_x} : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2({\mathcal{O}}(\mathcal{B}_x))$, dont l’évaluation en chaque $y\in \mathcal{B}_x$ est isomorphe à $V_y$. Comme $\mathcal{S}'$ est de type dénombrable, car $\mathcal{D}$ l’est, on peut extraire du recouvrement des $\mathcal{B}_x$ un recouvrement dénombrable $\{ \mathcal{B}_x \}_{x \in I}$ avec $I \simeq {\mathbf{Z}}_{{\geqslant}0}$. On pose alors $$\mathcal{S}'_{\rm dec} = \coprod_{x \in I} \mathcal{B}_x.$$ Il s’agit d’une “déconnexion” non canonique de $\mathcal{S}'$. On a par ailleurs une immersion ouverte surjective évidente $\mathcal{S}'_{\rm dec} \to \mathcal{S}'$ ainsi qu’une représentation galoisienne naturelle $\rho^{\mathcal{S}'_{\rm dec}} : G_{{\mathbf{Q}_p}} \to {\mathrm{GL}}_2({\mathcal{O}}(\mathcal{S}'_{\rm dec}))$ obtenue à partir des $\rho^{\mathcal{B}_x}$ pour $x \in I$. Notons enfin que via l’inclusion $\mathcal{S}' \subset \mathcal{D}$, l’opération consistant à fixer le polynôme de Sen et le déterminant en $p$ est encore parfaitement transparente, et que les lieux obtenus sont de type dénombrable et équi-dimensionnels de dimension $4-3=1$.
Il nous faut maintenant comprendre $\mathcal{S}\backslash \mathcal{S}'$. Remarquons que lorsque le polynôme de Sen d’une représentation trianguline $x \in \mathcal{S}$ est donné, on connaît les deux caractères ${\delta_{1,x}}_{|{\mathbf{Z}_p}^\times}$ et ${\delta_{2,x}}_{|{\mathbf{Z}_p}^\times}$ à des caractères d’ordre fini près. Si de plus on ne s’interesse qu’à des représentations dans $\mathcal{S}\backslash \mathcal{S}'$ dont le déterminant en $p$ est aussi donné, cela détermine un nombre dénombrable de $\delta_{i,x}(p)$ possibles, et donc de paires de $\delta_{i,x}$ possibles. Pour chacune de ces paires, il y a en fait une et une seule représentation trianguline associée dans $\mathcal{S}\backslash \mathcal{S}'$, à moins que $\delta_{1,x}/\delta_{2,x}$ ne soit de la forme $z \mapsto z^i|z|$ pour un entier $i{\geqslant}1$, d’après [@CTR Thm. 2.9].[^4] Mais dans ce cas, les représentations possibles sont des torsions par un caractère de représentations semi-stables, et sont donc du type (a).
Terminons enfin par le type (a). Le lieu de de Rham de $\mathcal{X}(r)$, s’il est non vide, est un fermé analytique (cf [@LB11]). En particulier, il est de type dénombrable. Sa dimension a été calculée par Kisin dans [@MK Thm. 3.3.8 ] : elle est toujours ${\leqslant}1$. Par torsion, on en déduit que pour tout caractère $\eta : G_{{\mathbf{Q}_p}} \to F^\times$ ($F$ étant une extension finie de $E$), le lieu des $x \in \mathcal{X}(r) \times_E F$ tels que $r_x \otimes \eta$ soit de de Rham est encore un fermé analytique de dimension ${\leqslant}1$.
Pour récapituler, nous avons démontré le résultat suivant:
Il existe un $E$-espace analytique $\mathcal{Y}$, ainsi qu’une représentation continue $\rho : G_{{\mathbf{Q}_p}} \rightarrow {\mathcal{O}}(\mathcal{Y})^\times$, tels que:
- $\mathcal{Y}$ est de type dénombrable et de dimension ${\leqslant}1$,
- le polynôme de Sen de $\rho$ est constant égal à $P_0$, et son déterminant en $p$ est constant égal à $\lambda_0$,
- pour tout point $x \in \mathcal{X}_0$ potentiellement triangulin, il existe $y \in \mathcal{Y}$, et des plongements de $K(x)$ et $K(y)$ dans ${\overline{\mathbf{Q}}_p}$, tels que $\rho_y \otimes_{K(y)} {\overline{\mathbf{Q}}_p}\simeq r_x \otimes_{K(x)} {\overline{\mathbf{Q}}_p}$,
- pour tout $y \in \mathcal{Y}$, la représentation $\rho_y$ est potentiellement trianguline.
En effet, on peut prendre pour $\mathcal{Y}$ la réunion disjointe de : l’ensemble dénombrable des points de type (b), l’ensemble dénombrable des points de type (c) dans $\mathcal{S}\backslash \mathcal{S}'$ qui ne sont pas de type (a), l’espace $\mathcal{S}'_{\rm dec}$, et pour chaque $\eta : G_{{\mathbf{Q}_p}} \to F^\times$ dans un ensemble dénombrable, du lieu de de Rham de $\mathcal{X}(r) \times_E F$.
Le théorème \[gchu\] est maintenant immédiat : soit $\mathcal{Y}(r) \subset \mathcal{Y}$ l’ouvert fermé sur lequel la représentation $\rho$ ci-dessus est résiduellement constante et isomorphe à $r$. Par la propriété universelle de $\mathcal{X}(r)$, et donc de $\mathcal{X}_0$, la représentation $\rho$ correspond à un morphisme analytique $f : \mathcal{Y}(r) \to \mathcal{X}_0$. De plus, $f(\mathcal{Y}(r))$ est exactement l’ensemble des points potentiellement triangulins de $\mathcal{X}_0$. Il vient que $f(\mathcal{Y}(r))$ est fine dans $\mathcal{X}_0$ car chaque composante irréductible de $\mathcal{X}_0$ est de dimension ${\geqslant}2$.
[^1]: Nous remercions Laurent Fargues de nous avoir indiqué cette référence.
[^2]: Supposons $r \simeq r(1)$. Si $p=2$ ou $p=3$, on peut voir qu’en fait $\mathcal{X}(r)$ est la réunion de $p$ boules unités ouvertes sur ${\mathbf{Q}_q}$ ; mieux, on a $R(r) \simeq
\mathbb{Z}_q{[\![ t_0,t_1,\hdots,t_4 ]\!]}[\mu_p]$. En effet, c’est une observation du second auteur quand $p=2$, utilisant le morphisme naturel $\mu_2 \rightarrow G_{\mathbb{Q}_2}^{\rm ab} \rightarrow R(r)^\times$ ; le cas $p=3$ est plus subtil et a été récemment obtenu par G. Böckle.
[^3]: Un argument de torsion permet de voir que ces composantes irréductibles sont de dimension $3$ au plus. Il est probable qu’elles soient toutes de dimension exactement $2$, mais ceci est inutile pour la suite.
[^4]: Notons que l’autre cas a priori exceptionnel, où $\delta_{1,x}/\delta_{2,x}$ est de la forme $z \mapsto z^{-i}$ pour $i{\geqslant}0$, ne se produit pas pour $x \in \mathcal{S}$, car il ne correspond pas à un $(\varphi,\Gamma_{{\mathbf{Q}_p}})$-module étale si $i\neq 0$, et qu’il est réductible si $i=0$.
|
---
abstract: 'In this paper we study the linear series $|L-3p|$ of hyperplane sections with a triple point $p$ on a surface $S$ embedded via a very ample line bundle $L$ for a *general* point $p$. If this linear series does not have the expected dimension we call $(S,L)$ *triple-point defective*. We show that on a triple-point defective surface through a general point every hyperplane section has either a triple component or the surface is rationally ruled and the hyperplane section contains twice a fibre of the ruling.'
address:
- |
Universitá degli Studi di Siena\
Dipartimento di Scienze Matematiche e Informatiche\
Pian dei Mantellini, 44 I – 53100 Siena
- |
Universität Kaiserslautern\
Fachbereich Mathematik\
Erwin-Schrödinger-Straße\
D – 67663 Kaiserslautern
author:
- Luca Chiantini
- Thomas Markwig
date: '30th July, 2009.'
title: 'Triple-Point Defective Surfaces'
---
[^1]
Introduction {#sec:tpd}
============
Throughout this note, $S$ will be a smooth projective surface, $K=K_S$ will denote the canonical class and $L$ will be a divisor class on $S$ such that $L$ is *very ample* and $L-K$ is *ample and base-point-free*.
The classical *interpolation problem* for the pair $(S,L)$ is devoted to the study of the varieties: $$V^{gen}_{m_1,\dots,m_n}=\big\{C\in |L|\;\big|\; p_1,\ldots,p_n\in
S\mbox{ general},\;\operatorname{mult}_{p_i}(C)
\geq m_i\big\}.$$
In a more precise formulation, we start from the incidence variety: $${{\mathcal L}}_{m_1,\dots,m_n}=\{(C,(p_1,\ldots,p_n))\in|L|\times
S^n\;|\;\operatorname{mult}_{p_i}(C)\geq m_i\}$$ together with the canonical projections: $$\label{eq:alphabeta}
\xymatrix{
{{\mathcal L}}_{m_1,\dots,m_n}\ar[r]^\alpha\ar[d]_\beta & S^n\\
|L|={{\mathds P}}(H^0(L)^*)
}$$ As for the map $\alpha$, the fibre over a fixed point $(p_1,\dots,p_n)\in S^n$ is just the linear series $|L-m_1p_1-\dots-m_np_n|$ of effective divisors in $|L|$ having a point of multiplicity at least $m_i$ at $p_i$. These fibres being irreducible, we deduce that if $\alpha$ is *dominant* then ${{\mathcal L}}_{m_1,\dots,m_n}$ has a unique irreducible component, say ${{\mathcal L}}_{m_1,\dots,m_n}^{gen}$, which dominates $S^n$. The closure of its image $$\label{VVV}
V_{m_1,\dots,m_n}:=V_{m_1,\dots,m_n}(S,L):=\overline{\beta({{\mathcal L}}_{m_1,\dots,m_n}^{gen})}$$ under $\beta$ is an irreducible closed subvariety of $|L|$, a *Severi variety* of $(S,L)$.
Imposing a point of multiplicity $m_i$ corresponds to killing $\binom{m_i+1}2$ partial derivatives, so that $$\dim|L-m_1p_1-\dots-m_np_n|\geq
\max\left\{-1,\dim|L|-\sum_{i=1}^n\binom{m_i+1}2\right\},$$ and one expects that the previous inequality is in fact an equality, for the choice of general points $p_1,\dots,p_n\in S$.
When this is not the case, then the pair $(S,L)$, is called *defective* and is endowed with some special structure. By abuse of notation we sometimes call the surface $S$ defective, if $L$ is understood.
The case when $m_i=2$ for all $i$ has been classically considered (and solved) by Terracini, who classified in [@Ter22] double–point defective surfaces. If $S$ is double-point defective, then a general curve $C\in|L-2p_1-\dots-2p_n|$ has a double component passing through each point $p_i$, and the map $\beta$ in Diagram has positive dimensional fibres.
When the multiplicities grow, the situation becomes much more complicated. Even in the case $S={{\mathds P}}^2$, the situation is not understood and there are several, still unproved conjectures on the structure of defective embeddings (see [@Cil01] for an introductory survey).
Let us point out a first difference between the case of multiplicity two and the case of higher multiplicity. It is easy to show that imposing on $|L|$ multiplicity two at [*one*]{} general point always yields three independent conditions, so that $\dim|L-2p|=\dim |L|-3$, the expected dimension. Contrary to this, on some surfaces, it turns out that even imposing [*just one*]{} point of multiplicity $3$, one may obtain a defective behaviour.
\[ex:hirzebruch\] Let $S={{\mathds F}}_e\stackrel{\pi}{\longrightarrow}{{\mathds P}}^1$ be a Hirzebruch surface, $e\geq 0$. We denote by $F$ a fibre of $\pi$ and by $C_0$ the section of $\pi$ of minimal self intersection $C_0^2=-e$ – both of which are smooth rational curves. The general element $C_1$ in the linear system $|C_0+eF|$ will be a section of $\pi$ which does not meet $C_0$ (see e.g. [@Har77], Theorem 2.17).
Consider now the divisor $L=(2+b)\cdot F+C_1=(2+b+e)\cdot
F+C_0$ for some fixed $b\geq 0$. Then for a general $p\in S$ there are curves $D_p\in|bF+C_1-p|$ and there is a unique curve $F_p\in |F-p|$, in particular $p\in F_p\cap D_p$. For each choice of $D_p$ we have $$2F_p+D_p \in|L-3p|.$$ Since $F{{\cdot}}L=1=F{{\cdot}}(L-F)$ we see that every curve in $|L-3p|$ must contain $F_p$ as a double component, i.e. $$|L-3p|=2F_p+|bF+C_1-p|.$$ Moreover, since $bF+C_1$ is base-point free (see [@Har77], Theorem 2.17) we have (see [@FP05], Lemma 35) $$\begin{gathered}
\dim|bF+C_1-p|=\dim|bF+C_1|-1\\=h^0\big({{\mathds P}}^1,{{\mathcal O}}_{{{\mathds P}}^1}(b+e)\big)+
h^0\big({{\mathds P}}^1,{{\mathcal O}}_{{{\mathds P}}^1}(b)\big)-2=2b+e
\end{gathered}$$ and, using the notation from above, $$\dim(V_3)\geq \dim|bF+C_1-p|+2=2b+e+2.$$ However, $$\dim|L|= h^0\big({{\mathds P}}^1,{{\mathcal O}}_{{{\mathds P}}^1}(2+b+e)\big)
+h^0\big({{\mathds P}}^1,{{\mathcal O}}_{{{\mathds P}}^1}(2+b)\big)-1
=2b+e+5,$$ and thus $$\operatorname{expdim}(V_3)=\dim|L|-4=2b+e+1<2b+e+2=\dim(V_3).$$ We say, $({{\mathds F}}_e,L)$ is *triple-point defective*, see Definition \[def:tpd\].
Note, moreover, that $L$ is very ample, as is $L-K_S$ for $b\geq\max\{0,e-3\}$ (see [@Har77], Corollary 2.18), and that $$(L-K)^2=\big((4+2e+b)\cdot F+3\cdot C_0)^22=24+6b+3e>16.$$ $\Box$
It is interesting to observe that, even though, in the previous example, the general element of $|L-3p|$ is non-reduced, still the map $\beta$ of Diagram has finite general fibres, since the general element of $|L-3p|$ has no triple components.
The aim of this note is to investigate the structure of pairs $(S,L)$ for which the linear system $|L-3p|$ for $p\in S$ general has dimension bigger that the expected value $\dim|L|-6$. We will show that, under some assumptions, ruled surfaces are the only case of triple point defective surfaces.
\[def:tpd\] We say that the pair $(S,L)$ is *triple-point defective* or, in classical notation, that *$(S,L)$ satisfies one Laplace equation* if $$\dim|L-3p|>\max\{-1,\dim|L|-6\}=\operatorname{expdim}|L-3p|$$ for $p\in S$ general.
\[L3\] Going back to Diagram , one sees that $(S,L)$ is triple-point defective if and only if either:
- $\dim|L|\leq 5$ and the projection $\alpha:{{\mathcal L}}_3\rightarrow S$ dominates, or
- $\dim|L|>5$ and the general fibre of the map $\alpha$ has dimension at least $\dim|L|-5$.
In particular, $(S,L)$ is triple-point defective if and only if the map $\alpha$ is *dominant* and $$\dim({{\mathcal L}}_3^{gen})>\dim|L|-4.$$
The particular case in which the general fibre of the map $\beta$ in Diagram is positive-dimensional, (i.e. the general member of $V_3$ contains a triple component through $p$) has been studied in [@Cas22], [@FI01], and [@BC05]. We will recall the classification of such surfaces in Theorem \[thm:notfinite\] below. Notice that these surfaces are almost always singular (i.e. $L$ is not very ample), so that they do not appear in the statement of our main theorem, [*where, indeed, we make no assumptions on the dimension of the fibres of $\beta$*]{}.
One of the major subjects in algebraic interpolation theory, namely Segre’s conjecture on defective linear systems *in the plane*, suggests in our situation that, when $(S,L)$ is triple-point defective, then the general element of $|L-3p|$ must be non-reduced, with a double component through $p$.
We will show here, under some assumptions, that this extension of Segre’s conjecture for triple point defectivity holds for a single triple point.
Our main result is:
\[thm:aim1\] Let $L$ be a very ample line bundle on $S$, such that $L-K$ is ample and base-point-free. Assume $(L-K)^2>16$ and $(S,L)$ is triple-point defective.
Then $S$ is ruled in the embedding defined by $L$. Moreover a general curve $C\in|L-3p|$ contains the fibre of the ruling through $p$ as fixed component with multiplicity at least two.
In the paper [@CM07a] we classify all triple-point defective linear systems $L$ on ruled surfaces satisfying the assumptions of Theorem \[thm:aim1\], and it follows from this classification that the linear system $|L-3p|$ will contain the fibre of the ruling through $p$ precisely with multiplicity two as a fixed component. In particular, the map $\beta$ will automatically be generically finite.
Our method is an application of Reider’s analysis of rank $2$ bundles arising from triple points which do not impose independent conditions. Under the assumption that $(L-K)^2>16$, the bundle is Bogomolov unstable, and we show that from the destabilising divisors $A$ and $B=L-K-A$ one gets the multiple fibre. We point out that we obtain in this way a natural geometric construction for the non–reduced divisor which must be part of any defective linear system.
This application of Reider’s construction for the investigation of defective surfaces was introduced by Beltrametti, Francia and Sommese in [@BFS89]. We will refer to [@BFS89] for the first main properties of the destabilising divisors $A$ and $B$ (see Section 4 below).
Then, we will use the assumption“$L-K$ ample and base-point-free” to control curves of low degree on $S$. The freeness of $L-K$ seems unavoidable in the argument of the crucial Lemma 14, which in turn implies that $B$ is fixed part free and determines a ruling on $S$.
Let us finish by pointing out in the following corollary what happens if we apply our result to ${{\mathds P}}^2$ and its blow ups, and notice that, combining results in [@Xu95] and [@Laz97] Corollary 2.6, one can give purely numerical conditions on $r$ and the $m_i$ such that $L-K$ there is ample and base-point-free.
Fix multiplicities $m_1\leq m_2\leq \dots\leq m_n$. Let $H$ denote the class of a line in ${{\mathds P}}^2$ and assume that, for $p_1,\dots,p_n$ general in ${{\mathds P}}^2$, the linear system $M=rH-m_1p_1-\dots-m_np_n$ is defective, i.e. $$\dim|M|>
\max\left\{-1,\binom{r+2}2-\sum_{i=1}^n\binom{m_i+1}2\right\}.$$ Let $\pi:S\longrightarrow{{\mathds P}}^2$ be the blowing up of ${{\mathds P}}^2$ at the points $p_2,\dots,p_n$ and set $L:= r\pi^*H-m_2E_2-\dots-m_nE_n$, where $E_i=\pi^*(p_i)$ is the i-th exceptional divisor. Assume that $L$ is very ample on $S$, of the expected dimension $\binom{r+2}2-\sum_{i=2}^n\binom{m_i+1}2$, and that $L-K$ is ample and base-point-free on $S$, with $(L-K)^2>16$. Assume, finally, $m_1\leq 3$.
Then $m_1=3$ and the general element of $M$ is non-reduced. Moreover $L$ embeds $S$ as a ruled surface.
Just apply the Main Theorem \[thm:aim1\] to the pair $(S,L)$.
The reader can easily check that the previous result is exactly the translation of Segre’s and Harbourne–Hirschowitz’s conjectures on defective linear systems in the plane, for the case of a *minimally* defective system with lower multiplicity $3$. The $(-1)$–curve predicted by the Harbourne–Hirschowitz conjecture, in this situation, is just the pull-back of a line of the ruling. Thus, although in a partial situation, we get new evidence for the conjecture, at least when the [*minimal*]{} multiplicity imposed at the points is $3$.
The paper is organised as follows.
The case where $\beta$ is not generically finite is pointed out in Theorem \[thm:notfinite\] in Section \[sec:triplecomponents\]. In Section \[sec:equimultiple\] we reformulate the problem as an $h^1$-vanishing problem. The Sections \[sec:construction\] to \[sec:ruled\] are devoted to the proof of the main result: in Section \[sec:construction\] we use Serre’s construction and Bogomolov instability in order to show that triple-point defectiveness leads to the existence of very special divisors $A$ and $B$ on our surface; in Section \[sec:zero\] we show that $|B|$ has no fixed component; in Section \[sec:generalcase\] we then list properties of $B$ and we use these in Section \[sec:ruled\] to classify the triple-point defective surfaces.
The authors wish to thank the referee, who pointed out the possibility of weakening one assumption in a preliminary version of the main theorem.
Triple Components {#sec:triplecomponents}
=================
In this section, we consider what happens when, in Diagram , the general fibre of $\beta$ is positive-dimensional, in other words, when the general member of $V_3$ contains a triple component through $p$.
This case has been investigated (and essentially solved) in [@Cas22], and then rephrased in modern language in [@FI01] and [@BC05].
Although not strictly necessary for the sequel, as our arguments do not make any use of the generic finiteness of $\beta$, (and so we will not assume this), for the sake of completeness we recall in this section some example and the classification of pairs $(S,L)$ which are triple-point defective, and such that a general curve $L_p\in|L-3p|$ has a triple component through $p$.
The family ${{\mathcal L}}_3$ of pairs $(L,p)\in |L|\times S$ where $L\in |L-3p|$ has dimension bounded below by $\dim|L|-4$, and in Remark \[L3\] it has been pointed out that $(S,L)$ is triple-point defective exactly when $\alpha$ is dominant and the bound is not attained.
Notice however that $\dim|L|-4$ is [*not*]{} necessarily a bound for the dimension of the subvariety $V_3\subset |L|$, the image of ${{\mathcal L}}_3$ under $\beta$. The following example (exploited in [@LM02]) shows that one may have $\dim(V_3)<\dim|L|-4$ even when $(S,L)$ is *not* triple-point defective.
Let $S$ be the blowing up of ${{\mathds P}}^2$ at $8$ general points $q_1,\dots,q_8$ and $L$ corresponds to the system of curves of degree nine in ${{\mathds P}}^2$, with a triple point at each $q_i$.
$\dim|L|=6$, but for $p\in S$ general, the unique divisor in $|L-3p|$ coincides with the cubic plane curve through $q_1,\dots,q_8,p$, counted three times. As there exists only a (non-linear) one-dimensional family of such divisors in $|L|$, then $\dim(V_3)=1<\dim|L|-4$. On the other hand, these divisors have a triple component, so that the general fibre of $\beta$ has dimension one, hence $\dim({{\mathcal L}}_3)=2=\dim|L|-4$.
The classification of triple-point defective pairs $(S,L)$ for which the map $\beta$ is not generically finite is the following.
\[thm:notfinite\] Suppose that $(S,L)$ is triple-point defective. Then for $p\in S$ general, the general member of $|L-3p|$ contains a triple component through $p$ if and only if $S$ lies in a three dimensional scroll $W$ containing a one dimensional family of planes, and moreover $W$ is developable, i.e. the tangent space to $W$ is constant along the planes.
First, since we assume that $S$ is triple-point defective and embedded in ${{\mathds P}}^r$ via $L$, then the hyperplanes $\pi$ that meet $S$ in a divisor $H=S\cap \pi$ with a triple point at a general $p\in S$, intersect in a ${{\mathds P}}^4$. Thus we may project down $S$ to ${{\mathds P}}^5$ and work with the corresponding surface.
In this setting, through a general $p\in S$ one has only one hyperplane $\pi$ with a triple contact, and $\pi$ has a triple contact with $S$ along the fibre $C$ of $\beta$. Thus $V_3$ is a curve.
If $H', H''$ are two consecutive infinitesimally near points to $H$ on $V_3$, then $C$ also belongs to $H\cap H'\cap H''$. Thus $C$ is a plane curve and $S$ is fibred by a $1$-dimensional family of plane curves. This determines the three dimensional scroll $W$.
The tangent line to $V_3$ determines in $({{\mathds P}}^5)^*$ a pencil of hyperplanes which are tangent to $S$ at any point of $C$, since this is the infinitesimal deformation of a family of hyperplanes with a triple contact along any point of $C$. Thus there is a ${{\mathds P}}^4=H_C$ which is tangent to $S$ along $C$.
Assume that $C$ is not a line. Then $C$ spans a ${{\mathds P}}^2=\pi_C$ fibre of $W$, moreover the tangent space to $W$ at a general point of $C$ is spanned by $\pi_C$ and $T_{S,P}$, hence it is constantly equal to $H_C$. Since $C$ spans $\pi_C$, then it turns out that the tangent space to $W$ is constant at any point of $\pi_C$, i.e. $W$ is developable.
When $C$ is a line, then arguing as above one finds that all the tangent planes to $S$ along $C$ belong to the same ${{\mathds P}}^3$. This is enough to conclude that $S$ sits in some developable $3$-dimensional scroll.
Conversely, if $S$ is contained in the developable scroll $W$, then at a general point $p$, with local coordinates $x,y$, the tangent space $t$ to $W$ at $p$ contains the derivatives $p, p_x, p_y, p_{xx}, p_{xy}$ (here $x$ is the direction of the tangent line to $C$). Thus the ${{\mathds P}}^4$ spanned by $t,p_{yy}$ intersects $S$ in a triple curve along $C$.
The Equimultiplicity Ideal {#sec:equimultiple}
==========================
If $L_p$ is a curve in $|L-3p|$ we denote by $f_p\in{{\mathds C}}\{x_p,y_p\}$ an equation of $L_p$ in local coordinates $x_p$ and $y_p$ at $p$. If $\operatorname{mult}_p(L_p)=3$, the ideal sheaf ${{\mathcal J}}_{Z_p}$ whose stalk at $p$ is the equimultiplicity ideal $${{\mathcal J}}_{Z_p,p}=\left\langle\frac{\partial
f_p}{\partial x_p},\frac{\partial f_p}{\partial
y_p}\right\rangle + \langle x_p,y_p\rangle^3$$ of $f_p$ defines a zero-dimensional scheme $Z_p=Z_p(L_p)$ concentrated at $p$, and the tangent space $T_{(L_p,p)}({{{\mathcal L}}_3})$ of ${{\mathcal L}}_3$ at $(L_p,p)$ satisfies (see [@Mar06] Example 10) $$T_{(L_p,p)}({{{\mathcal L}}_3})\cong \big(H^0\big(S,{{\mathcal J}}_{Z_p}(L_p)\big)/H^0(S,{{\mathcal O}}_S)\big)\oplus{{\mathcal K}},$$ where ${{\mathcal K}}$ is zero unless $L_p$ is unitangential at $p$, in which case ${{\mathcal K}}$ is a one-dimensional vector space.
In particular, ${{\mathcal L}}_3$ is smooth at $(L_p,p)$ of the expected dimension (see [@Mar06] Proposition 11) $$\operatorname{expdim}({{\mathcal L}}_3)=\dim|L|-4$$ as soon as $$h^1\big({S},{{\mathcal J}}_{Z_p}(L)\big)=0.$$ We thus have the following proposition.
\[prop:h1vanishing\] Suppose that $\alpha$ is surjective, then $(S,L)$ is not triple-point defective if $$h^1\big({S},{{\mathcal J}}_{Z_p}(L)\big)=0$$ for general $p\in S$ and $L_p\in|L|$ with $\operatorname{mult}_p(L_p)=3$.
Moreover, if $L$ is non-special, i.e. if $h^1(S,L)=0$, the above $h^1$-vanishing is also necessary for the non-triple-point-defectiveness of $(S,L)$.
Note that by Kodaira vanishing $L$ is non-special whenever $L-K$ is ample.
The Basic Construction {#sec:construction}
======================
Then by Serre’s construction for a subscheme $Z'_p\subseteq Z_p$ with ideal sheaf ${{\mathcal J}}_p={{\mathcal J}}_{Z_p'}$ of minimal length such that $h^1\big(S,{{\mathcal J}}_p(L)\big)\not=0$ there is a rank two bundle ${{\mathcal E}}_p$ on $S$ and a section $s\in H^0(S,{{\mathcal E}}_p)$ whose $0$-locus is $Z'_p$, giving the exact sequence $$\label{eq:vectorbundle}
0\rightarrow
{{\mathcal O}}_S\rightarrow{{\mathcal E}}_p\rightarrow{{\mathcal J}}_p(L-K)\rightarrow 0.$$ The Chern classes of ${{\mathcal E}}_p$ are $$c_1({{\mathcal E}}_p)=L-K\;\;\;\mbox{ and }\;\;\;
c_2({{\mathcal E}}_p)=\operatorname{length}(Z'_p).$$ Moreover, $Z'_p$ is automatically a complete intersection.
We would now like to understand what ${{\mathcal J}}_p$ is depending on $\operatorname{jet}_3(f_p)$, which in suitable local coordinates will be one of those in Table . For this we first of all note that the very ample divisor $L$ separates all subschemes of $Z_p$ of length at most two. Thus $Z'_p$ has length at least $3$, and due to Lemma \[lem:ideals\] below we are in one of the following situations:
$$\label{eq:3jets}
\begin{array}{|c|c|c|c|c|}
\hline
\operatorname{jet}_3(f_p)&{{\mathcal J}}_{Z_p,p}&\operatorname{length}(Z_p)&{{\mathcal J}}_p={{\mathcal J}}_{Z'_p,p}&c_2({{\mathcal E}}_p)\\
\hline\hline
x_p^3-y_p^3 & \langle x_p^2,y_p^2 \rangle & 4 & \langle x_p^2,y_p^2 \rangle & 4\\\hline
x_p^2y_p &\langle x_p^2, x_py_p,y_p^3\rangle & 4 &\langle x_p,y_p^3 \rangle & 3\\\hline
x_p^3 & \langle x_p^2,x_py_p^2,y_p^3\rangle & 5&\langle x_p^2,y_p^2 \rangle & 4\\\hline
x_p^3 & \langle x_p^2,x_py_p^2,y_p^3\rangle & 5&\langle x_p,y_p^3 \rangle & 3\\\hline
\end{array}$$
\[lem:ideals\] If $f\in R={{\mathds C}}\{x,y\}$ with $\operatorname{jet}_3(f)\in\{x^3-y^3,x^2y,x^3\}$, and if $I=\langle
g,h\rangle\lhd R$ such that $\dim_{{\mathds C}}(R/I)\geq 3$ and $\big\langle \frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\big\rangle+\langle
x,y\rangle^3\subseteq I$, then we may assume that we are in one of the following cases:
1. $I=\langle x^2,y^2\rangle$ and $\operatorname{jet}_3(f)\in\{x^3-y^3,x^3\}$, or
2. $I=\langle x,y^3\rangle$ and $\operatorname{jet}_3(f)\in\{x^2y,x^3\}$.
If $>$ is any *local degree* ordering on $R$, then the Hilbert-Samuel functions of $R/I$ and of $R/L_>(I)$ coincide, where $L_>(I)$ denotes the leading ideal of $I$ (see e.g. [@GP02] Proposition 5.5.7). In particular, $\dim_{{\mathds C}}(R/I)=\dim_{{\mathds C}}(R/L_>(I))$ and thus $$L_>(I)\in\big\{\langle x^2,xy^2,y^3\rangle,\langle x^2,xy,y^2\rangle,\langle
x^2,xy,y^3\rangle,\langle x^2,y^2\rangle,\langle
x,y^3\rangle\},$$ since $\langle x^2,xy^2,y^3\rangle\subset I$.
Taking $>$, for a moment, to be the local degree ordering on $R$ with $y>x$ we deduce at once that $I$ does not contain any power series with a linear term in $y$. For the remaining part of the proof $>$ will be the local degree ordering on $R$ with $x>y$.
$L_>(I)=\langle x^2,xy^2,y^3\rangle$ or $L_>(I)=\langle x^2,xy,y^2\rangle$. Thus the graph of the slope $H^0_{R/I}$ of the Hilbert-Samuel function of $R/I$ would be as shown in Figure \[fig:fp-histogram\], which contradicts the fact that $I$ is a complete intersection due to [@Iar77] Theorem 4.3.
cm 0.6 t:F l:0.5 w:0.3 (0 0) (0 1) (1 0) (0 1) (2 0) (0 -2) (0 0) (0 3) (0 0) (4 0) (0 0) (3 -0.6) h:C v:T (-0.8 2) h:R v:C
cm 0.6 t:F l:0.5 w:0.3 (0 0) (0 1) (1 0) (0 1) (1 0) (0 -2) (0 0) (0 3) (0 0) (4 0) (0 0) (2 -0.6) h:C v:T (-0.8 2) h:R v:C
$L_>(I)=\langle x^2,xy,y^3\rangle$. Then we may assume $$g=x^2+\alpha\cdot y^2+h.o.t.\;\;\;\mbox{ and }\;\;\; h=xy+\beta\cdot y^2+h.o.t..$$ Since $x^2\in I$ there are power series $a,b\in R$ such that $$x^2=a\cdot g+b\cdot h.$$ Thus the leading monomial of $a$ is one, $a$ is a unit and $g\in\langle x^2,h\rangle$. We may therefore assume that $g=x^2$. Moreover, since the intersection multiplicity of $g$ and $h$ is $\dim_{{\mathds C}}(R/I)=4$, $g$ and $h$ cannot have a common tangent line in the origin, i. e. $\beta\not=0$. Thus, since $g=x^2$, we may assume that $h=xy+y^2\cdot u$ with $u=\beta+h.o.t$ a unit.
In new coordinates $\widetilde{x}=x\cdot\sqrt{u}$ and $\widetilde{y}=y\cdot\frac{1}{\sqrt{u}}$ we have $$I=\langle \widetilde{x}^2,\widetilde{x}\widetilde{y}+\widetilde{y}^2\rangle.$$ Note that by the coordinate change $\operatorname{jet}_3(f)$ only changes by a constant, that $\frac{\partial f}{\partial \widetilde{x}},\frac{\partial
f}{\partial \widetilde{y}}\in I$ and that $\langle
\widetilde{x},\widetilde{y}\rangle^3\subset I$, but $\widetilde{x}\widetilde{y},\widetilde{y}^2\not\in I$. Thus $\operatorname{jet}_3(f)=x^3$.
Setting now $\bar{x}=\widetilde{x}$ and $\bar{y}=\widetilde{x}+2\widetilde{y}$, then $\bar{y}^2=\widetilde{x}^2+4\cdot
(\widetilde{x}\widetilde{y}+\widetilde{y}^2)\in I$ and thus, considering colengths, $$I=\langle \bar{x}^2,\bar{y}^2\rangle.$$ Moreover, the $3$-jet of $f$ does not change with respect to the new coordinates, so that we may assume we worked with these from the beginning.
$L_>(I)=\langle x^2,y^2\rangle$. Then we may assume $$g=x^2+\alpha\cdot xy+h.o.t.\;\;\;\mbox{ and }\;\;\; h=y^2+h.o.t.$$ As in the second case we deduce that w.l.o.g. $g=x^2$ and thus $h=y^2\cdot u$, where $u$ is a unit. But then $I=\langle
x^2,y^2\rangle$.
$L_>(I)=\langle x,y^3\rangle$. Then we may assume $$g=x+h.o.t.\;\;\;\mbox{ and }\;\;\; h=y^3+h.o.t.$$ since there is no power series in $I$ involving a linear term in $y$. In new coordinates $\widetilde{x}=g$ and $\widetilde{y}=y$ we have $$I=\big\langle \widetilde{x},\widetilde{h}\big\rangle,$$ and we may assume that $\widetilde{h}=\widetilde{y}^3\cdot u$, where $u$ is a unit only depending on $\widetilde{y}$. Hence, $I=\langle
\widetilde{x},\widetilde{y}^3\rangle$. Moreover, the $3$-jet of $f$ does not change with respect to the new coordinates, so that we may assume we worked with these from the beginning.
Thus $$c_1({{\mathcal E}}_p)^2-4\cdot c_2({{\mathcal E}}_p)>0,$$ and hence ${{\mathcal E}}_p$ is Bogomolov unstable. The Bogomolov instability implies the existence of a unique divisor $A_p$ which destabilises ${{\mathcal E}}_p$. (See e. g. [@Fri98] Section 9, Corollary 2.) In other words, setting $B_p=L-K-A_p$, i. e. $$\label{eq:AB:0}
A_p+B_p=L-K,$$ there is an immersion $$0\rightarrow{{\mathcal O}}_S(A_p)\rightarrow{{\mathcal E}}_p$$ where $(A_p-B_p)^2\geq c_1({{\mathcal E}}_p)^2-4\cdot c_2({{\mathcal E}}_p)>0$ and $(A_p-B_p){{\cdot}}H>0$ for every ample $H$. The same construction was considered in [@BFS89] and with their Proposition 1.4 it follows:
1. ${{\mathcal E}}_p(-A_p)$ has a global section that vanishes along a subscheme $\widetilde{Z}_p$ of codimension $2$ and which gives rise to a short exact sequence: $$\label{eq:AB:1}
0\rightarrow{{\mathcal O}}_S(A_p)\rightarrow{{\mathcal E}}_p\rightarrow{{\mathcal J}}_{\widetilde{Z}_p}(B_p)\rightarrow 0.$$
2. The divisor $B_p$ is effective and we may assume that $Z'_p\subset B_p$.
3. The divisors $A_p$ and $B_p$ satisfy the following numerical condition: $$\label{eq:AB:2}
\operatorname{length}(Z'_p)\geq A_p{{\cdot}}B_p \geq B_p^2+1.$$
4. $A_p-B_p$ and $A_p$ are big.
Now let $p$ move freely in $S$. Accordingly the scheme $Z'_p$ moves, hence the effective divisor $B_p$ containing $Z'_p$ moves in an algebraic family ${{\mathcal B}}\subseteq |B|_a$ which is the closure of $\{B_p\;|\;p\in S, L_p\in|L-3p|, \mbox{ both general}\}$ and which covers $S$. A priori this family ${{\mathcal B}}$ might have a *fixed part* $C$, \[page:fixedpart\] so that for general $p\in S$ there is an effective divisor $D_p$ moving in a fixed-part free algebraic family ${{\mathcal D}}\subseteq |D|_a$ such that $$B_p=C+D_p.$$ Whenever we only refer to the algebraic class of $A_p$ respectively $B_p$ respectively $D_p$ we will write $A$ respectively $B$ respectively $D$ for short.
$C=0$. {#sec:zero}
======
Our first aim is to show that actually $C=0$ (see Lemma \[lem:C\]). But in order to do so we first have to consider the boundary case that $A_p{{\cdot}}B_p=\operatorname{length}(Z_p')$.
\[prop:splitting\] If $A_p{{\cdot}}B_p=\operatorname{length}(Z'_p)$, then there exists a non-trivial global section $0\not=s\in
H^0\big(B_p,{{\mathcal J}}_{Z_p'/B_p}(A_p)\big)$ whose zero-locus is $Z_p'$.
In particular, $A_p{{\cdot}}D_p=A_p{{\cdot}}B_p=\operatorname{length}(Z_p')$ and $A_p{{\cdot}}C=0$.
By Sequence we have $$A_p{{\cdot}}B_p=\operatorname{length}(Z'_p)=c_2({{\mathcal E}}_p)=A_p{{\cdot}}B_p+\operatorname{length}\big(\widetilde{Z}_p\big).$$ Thus $\widetilde{Z}_p=\emptyset$.
If we merge the sequences , , and the structure sequence of $B$ twisted by $B$ we obtain the exact commutative diagram in Figure \[fig:AB\],
$$\xymatrix{
&& 0 \ar[d] & 0 \ar[d] &\\
&0\ar[r]\ar[d]& {{\mathcal O}}_S \ar[r]\ar[d] & {{\mathcal O}}_S \ar[r]\ar[d] &0 \\
0\ar[r]& {{\mathcal O}}_S(A_p) \ar[r]\ar[d] & {{\mathcal E}}_p \ar[r]\ar[d]&{{\mathcal O}}_S(B_p)\ar[r]\ar[d] &0 \\
0\ar[r]& {{\mathcal O}}_S(A_p) \ar[r]\ar[d] & {{\mathcal J}}_{Z_p'/S}(A_p+B_p) \ar[r]\ar[d]&{{\mathcal O}}_{B_p}(B_p)\ar[r]\ar[d] &0 \\
&0&0&0
}$$
where ${{\mathcal O}}_{B_p}(B_p)={{\mathcal J}}_{Z_p'/B_p}(A_p+B_p)$, or equivalently ${{\mathcal O}}_{B_p}={{\mathcal J}}_{Z_p'/B_p}(A_p)$. Thus from the rightmost column we get a non-trivial global section, say $s$, of this bundle which vanishes precisely at $Z_p'$, since $Z_p'$ is the zero-locus of the monomorphism of vector bundles ${{\mathcal O}}_S\hookrightarrow {{\mathcal E}}_p$. However, since $p$ is general we have that $p\not\in C$ and thus the restriction $0\not=s_{|D_p}\in H^0\big(D_p,{{\mathcal J}}_{Z_p'/D_p}(A_p)\big)$ and it still vanishes precisely at $Z_p'$. Thus $A_p{{\cdot}}D_p=\operatorname{length}(Z_p')=A_p{{\cdot}}B_p$, and $A_p{{\cdot}}C=A_p{{\cdot}}B_p-A_p{{\cdot}}D_p=0$.
We next want to show that positive self-intersection of $B$ imposes hard restrictions.
\[lem:B\^2positive\] $B^2\leq 2$ and if $B^2\in \{1,2\}$ then $A{{\cdot}}B=\operatorname{length}(Z'_p)=4$.
We may suppose that $B^2>0$. By we know that $4\geq
A{{\cdot}}B>B^2$ and by assumption $(A+B)^2\geq 17$, so that $$A^2=(A+B)^2-2\cdot A{{\cdot}}B-B^2\geq 17-8-3>0$$ and the Hodge Index Theorem gives $$(A{{\cdot}}B)^2\geq A^2\cdot B^2
\geq (17-2\cdot A{{\cdot}}B-B^2)\cdot B^2.$$ But then $B^2\geq 3$ leads to the contradiction $16\geq
18$. Similarly, $A{{\cdot}}B\leq 3$ leads to $9\geq (11-B^2)\cdot B^2$ which is neither for $B^2=1$ nor for $B^2=2$ fulfilled. This shows that $A{{\cdot}}B=4$, and thus by also $\operatorname{length}(Z'_p)=4$.
Even though we do not know whether ${{\mathcal B}}$ has a fixed part or not, we can get some information about the moving part ${{\mathcal D}}$.
\[lem:DD\] Let $p\in S$ be general and suppose $\operatorname{length}(Z'_p)=4$.
1. If $D_p$ is irreducible, then $\dim({{\mathcal D}})\geq 2$ and $D_p^2\geq 3$.
2. If $D_p$ is reducible but the part containing $p$ is reduced, then either $D_p$ has a component singular in $p$ and $D_p^2\geq 5$ or at least two components of $D_p$ pass through $p$ and $D_p^2\geq 2$.
3. If $D_p^2\leq 1$, then $D_p=k\cdot E_p$ where $k\geq 2$, $E_p$ is irreducible and $E_p^2=0$. In particular, $D_p^2=0$.
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1. If $D_p$ is irreducible, then $\dim({{\mathcal D}})\geq 2$, since $D_p$, containing $Z'_p$, is singular in $p$ by Table and since $p\in S$ is general. If through $p\in S$ general and a general $q\in D_p$ there is another $D'\in{{\mathcal D}}$, then due to the irreducibility of $D_p$ $$D_p^2=D_p{{\cdot}}D'\geq \operatorname{mult}_p(D_p)+\operatorname{mult}_q(D_p)\geq 3.$$ Otherwise, ${{\mathcal D}}$ is a two-dimensional involution whose general element is irreducible, so that by [@CC02] Theorem 5.10 ${{\mathcal D}}$ must be a linear system. This, however, contradicts the Theorem of Bertini, since the general element of ${{\mathcal D}}$ would be singular.
2. Suppose $D_p=\sum_{i=1}^k E_{i,p}$ is reducible but the part containing $p$ is reduced. Since $D_p$ has no fixed component and $p$ is general, each $E_{i,p}$ moves in an at least one-dimensional family. In particular $E_{i,p}^2\geq 0$.
If some $E_{i,p}$, say $i=1$, would be singular in $p$ for $p\in S$ general we could argue as above that $E_{1,p}^2\geq
3$. Moreover, either $E_{2,p}$ is algebraically equivalent to $E_{1,p}$ and $E_{2,p}^2\geq3$, or $E_{1,p}$ and $E_{2,p}$ intersect properly, since both vary in different, at least one-dimensional families. In any case we have $$D_p^2\geq (E_{1,p}+E_{2,p})^2\geq 5.$$ Otherwise, at least two components, say $E_{1,p}$ and $E_{2,p}$ pass through $p$, since $D_p$ is singular in $p$ and no component passes through $p$ with higher multiplicity. Hence, $E_{1,p}{{\cdot}}E_{2,p}\geq 1$ and therefore $$D_p^2\geq 2\cdot E_{1,p}{{\cdot}}E_{2,p}\geq 2.$$
3. From the above we see that $D_p$ is not reduced in $p$. Let therefore $D_p\equiv_a kE_p+E'$ where $k\geq 2$, $E_p$ passes through $p$ and $E'$ does not contain any component algebraically equivalent to $E_p$.
Suppose $E'\not=0$.\[eq:DD:0\] Since $D_p$ has no fixed component both, $E_p$ and $E'$ vary in an at least one dimensional family covering $S$ and must therefore intersect properly. In particular, $E_p{{\cdot}}E'\geq 1$ and $1\geq D_p^2\geq 2k\cdot
E_p{{\cdot}}E'\geq 4$. Thus, $E'=0$.
We therefore may assume that $D_p=kE_p$ with $k\geq 2$. Then $0\leq E_p^2=\frac{1}{k^2}\cdot
D_p^2\leq \frac{1}{4}$, which leaves only the possibility $E_p^2=0$, implying also $D_p^2=0$.
The following observations on the self intersection number of irreducible curves embedded via $L-K$ in our situation is an important tool in the proof that the fixed part $C$ does not exist.
\[lem:curves\] Suppose that $R\subset S$ is an irreducible curve, $L$ is very ample, and $L-K$ is base-point-free on $S$.
1. If $(L-K){{\cdot}}R=1$, then $R$ is smooth, rational and $R^2\leq -2$.
2. If $(L-K){{\cdot}}R=2$, then one of the following two cases occurs:
1. $R$ is smooth and rational with $R^2\leq -1$, or
2. $|L-K|$ induces a $\mathfrak{g}^1_2$ on $R$ and $L+R$ does not separate the points of this $\mathfrak{g}^1_2$.
In any case, if $R$ moves in a one dimensional algebraic family, then $R^2\not=0$.
Since $|L-K|$ is base-point-free it defines a morphism $$\varphi_{|L-K|}:S\longrightarrow {{\mathds P}}^n$$ and if $C=\varphi_{|L-K|}(R)$ and $\varphi:R\longrightarrow C$ denotes the restriction of $\varphi_{|L-K|}$ then $$\deg(\varphi)\cdot\deg(C)=(L-K){{\cdot}}R.$$ Moreover, by the adjunction formula we know that $$p_a(R)=\frac{R^2+R{{\cdot}}K}{2}+1,$$ and since $L$ is very ample we thus get $$\label{eq:curves:1}
1\leq L{{\cdot}}R =(L-K){{\cdot}}R+R{{\cdot}}K=(L-K){{\cdot}}R+2\cdot\big(p_a(R)-1\big)-R^2.$$
1. If $(L-K){{\cdot}}R=1$, then $C$ is a line in ${{\mathds P}}^n$ and $\varphi$ is a birational morphism from $R$ to $C$. It thus is an isomorphism, and $R$ must be a smooth, rational curve. We deduce from $$R^2\leq (L-K){{\cdot}}R-3=-2.$$
2. If $(L-K){{\cdot}}R=2$, then either the degree of $\varphi$ is one or two.
Suppose first that $\deg(\varphi)=1$. Then as above $\varphi$ is a birational morphism and hence an isomorphism. $C$ being an irreducible conic it is smooth and rational, and so is $R$. We deduce from $$R^2\leq (L-K){{\cdot}}R-3=-1.$$
Consider now the case $\deg(\varphi)=2$. $|L-K|$ cuts out a $\mathfrak{g}^1_2$ on $R$ which induces the morphism $\varphi$. Even if $R$ is singular the dualizing sheaf on $R$ is given by the restriction of $K+R$, and it satisfies the Riemann-Roch formula (see e.g. [@Har77 Ex. IV.I.9]), i.e. if $\mathfrak{d}$ is any divisor on $C$ we have $$\label{eq:rr}
h^0(\mathfrak{d})-h^0\big((K+R)_{|R}-\mathfrak{d})=\deg(\mathfrak{d})+1-p_a(R).$$ Suppose now that $P+Q\in \mathfrak{g}^1_2$ with $P$ and $Q$ in the smooth part of $C$. Then $$h^0\big((K+R)_{|R}-(L+R)_{|R}+P\big)=h^0\big(P-(L-K)_{|R}\big)=h^0(-Q)=0$$ and $$h^0\big((K+R)_{|R}-(L+R)_{|R}+P+Q\big)=h^0\big(P+Q-(L-K)_{|R}\big)=h^0({{\mathcal O}}_R)=1.$$ The Theorem of Riemann-Roch thus gives $$h^0\big((L+R)_{|R}-P\big)=(L+R).R-1+1-p_a(R)$$ and $$h^0\big((L+R)_{|R}-P-Q\big)-1=(L+R).R-2+1-p_a(R).$$ Hence $$h^0\big((L+R)_{|R}-P\big)=h^0\big((L+R)_{|R}-P-Q\big),$$ i.e. each divisor in the linear series induced by $L+R$ on $R$ which contains $P$ contains automatically also $Q$. The divisors in $|L+R|$ thus do not seperate the points $P$ and $Q$.
Suppose now that $\dim|R|_a\geq 1$ and $R^2=0$. Then $|R|_a$ is pencil and induces a fibration of $S$ whose fibres are the elements of $|R|_a$ (see [@Kei01] App. B.1). But then ${{\mathcal O}}_R(R)$ is trivial (see e.g. [@BHPV04 Lem. 8.1]) and thus ${{\mathcal O}}_R(L+R)={{\mathcal O}}_R(L)$ is very ample, which contradicts the fact that it does not separate the points of the $\mathfrak{g}^1_2$.
\[lem:C\] The family $\mathcal{B}$ introduced on page has no fixed part. I.e. under the assumptions of Section \[sec:construction\] and with the notation there, we have $C=0$.
Suppose $C\not=0$ and $r$ is the number of irreducible components of $C$. Since ${{\mathcal D}}$ has no fixed component and $A-B$ is big we know that $(A-B){{\cdot}}D>0$, so that $$\label{eq:c:0}
A{{\cdot}}D\geq B{{\cdot}}D+1=D{{\cdot}}C+D^2+1$$ or equivalently $$\label{eq:c:1}
D{{\cdot}}C\leq A{{\cdot}}D-D^2-1.$$ Moreover, since $A+B$ is ample we have $r\leq(A+B){{\cdot}}C=A{{\cdot}}C+D{{\cdot}}C+C^2$ and thus $$\label{eq:c:2}
A{{\cdot}}C+D{{\cdot}}C=(A+B){{\cdot}}C-C^2\geq r-C^2.$$
**1st Case:** $C^2\leq 0$. Then together with gives $$\label{eq:c:3}
A{{\cdot}}B=A{{\cdot}}C+A{{\cdot}}D\geq A{{\cdot}}C+D{{\cdot}}C+D^2+1\geq r+(-C^2)+D^2+1\geq 2,$$ or the slightly stronger inequality $$\label{eq:c:4}
A{{\cdot}}B\geq (A+B){{\cdot}}C +(-C^2)+D^2+1.$$
**2nd Case:** $C^2>0$. Then necessarily $B^2>0$ and by Lemma \[lem:B\^2positive\] we have $A{{\cdot}}B=\operatorname{length}(Z'_p)=4$ and $$\label{eq:c:5}
2\geq B^2=D^2+2\cdot C{{\cdot}}D+C^2\geq 1.$$
Since all the summands involved in the right hand side of and all summands in are non-negative, and since by Lemma \[lem:DD\] the case $D^2=1$ cannot occur when $\operatorname{length}(Z_p')=4$, and since by Lemma \[lem:B\^2positive\] $B^2>0$ is impossible when $\operatorname{length}(Z'_p)=3$, we are left considering the cases shown in Figure \[fig:c\], where for the additional information (the last four columns) we take Proposition \[prop:splitting\], Lemma \[lem:B\^2positive\] and Lemma \[lem:DD\] into account.
$$\begin{array}{|c|c|c|c|c|c||c|c|c|c|}
\hline
& \operatorname{length}(Z'_p) & D^2 & C^2 & C{{\cdot}}D & r & A{{\cdot}}B & A{{\cdot}}D & A{{\cdot}}C & D
\\\hline\hline
1)& 4 & 0 & -2 & & 1 & 4 & 4 & 0 & kE,k\geq 2 \\\hline
2)& 4 & 0 & -1 & & 2 & 4 & 4 & 0 & kE,k\geq 2 \\\hline
3)& 4 & 0 & 0 & & 3 & 4 & 4 & 0 & kE,k\geq 2 \\\hline
4)& 4 & 0 & -1 & & 1 & 3,4 & & & kE,k\geq 2 \\\hline
5)& 4 & 2 & 0 & & 1 & 4 & 4 & 0 & \\\hline
6)& 4 & 0 & 0 & & 2 & 3,4 & & & kE,k\geq 2 \\\hline
7)& 4 & 0 & 0 & & 1 &2,3,4& & & kE,k\geq 2 \\\hline
8)& 3 & 0 & -1 & & 1 & 3 & 3 & 0 & \\\hline
9)& 3 & 0 & 0 & & 2 & 3 & 3 & 0 & \\\hline
10)& 3 & 0 & 0 & & 1 & 2,3 & & & \\\hline\hline
11)& 4 & 0 & 1 & 0 & & 4& 4 & 0 & kE,k\geq 2 \\\hline
12)& 4 & 0 & 2 & 0 & & 4 & 4 & 0 & kE,k\geq 2 \\\hline
\end{array}$$
Let us first and for a while consider the situation $\operatorname{length}(Z_p')=4$ and $D^2=0$, so that by Lemma \[lem:DD\] $D=kE$ for some irreducible curve $E$ with $k\geq 2$ and $E^2=0$. Applying Lemma \[lem:curves\] to $E$ we see that $(A+B){{\cdot}}E\geq 3$, and thus $$\label{eq:c:6a}
6\leq 3k\leq (A+B){{\cdot}}D=A{{\cdot}}D+C{{\cdot}}D.$$
If in addition $A{{\cdot}}D\leq 4$, then leads to $$6\leq 3k\leq A{{\cdot}}D+C{{\cdot}}D\leq 4+C{{\cdot}}D\leq 7,$$ which is only possible for $k=2$, $C{{\cdot}}E=1$ and $$C{{\cdot}}D=k\cdot C{{\cdot}}E=2.\label{eq:c:6}$$
In Cases 1, 2 and 3 we have $A{{\cdot}}D=4$, and we can apply , which by then gives the contradiction $$2=A{{\cdot}}C+C{{\cdot}}D\geq r-C^2=3.$$
If, still under the assumption $\operatorname{length}(Z_p')=4$ and $D^2=0$, we moreover assume $2\geq C^2\geq 0$ then by Lemma \[lem:B\^2positive\] $$2\geq B^2=2\cdot C{{\cdot}}D+C^2\geq 2\cdot C{{\cdot}}D\geq 0,$$ and thus $C{{\cdot}}D\leq 1$ and $C{{\cdot}}D+C^2\leq 2$, which due to implies $A{{\cdot}}D\geq 5$. But then by Proposition \[prop:splitting\] we have $A{{\cdot}}B\leq 3$ and hence $A{{\cdot}}C=A{{\cdot}}B-A{{\cdot}}D\leq -2$, which leads to the contradiction $$\label{eq:c:7}
(A+B){{\cdot}}C=A{{\cdot}}C+D{{\cdot}}C+C^2\leq 0,$$ since $A+B$ is ample. This rules out the Cases 6, 7, 11 and 12.
In Case 4 Lemma \[lem:curves\] applied to $C$ shows $$\label{eq:c:8}
2\leq (A+B){{\cdot}}C=A{{\cdot}}C+D{{\cdot}}C+C^2.$$ Lemma \[lem:B\^2positive\] implies $$2\geq B^2=2\cdot C{{\cdot}}D+C^2= 2k\cdot C{{\cdot}}E-1\geq 4\cdot C{{\cdot}}E-1 \geq -1,$$ which is only possible for $C{{\cdot}}E=C{{\cdot}}D=0$. But then implies $A{{\cdot}}C\geq 3$, and since $A$ is big and $E$ is irreducible with non-negative self intersection we get the contradiction $$2\leq k\cdot A{{\cdot}}E\leq A{{\cdot}}D=A{{\cdot}}B-A{{\cdot}}C\leq 1.$$
This finishes the cases where $\operatorname{length}(Z_p')=4$ and $D^2=0$.
In Cases 5 and 10 we apply Lemma \[lem:curves\] to the irreducible curve $C$ with $C^2=0$ and find $$(A+B){{\cdot}}C\geq 2.$$ In Case 5 Equation then gives the contradiction $$4= A{{\cdot}}B \geq 2-C^2+D^2+1=5.$$ In Case 10 we get $$3\geq A{{\cdot}}B \geq (A+B).C-C^2+D^2+1=(A+B).C+1,$$ which shows that $$\label{eq:c:corr}
2=(A+B).C=A.C+D.C+C^2$$ and that $A.B=3=\operatorname{length}(Z_p')$. Then by Proposition \[prop:splitting\] we get $A.C=0$ and by $$D.C=2-A.C-C^2=2,$$ which due to Lemma \[lem:B\^2positive\] leads to the contradiction $$2\geq B^2=D^2+2\cdot D.C+C^2=4.$$ In very much the same way we get in Case 8 by Lemma \[lem:curves\] $$(A+B){{\cdot}}C\geq 2$$ and the contradiction $$3= A{{\cdot}}B\geq 2-C^2+D^2+1=4.$$
It remains to consider Case 9. Here we deduce from that $$2\geq (A+B){{\cdot}}C\geq r=2,$$ and hence $$2=(A+B){{\cdot}}C=A{{\cdot}}C+D{{\cdot}}C+C^2=D{{\cdot}}C.$$ But then Lemma \[lem:B\^2positive\] leads to the final contradiction $$2\geq B^2=D^2+2\cdot D{{\cdot}}C+C^2=4.$$
It follows that $B_p=D_p$, ${{\mathcal B}}={{\mathcal D}}$, and that $B_p$ is nef.
The General Case {#sec:generalcase}
================
Let us review the situation and recall some notation. We are considering a divisor $L$ such that $L$ is very ample and $L-K$ is ample and base-point-free with $(L-K)^2>16$, and such that for a general point $p\in S$ the general element $L_p\in |L-3p|$ has no triple component through $p$ and that the equimultiplicity ideal of $L_p$ in $p$ in suitable local coordinates is one of the ideals in Table – and for all $p$ the ideals have the same length. Moreover, we know that there is an algebraic family ${{\mathcal B}}=\overline{\{B_p\;|\;p\in S\}}\subset |B|_a$ without fixed component such that for a general point $p\in S$ $$B_p\in |{{\mathcal J}}_{Z'_p/S}(L-K-A_p)|,$$ where $Z'_p$ is contained in the equimultiplicity scheme $Z_p$ of $L_p$ and $A_p$ is the unique divisor linearly equivalent to $L-K-B_p$ such that $B_p$ and $A_p$ destabilise the vector bundle ${{\mathcal E}}_p$ in . Keeping these notations in mind we can now consider the two cases that either $\operatorname{length}(Z'_p)=4$ or $\operatorname{length}(Z'_p)=3$.
\[prop:length4\] With the above notation and assumptions, it is impossible that for a general point $p\in S$ the length of $Z'_p$ is $\operatorname{length}(Z'_p)=4$.
In Section \[sec:zero\] we have shown that $B=D$ is nef, and thus Lemma \[lem:B\^2positive\] shows $$\label{eq:length4:1}
0\leq B^2\leq 2.$$ Then, however, Lemma \[lem:DD\] implies that $B_p$ must be reducible.
Let us first consider the case that the part of $B_p$ through $p$ is reduced. Then by Lemma \[lem:DD\] and Equation we know that $B_p=E_p+F_p+R$, where $E_p$ and $F_p$ are irreducible and smooth in $p$. In particular, $E_p{{\cdot}}F_p\geq 1$, and thus $$\begin{gathered}
2= B^2=E_p^2+2\cdot E_p{{\cdot}}F_p+F_p^2+2\cdot(E_p+F_p){{\cdot}}R+R^2\\
\geq 2+2\cdot(E_p+F_p){{\cdot}}R.
\end{gathered}$$ Since $E_p{{\cdot}}F_p\geq 1$ and since the components $E_p$ and $F_p$ vary in at least one-dimensional families and $R$ has no fixed component, $(E_p+F_p){{\cdot}}R\geq 1$, unless $R=0$. This would however give a contradiction, so $R=0$. Therefore necessarily, $B_p=E_p+F_p$, $E_p{{\cdot}}F_p=1$, and $E_p^2=F_p^2=0$. Then by Lemma \[lem:curves\] $(A+B){{\cdot}}E_p\geq 3$ and $(A+B){{\cdot}}F_p\geq 3$, so that $$4\geq A{{\cdot}}B\geq (A+B){{\cdot}}E_p+(A+B){{\cdot}}F_p-B^2\geq 4$$ implies $E_p{{\cdot}}A_p=2=F_p{{\cdot}}A_p$ and $(A+B){{\cdot}}E_p=3=(A+B){{\cdot}}F_p$.
Since $E_p^2=0$ the family $|E|_a$ is a pencil and induces a fibration on $S$ (see [@Kei01] App. B.1). In particular, the generic element $E_p$ in $|E|_a$ must be smooth (see e.g. [@BHPV04] p. 110).
We claim that in $p$ the curve $L_p$ can share at most with one of $E_p$ or $F_p$ a common tangent, and it can do so at most with multiplicity one. For this consider local coordinates $(x_p,y_p)$ as in the Table . Since $\operatorname{length}(Z_p')=4$ we know that ${{\mathcal J}}_{Z_p',p}=\langle x_p^2,y_p^2\rangle$ does not contain $x_py_p$, and since $B_p=E_p+F_p\in|{{\mathcal J}}_{Z_p'}(L-K-A)|$, where $E_p$ and $F_p$ are smooth in $p$, we deduce that in local coordinates their equations are $$x_p+a\cdot y_p+h.o.t.\;\;\;\mbox{ respectively }\;\;\;x_p-a\cdot y_p+h.o.t.,$$ where $a\not=0$. By Table the local equation $f_p$ of $L_p$ has either $\operatorname{jet}_3(f_p)=x_p^3$ and has thus no common tangent with either $E_p$ or $F_p$, or $\operatorname{jet}_3(f_p)=x_p^3-y_p^3$ and it is divisible at most once by one of $x_p-ay_p$ or $x_p+ay_p$ .
In particular, $E_p$ can at most once be a component of $L_p$, and we deduce $$E_p{{\cdot}}K_S=E_p{{\cdot}}L_p-E_p{{\cdot}}A_p-E_p{{\cdot}}B_p=E_p{{\cdot}}L_p-3\geq
\left\{
\begin{array}{ll}
0,&\mbox{ if } E_p\not\subset L_p,\\
-1, &\mbox{ if } E_p\subset L_p.
\end{array}
\right.$$ But then, since the genus is an integer, $$\label{eq:length4:genus}
p_a(E_p)=\frac{E_p^2+E_p{{\cdot}}K_S}{2}+1=\frac{E_p{{\cdot}}K_S}{2}+1\geq 1.$$
Fix a general point $p$ in $S$ and a general point $q$ on $E_p$, then $E_p=E_q$ since $|E|_a$ is a pencil. Hence, $$A_p+F_p\sim_l L-K-E_p=L-K-E_q \sim_l A_q+F_q.$$ Since $A_q.B_q=4=\operatorname{length}(Z'_q)$ by Proposition \[prop:splitting\] there is a global section $s_q\in
H^0\big(B_q,{{\mathcal J}}_{Z'_q/B_q}(A_q)\big)$ whose zero locus is $Z'_q$. Restricting $s_q$ to $E_q$ we get a global section of ${{\mathcal O}}_{E_p}(A_q)={{\mathcal O}}_{E_q}(A_q)$ which cuts out $2q$ on $E_p=E_q$. Moreover, ${{\mathcal O}}_{E_p}(F_q)={{\mathcal O}}_{E_q}(F_q)$ has a global section which cuts out $q$. Thus ${{\mathcal O}}_{E_p}(A_p+F_p)={{\mathcal O}}_{E_p}(A_q+F_q)$ has for infinitely many points $q$ on $E_p$ a global section which cuts out $3q$. The linear system $|{{\mathcal O}}_{E_p}(A_p+F_p)|$ thus has degree three and contains the divisor $3q$ for infinitely many points $q$, and it hence has no base point. So it defines a morphism to ${{\mathds P}}^k$, where $k$ is the dimension of the linear system. $k$ cannot be one, since otherwise the morphism would have infinitely many ramification points. If the dimension $k$ is two, the morphism maps the curve $E_p$ to the plane. Then either the morphism has degree three and the image is a line, which leads to the same contradiction, or the morphism is an isomorphism and the image is a cubic which has infinitely many reflection points, which is also impossible. It remains the case that the dimension $k$ is three, but then $E_p$ has a $\mathfrak{g}_3^3$ and is rational, in contradiction to .
This finishes the case that the part of $B_p$ through $p$ is reduced.
It remains to consider the case that $B_p$ is not reduced in $p$. Using the notation of the proof of Lemma \[lem:DD\] we write $B_p\equiv
k\cdot E_p+E'$ with $k\geq 2$, $E_p$ irreducible passing through $p$ and $E'$ not containing any component algebraically equivalent to $E_p$. We have seen there (see p. ) that $E'\not=0$ implies $B_p^2\geq 4$ in contradiction to . We may therefore assume $B_p=k\cdot E_p$ with $E_p^2\geq 0$. If $E_p^2\geq 1$, then again $B_p^2\geq 4$. Thus $E_p^2=0$. Applying Lemma \[lem:curves\] to $E_p$ we get $$3\leq (A+B){{\cdot}}E_p=A{{\cdot}}E_p,$$ and hence the contradiction $$4\geq A{{\cdot}}B=k\cdot A{{\cdot}}E_p\geq 6.$$ This finishes the case that $B_p$ is not reduced in $p$, and shows thus that the case $\operatorname{length}(Z'_p)=4$ cannot occur.
\[prop:length3\] Let $p\in S$ be general and suppose that $\operatorname{length}(Z'_p)=3$. Then $B_p$ is an irreducible, smooth, rational curve in the pencil $|B|_a$ with $B^2=0$, $A{{\cdot}}B=3$ and $\exists\;s\in H^0\big(B_p,{{\mathcal O}}_{B_p}(A_p)\big)$ such that $Z_p'$ is the zero-locus of $s$.
In particular, $S\rightarrow |B|_a$ is a ruled surface and $2B_p$ is a fixed component of $|L-3p|$.
Since in Section \[sec:zero\] we have shown that $B$ is nef, Lemma \[lem:B\^2positive\] implies $$\label{eq:length3:0}
B^2=0.$$ Once we have shown that $B_p$ is irreducible and reduced, we then know that $|B|_a$ is a pencil and induces a fibration on $S$ whose fibres are the elements of $|B|_a$ (see [@Kei01] App. B.1). In particular, the general element of $|B|_a$, which is $B_p$, is smooth (see [@BHPV04] p. 110).
Let us therefore first show that $B_p$ is irreducible and reduced. Since ${{\mathcal B}}$ has no fixed component we know for each irreducible component $B_i$ of $B_p=\sum_{i=1}^rB_i$ that $B_i^2\geq 0$, and hence by Lemma \[lem:curves\] that $(A+B){{\cdot}}B_i\geq
2$. Thus by and $$2\cdot r\leq (A+B){{\cdot}}B=A{{\cdot}}B+B^2=A{{\cdot}}B\leq 3,$$ which shows that $B_p$ is irreducible and reduced and that $A{{\cdot}}B=3$.
Since $A{{\cdot}}B=3=\operatorname{length}(Z'_p)$ Proposition \[prop:splitting\] implies that there is a section $s_p\in H^0\big(B_p,{{\mathcal O}}_{B_p}(A_p)\big)$ such that $Z_p'$ is the zero-locus of $s_p$, which is just $3p$. Note that for $p\in S$ general and $q\in B_p$ general we have $B_p=B_q$ since $|B|_a$ is a pencil, and thus by the construction of $B_p$ and $B_q$ we also have $$A_p\sim_l L-K-B_p=L-K-B_q\sim_l A_q.$$ But if $A_p$ and $A_q$ are linearly equivalent, then so are the divisors $s_p$ and $s_q$ induced on the curve $B_p=B_q$. The curve $B_p$ therefore contains a linear series $|{{\mathcal O}}_{B_p}(A_p)|$ of degree three which contains $3q$ for a general point $q\in
B_p$. In particular, the linear series has no base point and induces a morphism $\varphi:B_p\longrightarrow{{\mathds P}}^k$ where $k$ is the dimension of the linear series.
Suppose that $k$ was one, then $\varphi$ would be a morphism of degree three from the curve $B_p$ to a line and it would have infinitely many ramification points $q$, which is clearly not possible. If $k$ is two, then either $\varphi$ has degree three and its image is a line, which leads to the same contradiction, or $\varphi$ has degree one and the image is a plane cubic. In that case $\varphi$ is a birational morphism and either $B_p$ is rational (if $\operatorname{Im}(\varphi)$ is singular) or $B_p$ is elliptic (if $\operatorname{Im}(\varphi)$ is smooth). If $B_p$ was an elliptic curve, then the general point $q$ of the cubic $\operatorname{Im}(\varphi)$ embedded via the $\mathfrak{g}_3^2=|{{\mathcal O}}_{B_p}(A_p)|$ would be an inflexion point. But that is clearly not possible. Finally, if $k$ is three, then $B_p$ has a $\mathfrak{g}_3^3$ and is thus rational. Alltogether we have shown that $$p_a(B_p)=0,$$ and by the adjunction formula we get $$\label{eq:length3:2}
K{{\cdot}}B=2\cdot p_a(B)-2-B^2=-2.$$
Note also, that $Z'_p\subset B_p$ in view of Table implies that $B_p$ and $L_p$ have a common tangent in $p$. Suppose that $B_p$ and $L_p$ have no common component, i. e. $B_p\not\subset L_p$, then $$3\leq\operatorname{mult}_p(B_p)\cdot\operatorname{mult}_p(L_p)<
L{{\cdot}}B=A{{\cdot}}B+B^2+K{{\cdot}}B=3+K{{\cdot}}B=1,$$ which contradicts . Thus, $B_p$ is at least once contained in $L_p$. Moreover, if $2B_p\not\subset L_p$ then by Table $L'_p:=L_p-B_p$ has multiplicity two in $p$, and it still has a common tangent with $B_p$ in $p$, so that $$\label{eq:length3:3}
3\leq L'_p{{\cdot}}B_p=L{{\cdot}}B-B^2=A{{\cdot}}B+K{{\cdot}}B=3+K{{\cdot}}B=1$$ again is impossible. We conclude finally, that $B_p$ is at least twice contained in $L_p$
Note finally, since $\dim|B|_a=1$ there is a unique curve $B_p$ in $|B|_a$ which passes through $p$, i. e. it does not depend on the choice of $L_p$, so that in these cases $B_p$ respectively $2B_p$ is actually a fixed component of $|L-3p|$.
Triple-Point Defective Surfaces are Ruled {#sec:ruled}
=========================================
The considerations of the previous sections prove the following theorem.
\[thm:ruled\] More precisely, let $L$ be a line bundle on $S$ such that $L$ is very ample and $L-K$ is ample and base-point-free. Suppose that $(L-K)^2>16$ and that for a general $p\in S$ the linear system $|L-3p|$ contains a curve $L_p$ which has no triple component through $p$, but such that $h^1\big(S,{{\mathcal J}}_{Z_p}(L)\big)\not=0$ where $Z_p$ is the equimultiplicity scheme of $L_p$ at $p$.
Then there is a ruling $\pi:S\rightarrow C$ of $S$ such that $L_p$ contains the fibre over $\pi(p)$ with multiplicity two.
In view of Propositon \[prop:h1vanishing\] this proves Theorem \[thm:aim1\].
[10]{}
Wolf Barth, Klaus Hulek, Christian Peters, and Antonius [Van de Ven]{}. , volume 4 of [*Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge*]{}. Springer, 2nd edition, 2004.
Mauro C. Beltrametti, Paolo Francia, and Andrew J. Sommese. On [Reider’s]{} method and higher order embeddings. , 58(2), 1989.
Cristiano Bocci and Luca Chiantini. Triple points imposing triple divisors and the defective hierarchy. In [*Projective Varieties with Unexpected Properties*]{}, pages 35–49. De Gruyter, 2005.
Maria Castellani. Sulle superficie i cui spazi osculatori sono biosculatori. , XXXI:347–350, 1922.
Luca Chiantini and Ciro Ciliberto. Weakly defective varieties. , 354:151–178, 2002.
Luca Chiantini and Thomas Markwig. Triple-point defective ruled surfaces. , 212:1337–1346, 2008.
Ciro Ciliberto. Linear systems with multiple assigned base points. To appear in: Proceedings of the Conference Zero Dimensional Schemes and Applications, Feb. 2000, Naples, 2001.
Davide Franco and Giovanna Ilardi. On multiosculating spaces. , 29(7):2961–2976, 2001.
Robert Friedman. . Springer, 1998.
Luis Fuentes Garcia and Manuel Pedreira. The projective theory of ruled surfaces. , 24(1):25–63, 2005.
Greuel and Gerhard Pfister. . Springer, 2002.
Robin Hartshorne. . Springer, 1977.
Anthony Iarrobino. Punctual [Hilbert]{} schemes. , 10(188):1–97, 1977.
Thomas Keilen. . PhD thesis, Universität Kaiserslautern, 2001. http:// www. mathematik. uni-kl. de/ \~keilen/ download/ Thesis/ thesis.ps.gz.
Antonio Lanteri and Raquel Mallavibarrena. Jets of antimulticanonical bundles on [Del Pezzo]{} surfaces of degree $\leq 2$. In [*Algebraic Geometry, A Volume in Memory of [Paolo Francia]{}*]{}, pages 258–276. De Gruyter, 2002.
Robert Lazarsfeld. Lectures on linear series. In János Kollár, editor, [*Complex Algebraic Geometry*]{}, number 3 in IAS/Park City Mathematics Series, pages 161–219. Amer. Math. Soc., 1997.
Thomas Markwig. A note on equimultiple deformations. arXiv:0705.3911, 2006.
Alessandro Terracini. Su due problemi concernenti la determinazione di alcune classi di superficie, considerate da [G. Scorza]{} e [F. Platini]{}. , 6:3–16, 1921–22.
Geng Xu. Ample line bundles on smooth surfaces. , 469:199–209, 1995.
[^1]: The second author was supported by the EAGER node of Torino, and by the Institute for Mathematics and its Applications (IMA), University of Minnesota.
|
---
abstract: 'We propose a new chirality-imbalance phenomenon arising in baryonic/high dense matters under a magnetic field. A locally chiral-imbalanced (parity-odd) domain can be created due to the electromagnetically induced $U(1)_A$ anomaly in high-dense matters. The proposed local-chiral imbalance generically possesses a close relationship to a spacial distribution of an inhomogeneous chiral (pion)-vector current coupled to the magnetic field. To demonstrate such a nontrivial correlation, we take the skyrmion crystal approach to model baryonic/high dense matters. Remarkably enough, we find the chirality-imbalance distribution takes a wave form in a high density region (dobbed “[*chiral-imbalance density wave*]{}”), when the inhomogeneous chiral condensate develops to form a chiral density wave. This implies the emergence of a nontrivial density wave for the explicitly broken $U(1)_A$ current simultaneously with the chiral density wave for the spontaneously broken chiral-flavor current. We further find that the topological phase transition in the skyrmion crystal model (between skyrmion and half-skyrmion phases) undergoes the deformation of the chiral-imbalance density wave in shape and periodicity. The emergence of this chiral-imbalance density wave could give a crucial contribution to studies on the chiral phase transition, as well as the nuclear matter structure, in compact stars under a magnetic field.'
author:
- 'Mamiya Kawaguchi[^1]'
- 'Shinya Matsuzaki[^2]'
title: ' Chiral-Imbalance Density Wave in Baryonic Matters '
---
Introduction
============
Exploring the properties of QCD under an extreme environment has attracted a lot of attentions to extract a novel insight of the nonperturbative nature of QCD involving the chiral-symmetry breaking structure. Particularly, it would be important to ask how much the $U(1)_A$ breaking (anomaly) can serve as a source for the baryonic matter structure, while competing with contributions from the spontaneous breaking of the chiral symmetry, which would also be linked to the origin of the [[nucleon]{}]{} mass.
Regarding a nontrivial issue on the $U(1)_A$ anomaly under exotic environments, [[it has been expected [@Kharzeev:2001ev; @Kharzeev:2007tn; @Kharzeev:2007jp; @Fukushima:2008xe; @Son:2009tf; @Kharzeev:2010gd; @Burnier:2011bf; @Hongo:2013cqa; @Yee:2013cya; @Hirono:2014oda; @Adamczyk:2015eqo; @Yin:2015fca; @Huang:2015oca; @Kharzeev:2015znc; @Hattori:2016emy; @Shi:2017cpu] in the hot QCD system]{}]{} that the local-parity-odd domain would show up due to the nonzero chirality (sometimes called the chiral-charge separation) induced by the anomalous $U(1)_A$ current coupled to the topological gluon configuration (sphaleron [@Manton:1983nd; @Klinkhamer:1984di]). This local $P$-odd domain, called the chiral-imbalance medium [@McLerran:1990de], is expected to be observed as the nontrivial consequences for the presence of the $U(1)_A$ anomaly (and/or strong CP violation) in heavy ion collisions, although being metastable to be gone after the typical time scale of QCD pasts [@McLerran:1990de; @Moore:1997im; @Moore:1999fs; @Bodeker:1999gx; @deSouza:2015ena]. Such a chiral imbalance medium is characterized by the chiral (axial) chemical potential (often denoted by $\mu_5$), and has so far extensively been searched based on various arguments in hot QCD matter applied to heavy ion collision experiments, [[e.g. [@Kharzeev:2001ev; @Kharzeev:2007tn; @Kharzeev:2007jp; @Fukushima:2008xe; @Son:2009tf; @Kharzeev:2010gd; @Burnier:2011bf; @Hongo:2013cqa; @Yee:2013cya; @Hirono:2014oda; @Adamczyk:2015eqo; @Yin:2015fca; @Huang:2015oca; @Kharzeev:2015znc; @Hattori:2016emy; @Shi:2017cpu] and]{}]{} [@Andrianov:2012hq; @Andrianov:2012dj; @Kawaguchi:2016avk; @Andrianov:2017hbf; @Andrianov:2017ilv; @Andrianov:2018gim; @Putilova:2018qli; @Andrianov:2019you].
In place of invoking the gluonic $U(1)_A$ anomaly, a local chiral-imbalance medium can actually be created even in other specific environments apart from QCD. For instance, weak interactions generically [[break]{}]{} the parity and [[lead]{}]{} to the chirality imbalance. [[Indeed, it has recently been shown [@Charbonneau:2009ax; @Grabowska:2014efa; @Kaminski:2014jda; @Sigl:2015xva; @Yamamoto:2015gzz; @Masada:2018swb] that]{}]{} the chirality imbalance for weakly interacting leptons can be generated in a process of supernova explosions, [[through the chiral transport mechanism acting on neutrinos]{}]{}. Moreover, a signal induced from the chiral imbalance medium (so-called the chiral magnetic effect) has actually been observed in a condense matter system, called Weyl semimetals [@Li:2014bha; @Lv:2015pya; @Xu:2015cga]. Thus, understanding of the chirality imbalance as well as the $U(1)_A$ anomaly has recently [[been developing]{}]{} and is currently getting interdisciplinary.
In this paper, we propose a novel possibility to create a (stable) chiral-imbalance medium in a high dense QCD with a strong magnetic field. Our central idea is built on a simple observation as follows.
In zero-temperature environment, the topologically nontrivial gluon configuration will not survive longer enough to stay in the QCD time scale, due to the gigantic exponential suppression form of the instanton configuration. Thereby, the $U(1)_A$ anomaly of the gluon-field strength form $\sim \epsilon_{\mu\nu\rho \sigma }G^{\mu\nu} G^{\rho \sigma}$ is supposed to undetectable, in contrast to the finite temperature case with the QCD sphaleron configuration. Instead of the gluonic contribution, the chiral imbalance for quarks can be induced by the electromagnetic $U(1)_A$ anomaly, where the $U(1)_A$ symmetry is explicitly broken by coupling the chiral quarks to the electromagnetic current, yielding the anomaly form like $\epsilon_{\mu\nu\rho \sigma }F^{\mu\nu} F^{\rho \sigma}$, where $F_{\mu\nu}$ stands for the electromagnetic filed strength.
Now, consider a high-dense matter system under a magnetic field, inside of which charged pions form the pion-vector current $J_\mu^\pi = i (\partial_\mu \pi^+ \pi^- - \pi^+ \partial_\mu \pi^-)$, coupled to the electromagnetic field, and then might also couple to the electromagnetic $U(1)_A$ anomaly as above. Of importance here is to note that as discussed in [@Nishiyama:2015fba; @Abuki:2016zpv; @Abuki:2018wuv; @Kawaguchi:2018xug], in a high-dense medium with a strong magnetic field, the pions can locally form condensates, (so-called inhomogeneous chiral condensates), so that the pion-vector current $J_\mu^\pi$ can also have a locally nontrivial distribution in the medium. Suppose the medium to be so high-dense like that, in such a way that the dense matter can be highly compressed to be almost static. In that case, we may expect to have a local-chiral density ($\rho_5(\vec{x})$ with $\vec{x}$ being three-dimensional spatial vector) associated with the $U(1)_A$ anomaly in the high-dense medium, like $$\begin{aligned}
\rho_5(\vec{x})
& \sim \epsilon^{0 \mu\nu\rho} J_\mu^\pi(\vec{x}) F_{\nu\rho}(\vec{x})/f_\pi^2
\notag \\
&\sim \vec{J}^\pi(\vec{x}) \cdot \vec{B}(\vec{x})/f_\pi^2
\,. \label{generic-rho5}\end{aligned}$$
The nonzero $\rho_5$ in Eq.(\[generic-rho5\]) manifestly implies the emergence of a chirality-imbalance driven in the presence of a magnetic field and a nontrivial spatial distribution for a pion-vector current. In fact, [[one could immediately find the existence of the interaction terms as in Eq.(\[generic-rho5\]), once writing down the conventional chiral Lagrangian including the chiral anomaly part:]{}]{} the interaction terms of this type can be generated via the Wess-Zumino-Witten (WZW) term [@Wess:1971yu; @Witten:1983tw], when the $U(1)_A$ charge and the isospin charge plus baryon number are gauged to be identified with the chiral chemical potential $\mu_5$ and the electromagnetic charge for quarks, respectively. Then the functional derivative of the chiral Lagrangian with respect to the $\mu_5$ will immediately lead to the chiral-imbalance density in Eq.(\[generic-rho5\]) [^3]. Therefore, might it sound trivial? — the answer is “No.”. [[From]{}]{} Eq.(\[generic-rho5\]) one would expect a nontrivial correlation between the presence of the chirality imbalance and the inhomogeneity of the pion-vector current (arising from the inhomogeneous chiral condensate) in medium. That is, Eq.(\[generic-rho5\]) would provide a significant possibility to examine [*how much the $U(1)_A$ anomaly can be correlated with the spontaneously broken chiral symmetry, by setting the chiral dynamics in a high-dense medium with use of a magnetic field as the probe*]{}.
To monitor a nontrivial physics derived from the $\rho_5$ in Eq.(\[generic-rho5\]), in this paper we take the skyrmion crystal approach [@Skyrme:1962vh; @Klebanov:1985qi] as a candidate effective model for describing the baryonic/high-dense matter. In the skyrmion crystal approach, the baryonic matter is described by putting the skyrmions on lattice vertices of a crystal structure. Actually, the large-$N_c$ QCD supports that a topologically-static soliton arising as a skyrmion can be regarded as a baryon [@Witten:1983tw; @Witten:1983tx]. By using the skyrmion crystal approach, the baryonic matter can also be described as the topologically static-object. In high density region, where baryons is so compressed to be a static object, the skyrmion crystal is a powerful approach for the [[qualitative-baryon description]{}]{} [^4] as if they could form crystals [@Lee:2003aq; @Lee:2003rj; @Ma:2013ooa; @Ma:2013ela; @book; @Ma:2016npf; @Ma:2016gdd; @Harada:2015lma; @Kawaguchi:2018xug; @Kawaguchi:2018fpi].
The skyrmion crystal picture is indeed in accord with the desired setup for the emergence of a local-chirality imbalance proposed in Eq.(\[generic-rho5\]). Besides, the skyrmion crystal approach predicts a characteristic phenomena which is called “topological phase transition" [[(for reviews see e.g., Refs. [@book; @Ma:2016npf; @Ma:2016gdd])]{}]{}. If we choose the underlying structure as the face-centered-cubic (FCC) crystal, the crystal configuration is changed from a FCC crystal to a cubic-centered crystal (CC). [[ Actually, [[it has been indicated]{}]{} that the results from effective field theories, in which such a topological phase transition is encoded, [[can be]{}]{} consistent with the present observation of neutron star physics [@Paeng:2017qvp; @Ma:2018jze; @Ma:2018xjw]. ]{}]{} [[Those facts]{}]{} would give us another interesting chance to investigate some correlations between the chirality imbalance and the baryonic matter structure.
We demonstrate that a nontrivial correlation between the chiral imbalance distribution and the baryonic/high-dense matter structure actually shows up: it turns out that the $\rho_5$ as in Eq.(\[generic-rho5\]) emerges to form a density wave, simultaneously with a chiral density wave for a pion-vector current $J^\pi_\mu$. It is dobbed “[*chiral-imbalance density wave*]{}”. We also observe that the periodicity of the chiral-imbalance density wave harmonizes to almost coincide with the previously proposed inhomogeneous-chiral condensate distributions induced on the skyrmion crystal [@Kawaguchi:2018xug].
The emergence of this chiral-imbalance density wave would significantly contribute to studies on the chiral phase transition under a magnetic field with the inhomogeneous chiral condensates (chiral density waves) incorporated, as has been discussed in [@Nishiyama:2015fba; @Abuki:2016zpv; @Abuki:2018wuv] in different setup for chiral effective models, and also on the nuclear matter structure, in compact stars holding a magnetic field.
This paper is organized as follows: In Sec. II we start with introduction of our target model-setup, by reviewing the skyrmion crystal approach for modeling of baryonic matters. This part also includes introduction of a magnetic field and a chiral chemical potential in the chiral effective model, so as to examine the chirality imbalance induced by a magnetic field. Then we explicitly see that the chiral imbalance density as proposed in Eq.(\[generic-rho5\]) indeed shows up in the present setup. Quantities related to the inhomogeneous chiral condensate as well as the baryon number density are also derived there. Sec. III provides the numerical analysis on the chiral imbalance distributions in the skyrmion crystal, and shows the emergence of the chira-imbalance density wave, and several related phenomena, such as correlation with the inhomogeneity of the chiral condensate. [[Conclusion]{}]{} for the present paper is given in Sec.IV. Appendix A compensates knowledge on symmetry properties for the chiral-imbalance distribution and the inhomogeneous chiral condensate, in the skyrmion crystal.
Chirality imbalance in skyrmion crystal under a magnetic field
==============================================================
In this section, we first introduce preliminary setups in studying the magnetic properties of the skyrmion crystal, such as the basic construction in the chiral limit (Part A) and see how the baryon number density is modified by the presence of a magnetic field which explicitly breaks the chiral symmetry (Part B). And then, we discuss how the magnetic-field driven $U(1)_A$ anomaly induces the chirality imbalance (Part C).
Skyrmion crystal approach
-------------------------
The Skyrme model [@Skyrme:1962vh] based on the 2-flavor chiral symmetry is described in the chiral limit by the following Lagrangian: $$\begin{aligned}
{\cal L}_{\rm Skyr} & = & \frac{f_\pi^2}{4}{\rm tr}\left[\partial_\mu U \partial^\mu U^\dagger\right] + \frac{1}{32g^2}{\rm tr}\left[U^\dagger \partial_\mu U,U^\dagger \partial_\nu U\right]^2,
\nonumber\\
\label{Lag1}\end{aligned}$$ where $U$ is the chiral field parameterized by the pion fields, $ f_\pi$ the pion decay constant, and $g$ the dimensionless coupling constant. In the skyrmion crystal approach, the chiral field $U$ is parameterized as $$\begin{aligned}
U=\phi_0+i\tau_a\phi_a,
\label{paraU}\end{aligned}$$ with $a = 1,2,3$, $\tau^a$ being the Pauli matrices, and the unitary constraint $(\phi_0)^2+(\phi_a)^2=1$. For later convenience, we also introduce unnormalized fields $\bar \phi_\alpha\;(\alpha=0,1,2,3)$, which are related to the corresponding normalized ones through $$\begin{aligned}
\phi_\alpha=
\frac{\bar\phi_\alpha}{\sqrt{\sum_{\beta=0}^3\bar\phi_\beta\bar\phi_\beta}}.\end{aligned}$$
We will consider the static skyrmion crystal formed by the static pion fields, $\phi_\alpha(t,x,y,z)=\phi_\alpha(x,y,z)$. In a crystal lattice with a periodicity of $2L$ (the size of the unit cell for a single crystal), the static pion fields can be expanded in terms of the Fourier series [@Lee:2003aq]: $$\begin{aligned}
\bar \phi_0(x,y,z)&=\sum_{a,b,c}\bar \beta_{abc}\cos(a\pi x/L)\cos(b\pi y/L)\cos(c\pi z/L)\nonumber\\
\bar \phi_1(x,y,z)&=\sum_{h,k,l}\bar \alpha_{hkl}^{(1)}\sin(h\pi x/L)\cos(k\pi y/L)\cos(l\pi z/L)\nonumber\\
\bar \phi_2(x,y,z)&=\sum_{h,k,l}\bar \alpha_{hkl}^{(2)}\cos(l\pi x/L)\sin(h\pi y/L)\cos(k\pi z/L)\nonumber\\
\bar \phi_3(x,y,z)&=\sum_{h,k,l}\bar \alpha_{hkl}^{(3)}\cos(k\pi x/L)\cos(l\pi y/L)\sin(h\pi z/L)\,,
\label{ansatz_1}\end{aligned}$$ where $h, k$ and $l$ are taken to be integers. In the present study, we shall construct the FCC crystal from the skyrmion approach. Then we need to impose some constraint conditions on the Fourier coefficient $\bar{\alpha}$ and $\bar{\beta}$ (for more on this, see [@Lee:2003aq]).
It is also interesting to note that in the skyrmion crystal approach the configuration of the $\phi_\alpha(x,y,z)$ can be rephrased as the inhomogeneous quark condensates like, $$\begin{aligned}
\phi_0(x,y,z) &\sim& \langle 0|\bar q q |0\rangle(x,y,z)\nonumber \\
\phi_{a} (x,y,z) &\sim&
\langle 0|\bar q i\gamma_5\tau_{a} q |0\rangle(x,y,z).
\label{qcond}\end{aligned}$$
The skyrmion crystal approach has the characteristic phenomena which is so-called topological phase transition. At some critical lattice size, this phase transition occurs in the skyrmion crystal. As a result, the FCC crystal structure changes to be the CC form. In terms of the phase transition, the low density region with the skyrmion crystal realized as the FCC form is called “skyrmion phase", while the high density region where the CC structure is manifest, called “half-skyrmion phase" [@Lee:2003aq; @Lee:2003rj; @Ma:2013ooa; @Ma:2013ela; @book; @Ma:2016npf; @Ma:2016gdd; @Harada:2015lma; @Kawaguchi:2018xug; @Kawaguchi:2018fpi].
Baryon number density in a magnetic field
-----------------------------------------
To consider the magnetic effect on the skyrmion crystal, we introduce the external vector field ${\cal V}_\mu$ and the external axial field ${\cal A}_\mu$, by gauging the chiral symmetry, $$\begin{aligned}
D_\mu U=\partial_\mu U-i[{\cal V}_\mu, U]+i\{{\cal A}_\mu,U\}.\end{aligned}$$ The magnetic field ($B$), the baryon chemical potential ($\mu_B$) and the chiral chemical potential ($\mu_5$) are embedded into the ${\cal V}_\mu$ and ${\cal A}_\mu$, $$\begin{aligned}
{\cal V}_\mu&=& Q_B\, \mu_B \delta_{\mu0}
+eQ_{\rm em} A_\mu\nonumber\\
{\cal A}_\mu&=&\mu_5 \delta_{\mu0}
{\bm 1}_{2\times 2},$$ where $Q_{B}=\frac{1}{3}{\bm 1}_{2\times 2}$ is the baryon number charge matrix, $Q_{\rm em}=\frac{1}{6}{\bm 1}_{2\times 2}+\frac{1}{2}\tau_3$ the electric charge matrix, and the magnetic field $B$ is incorporated into $A_\mu$. In this study, we consider a constant magnetic field $(B)$ along the z axis. Then, the magnetic field breaks the $O(3)$ symmetry down to the $O(2)$ symmetry. To respect the residual $O(2)$ symmetry, we choose the following symmetric gauge, $$\begin{aligned}
A_\mu=
-\frac{1}{2}B y\delta_\mu^{\;\;1}+\frac{1}{2}Bx \delta^{\;\;2}_{\mu}.
\label{symgauge}\end{aligned}$$
The covariantized Wess-Zumino-Witten (WZW) action ($\Gamma_{\rm WZW}=\int d^4x {\cal L}_{\rm WZW}$), which corresponds to the solution for the non-Abelian $U(2)_L \times U(2)_R$ anomaly equation, makes the baryon number density $\rho_B$ coupled to the baryon chemical potential $\mu_B$. Hence, the $\rho_B$ is given by [@Son:2004tq; @Son:2007ny; @He:2015zca] $$\begin{aligned}
\rho_B&=&\frac{\partial{\cal L}_{\rm wzw}}{\partial \mu_B}\Biggl|_{\mu_B=0}
=
\rho_W+\tilde\rho_{eB}
\end{aligned}$$ with $$\begin{aligned}
\rho_W&=
\frac{1}{24\pi^2}\epsilon^{0\nu\rho\sigma}{\rm tr}
\left[
(\partial_\nu U\cdot U^\dagger)(\partial_\rho U\cdot U^\dagger)(\partial_\sigma U\cdot U^\dagger)
\right]
\notag\\
\tilde\rho_{eB}
&=
\frac{i e}{16\pi^2}\epsilon^{0\nu\rho\sigma}
\partial_\sigma \left(
{\rm tr} [
A_\nu Q_{\rm em}
\{ \partial_\rho U, U^\dag \}
]
\right)
\,,
\label{rhoeB}\end{aligned}$$ where $\rho_W$ denotes the winding-number density and $\tilde\rho_{eB}$ is the induced baryon number density. By taking the symmetric gauge in Eq.(\[symgauge\]), $ \rho_B$ is evaluated as a function of a set of the topological $\phi_\alpha$ fields (Eq.(\[paraU\])) as follows: $$\begin{aligned}
\rho_B&=\frac{1}{24\pi^2}\epsilon^{0\nu\rho\sigma}{\rm tr}
\left[
(\partial_\nu U\cdot U^\dagger)(\partial_\rho U\cdot U^\dagger)(\partial_\sigma U\cdot U^\dagger)
\right] \Bigg|_{\phi_{\alpha}}
\nonumber\\
&
- \frac{eB}{4\pi^2}\left[(\partial_z\phi_3)\phi_0-(\partial_z\phi_0)\phi_3 \right]\nonumber\\
&
+ \frac{eB}{8\pi^2} \bigl(\left\{
[y\partial_y\phi_3]_{\rm disc}(\partial_z\phi_0)-[y\partial_y\phi_0]_{\rm disc}(\partial_z\phi_3)\right\}
\notag \\
& -
\left\{
(\partial_z\phi_3)[x\partial_x\phi_0]_{\rm disc}-(\partial_z\phi_0)[x\partial_x\phi_3]_{\rm disc}
\right\}\bigl)
\,.
\label{discsymmetricBD}\end{aligned}$$ Here we have used discretized form with a derivative operator such as $[y\partial_y\phi_3]_{\rm disc}$ to hold the translational invariance in the skyrmion crystal (the explicit expressions and more details are supplied in [@Kawaguchi:2018xug]).
In the skyrmion crystal approach, the skyrmion crystal configuration can be visualized through the baryon-number density distribution $\rho_B(x,y,z)$. Therefore the external magnetic field deforms the skyrmion configuration due to the existence of the induced baryon number density, as discussed in [@Kawaguchi:2018xug].
Induced chiral-imbalance density
--------------------------------
Now we discuss how the chirality imbalance shows up under a magnetic field through the $U(1)_A$ anomaly. We first note that in a way similar to the baryon number density, the covariantized WZW action regarding the $U(2)_L \times U(2)_R$ anomaly makes it possible to couple the chirality imbalance functional $\rho_5$ (hereafter called chiral imbalance distribution) with the chiral chemical potential $\mu_5$. It arises from the ${\cal V}-{\cal V}-{\cal A}$ type interaction terms, to be $$\begin{aligned}
\rho_5&=\
\frac{\partial{\cal L}_{\rm wzw}}{\partial \mu_5}\Biggl|_{\mu_5=0}
=
\frac{ie }{16\pi^2}\epsilon^{0\nu\rho\sigma}
\partial_\nu \left( {\rm tr} [A_\rho Q_{\rm em} [\partial_\sigma U, U^\dag]] \right)
\notag \\
&=
\frac{e }{8 \pi^2}
{\rm tr} \left[ Q_{\rm em} \left\{
\tilde{F}^{0\mu} H_\mu(U) + \tilde{F}^{0\mu}(U) A_\mu \right\} \right]
\,,
\label{rho5:rewritten}
\end{aligned}$$ where $H_\mu(U) = i [\partial_\mu U, U^\dag]$ denotes a pion-vector current field (sometimes called a chiral connection field) and $\tilde{F}^{0\mu} \equiv \frac{1}{2} \epsilon^{0\mu\nu\rho} F_{\nu \rho}$ is a generic magnetic field strength. Thus the chiral imbalance density $\rho_5$ is supplied by the pionic vector current coupled to the external magnetic field.
At this moment, we have arrived at our central formula, Eq.(\[rho5:rewritten\]), which is a generalization form for Eq.(\[generic-rho5\]). Indeed, this is in accord with the form of the $U(1)_A$ anomaly $\sim \tilde{F}_{\mu\nu} F^{\mu\nu}$, where one of the field strength $F_{\mu\nu}$ is replaced by the one constructed from [[the pionic-vector current $H_\mu(U)$]{}]{}.
Projecting the generic formula of Eq.(\[rho5:rewritten\]) onto the skyrmion crystal approach, we see that the $\rho_5$ is expressed by a set of skyrmion configurations parametrized by the topological $\phi_\alpha$ fields, as in Eq.(\[paraU\]). Under the symmetric gauge in Eq.(\[symgauge\]), the chiral imbalance distribution $\rho_5$ is thus evaluated as $$\begin{aligned}
\rho_5&=
\frac{eB}{4\pi^2}\bigl\{(\partial_z\phi_1)\phi_2-(\partial_z\phi_2)\phi_1\bigl\}
\notag\\
&
+ \frac{eB}{8\pi^2}\bigl(
[x\partial_x\phi_1]_{\rm disc}
+[y\partial_y\phi_1]_{\rm disc}
\bigl)(\partial_z\phi_2)
\notag \\
& -
\frac{ eB}{8\pi^2}\bigl(
[x\partial_x\phi_2]_{\rm disc}
+[y\partial_y\phi_2]_{\rm disc}
\bigl)(\partial_z\phi_1)
\,.
\label{symmetricCID}\end{aligned}$$ Here we have used the discretized form in similar way to Eq.(\[discsymmetricBD\]). Crucial to note here is that no chirality imbalance distribution emerges without nonzero magnetic field. Note also that the space averaged value of $\rho_5$ goes to zero: $\int_{\rm cube} d^3 x \rho_5=0$. This is because of the parity-odd property. (Or, more generically, it is due to the fact that no nontrivial configuration is presented for the external gauge field $A_\mu$ at the boundary of the target cube. See Eq.(\[rho5:rewritten\]).) Hence, these facts imply that the skyrmion crystal is turned into the local-chiral imbalance medium by the presence of a magnetic field, in which the local-chiral imbalance distribution would be expected to have a nontrivial correlation with the local-inhomogeneous chiral condensates in Eq.(\[qcond\]), as well as the local-baryon number density distribution given by $\rho_B$ in Eq.(\[discsymmetricBD\]).
Numerical results
=================
In this section we numerically examine the chiral imbalance distribution $\rho_5$ in Eq.(\[symmetricCID\]) and make an attempt to find its nontrivial correlation with the baryon matter structure, based on the skyrmion crystal approach under a magnetic field. The baryon matter structure can be monitored by examining the position dependence of $\rho_B$ in Eq. (\[discsymmetricBD\]), and the chiral imbalance distribution $\rho_5$ by Eq.(\[symmetricCID\]). We then note that the baryon number density $\rho_B$ and the chiral imbalance distribution $\rho_5$ are expressed as the function of the Fourier coefficients $\bar\beta_{abc}$ and $\bar\alpha_{hkl}^{(i)}$, as seen from Eq. (\[ansatz\_1\]). There the Fourier coefficients are determined by minimizing the per-baryon energy $
E/N_ B=-
\frac{1}{4}\int_{\rm cube}d^3x {\cal L}_{\rm Skyr},
$ with $N_B$ having been taken to be 4, and $\int_{\rm cube}=\int_{-L}^{L}dx\int_{-L}^{L}dy\int_{-L}^{L}dz$. (Note that the magnetically induced $\tilde{\rho}_{\rm eB}$ in Eq.(\[rhoeB\]) vanishes in the integral, because of the trivial configuration for the gauge field $A_\mu$ at the boundary of the target cube, hence it does not affect the total baryon number at all.) Then, the per-baryon energy is also expressed as a function of the Fourier coefficients $\bar\beta_{abc},\bar\alpha_{hkl}^{(i)}$ which are used as variational parameters in the numerical calculation. Once the strength of a magnetic field is fixed, the Fourier coefficients for a given set of crystal size $L$ is determined by minimizing the per-baryon energy. Thus, the Fourier coefficients $\bar\beta_{abc},\bar\alpha_{hkl}^{(i)}$ depend on the crystal size $L$ and a magnetic field scale $eB$. For numerical computations, we take $f_\pi=92.4\,{\rm MeV}$ and $g=5.93$ as inputs [@Ma:2016npf].
Chiral imbalance distribution on the skyrmion crystal: chiral-imbalance density wave
------------------------------------------------------------------------------------
In this subsection we examine the [[chirality imbalance]{}]{} on the skyrmion crystal configuration in the presence of a magnetic field, which can be visualized through the chiral imbalance distribution, $\rho_5$, and the baryon-number density distribution, $\rho_B$, respectively.
In Figs. \[eB400L200\] and \[eB800L200\], we plot the skyrmion crystal configurations and the chiral imbalance distribution $\rho_5$ in a low density region where $L=2.0\,{\rm fm}$ (corresponding to the skyrmion phase), with the magnetic field scale fixed to 400 MeV and 800 MeV, respectively. [[The magnitude of $\rho_5$ has been amplified by multiplying a factor of 10, because of its smallness compared to the baryon number density $\rho_B$]{}]{}.
First, see the left panels in the figures (the panels (a)), showing the skyrmion crystal configurations characterized by the baryon number density $\rho_B$ under a magnetic field. We then find that even in the presence of a magnetic field, the skyrmion crystal keeps the FCC structure, as was discussed in [@Kawaguchi:2018xug].
Looking at the middle panels (b) in Figs. \[eB400L200\] and \[eB800L200\]: one realizes that remarkably nontrivial phenomenon has been emergent. These panels display the chiral imbalance distributions in the skyrmion crystal, and show that the chirality imbalance shows up on the skyrmion crystal to be locally distributed on the FCC crystal configuration. Interesting enough, such a chiral imbalance distribution looks like forming a wave (with odd parity). This can be dubbed “[*chiral-imbalance density wave*]{}”, which flows quite differently from the baryon number density on the crystal.
From the left panels in the figures (panels (c)), we also see that the magnitude of the chiral imbalance distribution gets bigger, as a magnetic field increases, which is simply expected from the form of the functional of $\rho_5$ in Eq.(\[symmetricCID\]).
Next, in Figs. \[eB400L100\] and \[eB800L100\], we draw the skyrmion configurations and the chiral imbalance distribution in a high density region where $L=1.0\,{\rm fm}$ (corresponding to the half-skyrmion phase), with $\sqrt{eB}=400$ and $800$ MeV, respectively. At the first glance, one immediately finds that as a magnetic field gets bigger, the CC configuration dramatically becomes distorted (Figs. \[eB400L100\](a) and \[eB800L100\](a)), as was observed in [@Kawaguchi:2018xug].
As for the chiral imbalance distribution, again, the configuration of $\rho_5$ forms a wave (the chiral-imbalance density wave), which, however, looks quite similar to the skyrmion configuration except for the parity property. This is in contrast to the case of the skyrmion phase, as depicted in Figs. \[eB400L100\](b) and \[eB800L100\](b). This observation indicates a nontrivial consequence that the topological phase transition leads to the change of the chiral-imbalance density wave in shape.
Similarly to the case for the skrymion phase, Figs. \[eB400L100\](c) and \[eB800L100\](c) also show that the chiral imbalance distribution grows up as the baryonic matter approaches a higher-intense object influenced by a strong magnetic field (with $\sqrt{eB}=800\,{\rm MeV}$).
[cc]{}
![ The chiral imbalance distribution on the skyrmion configuration at $L=2.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.25[{\rm fm}]$.[]{data-label="eB400L200"}](Baryon_eB400_L200_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=2.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.25[{\rm fm}]$.[]{data-label="eB400L200"}](imbalance_eB400_L200_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=2.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.25[{\rm fm}]$.[]{data-label="eB400L200"}](BvsIeB400_L200_x.eps){width="5.7cm"}
[cc]{}
![ The chiral imbalance distribution on the skyrmion configuration at $L=2.0[{\rm fm}]$ for $\sqrt{eB}=800[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.25[{\rm fm}]$.[]{data-label="eB800L200"}](Baryon_eB800_L200_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=2.0[{\rm fm}]$ for $\sqrt{eB}=800[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.25[{\rm fm}]$.[]{data-label="eB800L200"}](imbalance_eB800_L200_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=2.0[{\rm fm}]$ for $\sqrt{eB}=800[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.25[{\rm fm}]$.[]{data-label="eB800L200"}](BvsIeB800_L200_x.eps){width="5.7cm"}
[cc]{}
![ The chiral imbalance distribution on the skyrmion configuration at $L=1.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.125[{\rm fm}]$.[]{data-label="eB400L100"}](Baryon_eB400_L100_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=1.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.125[{\rm fm}]$.[]{data-label="eB400L100"}](imbalance_eB400_L100_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=1.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.125[{\rm fm}]$.[]{data-label="eB400L100"}](BvsIeB400_L100_x.eps){width="5.7cm"}
[cc]{}
![ The chiral imbalance distribution on the skyrmion configuration at $L=1.0[{\rm fm}]$ for $\sqrt{eB}=800[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.125[{\rm fm}]$.[]{data-label="eB800L100"}](Baryon_eB800_L100_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=1.0[{\rm fm}]$ for $\sqrt{eB}=800[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.125[{\rm fm}]$.[]{data-label="eB800L100"}](imbalance_eB800_L100_xy.eps){width="5.5cm"}
![ The chiral imbalance distribution on the skyrmion configuration at $L=1.0[{\rm fm}]$ for $\sqrt{eB}=800[{\rm MeV}]$. (a) The contour plot of the skyrmion configuration on the x-y plane, $\rho_B(x, y, L/8)$. (b) The contour plot of the chiral imbalance distribution on the x-y plane, $\rho_5(x, y, L/8)$. (c) The distribution of the skyrmion and the chiral imbalance along the x axis specified at $y=z=L/8=0.125[{\rm fm}]$.[]{data-label="eB800L100"}](BvsIeB800_L100_x.eps){width="5.7cm"}
Chiral-imbalance density wave and chiral density wave
-----------------------------------------------------
In the previous work done by authors [@Kawaguchi:2018xug], a possible correlation between the inhomogeneous quark condensate and the deformation of the skyrmion crystal form was addressed. In this subsection, we shall further show a remarkable presence of a nontrivial correlation between the chiral imbalance distribution, i.e., the chiral-imbalance density wave, and the inhomogeneous-chiral condensate distribution, i.e., a chiral density wave, as was observed in [@Kawaguchi:2018xug]. As to the latter distribution, we may select the inhomogeneity of $\phi_1$ associated with the inhomogeneous quark condensate (see Eq.(\[qcond\])). This is because among the inhomogeneous chiral condensates $\sim \phi^\alpha$ in Eq.(\[qcond\]), only the $\phi_1$ has the same parity property as the $\rho_5$ along the $x$-direction (see Eq.(\[ansatz\_1\])), hence it is most convenient to compare the $\rho_5$ distribution with that of the $\phi_1$ along the $x$-axis as has been depicted in the distribution plots so far.
Before proceeding the comparison between the $\rho_5$ and the $\phi_1$, we first note from Figs \[eB400L200\], \[eB800L200\], \[eB400L100\], and \[eB800L100\] that the magnitude of the chiral-inhomogeneous density wave is much smaller (by a factor of 10) than the baryon number distribution (skyrmion configuration) in the system. Though it implies a small $U(1)_A$ anomaly effect, we may try to see how the local chirality imbalance is responsible for the nonzero inhomogeneity of $\phi_1$, in both the skyrmion and half-skyrmion phases, by comparing the wave forms and the peak structures for the chiral-inhomogeneous density wave and a chiral density wave. The observation like this would deduce a quantitative understanding on how much the $U(1)_A$ anomaly effect is encoded in the chiral condensate. To easily see and visually grasp a nontrivial correlation in such peak structures by eyes, we may amplify the amplitude of $\rho_5$ in the (half-) skyrmion phase, by a factor defined as $$\begin{aligned}
C_{400(800)}^{{\rm (h-)skyr}}
\equiv
\frac{\phi_1^{\rm max}(\bar{x},y=z=L/8)|_{\sqrt{eB}=400(800)\, {\rm MeV}} }
{\rho_5^{\rm max} (\bar{x}, y=z=L/8)|_{\sqrt{eB}=400(800) {\rm MeV}} }
\,,
\label{dimensionless}\end{aligned}$$ where “max” denotes the maximum value realized at $(x, y, z)=(\bar{x}, L/8, L/8)$ for given $\sqrt{e B}$ (in which the $\bar{x}$ is just a number, to be read off). Thus, the amplified chiral-imbalance distribution, ($\rho_5\times C_{400(800)}^{\rm (h-)skyr}$), is set to a dimensionless quantity as well as the $\phi_1$, and can have the same order of magnitude as what the $\phi_1$ can have.
In Fig. \[L200phi1rho5\] we plot the amplified chiral-imbalance distributions, ($\rho_5\times C_{400(800)}^{\rm skyr}$), and the inhomogeneity distributions for $\phi_1$, in the skyrmion and half-skyrmion phases, respectively. From this figure, we find the following features:
- As the strength of a magnetic field increases, the inhomogeneity of $\phi_1$ tend to be localized and the amplitude of $\phi_1$ becomes small, as discussed in [@Kawaguchi:2018xug].
- As for the correlation between the chiral-imbalance density wave and a chiral density wave depicted by the inhomogeneity of $\phi_1$, the peak point for the former does not match with that of the $\phi_1$-chiral density wave.
Moving on to the half-skyrmion phase, we make plots of the amplified chiral-imbalance distribution, $\rho_5\times C_{400(800)}^{\rm h-skyr}$, and the inhomogeneities of $\phi_1$ in Fig. \[L100phi1rho5\]. The figure tells us the following characteristic properties:
- In contrast to the skyrmion phase, the magnetic effect is insensitive to the inhomogeneous configuration of $\phi_1$, as discussed in [@Kawaguchi:2018xug].
- The periodicity of the chiral-imbalance density wave ($\rho_5\times C_{400(800)}^{\rm h-skyr}$) synchronizes with a chiral density wave formed by the $\phi_1$ inhomogeneity for any strength of a magnetic field.
It is of particular interest to note from the second item that in the half-skyrmion phase the presence of a magnetic field makes a nontrivial correlation between the chirality imbalance and the inhomogeneous quark condense, in terms of the periodicity for density wave distributions. Actually, the coincidence of the periodicity in the half-skrymion phase can analytically be understood as by the symmetry properties for the $\rho_5$ and $\phi_1$ on crystals given in Eqs. (\[rho5trans:1\]) - (\[phi1trans:2\]), in Appendix A.
Those nontrivial-wave correlations having a different aspect between the skrymion and half-skyrmion phases would provide us with [[a novel possibility: the presence of the chiral-imbalance density wave as a consequence of the $U(1)_A$ anomaly]{}]{} would be an important probe for the phase boundary between the skyrmion and half-skyrmion phases in the high-dense baryonic matter under a magnetic field.
[cc]{}
![ The distribution of $\phi_1(x,L/8,L/8)$ and $\rho_5(x, L/8, L/8)\times C_{400(800)}^{\rm skyr}$in the skyrmion phase where $L=2.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$ (a) and $\sqrt{eB}=800[{\rm MeV}]$ (b). []{data-label="L200phi1rho5"}](phi1rho5eB400L200v1.eps){width="5.5cm"}
![ The distribution of $\phi_1(x,L/8,L/8)$ and $\rho_5(x, L/8, L/8)\times C_{400(800)}^{\rm skyr}$in the skyrmion phase where $L=2.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$ (a) and $\sqrt{eB}=800[{\rm MeV}]$ (b). []{data-label="L200phi1rho5"}](phi1rho5eB800L200v1.eps){width="5.5cm"}
[cc]{}
![ The distribution of $\phi_1(x,L/8,L/8)$ and $\rho_5(x, L/8, L/8)\times C_{400(800)}^{\rm (h-)skyr}$ in the half-skyrmion phase where $L=1.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$ (a) and $\sqrt{eB}=800[{\rm MeV}]$ (b). []{data-label="L100phi1rho5"}](phi1rho5eB400L100v1.eps){width="5.5cm"}
![ The distribution of $\phi_1(x,L/8,L/8)$ and $\rho_5(x, L/8, L/8)\times C_{400(800)}^{\rm (h-)skyr}$ in the half-skyrmion phase where $L=1.0[{\rm fm}]$ for $\sqrt{eB}=400[{\rm MeV}]$ (a) and $\sqrt{eB}=800[{\rm MeV}]$ (b). []{data-label="L100phi1rho5"}](phi1rho5eB800L100v1.eps){width="5.5cm"}
Conclusions
===========
In summary, we proposed a novel possibility to create a chiral-imbalance medium in a high dense baryonic matter under a magnetic field. It is a chirality-imbalance that can be emerged due to the magnetic $U(1)_A$ anomaly coupled with a local-nontrivial inhomogeneity of a pion-vector current arising in the high-dense matter system, as was roughly sketched in the introductory part of the present paper (Eq.(\[generic-rho5\])). This imbalance is in contrast to the conventional one generated by the gluonic $U(1)_A$ anomaly in the case of hot QCD matter. Hence it would provide a new chance to examine how much the $U(1)_A$ anomaly can be relevant to the net chiral asymmetry, compared to the spontaneously broken chiral symmetry.
To demonstrate the crucial contribution of the proposed chiral-imbalance in Eq.(\[generic-rho5\]), in the present paper we have taken the skyrmion crystal approach to make a model description for baryonic/high dense matters, to explicitly show that a nontrivial chiral imbalance distribution can indeed be induced in the modeled skyrmion crystal, due to the presence of a magnetic field. Interestingly enough, the chiral imbalance distribution turned out to take a wave form in a high density region, when the inhomogeneous chiral condensate develops to form a chiral density wave. This implies the emergence of a nontrivial density wave for the explicitly broken $U(1)_A$ current simultaneously with the chiral density wave for the spontaneously broken chiral-flavor current. This emergent wave was dobbed “[*chiral-imbalance density wave*]{}”.
We further observed that the topological phase transition in the skyrmion crystal model (between the skyrmion and half-skyrmion phases) leads to the change of the chiral-imbalance density wave in shape. In particular, it was shown that in the half-skyrmion phase, the periodicity of the chirality-imbalance distribution synchronizes with the the inhomogeneous chiral condensate, in contrast to the case of the skyrmion phase where the chiral-imbalance density wave flows with a different periodicity from a chiral density wave.
The emergence of the chiral-imbalance density wave in dense matters could give a crucial contribution to studies on the chiral phase transition, as well as the nuclear matter structure, in compact stars under a magnetic field, and would give a significant impact on analyses regarding the inhomogeneous chiral condensate through introducing the chiral density wave, as was mentioned in Introduction of the present paper.
Also, our findings would make an important step to make deeper understanding of the role of the $U(1)_A$ anomaly in a sense of the origin of baryon mass as well as the baryon matter structure, and would give some impacts on an interdisciplinary physics like those raised in Introduction of this paper, e.g. the chirality imbalance for chiral neutrinos in supernova explosions and similar related chiral transport physics in dense matter systems.
[[ We are grateful to Yong-Liang Ma for several useful comments. This work was supported in part by the JSPS Grant-in-Aid for Young Scientists (B) No. 15K17645 (S.M.), National Science Foundation of China (NSFC) under Grant No. 11747308 (S.M.), and the Seeds Funding of Jilin University (S.M.). K.M. was also partially supported by the JSPS Grant-in-Aid for JSPS Research Fellow No. 18J1532. ]{}]{}
Translational symmetries for $\rho_5$ and $\phi_1$
==================================================
In this Appendix, we provide a supplement on the translational symmetry properties for $\rho_5$ and $\phi_1$, which can help understand the coincidence for the periodicity among them in the half-skyrmion phase, as has been observed in Fig. \[L100phi1rho5\].
Under the translational symmetry, $\rho_5$ transforms in the following way: in skyrmion phase with the FCC structure, we have $$\begin{aligned}
&\rho_5(x,y,z)=\rho_5(x+L,y+,L,z)
\notag \\
& =-\rho_5(x+L,y,z+L)=-\rho_5(x,y+L,z+L)
\,,
\label{rho5trans:1}\end{aligned}$$ while in half-skyrmion phase with the CC structure, we have $$\begin{aligned}
&\rho_5(x,y,z)=-\rho_5(x+L,y,z)
\notag \\
& =-\rho_5(x,y+L,z)
=\rho_5(x,y,z+L).
\label{rho5trans:2}\end{aligned}$$
Under the translational symmetry, $\phi_1$ transforms as follows: in skyrmion phase with the FCC structure, we have $$\begin{aligned}
&\phi_1(x,y,z)=-\phi_1(x+L,y+L,z)
\notag \\
& =-\phi_1(x+L,y,z+L)=\phi_1(x,y+L,z+L)\,,
\label{phi1trans:1}\end{aligned}$$ while in half-skyrmion phase with the CC structure, we have $$\begin{aligned}
&\phi_1(x,y,z)=-\phi_1(x+L,y,z)
\notag \\
& =\phi_1(x,y+L,z)=\phi_1(x,y,z+L)
\,.
\label{phi1trans:2}\end{aligned}$$
[99]{}
D. Kharzeev, A. Krasnitz and R. Venugopalan, Phys. Lett. B [**545**]{}, 298 (2002) doi:10.1016/S0370-2693(02)02630-8 \[hep-ph/0109253\]. D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A [**797**]{}, 67 (2007) doi:10.1016/j.nuclphysa.2007.10.001 \[arXiv:0706.1026 \[hep-ph\]\]. D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A [**803**]{}, 227 (2008) doi:10.1016/j.nuclphysa.2008.02.298 \[arXiv:0711.0950 \[hep-ph\]\]. K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D [**78**]{}, 074033 (2008) doi:10.1103/PhysRevD.78.074033 \[arXiv:0808.3382 \[hep-ph\]\].
D. T. Son and P. Surowka, Phys. Rev. Lett. [**103**]{}, 191601 (2009) doi:10.1103/PhysRevLett.103.191601 \[arXiv:0906.5044 \[hep-th\]\].
D. E. Kharzeev and H. U. Yee, Phys. Rev. D [**83**]{}, 085007 (2011) doi:10.1103/PhysRevD.83.085007 \[arXiv:1012.6026 \[hep-th\]\]. Y. Burnier, D. E. Kharzeev, J. Liao and H. U. Yee, Phys. Rev. Lett. [**107**]{}, 052303 (2011) doi:10.1103/PhysRevLett.107.052303 \[arXiv:1103.1307 \[hep-ph\]\].
M. Hongo, Y. Hirono and T. Hirano, Phys. Lett. B [**775**]{}, 266 (2017) doi:10.1016/j.physletb.2017.10.028 \[arXiv:1309.2823 \[nucl-th\]\].
H. U. Yee and Y. Yin, Phys. Rev. C [**89**]{}, no. 4, 044909 (2014) doi:10.1103/PhysRevC.89.044909 \[arXiv:1311.2574 \[nucl-th\]\]. Y. Hirono, T. Hirano and D. E. Kharzeev, arXiv:1412.0311 \[hep-ph\].
L. Adamczyk [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett. [**114**]{}, no. 25, 252302 (2015) doi:10.1103/PhysRevLett.114.252302 \[arXiv:1504.02175 \[nucl-ex\]\]. Y. Yin and J. Liao, Phys. Lett. B [**756**]{}, 42 (2016) doi:10.1016/j.physletb.2016.02.065 \[arXiv:1504.06906 \[nucl-th\]\]. X. G. Huang, Rept. Prog. Phys. [**79**]{}, no. 7, 076302 (2016) doi:10.1088/0034-4885/79/7/076302 \[arXiv:1509.04073 \[nucl-th\]\].
D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang, Prog. Part. Nucl. Phys. [**88**]{}, 1 (2016) doi:10.1016/j.ppnp.2016.01.001 \[arXiv:1511.04050 \[hep-ph\]\].
K. Hattori and X. G. Huang, Nucl. Sci. Tech. [**28**]{}, no. 2, 26 (2017) doi:10.1007/s41365-016-0178-3 \[arXiv:1609.00747 \[nucl-th\]\].
S. Shi, Y. Jiang, E. Lilleskov and J. Liao, Annals Phys. [**394**]{}, 50 (2018) doi:10.1016/j.aop.2018.04.026 \[arXiv:1711.02496 \[nucl-th\]\].
N. S. Manton, Phys. Rev. D [**28**]{}, 2019 (1983). doi:10.1103/PhysRevD.28.2019
F. R. Klinkhamer and N. S. Manton, Phys. Rev. D [**30**]{}, 2212 (1984). doi:10.1103/PhysRevD.30.2212
L. D. McLerran, E. Mottola and M. E. Shaposhnikov, Phys. Rev. D [**43**]{}, 2027 (1991). doi:10.1103/PhysRevD.43.2027
G. D. Moore, Phys. Lett. B [**412**]{}, 359 (1997) doi:10.1016/S0370-2693(97)01046-0 \[hep-ph/9705248\].
G. D. Moore and K. Rummukainen, Phys. Rev. D [**61**]{}, 105008 (2000) doi:10.1103/PhysRevD.61.105008 \[hep-ph/9906259\].
D. Bodeker, G. D. Moore and K. Rummukainen, Phys. Rev. D [**61**]{}, 056003 (2000) doi:10.1103/PhysRevD.61.056003 \[hep-ph/9907545\].
R. Derradi de Souza, T. Koide and T. Kodama, Prog. Part. Nucl. Phys. [**86**]{}, 35 (2016) doi:10.1016/j.ppnp.2015.09.002 \[arXiv:1506.03863 \[nucl-th\]\].
A. A. Andrianov, V. A. Andrianov, D. Espriu and X. Planells, Phys. Lett. B [**710**]{}, 230 (2012) doi:10.1016/j.physletb.2012.02.072 \[arXiv:1201.3485 \[hep-ph\]\].
A. A. Andrianov, D. Espriu and X. Planells, Eur. Phys. J. C [**73**]{}, no. 1, 2294 (2013) doi:10.1140/epjc/s10052-013-2294-0 \[arXiv:1210.7712 \[hep-ph\]\].
M. Kawaguchi, M. Harada, S. Matsuzaki and R. Ouyang, Phys. Rev. C [**95**]{}, no. 6, 065204 (2017) doi:10.1103/PhysRevC.95.065204 \[arXiv:1612.00616 \[nucl-th\]\]. A. A. Andrianov, V. A. Andrianov, D. Espriu, A. E. Putilova and A. V. Iakubovich, Acta Phys. Polon. Supp. [**10**]{}, 977 (2017) doi:10.5506/APhysPolBSupp.10.977 \[arXiv:1707.06150 \[hep-ph\]\]. A. A. Andrianov, V. A. Andrianov, D. Espriu, A. V. Iakubovich and A. E. Putilova, EPJ Web Conf. [**158**]{}, 03012 (2017) doi:10.1051/epjconf/201715803012 \[arXiv:1710.01760 \[hep-ph\]\].
A. A. Andrianov, V. A. Andrianov, D. Espriu, A. V. Iakubovich and A. E. Putilova, Phys. Part. Nucl. Lett. [**15**]{}, no. 4, 357 (2018). doi:10.1134/S1547477118040027
A. E. Putilova, A. V. Iakubovich, A. A. Andrianov, V. A. Andrianov and D. Espriu, EPJ Web Conf. [**191**]{}, 05014 (2018). doi:10.1051/epjconf/201819105014
A. A. Andrianov, V. A. Andrianov, D. Espriu, A. V. Iakubovich and A. E. Putilova, arXiv:1902.10705 \[hep-ph\].
J. Charbonneau and A. Zhitnitsky, JCAP [**1008**]{}, 010 (2010) doi:10.1088/1475-7516/2010/08/010 \[arXiv:0903.4450 \[astro-ph.HE\]\]. D. Grabowska, D. B. Kaplan and S. Reddy, Phys. Rev. D [**91**]{}, no. 8, 085035 (2015) doi:10.1103/PhysRevD.91.085035 \[arXiv:1409.3602 \[hep-ph\]\]. M. Kaminski, C. F. Uhlemann, M. Bleicher and J. Schaffner-Bielich, Phys. Lett. B [**760**]{}, 170 (2016) doi:10.1016/j.physletb.2016.06.054 \[arXiv:1410.3833 \[nucl-th\]\]. G. Sigl and N. Leite, JCAP [**1601**]{}, no. 01, 025 (2016) doi:10.1088/1475-7516/2016/01/025 \[arXiv:1507.04983 \[astro-ph.HE\]\].
N. Yamamoto, Phys. Rev. D [**93**]{}, no. 6, 065017 (2016) doi:10.1103/PhysRevD.93.065017 \[arXiv:1511.00933 \[astro-ph.HE\]\]. Y. Masada, K. Kotake, T. Takiwaki and N. Yamamoto, Phys. Rev. D [**98**]{}, no. 8, 083018 (2018) doi:10.1103/PhysRevD.98.083018 \[arXiv:1805.10419 \[astro-ph.HE\]\].
Q. Li [*et al.*]{}, Nature Phys. [**12**]{}, 550 (2016) doi:10.1038/nphys3648 \[arXiv:1412.6543 \[cond-mat.str-el\]\]. B. Q. Lv [*et al.*]{}, Phys. Rev. X [**5**]{}, no. 3, 031013 (2015) doi:10.1103/PhysRevX.5.031013 \[arXiv:1502.04684 \[cond-mat.mtrl-sci\]\].
S. Y. Xu [*et al.*]{}, Science [**349**]{}, 613 (2015). doi:10.1126/science.aaa9297
K. Nishiyama, S. Karasawa and T. Tatsumi, Phys. Rev. D [**92**]{}, 036008 (2015) doi:10.1103/PhysRevD.92.036008 \[arXiv:1505.01928 \[nucl-th\]\].
H. Abuki, EPJ Web Conf. [**129**]{}, 00036 (2016) doi:10.1051/epjconf/201612900036 \[arXiv:1609.04605 \[hep-ph\]\].
H. Abuki, JPS Conf. Proc. [**20**]{}, 011017 (2018). doi:10.7566/JPSCP.20.011017
M. Kawaguchi, Y. L. Ma and S. Matsuzaki, Phys. Rev. C [**98**]{}, no. 3, 035803 (2018) doi:10.1103/PhysRevC.98.035803 \[arXiv:1804.09015 \[nucl-th\]\].
J. Wess and B. Zumino, Phys. Lett. [**37B**]{}, 95 (1971). doi:10.1016/0370-2693(71)90582-X
E. Witten, Nucl. Phys. B [**223**]{}, 422 (1983). doi:10.1016/0550-3213(83)90063-9
K. Fukushima and K. Mameda, Phys. Rev. D [**86**]{}, 071501 (2012) doi:10.1103/PhysRevD.86.071501 \[arXiv:1206.3128 \[hep-ph\]\].
G. Cao and X. G. Huang, Phys. Lett. B [**757**]{}, 1 (2016) doi:10.1016/j.physletb.2016.03.066 \[arXiv:1509.06222 \[hep-ph\]\].
T. H. R. Skyrme, “A Unified Field Theory of Mesons and Baryons,” Nucl. Phys. [**31**]{}, 556 (1962). doi:10.1016/0029-5582(62)90775-7
I. R. Klebanov, Nucl. Phys. B [**262**]{}, 133 (1985). doi:10.1016/0550-3213(85)90068-9
E. Witten, Nucl. Phys. B [**223**]{}, 433 (1983). doi:10.1016/0550-3213(83)90064-0 C. Naya and P. Sutcliffe, Phys. Rev. Lett. [**121**]{}, no. 23, 232002 (2018) doi:10.1103/PhysRevLett.121.232002 \[arXiv:1811.02064 \[hep-th\]\].
H. J. Lee, B. Y. Park, D. P. Min, M. Rho and V. Vento, “A Unified approach to high density: Pion fluctuations in skyrmion matter,” Nucl. Phys. A [**723**]{}, 427 (2003) doi:10.1016/S0375-9474(03)01452-0 \[hep-ph/0302019\].
H. J. Lee, B. Y. Park, M. Rho and V. Vento, Nucl. Phys. A [**741**]{}, 161 (2004) doi:10.1016/j.nuclphysa.2004.06.010 \[hep-ph/0307111\]. Y. L. Ma, M. Harada, H. K. Lee, Y. Oh, B. Y. Park and M. Rho, Phys. Rev. D [**88**]{}, no. 1, 014016 (2013) Erratum: \[Phys. Rev. D [**88**]{}, no. 7, 079904 (2013)\] doi:10.1103/PhysRevD.88.014016, 10.1103/PhysRevD.88.079904 \[arXiv:1304.5638 \[hep-ph\]\]. Y. L. Ma, M. Harada, H. K. Lee, Y. Oh, B. Y. Park and M. Rho, Phys. Rev. D [**90**]{}, no. 3, 034015 (2014) doi:10.1103/PhysRevD.90.034015 \[arXiv:1308.6476 \[hep-ph\]\]. B. Y. Park and V. Vento, “Skyrmion approach to finite density and temperature” in [*The Multifaceted Skyrmion (2nd edition)*]{} (World Scientific, Singapore, 2017), ed. M. Rho and I. Zahed.
Y. L. Ma and M. Harada, arXiv:1604.04850 \[hep-ph\].
Y. L. Ma and M. Rho, Sci. China Phys. Mech. Astron. [**60**]{}, no. 3, 032001 (2017) doi:10.1007/s11433-016-0497-2 \[arXiv:1612.06600 \[nucl-th\]\].
M. Harada, H. K. Lee, Y. L. Ma and M. Rho, “Inhomogeneous quark condensate in compressed Skyrmion matter,” Phys. Rev. D [**91**]{}, no. 9, 096011 (2015) doi:10.1103/PhysRevD.91.096011 \[arXiv:1502.02508 \[hep-ph\]\].
M. Kawaguchi, Y. L. Ma and S. Matsuzaki, arXiv:1810.12880 \[nucl-th\].
W. G. Paeng, T. T. S. Kuo, H. K. Lee, Y. L. Ma and M. Rho, Phys. Rev. D [**96**]{}, no. 1, 014031 (2017) doi:10.1103/PhysRevD.96.014031 \[arXiv:1704.02775 \[nucl-th\]\].
Y. L. Ma, H. K. Lee, W. G. Paeng and M. Rho, arXiv:1804.00305 \[nucl-th\]. Y. L. Ma and M. Rho, Phys. Rev. D [**99**]{}, no. 1, 014034 (2019) doi:10.1103/PhysRevD.99.014034 \[arXiv:1810.06062 \[nucl-th\]\].
D. T. Son and A. R. Zhitnitsky, Phys. Rev. D [**70**]{}, 074018 (2004) doi:10.1103/PhysRevD.70.074018 \[hep-ph/0405216\].
D. T. Son and M. A. Stephanov, Phys. Rev. D [**77**]{}, 014021 (2008) doi:10.1103/PhysRevD.77.014021 \[arXiv:0710.1084 \[hep-ph\]\].
B. R. He, Phys. Rev. D [**92**]{}, no. 11, 111503 (2015) doi:10.1103/PhysRevD.92.111503 \[arXiv:1510.04683 \[hep-ph\]\].
[^1]: mkawaguchi@hken.phys.nagoya-u.ac.jp
[^2]: synya@jlu.edu.cn
[^3]: Several discussions on chirality-imbalance effects, induced with a strong magnetic field for hadron physics, arising through the WZW term, have been made [[so far [@Fukushima:2012fg; @Cao:2015cka]]{}]{}. However, to our best knowledge, no references have picked up the interaction term as in Eq.(\[generic-rho5\]), which involves the pion-vector current part, instead of the external photon field.
[^4]: [[Going beyond such qualitative arguments on the baryon description, it has recently been suggested that the skyrmion approach could quantitatively be consistent with realistic light nuclei with a desirable size of the binding energy [@Naya:2018kyi]. ]{}]{}
|
---
abstract: 'We analyze the branching of center vortices in $SU(3)$ Yang-Mills theory in maximal center gauge. When properly normalized, we can define a branching probability that turns out to be independent of the lattice spacing (in the limited scaling window studied here). The branching probability shows a rapid change at the deconfinement phase transition which is much more pronounced in space slices of the lattice as compared to time slices. Though not a strict order parameter (in the sense that it vanishes in one phase) the branching probability is thus found to be a reliable indicator for both the location of the critical temperature and the geometric re-arrangement of vortex matter across the deconfinement phase transition.'
author:
- 'Felix Spengler, Markus Quandt and Hugo Reinhardt'
title: 'Branching of Center Vortices in $SU(3)$ Lattice Gauge Theory'
---
Introduction {#Abschn: Einleitung}
============
The center vortex picture is one of the most intuitive and prolific explanation of colour confinement in strong interactions. It was first proposed by Mack and Petkova [@mack], but lay dormant until the advent of new gauge fixing techniques which permitted the detection of center vortex structures directly within lattice Yang-Mills configurations [@DelDebbio]. These numerical studies have revealed a large mount of evidence in favour of a center vortex picture of confinement: The center vortex density detected on the lattice in the maximual center gauge after center projection properly scales with the lattice constant in the continuum limit and therefore must be considered a physical quantity [@Langfeld:1997jx]. When center vortices are removed from the ensemble of gauge field configurations the string tension is lost in the temporal Wilson loop. Conversely, keeping the center vortex configurations only, the static quark potential extracted from the temporal Wilson loop is linearly rising at all distances [@DelDebbio]. Center vortices also seem to carry the non-trivial topological content of gauge fields: the Pontryagin index can be understood as self-intersection number of center vortex sheets in four Euclidean dimensions [@Engelhardt:1999xw; @Reinhardt:2001kf] or in terms of the writhing number of their 3-dimensional projection which are loops [@Reinhardt:2001kf]. For the colour group $SU(2)$, attempts to restore the structure of the underlying (fat) vortices suggest that the topological charge also receives contributions from the colour structure of self-intersection regions of such fat vortices [@Nejad:2015aia; @Nejad:2016fcl]. Removing the center vortex content of the gauge fields makes the field configuration topological trivial and simultaneously restores chiral symmetry. The Pontryagin index [@Bertle:2001xd] as well as the quark condensate [@Gattnar:2004gx; @Hollwieser:2008tq] are both lost when center vortices are removed, see also [@Reinhardt:2003ku; @*Reinhardt:2002cm]. In the case of $SU(3)$, this link of center vortices to both confinement and chiral symmetry breaking has also been observed directly in lattice simulations of the low lying hadron spectrum [@OMalley:2012]. Finally, the center vortex picture also gives a natural explanation of the deconfinement phase transition which appears as a depercolation transition from a confined phase of percolating vortices to a smoothly interacting gas of small vortices winding dominantly around the compactified Euclidean time axis [@Engelhardt:1999fd].
Center vortices detected on the lattice after center projection form loops in $D = 3$ dimensions and surfaces in $D = 4$; in both cases, they live on the *dual* lattice and are closed due to Bianchi’s identity. While a gas of closed loops can be treated analytically, see e.g. [@Oxman:2017tel], an ensemble of closed sheets is described by string theory, which has to be treated numerically. The main features of $D = 4$ center vortices detected on the lattice after center projection, such as the emergence of the string tension or the order of the deconfinement transition, can all be reproduced in an effective *random center vortex model*: in this approach, vortices are described on a rather coarse dual lattice (to account for the finite vortex thickness), with the action given by the vortex area (Nambu-Goto term) plus a penalty for the curvature of the vortex sheets to account for vortex stiffness [@Engelhardt:1999wr; @engelhardt; @Quandt:2004gy]. The model was originally formulated for the gauge group $SU(2)$ [@Engelhardt:1999wr] and later extended to $SU(3)$ in Ref. [@engelhardt].
The $SU(3)$ group has two non-trivial center elements $z_{1/2} = e^{\pm i 2 \pi/3}$ which are related by $z^2_1 = z_2 \, , \, z^2_2 = z_1$. Due to this property two $z_1$ center vortices can fuse to a single $z_2$ vortex sheet and vice versa (see Fig. \[fig:2\] below). This vortex branching is a new element absent in the gauge group $SU(2)$. In Ref. [@engelhardt] it was found within the random center vortex model that the deconfinement phase transition is accompanied with a strong reduction of the vortex branching and fusion. In the present paper we investigate the branching of center projected lattice vortices found in the maximal center gauge.
This paper is organized as follows: In section \[sec:branch\] we describe the geometrical and physical properties of vortex branching and develop the necessary quantities to study this new phenomenon on the lattice. Section \[sec:setup\] gives details on our numerical setup and the lattice parameters and techniques used in the simulations. The results are presented and discussed in section \[sec:results\], and we close with a short summary and an outlook to future investigations.
Center vortex branching points {#sec:branch}
==============================
On the lattice, center vortices are detected by first fixing all links $U_\mu(x)$ to a suitable center gauge, preferably the so-called *maximal center gauge* (MCG), cf. eq. (\[mcg\]) below. This condition attempts to find a gauge transformation which brings each link, on average, as close as possible to a center element. The transformed links are then projected on the nearest center-element, $U_\mu(x) \to Z_\mu(x) \in \mathbb{Z}_N$, and since it was already close, we can hope that the resulting $\mathbb{Z}_N$ theory preserves the relevant features of the original Yang-Mills theory. In fact, it has been shown that the string tension is retained to almost $100\%$ under center projection for the colour group $G=SU(2)$ and still to about $62\%$ for $G=SU(3)$ [@langfeld], while the string tension disappears for all $G$ if vortices are removed [@DelDebbio; @forcrand]. Also the near-zero modes of the Dirac operator relevant for chiral symmetry breaking disappear if vortices are removed from the physical ensemble [@Gattnar:2004gx; @Hollwieser:2008tq].
The center projected theory is much simpler to analyze. Since all links are center-valued after projection, so are the plaquettes. If such a center-valued plaquette happens to be non-trivial, it is said to be pierced by a center vortex, i.e. the corresponding *dual* plaquette is considered part of a center vortex world sheet. For $G=SU(3)$, in particular, we associate a center projected plaquette $Z_{\mu\nu}(x)$ in the original lattice with a *triality* $q_{\alpha\beta}(x^\ast) \in \{0,1,2\}$ on the dual lattice, where $$\begin{aligned}
Z_{\mu\nu}(x) = \exp\left[i\,\frac{\pi}{3}\,\epsilon_{\mu\nu\alpha\beta}\,
q_{\alpha\beta}(x^\ast)\right]\,.
\label{triality}\end{aligned}$$ Here, the usual sum convention over greek indices is in effect, and the footpoint of the dual plaquette is defined as $x^\ast = x + (\mathbf{e}_\mu + \mathbf{e}_\nu -
\mathbf{e}_\alpha-\mathbf{e}_\beta)/2$. As the reader may convince herself, this assignment is such that the initial and dual plaquette link with each other. The triality can be viewed as a quantized flux of field strength flowing through the original plaquette. It is, however, only defined modulo $N=3$ so that a $q=1$ vortex is equivalent to $q=-2$, which in turn is a $q=2$ vortex with opposite direction of flux. This ambiguity gives rise to different geometrical interpretations (see. Fig. \[fig:2\]), but it does not affect the quantities studied in the present work. The vortex world sheet itself is now composed of all connected non-trivial dual plaquettes. This world sheet may *branch* along links of the dual lattice where three or more vortex plaquettes join, cf. the left panel of Fig. \[fig:1\].
For the actual measurement, we study the branching in the orignal lattice, where the branching link is dual to an elementary cube, while the plaquettes attached to the branching link are dual to the plaquettes on the surface of the cube. Geometrically, this can be visualized in the 3D slice[^1] of the original lattice which contains the cube, cf. Fig. \[fig:1\]: in this slice, the vortex plaquettes are projected onto links which are dual to the non-trivial plaquettes and represent the center flux through the plaquettes. Vortex matter thus appears as a network of closed lines composed of non-trivial dual links. These thin lines are the projection vortices in which the center flux of the unprojected (thick) vortex is compressed into a narrow tube with a cross section of only a single plaquette.
Vortex branching in a 3D slice occurs at *branching points* which are the projection of the branching links in the 4D lattice. Geometrically, the branching points are located in the middle of the cubes dual to the branching links as illustrated in Fig. \[fig:1\]: the vortex lines entering an elementary cube must pierce the plaquettes on its surface, and so up to six vortices can join at any given point of the dual 3D slice.[^2] We call this number $\nu(x^\ast) \in \{0,\ldots,6\}$ of vortex lines joining at a site $x^\ast$ of the dual 3D slice its *branching genus*. Clearly, $\nu=0$ means that no vortex passes through $x^\ast$, while $\nu=2$ means that a vortex goes in and out without branching (but possibly changing its direction). The cases $\nu=4$ and $\nu=6$ correspond to vortex self-intersections (or osculation points), which are also present in the case of $G=SU(2)$. The odd numbers $\nu=3$ and $\nu=5$, however, are genuine vortex branchings which cannot be observed in $SU(2)$ and are thus a new feature of the center projected theory for the more complex colour group $SU(3)$. In the present study, we investigate the distribution of branching points in 3D slices across the deconfinement phase transition.
![Illustration of vortex branching. The single and double arrows on the lines represent triality $q=1$ and $q=2$, respectively. The left figure represents a $\nu=3$ vortex branching in the full $4D$ lattice. The graphic in the middle shows the same situation from a 3D slice, where the vortex plaquettes are replaced by three flux tubes joining at a branching point $x^\ast$. The tubes enter the elementary cube surrounding $x^\ast$ by piercing three of its six surface plaquettes. The right figure gives a simplified picture where only the branching vortex lines are displayed.[]{data-label="fig:1"}](branch3D "fig:"){width="4cm"} ![Illustration of vortex branching. The single and double arrows on the lines represent triality $q=1$ and $q=2$, respectively. The left figure represents a $\nu=3$ vortex branching in the full $4D$ lattice. The graphic in the middle shows the same situation from a 3D slice, where the vortex plaquettes are replaced by three flux tubes joining at a branching point $x^\ast$. The tubes enter the elementary cube surrounding $x^\ast$ by piercing three of its six surface plaquettes. The right figure gives a simplified picture where only the branching vortex lines are displayed.[]{data-label="fig:1"}](cube "fig:"){width="4cm"} ![Illustration of vortex branching. The single and double arrows on the lines represent triality $q=1$ and $q=2$, respectively. The left figure represents a $\nu=3$ vortex branching in the full $4D$ lattice. The graphic in the middle shows the same situation from a 3D slice, where the vortex plaquettes are replaced by three flux tubes joining at a branching point $x^\ast$. The tubes enter the elementary cube surrounding $x^\ast$ by piercing three of its six surface plaquettes. The right figure gives a simplified picture where only the branching vortex lines are displayed.[]{data-label="fig:1"}](branch2b "fig:"){width="3cm"}
It should also be mentioned that the case $\nu=1$ would represent a vortex end-point which is forbidden by Bianchi’s identity, i.e. flux conservation modulo 3. More precisely, Bianchi’s identity in the present case states that the sum of the trialities of all plaquettes in an elementary cube of a 3D slice must vanish modulo $N$ (the number of colours). This holds even for cubes on the edge of the lattice if periodic boundary conditions are employed. Clearly, this rule is violated if the cube has only $\nu=1$ non-trivial plaquette, which is hence forbidden. In our numerical study, the number of $\nu=1$ branching points must then be exactly zero, which is a good test on our algorithmical book-keeping.
Finally, we must also stress that $\nu=6$ branchings for the colour group $G=SU(2)$ are *always* self-intersections or osculation points, while they can also be interpreted as *double vortex branchings* in the case of $G=SU(3)$. With the present technique, we cannot keep track of the orientation of vortices (i.e. the direction of vortex flux), and hence are unable to distinguish double branchings from complex self-intersections. Fortunately, $\nu=6$ branching points are so extremely rare that they can be neglected entirely for our numerical analysis. If we speak of vortex branching, we thus always mean the cases $\nu=3$ and $\nu=5$, which only exist for $G=SU(3)$, and for which all possible interpretations involve a single vortex branching. Table \[tab:2\] summarizes again the different sorts of vortex branchings and their geometrical meaning.
[c|l]{} $\nu=0$ & no vortex\
$\nu=1$ & vortex endpoint, forbidden by Bianchi’s identity\
$\nu=2$ & non-branching vortex\
$\nu=3$ & simple vortex branching\
$\nu=4$ & vortex self-intersection/osculation\
$\nu=5$ & complex vortex branching\
$\nu=6$ & complex vortex self-intersection/osculation/double branching\
Numerical setup {#sec:setup}
===============
We simulate $SU(3)$ Yang-Mills theory on a hypercubic lattice using the standard Wilson action as a sum over all plaquettes $U_P \equiv U_{\mu \nu}(x)$ $$\begin{aligned}
S=\sum_P \left[1-\frac{1}{2N}\,\mathrm{tr}(U_P + U_P^\dagger )\right].\end{aligned}$$ Configurations are updated with the pseudo-heatbath algorithm due to Cabibbo and Marinari [@su3heatbath] applied to a full set of $SU(2)$ subgroups. To study finite temperature, we reduce the extent $L_t$ of the Euclidean time direction, while keeping the spatial extent $L_s \gg L_t$ to eliminate possible finite size effects, $$\begin{aligned}
T = \frac{1}{a(\beta)\,L_t}\,.\end{aligned}$$ Since the variation of $L_t$ only allows for a rather coarse temperature grid, we have also varied the lattice spacing $a(\beta)$ by considering three different couplings $\beta$ within the scaling window.[^3] Table \[tab:1\] lists the lattice extents and coupling constants used in our simulations.
For each run, the lattice was thermalized using at least 100 heatbath sweeps, and measurements were then taken on $70$ to $200$ thermalized configurations (depending on $L_t$), with $10$ sweeps between measurements to reduce auto-correlations. For each measurement, the following sequence of steps was performed:
![Ambiguities in the interpretation of $SU(3)$ vortex branching. In the top line, the simple branching of a $q=2$ vortex on the left can be equivalently described as three $q=1$ vortices emenating from a common source, i.e. as $\mathbb{Z}_3$ center monopole. Similarly, the self-intersection of a $q=1$ vortex in the bottom line (left), is equivalent to an osculation point of e.g. two $q=1$ vortices (middle) or a $q=1$ and a $q=2$ vortex (right).[]{data-label="fig:2"}](branch2b "fig:"){width="2.5cm"} ![Ambiguities in the interpretation of $SU(3)$ vortex branching. In the top line, the simple branching of a $q=2$ vortex on the left can be equivalently described as three $q=1$ vortices emenating from a common source, i.e. as $\mathbb{Z}_3$ center monopole. Similarly, the self-intersection of a $q=1$ vortex in the bottom line (left), is equivalent to an osculation point of e.g. two $q=1$ vortices (middle) or a $q=1$ and a $q=2$ vortex (right).[]{data-label="fig:2"}](branch2a "fig:"){width="2.5cm"}\
![Ambiguities in the interpretation of $SU(3)$ vortex branching. In the top line, the simple branching of a $q=2$ vortex on the left can be equivalently described as three $q=1$ vortices emenating from a common source, i.e. as $\mathbb{Z}_3$ center monopole. Similarly, the self-intersection of a $q=1$ vortex in the bottom line (left), is equivalent to an osculation point of e.g. two $q=1$ vortices (middle) or a $q=1$ and a $q=2$ vortex (right).[]{data-label="fig:2"}](branch4a "fig:"){width="2.5cm"} ![Ambiguities in the interpretation of $SU(3)$ vortex branching. In the top line, the simple branching of a $q=2$ vortex on the left can be equivalently described as three $q=1$ vortices emenating from a common source, i.e. as $\mathbb{Z}_3$ center monopole. Similarly, the self-intersection of a $q=1$ vortex in the bottom line (left), is equivalent to an osculation point of e.g. two $q=1$ vortices (middle) or a $q=1$ and a $q=2$ vortex (right).[]{data-label="fig:2"}](branch4b "fig:"){width="2.5cm"} ![Ambiguities in the interpretation of $SU(3)$ vortex branching. In the top line, the simple branching of a $q=2$ vortex on the left can be equivalently described as three $q=1$ vortices emenating from a common source, i.e. as $\mathbb{Z}_3$ center monopole. Similarly, the self-intersection of a $q=1$ vortex in the bottom line (left), is equivalent to an osculation point of e.g. two $q=1$ vortices (middle) or a $q=1$ and a $q=2$ vortex (right).[]{data-label="fig:2"}](branch4c "fig:"){width="2.5cm"}
#### Gauge fixing to maximal center gauge (MCG):
This is achieved by maximizing the functional $$\begin{aligned}
F=\frac{1}{V} \sum\limits_{\{x,\mu\}} \left|\frac{1}{N}\,\mathrm{tr}\,U_\mu(x)\right|^2\,,
\label{mcg}\end{aligned}$$ under gauge rotations, where $N=3$ is the number of colours and $V=\prod_\mu L_\mu$ is the lattice volume. The main gauge fixing algorithm used in this study is iterated overrelaxation [@overrelax] in which the local quantity $$\begin{aligned}
F_x = \sum_\mu \left( \bigl|\mathrm{tr}\, \big\{\Omega (x) U_\mu (x)\big\} \bigr|^2 +
\bigl|\mathrm{tr} \, \big\{U_\mu (x-\hat{\mu})\Omega^\dagger (x)\big\} \bigr|^2\right)\end{aligned}$$ is maximized with respect to a local gauge rotation $\Omega(x) \in SU(3) $ at each lattice site $x$. We stop this process when the largest relative change of $F_x$ at all sites $x$ falls below $10^{-6}$. More advanced g.f. techniques such as simulated annealing [@gf_anneal] or Landau gauge preconditioners [@gf_landau] from multiple random initial gauge copies have also been tested. While such methods are known to have a significant effect on the propagators of the theory in any gauge [@gf_green; @gf_green2; @*gf_green3], we found that they have very little effect, at our lattice sizes, on the gauge fixing functional and the vortex geometry investigated here. For the production runs, we have therefore reverted to simple overrelaxation with random starts.
#### Center projection:
Once a configuration is fixed to MCG, each link is projected to its closest center element $U_\mu(x) \rightarrow Z_\mu (x)$ by first splitting off the phase $$\mathrm{tr}\, U_\mu(x) = \left|\mathrm{tr}\, U_\mu(x)\right| \cdot
e^{ 2 \pi i \delta_\mu / N},$$ which defines $\delta_\mu \in \mathbb{R}$ modulo $N$. After rounding $(\delta_\mu \,\mathrm{mod}\, N)$ to the closest integer $q_\mu \in [0,N-1]$, we can then extract the center projected link as $$\begin{aligned}
Z_\mu(x) \equiv \exp\left(i\,\frac{2\pi}{N}\,q_\mu\right){\mathbb{1}}\in \mathbb{Z}_N\,.\end{aligned}$$ In the case of $SU(3)$, we will call the integer $q_\mu \in \{0,1,2\} $ the *triality* of a center element. As mentioned earlier, the triality is only defined modulo 3, i.e. $q_\mu= -2$ is identical to $q_\mu=1$. While this ambiguity alters the geometric interpretation of a given vortex distribution (cf. Fig. \[fig:2\]), both the existence of a vortex branching point and its genus (the number of vortex lines meeting at the point) are independent of the triality assignment.
#### Vortex identification:
After center projection, all links are center valued, and so are the projected plaquettes. If such a center-valued plaquette happens to be non-trivial, we interpret this as a center vortex piercing the plaquette, i.e. the corresponding dual plaquette is part of the center vortex world sheet. The exact formula for the triality assignment of the vortex plaquettes was given in eq. (\[triality\]) above. For the computation of the area density of vortices, it is sufficient to consider a 2D plane in the original lattice and count the number of non-trivial plaquettes after center projection.
#### branching points:
As explained earlier, center vortices appear within a time or space slice as a network of links on the lattice dual to the slice. At each point $x^\ast$ of this dual 3D slice, between $\nu=0,2,\ldots,6$ vortex lines may join. Since the point $x^\ast$ is the center of an elementary cube of the original time or space slice, the vortices joining in $x^\ast$ must enter or exit the cube and hence pierce some or all of the six plaquettes on its surface. We can thus determine $\nu(x^\ast)$ simply by counting the number of non-trivial plaquettes on elementary cubes in 3D slices of the lattice, and assign it to the possible branching point $x^\ast$ in the middle of the cube.
[c||ccccc|ccccc|ccccc]{} $\beta$& & &\
$ L_t$ & 3 & 4 & 5 & 6 & 9 & 4 & 5 & 6 & 7 & 10 & 4 & 5 & 6 & 7 & 10\
\# configs & 140 & 102 & 106 & 90 & 92 & 122 & 98 & 80 & 73 & 73 & 119 & 107 & 75 & 74 & 78\
Results {#sec:results}
=======
The vortex area density is known to be a physical quantity in the sense that it scales properly with the lattice spacing $a(\beta)$ (see below) [@Langfeld:1997jx]. This entails that the overall amount of vortex matter quickly decays with increasing coupling $\beta$. To improve the statistics, we therefore choose coupling constants $\beta$ near the lower end of the scaling window $ 5.7 \lesssim\beta \lesssim 7 $, cf. table \[tab:1\]. Since this implies a rather coarse lattice, we must ensure that the lattice size in the short time direction does not become too small. For the values of $\beta$ chosen in our simulation, $L_t = \big[a(\beta)\,T\big]^{-1} \gg 1$ for temperatures at least up to $T\lesssim 2 T^\ast$, which is entirely sufficient for the present purpose. We have also checked that increasing the spatial volume from $L_s=16$ to $L_s=24$ has only marginal effects on the results, so that finite volume errors are also under control. In the final results, we only include the findings for the larger lattice extent $L_s = 24$.
The properties of vortex matter are intimately related to the choice and implementation of the gauge condition, as well as the absence of lattice artifacts. In particular, the vortex area density only survives the continuum limit if MCG is chosen and implemented accurately, and the lattice spacing is sufficiently small to suppress artifacts. As an independent test of these conditions, we have therefore re-analyzed the area density $\rho$ of vortex matter. In lattice units, this is defined as the ratio $$\begin{aligned}
\hat{\rho}(\beta) = a(\beta)^2\,\rho =
\frac{\# \text{non-trivial\, center plaquettes}}{\# \text{total\, plaquettes}}
\label{xvdens}\end{aligned}$$ in every 2D plane within the lattice. (We average over all planes in the full lattice or in appropriate 3D slices in order to improve the statistics.) After gauge fixing and center projection, the measurement of the vortex density is therefore a simple matter of counting non-trivial plaquettes. If we assume that the vortex area density is a physical quantity that survives the continuum limit, we should have $\rho = c\,\sigma$, where $\sigma$ is the physical string tension and $c$ is a dimensionless numerical constant. A random vortex scenario [@mack] entails $\sigma = \frac{3}{2} \rho$ for $G=SU(3)$ which corresponds to $c=0.67$. Previous lattice studies found a somewhat smaller value of about $c=0.5$ instead, indicating that the random vortex picture for MCG vortices at $T=0$ is not always justified [@langfeld]. In lattice units, these findings translate into $$\begin{aligned}
\frac{\hat{\rho}(\beta)}{\hat{\sigma}(\beta)} =
\frac{a(\beta)^2\,\rho}{a(\beta)^2\,\sigma} = \frac{\rho}{\sigma} = c \simeq 0.5
\qquad\qquad\text{indep.~of $\beta$ in scaling window}\,.
\label{vdens}\end{aligned}$$ For our values of the coupling as in table \[tab:1\], we have not measured the area density $\hat{\rho}(\beta)$ at $T=0$ directly, but instead took the data from the largest temporal extent $L_t = 10$ which corresponds to a temperature $T/T^\ast \approx 0.55$ deep within the confined phase. Since the string tension and the vortex density do not change significantly until very close to the phase transition, the $L_t=10$ data should still be indicative for the values at $T=0$. From these results and the string tension data $\hat{\sigma}(\beta)$ in Ref. [@lucini], the ratio (\[vdens\]) can then be determined as follows:
[c|ccc]{} $\beta$& $5.8 $ & $ 5.85 $ & $ 5.9 $\
$c$ & $0.558$ & $0.573$ & $0.591$\
\[tab:3\]
As can be seen from this chart, the ratio (\[vdens\]) is indeed roughly constant in the considered coupling range, and also in fair agreement with previous lattice studies [@langfeld], given the fact that we did not really make a $T=0$ simulation. In addition, inadequately gauge fixed configurations would show increased randomness which would lead to a significant drop in the vortex density as compared to the string tension data from Ref. [@lucini], and hence a much smaller value of $c$. We thus conclude that our chosen lattice setup and gf. algorithm are sufficient for the present investigation.
Next we study the finite temperature behaviour of vortex matter. The critical deconfinement temperature for $G=SU(3)$ is given by $T^\ast / \sqrt{\sigma} \approx 0.64$ [@lucini] . Since we do not measure the string tension independently, we can use eq. (\[vdens\]) $$\begin{aligned}
\frac{T^\ast}{\sqrt{\rho_0}} = \frac{T^\ast}{\sqrt{\sigma}}\,\sqrt{\frac{\sigma}{\rho_0}}
= \frac{T^\ast / \sqrt{\sigma}}{\sqrt{c}} \approx 0.90\end{aligned}$$ to determine the critical temperature in units of the zero-temperature vortex density $\rho_0 \equiv \rho(T=0)$ which sets the scale in our simulations. In absolute units, $$\begin{aligned}
\sqrt{\rho_0} = \sqrt{c\,\sigma} \approx 330\,\mathrm{MeV}\,.\end{aligned}$$ From the results in Fig. \[fig:3\] we see that there is roughly a 50% drop in the vortex density at the critical temperature, which is consistent with the findings of Ref. [@langfeld]. A complete loss of vortex matter at $T^\ast$ would mean that both the temporal *and* spatial string tension would vanish in the deconfined phase, contrary to lattice results [@sigma_spatial]. What happens instead is a *percolation phase transition* in which the geometric arrangement of vortices changes from a mostly random ensemble to a configuration in which most vortices are aligned along the short time direction [@Engelhardt:1999fd]. Since this leads to a nearly vanishing vortex density in space slices while the average density only drops mildly, the density in time slices and the associated spatial string tension must even increase for $T > T^\ast$.
This considerations imply that a good order parameter for confinement in the vortex picture should be sensitive to the randomness or order in the geometric arrangement of vortex matter and, as a consequence, should behave differently in temporal or spatial 3D slices of the lattice. A prime candidate in $SU(3)$ Yang-Mills theory is the 3-volume density of *branching points*, since it is directly defined in 3D slices and describes deviations of the vortex cluster from a straight aligned ensemble. This has previously been studied in the effective center vortex model [@engelhardt] where indeed a significant drop of vortex branching was observed in the deconfined phase, but not directly in lattice Yang-Mills theory.
![Scaling of the volume density of vortex branching in space slices of the lattice. The physical density (\[rbx\]) (*left*) shows no apparent scaling violations. For comparison, the dimensionless density (\[rb\]) (*right*) shows the amount of scaling violations to be expected for the present range of couplings. Error bars for the physical density are much larger since they also include uncertainties in the physical scale taken from Ref. [@lucini].[]{data-label="fig:4"}](rho_b_space "fig:"){width="8.7cm"} ![Scaling of the volume density of vortex branching in space slices of the lattice. The physical density (\[rbx\]) (*left*) shows no apparent scaling violations. For comparison, the dimensionless density (\[rb\]) (*right*) shows the amount of scaling violations to be expected for the present range of couplings. Error bars for the physical density are much larger since they also include uncertainties in the physical scale taken from Ref. [@lucini].[]{data-label="fig:4"}](rho_b_hut_space "fig:"){width="8.7cm"}
Since vortex branching implies a deviation from a straight vortex flow, we expect that it is suppressed in the deconfined phase where most vortices wind directly around the short time direction. In addition, the residual branching for $T > T^\ast$ should be predominantly in a space direction (since the vortices are already temporally aligned) and should hence be mostly visible in *time slices*, where the vortex matter is expected to still form large percolating clusters. In space slices, by contrast, vortices are mostly aligned (along the time axis) in the deconfined phase, and the suppression of the remnant branching for $T > T^\ast$ should be much more pronounced.
To test these expectations, we have measured the (dimensionless) volume density of branching points $$\begin{aligned}
\hat{\rho}_B \equiv \frac{\text{\# branching points in lattice dual to 3D slice}}
{\text{\# total sites in lattice dual to 3D slice}} =
\frac{\text{\# elementary cubes in 3D slice with $\nu \in \{3,5\}$ }}
{\text{\# all elementary cubes in 3D slice}}\,,
\label{rb}\end{aligned}$$ by assigning the vortex genus $\nu \in \{0,\ldots,6\} $ to all elementary cubes in a 3D slice, cf. section \[sec:branch\], and counting them. (To improve the statistics, we have averaged over space- and time slices separately using the same thermalized configurations.) Generally, we find
1. vortex endpoints with $\nu=1$ do not appear, i.e. vortices are closed in accordance with Bianchi’s identity;
2. vortex branchings are rare as compared to $\nu=2$ non-branching vortex matter;
3. complex vortex branchings with $\nu=5$ are very rare and significantly reduced as compared to the simple branchings with $\nu=3$; numerically, the $\nu=5$ branchings contribute with only $0.1\ldots 1.0 \%$ to the total branching probability.
To construct a quantity which has the chance of scaling to the continuum, we must express the branching density in physical units, $$\begin{aligned}
\rho_B(T,\beta) \equiv \frac{\hat{\rho}_B(T,\beta)}{a(\beta)^3}\,,
\label{rbx}\end{aligned}$$ where $a(\beta)$ is the lattice spacing at coupling $\beta$, which we take from Ref. [@lucini]. Eq. (\[rbx\]) is indeed a physical quantity as can be be seen directly from the result in Fig. \[fig:4\] where the data for all $\beta$ considered here fall on a common curve. Since we only considered a limited range of couplings $\beta$, one could be worried that possible scaling violations in $\rho_B$ would not be very pronounced. As can be seen from the right panel of Fig. \[fig:4\], this is not the case: the dimensionless density (\[rb\]), for instance, exhibits large scaling violations which are clearly visible even for our restricted range of couplings. This gives a strong indication that the branching density $\rho_B(T)$ really survives the continuum limit, even though further simulations at large couplings would be helpfull to corroborate this fact.
![Volume density of vortex branching points in physical units, measured in space slices (*left*) and time slices (*right*). Error bars include statistical errors and uncertainties in in the physical scale taken from Ref. [@lucini].[]{data-label="fig:5"}](rho_b_space "fig:"){width="8.7cm"} ![Volume density of vortex branching points in physical units, measured in space slices (*left*) and time slices (*right*). Error bars include statistical errors and uncertainties in in the physical scale taken from Ref. [@lucini].[]{data-label="fig:5"}](rho_b_time "fig:"){width="8.7cm"}
From Fig. \[fig:5\], the physical branching density indeed shows a rapid drop at the critical temperature $T=T^\ast$, while it stays roughly constant below and above $T^\ast$. In particular, the maximal value is expected at $T \to 0$. We have not made independent measurements at $T=0$, but the available data from $L_t=9$ and $L_t=10$ corresponding to $T/T^\ast = 0.55$ should still be indicative for the value at zero temperature since the vortex properties are known to show no significant change until very close to the phase transition. With this assumption, we find, in absolute units, $$\begin{aligned}
\rho_B(0) \approx 5.86 \,\mathrm{fm}^{-3} = (0.56\,\mathrm{fm})^{-3}\,.
\label{rb0}\end{aligned}$$ There is also a remnant branching density in the deconfined phase, but this is much smaller in space slices ($20\%$ of $\rho_B(0)$) than in time slices $(60\%$), in agreement with our geometrical discussion of vortex branching above. In fact, the branching density in time slices even increases slightly with the temperature within the deconfined phase.
Next, we want to demonstrate that the steep drop in the branching density is *not* due to an overall reduction of vortex matter itself, but rather signals a geometrical re-arrangement. Instead of studying $\rho_B / \rho$ directly, we make a small detour and first introduce the *branching probability* $$\begin{aligned}
q_B \equiv \frac{\text{\# elementary cubes in 3D slice with $\nu \in \{3,5\}$ }}
{\text{\# all elementary cubes in 3D slice with $\nu \neq 0$}}\,,
\label{qb}\end{aligned}$$ which gives the likelihood that a vortex which enters an elementary cube of edge length equal to the lattice spacing $a(\beta)$ will actually branch within that cube. The branching probability $q_B$ itself cannot be a physical quantity since it is expected to be proportional to the lattice spacing $a$ near the continuum limit.[^4] This entails that the *branching probability per unit length* $$\begin{aligned}
w_B(T,\beta) \equiv \frac{q_B(T,\beta)}{a(\beta)}
\label{wb}\end{aligned}$$ could be a physical quantity. As can be seen from Fig. \[fig:6\], this is indeed the case as the curves for $w_B$ for all available couplings fall on a common curve. The temperature dependence of the physical quantity $w_B(T)$ is very similar to the branching density in Fig. \[fig:4\], with the drop at $T=T^\ast$ being reduced from $75\%$ to about $50\%$. The qualitative features of the branching probability per unit length are, however, very similar to the branching point density, and both are physical quantities that scale to the continuum.
![Branching probability per unit length (\[wb\]) in physical units, measured in space slices (*left*) and time slices (*right*). Error bars include statistical errors and uncertainties in the physical scale taken from Ref. [@lucini].[]{data-label="fig:6"}](w_space "fig:"){width="8.7cm"} ![Branching probability per unit length (\[wb\]) in physical units, measured in space slices (*left*) and time slices (*right*). Error bars include statistical errors and uncertainties in the physical scale taken from Ref. [@lucini].[]{data-label="fig:6"}](w_time "fig:"){width="8.7cm"}
Next we want to show that the branching probability per unit length $w_B$ is actually related to the ratio $\rho_B / \rho$ of branching points and vortex matter density. To see this, we consider an arbitrary 3D slice containing $V$ sites and thus also $V$ elementary cubes. The number of cubes of branching genus $\nu$ is denoted by $N_\nu$, and obviously $\sum_{\nu=0}^6 N_\nu = V$. Then the dimensionless branching density (\[rb\]) can be expressed with eq. (\[qb\]) as $$\begin{aligned}
\hat{\rho}_B &= \frac{N_3 + N_5}{V}
= q_B \cdot \frac{\sum\limits_{\nu=2}^6 N_\nu}{V}
= 3 q_B\,\frac{\sum\limits_{\nu=2}^6 \big[\nu + (2-\nu)\big] N_\nu}{6V}
= 3 q_B\,\frac{\sum\limits_{\nu=2}^6 \nu N_\nu}{6V}\cdot \left \{ 1 -
\frac{\sum_{\nu=2}^6 (\nu-2) N_\nu}{\sum_{\nu=2}^6 \nu N_\nu}\right\}
= 3 q_B\,\hat{\rho}\,\lambda
\label{rel0}\end{aligned}$$ with the dimensionless factor $$\begin{aligned}
\lambda \equiv 1 - \frac{\sum\limits_{\nu=2}^6 (\nu-2) N_\nu}{\sum\limits_{\nu=2}^6 \nu N_\nu}
\in [0,1]\,.
\label{lambda}\end{aligned}$$ In the last step in eq. (\[rel0\]), we have used the fact that a cube with branching genus $\nu$ has $\nu$ non-trivial plaquettes on its surface, each of which is shared with an adjacent cube. Thus, the sum $\sum_\nu \nu N_\nu$ counts every non-trivial plaquette twice, and the dimensionless vortex area density eq. (\[xvdens\]) becomes, after averaging over all planes in the 3D slice, $$\hat{\rho} = \frac{\frac{1}{2}\,\sum\limits_{\nu=0}^6 \nu N_\nu}{3 V}
= \frac{\sum\limits_{\nu=2}^6 \nu N_\nu}{6V}\,,$$ since a 3D slice with $V$ sites and periodic boundary conditions contains a total of $3V$ plaquettes. After inserting appropriate factors of the lattice spacing in eq. (\[rel0\]), we obtain the exact relation $$\begin{aligned}
\rho_B(T) = 3\,w_B(T)\,\rho(T) \,\lambda(T, a)\,.
\label{exa}\end{aligned}$$ As indicated, the coefficient $\lambda$ may depend on the temperature and the lattice spacing, but it must fall in the range $[0,1]$. As a consequence, we obtain an exact inequality between physical quantities, $$\begin{aligned}
\rho_B(T) \le 3 \,w_b(T) \rho(T)\,,
\label{ineq}\end{aligned}$$ which must be valid at all temperatures. Moreover, the deviation from unity in the coefficient $\lambda$ can be estimated, from eq. (\[lambda\]), $$\begin{aligned}
\lambda = 1 - \frac{\sum\limits_{\nu=2}^6 (\nu-2)N_\nu}{\sum\limits_{\nu=2}^6 \nu N_\nu}
= 1 - \frac{N_3 + N_5}{\sum\limits_{\nu=2}^6 \nu N_\nu} +
2 \,\frac{N_4 + N_5 + 2 N_6}{\sum\limits_{\nu=2}^6 \nu N_\nu} = 1 -
\frac{1}{6}\,\frac{\hat{\rho}_B}{\hat{\rho}} + \mathcal{O}\big(\frac{N_4}{N_2}\big)
= 1 - \frac{1}{6}\,\frac{\rho_B(T)}{\rho(T)}\,a + \mathcal{O}\big(\frac{N_4}{N_2}\big)\,.\end{aligned}$$ Here, the leading correction to unity vanishes in the continuum limit $a \to 0$ since both $\rho_B$ and $\rho$ are physical. Furthermore, the next-to-leading term has the simple branching $\nu=3$ removed and starts with the probability of self-intersection or osculation, which is small and presumably also proportional to $a$, by the same argument that led from eq. (\[qb\]) to eq. (\[wb\]) above. Thus, it is conceivable that $\lambda(T,a) = 1 + \mathcal{O}(a)$ and eq. (\[exa\]) turns into the relation $$\begin{aligned}
w_B(T) = \frac{1}{3}\,\frac{\rho_B(T)}{\rho(T)}
\label{conject}\end{aligned}$$ for $a \to 0$. This is renormalization group invariant. We have tested this conjecture numerically by computing the relevant coefficient $\lambda(T, a)$ from eq. (\[lambda\]). The result is presented in Fig. \[fig:7\], where we accumulate all available data for all temperatures and lattice spacings. As can be seen, $\lambda$ is indeed in the range $[0,1]$, independent of temperature and very close to unity. Since the overall statistical uncertainty is about $5\%$ and our calculations were all done at the lower end of the scaling window with a relatively large lattice spacing $a$, our numerics are at least compatible with $\lambda=1$ and hence eq. (\[conject\]) in the continuum limit. Further calculations with larger and finer lattices are clearly necessary to corroborate this conjecture.
![The ratio $\lambda$ of physical quantities from eq. (\[lambda\]). Data comprises all available couplings and temperatures. Statistical errors are generally at the $5\%$ level, but no error bars have been displayed to improve the readability of the plot.[]{data-label="fig:7"}](lambda){width="12cm"}
Eq. (\[conject\]) shows that the drop of the branching density $\rho_B$ at the phase transition is *not* due to an overall reduction of vortex matter $\rho$, since the branching probability per unit length, $w_B \sim \rho_B / \rho$ shows the same qualitative behaviour as $\rho_B$, even after scaling out the overall vortex density. The conclusion is that both the branching point density $\rho_B(T)$ from eq. (\[rbx\]) and the branching probability $w_B(T)$ per unit length eq. (\[wb\]) can be used as a reliable indicator for the phase transition, and as a signal for the change in geometrical order of the vortices at the deconfinement transition. Our findings in full Yang-Mills theory match the general expectations discussed above and also comply with the predictions made in the random vortex world-surface model [@engelhardt].
Conclusion
==========
In this work, we have studied the probability of center vortex branching within $SU(3)$ Yang-Mills theory on the lattice. The general expectation, confirmed only in models so far, was that the branching probability should be sensitive to the geometry of vortex clusters and thus provide an alternative indicator for the deconfinement phase transition. We were able to corroborate this conjecture: both the branching point density $\rho_B(T)$ and the branching probability per unit length $w_B(T)$ are independent of the lattice spacing and exhibits a steep drop at the critical temperature, though a remnant branching probability remains even above $T^\ast$. This effect is much more pronounced in space slices of the original lattice, which clearly indicates a dominant alignment of vortices along the short time direction within the deconfined phase. The same conclusion can be drawn from the renormalization group invariant relation $w_B \sim \rho_B / \rho$, which proves that the drop in the branching density is *not* due to an overall reduction of the vortex matter $\rho$, but instead must be caused by the change in the geometry of the vortex cluster.
In future studies, it would be interesting to directly control the branching of vortices and study its effect on the confinement and the chiral symmetry breaking e.g. through the Dirac spectrum in the background of such branching-free configurations. The control over vortex branching could also address the obvious conjecture that the different (first) order of the phase transition for $G=SU(3)$ as compared to the weaker second order transition of $G=SU(2)$ is a result of the new geometrical feature of vortex branching.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by Deutsche Forschungsgemeinschaft (DFG) under contract Re 856/9-2.
[9]{} G. Mack and V. B. Petkova, Annals Phys. [**125**]{} (1980) 117.
L. Del Debbio, M. Faber, J. Greensite and S. Olejnik, Phys. Rev. D [**55**]{} (1997) 2298 \[hep-lat/9610005\]
K. Langfeld, H. Reinhardt and O. Tennert, Phys. Lett. B [**419**]{}, 317 (1998) \[hep-lat/9710068\].
M. Engelhardt and H. Reinhardt, Nucl. Phys. B [**567**]{}, 249 (2000) \[hep-th/9907139\].
H. Reinhardt, Nucl. Phys. B [**628**]{}, 133 (2002) \[hep-th/0112215\].
S. M. Hosseini Nejad and M. Faber, JHEP [**1709**]{} (2017) 068 S. M. H. Nejad, M. Faber and R. H[ö]{}llwieser, JHEP [**1510**]{} (2015) 108 \[arXiv:1508.01042 \[hep-lat\]\].
R. Bertle, M. Engelhardt and M. Faber, Phys. Rev. D [**64**]{}, 074504 (2001) \[hep-lat/0104004\].
J. Gattnar *et al.*, Nucl. Phys. B [**716**]{}, 105 (2005) \[hep-lat/0412032\].
R. Hollwieser *et al.*, Phys. Rev. D [**78**]{}, 054508 (2008) \[arXiv:0805.1846\].
H. Reinhardt and T. Tok, Phys. Rev. D [**68**]{}, 065004 (2003) \[hep-th/0302100\];
H. Reinhardt *et al.*, Phys. Rev. D [**66**]{}, 085004 (2002) \[hep-th/0203012\].
E. O’Malley, W. Kamleh, D. Leinweber and P. Moran, Phys. Rev. **D86** (2012) 054503 \[arXiv:1112.2490v2\] K. Langfeld, O. Tennert, M. Engelhardt and H. Reinhardt, Phys. Lett. **B452** (1999) 301;\
M. Engelhardt, K. Langfeld, H. Reinhardt and O. Tennert, Phys. Rev. D [**61**]{}, 054504 (2000) \[hep-lat/9904004\].
L. E. Oxman and H. Reinhardt, Eur. Phys. J. C [**78**]{}, no. 3, 177 (2018) \[arXiv:1712.08056 \[hep-th\]\].
M. Engelhardt and H. Reinhardt, Nucl. Phys. B [**585**]{}, 591 (2000) \[hep-lat/9912003\].
M. Engelhardt, M. Quandt and H. Reinhardt, Nucl. Phys. B [**685**]{} (2004) 227 \[hep-lat/0311029\]
M. Quandt, H. Reinhardt and M. Engelhardt, Phys. Rev. D [**71**]{}, 054026 (2005) \[hep-lat/0412033\].
K. Langfeld, Phys. Rev. D [**69**]{} (2004) 014503 \[hep-lat/0307030\]
Ph. de Forcrand and M. D’Elia, Phys. Rev. Lett. **82** (1999) 4582 \[hep-lat/9901020\]
N. Cabibbo and E. Marinari, Phys. Lett. **B119** (1982) 387
Jeffrey E. Mandula and Micheal Ogilvie, Phys. Lett. **B248** (1990) 156-158
V. Bornyakov, D. Komarov and M. Polykarpov, Phys. Lett. **B497** (2001) \[hep-lat/0009035\]
T. Kovacs and E. Tomboulis Phys. Lett. **B463** (1999) 104 \[hep-lat/9905029\]
A. Sternbeck *et al.*, Phys. Rev. **D72** (2005) 014507
G. Burgio, M. Quandt and H. Reinhardt Phys. Rev. Lett. **102** (2009) 032002 \[arXiv:0807.3291\];
G. Burgio *et al.*, Phys. Rev. **D95** (2017) 014503 \[arXiv:1608.05795\]
B. Lucini, M. Teper and U. Wenger, JHEP [**0401**]{} (2004) 061 \[hep-lat/0307017\]
G. S. Bali *et al.*, Int. J. Mod. Phys. **C04** (1993) 1179 \[hep-lat/9308003\]
[^1]: Such slices are obtained by holding either the Euclidean time coordinate $x_0$ (*time slice*) or a space coordinate $x_i$ (*space slice*) fixed.
[^2]: Equivalently, up to six vortex plaquettes in $D=4$ can join a common branching link.
[^3]: Finer temperature resolutions through the use of anisotropic lattices proved to be unnecessary for the present investigation.
[^4]: To see this, assume that the probability of branching in a cube of edge length $a \ll 1$ is $q \ll 1$, and consider a cube of length $n a$ composed of $n^3$ sub-cubes of length $a$. Since vortices are stiff, most non-branching vortices do not change their direction if $a \ll 1$ and just pass straight through $n$ sub-cubes. The probability of non-branching within the $n a$-cube is therefore $(1-q)^n$ at small spacing, so that the branching probability in the $n a$-cube becomes $1-(1-q)^n \approx n q$, i.e. it is proportional to the edge length of the cube.
|
---
abstract: 'A numerically cheap way to obtain structural information about clusters of rare gas atoms at low temperatures is developed. The semiclassical frozen Gaussian imaginary time propagator is extended such that it can account for the mean values of all inter-atomic distances in the cluster and their variances. To reduce the required numerical effort an approximation for the mean values is developed which preserves the quality of the results offered by the semiclassical ansatz. The method is applied to the $\text{Ar}_6$ cluster. It is found that the cluster dissociates almost in one step to six free atoms when the temperature is increased. Precursors of the dissociation are only observable in the distances of the atoms via the appearance of a second isomer. The process is almost classical. However, the method is able to resolve small differences in the temperatures at which the dissociation takes place and in the mean distances of the bound configuration.'
author:
- Holger Cartarius
title: 'Structural information about the $\text{Ar}_6$ cluster with the frozen Gaussian imaginary time propagator'
---
Introduction
============
At very low temperatures of a few Kelvin rare gas atoms may assemble and form clusters due to the van der Waals interaction. It has been found in a variety of numerical studies that these clusters exhibit a large number of interesting effects. In particular, structural changes of the systems for an increase or decrease of the temperature provide insight into the behavior and binding mechanisms of the quantum mechanical objects at finite temperatures. Thus, rare gas atomic clusters are a topic of ongoing research. Among the rich variety of their thermodynamic properties are a change from one packing of the atoms to another with increasing energy, or phase transitions from a solid-like behavior to a liquid-like arrangement of the atoms .
For the weak van der Waals interaction between rare gas atoms the thermal energy corresponding to a few Kelvin already suffices to lead to all impacts mentioned above. Thus, accurate quantum mechanical computations are essential to obtain reliable results. In numerical calculations the Boltzmann operator at inverse temperature $\beta$, $K = \exp(-\beta H)$, (due to its form also called imaginary time propagator) is the most important quantity. Its trace yields the partition function $Z(\beta)$, and the thermal averages of every observable $O$ follow from $\langle O \rangle_\beta =
\mathrm{Tr}(K(\beta)O)/Z(\beta)$. However, precise calculations for multi-dimensional systems are still a challenge for today’s numerical possibilities. For example, path-integral Monte Carlo methods have been used to investigate rare gas clusters . At low temperatures they become already too expensive for a few dozen atoms, and efficient but sufficiently accurate approximations are required. These approximations have to reproduce quantum effects correctly since they will be of eminent importance at low temperatures. Still many questions are open. For example $\mathrm{Ne}_{13}$ and $\mathrm{Ne}_{38}$ [@Predescu05a; @Frantsuzov06a; @Adjanor2006a; @Pahl2008a] might exhibit novel low temperature quantum effects, such as liquid-like zero temperature structures of $\mathrm{Ne}_{38}$ as compared to a solid-like structure predicted from classical mechanics [@Frantz92a].
Semiclassical methods help even with presently available computation capabilities to overcome the numerical drawbacks of numerically exact quantum mechanical algorithms and are still developed and applied in a wide context concerning thermodynamic properties . Important semiclassical approximations are based on the idea to restrict the quantum mechanical wave functions to a Gaussian shape. The Gaussian functions are defined completely by a low number of parameters as the position of the center, a momentum of the wave packet and a width matrix. The lowest numerical effort is achieved with a frozen Gaussian propagator, i.e. a Gaussian wave function of which the width matrix in the exponent is fixed for all imaginary times (or temperatures).
Small argon clusters are among those rare gas clusters which attracted large interest for a long time and again recently . Despite their apparent simplicity numerical calculations turned out to be nontrivial, and for the argon trimer even sophisticated path-integral Monte Carlo calculations could not distinguish a complete dissociation from structural changes within a bound system because the numerics suffered strongly from noise [@Perez10a]. The frozen Gaussian imaginary time propagator has proved to provide high quality results. It could solve this question and identified an almost classical dissociation effect [@Cartarius11a], which was confirmed [@Cartarius12a] with a first-order correction to the semiclassical imaginary time propagator [@Shao06a; @Zhang09a].
For the more complicated $\mathrm{Ar}_6$ cluster the specific heat and mean energy do not provide enough information to understand its whole structure. It is important to know which alignment of the atoms is present. A very detailed knowledge of the atomic structure is available via the inter-particle distances. To obtain their values it is very common to compute the radial pair correlation function between particles $i$ and $j$ at positions $\bm{r}_i$ and $\bm{r}_j$, respectively, $$\begin{gathered}
p_{ij}(r) = \langle \delta(|\bm{r}_i-\bm{r}_j|-r) \rangle_\beta \\
= \mathrm{Tr} \left [ K(\beta) \delta(|\bm{r}_i-\bm{r}_j|-r) \right ]
/Z(\beta),
\label{eq:pair_correlation}\end{gathered}$$ where $\langle \rangle_\beta$ is the thermal average at inverse temperature $\beta$ and $K(\beta) = \mathrm{e}^{-\beta H}$ represents the imaginary time propagator at the same temperature [@Frantsuzov04a]. This quantity provides information about the distribution of the distances occurring at every temperature.
It is the purpose of this paper to show that the frozen Gaussian method applied to atomic clusters in Ref. [@Cartarius11a] can be used to determine the distances of the atoms in clusters in a numerically cheap and easy way. To do so, an extension to the semiclassical frozen Gaussian method will be developed. The mean values and the variances of the distances $d_{ij}$ between atoms $i$ and $j$ can directly be accessed and provide a very clear information of how the atoms are arranged in the cluster. Since the distances $d_{ij}$ of the single pairs can be calculated in parallel with the same Monte Carlo sampling all of these values can be obtained with low numerical extra cost. This provides for $N$ atoms $N(N-1)/2$ independent values, whereas in the pair correlation function small differences in the mean distances might be hidden below broad distributions.
The example of the $\mathrm{Ar}_6$ cluster investigated in this article demonstrates the importance of the additional structural information even in this relatively simple system, which does not undergo a structural transformation in the bound state. The mean energy and the specific heat will show that the cluster dissociates with increasing temperature to a system of six free atoms in one step. However, precursors of the dissociation will only be observable in the distances between the atoms, which will indicate a loss of the ground state configuration at slightly lower temperatures. The variances of the distances will turn out to be very sensitive to the breakdown of the structure.
The further sections of this paper are organized as follows. In Sec.\[sec:method\] mean values and variances for inter-atomic distances with the frozen Gaussian method are introduced. For comparison the same values are introduced within the more flexible thawed Gaussian variant. Then the method is applied to the $\mathrm{Ar}_6$ cluster in Sec. \[sec:ar6\]. After a short introduction of the system (Sec. \[sec:system\]), the confining sphere (Sec. \[sec:sphere\]), and the proper choice of the Gaussian width matrix (Sec. \[sec:width\_matrix\]) derivatives of the partition function (Sec. \[sec:energy\]) and the structural information (Sec. \[sec:distances\]) are investigated. A discussion in Sec.\[sec:discussion\] concludes the paper.
The frozen Gaussian method {#sec:method}
==========================
Propagator
----------
The method is based on a semiclassical approximation of the thermal operator $$K(\beta) = e^{-\beta H} ,
\label{eq:imag_propagator}$$ where $\beta = 1/(kT)$ is the inverse temperature. The approximation consists of evaluating $K(\beta)$ by solving the Bloch equation $$-\frac{\partial}{\partial \tau} |\bm{q}_0,\tau \rangle
= H |\bm{q}_0,\tau \rangle
\label{eq:bloch_equation}$$ approximately for a frozen Gaussian coherent state in position space representation, $$\begin{gathered}
\langle \bm{x} | \bm{q}_0(\tau) \rangle
= \left ( \frac{\det(\bm{\Gamma})}{\pi^{3N}} \right )^{1/4}
\exp \biggl ( -\frac{1}{2} [\bm{x} - \bm{q}(\tau)]^\mathrm{T} \\
\times \bm{\Gamma}
[\bm{x} - \bm{q}(\tau)] +\frac{i}{\hbar} \bm{p}^\mathrm{T}(\tau)
\cdot [\bm{x}-\bm{q}(\tau)]\biggr ) ,
\label{eq:frozen_state}\end{gathered}$$ where the width matrix $\bm{\Gamma}$ is a free parameter and has to be adapted to the given problem as will be explained later for the cluster considered in this article. The propagation in Eq. is done in imaginary time $\tau$ up to the value $\tau = \beta$ one is interested in.
With Gaussian averages of the type $$\langle h(\bm{q}) \rangle = \int_{-\infty}^{\infty} d\bm{x}^{3N} \,
|\langle \bm{x} | \bm{q}_0(\tau) \rangle|^2 h(\bm{x})
\label{eq:Gaverage_general}$$ the symmetrized form of the frozen Gaussian propagator is given by $$\begin{gathered}
\langle \bm{x}' | K_\mathrm{FG}(\tau) | \bm{x} \rangle
= \det(\bm{\Gamma}) \exp \left ( -\frac{\hbar^2}{4} \mathrm{Tr}(\bm{\Gamma})
\tau \right ) \\
\times \sqrt{\det \left ( 2 \left [ \bm{1} - \exp (-\hbar^2
\bm{\Gamma} \tau) \right ]^{-1} \right )} \\
\times \exp \left ( -\frac{1}{4} [\bm{x}' - \bm{x}]^\mathrm{T} \bm{\Gamma}
[\tanh(\hbar^2 \bm{\Gamma} \tau/2)]^{-1} [\bm{x}'-\bm{x}] \right ) \\
\times \int_{-\infty}^\infty \frac{d\bm{q}^{3N}}{(2\pi)^{3N}}
\exp \biggl (-2 \int_0^{\tau/2} d\tau \langle V(\bm{q}(\tau)) \rangle \\
- [\bm{\bar{x}}-\bm{q}(\tau/2)]^\mathrm{T} \bm{\Gamma}
[\bm{\bar{x}}-\bm{q}(\tau/2)] \biggr ) ,
\label{eq:prop_FG}\end{gathered}$$ where $\bm{\bar{x}} = (\bm{x}' + \bm{x})/2$. The partition function is simply given by the trace [@Zhang09a] $$\begin{gathered}
Z_\mathrm{FG}(\tau) = \mathrm{Tr} \left [ K_\mathrm{FG}(\tau) \right ]
= \sqrt{\det(\bm{\Gamma})} \exp \left ( -\frac{\hbar^2}{4}
\mathrm{Tr}(\bm{\Gamma}) \tau \right ) \\
\times \sqrt{\det \left ( \left [ \bm{1}
- \exp (-\hbar^2 \bm{\Gamma} \tau) \right ]^{-1} \right )} \\
\times \int_{-\infty}^\infty \frac{d\bm{q}^{3N}}{(2\pi)^{N/2}}
\exp \left (-2 \int_0^{\tau/2} d\tau \langle V(\bm{q}(\tau)) \rangle
\right ) .
\label{eq:pf_FG}\end{gathered}$$ The whole dynamical information is contained in the imaginary time propagation of the variable $\bm{q}(\tau)$ and is governed by the $3N$ coupled equations of motion $$\frac{\partial \bm{q}(\tau)}{\partial \tau} = -\bm{\Gamma}^{-1}
\langle \nabla V(\bm{q}(\tau)) \rangle ,
\label{eq:FG_eqs_motion_q}$$ which are relatively simple and can be integrated with a standard integrator for ordinary differential equations. The remaining numerical task is a single position space integration for the initial positions $\bm{q}(0)$ in Eqs. or , which is done with a Monte-Carlo integration.
As shown previously [@Cartarius11a; @Cartarius12a] a reasonable choice of the width matrix is crucial for the quality of the semiclassical method. However, highly precise values can be obtained already with a very simple structure. Since all particles are identical and thus the pairwise interactions are the same for all combinations only the center of mass motion has to be distinguished. In center of mass coordinates,
$$\begin{aligned}
\bm{R}_\mathrm{cm} &= \frac{1}{N} \sum_{i=1}^{N} \bm{r}_i , \\
\bm{R}_i &= \bm{r}_i - \bm{r}_{i+1} , \quad i = 1\,\dots,N-1 ,
\end{aligned}$$
good semiclassical estimates are obtained with the matrix $$\bm{\Gamma}_\mathrm{cmc} = \begin{pmatrix}
\bm{D}_1 & \bm{0} & \cdots \\
\bm{0} & \bm{D}_2 & \\
\vdots & & \ddots
\end{pmatrix}
\label{eq:gamma_cmc}$$ and the $3\times 3$ diagonal matrices $\bm{D}_1$ and $\bm{D}_2$ describing the three spacial directions of the center of mass and the relative coordinates, respectively. Since also the spacial directions are equivalent the best choice are multiples of the unity matrix $\bm{1}_{3\times 3}$, such that two scalar coefficients $D_1$ and $D_2$ have to be chosen, i.e. $$\bm{D}_1 = D_1 \bm{1}_{3\times 3}, \quad \bm{D}_2 = D_2 \bm{1}_{3\times 3} .
\label{eq:submatrices}$$ The form of the propagator is represented in Cartesian coordinates and is the most efficient for the numerical evaluation. Consequently the Cartesian representation
\[eq:FG\_general\_matrix\] $$\bm{\Gamma} = \begin{pmatrix}
\bar{\bm{D}} & \Delta \bm{D} & \Delta \bm{D} & \\
\Delta \bm{D} & \bar{\bm{D}} & \Delta \bm{D} & \cdots \\
& & \ddots & \\
\end{pmatrix} ,$$ $$\begin{aligned}
\bar{\bm{D}} &= (\bm{D}_1+(N-1)\bm{D}_2)/N , \\
\Delta \bm{D} &= (\bm{D}_1-\bm{D}_2)/N
\end{aligned}$$
of the matrix is used.
Thermal averages of structural information
------------------------------------------
Simple thermal averages requiring only the partition function $Z(\beta)$ are the mean energy $E = \mathrm{k} T^2 \partial \ln Z/\partial T$ and the specific heat $C = \partial E/\partial T$. However, $K(\beta)$ provides access to the thermal average of any observable $O$ via $$\bar{O}^{\mathrm{(FG)}} = \frac{\mathrm{Tr} (K_\mathrm{FG}(\beta)O)}
{Z_\mathrm{FG}(\beta)} ,$$ which is exploited in this article to gain access to the structural information. A well suited property is the distance between two atoms, i.e.$$O = d_{ij} = | \bm{x}_i - \bm{x}_j | .
\label{eq:operator_distances_full}$$ With the frozen Gaussian propagator this leads to the expression
$$\begin{gathered}
\bar{d}_{ij}^\mathrm{(FG)} = \frac{1}{Z_\mathrm{FG}(\beta)}
\mathrm{Tr}(K_\mathrm{FG}(\beta) \, | \bm{x}_i - \bm{x}_j |)
= \frac{1}{Z_\mathrm{FG}(\beta)} \det(\bm{\Gamma})
\exp \left ( -\frac{\hbar^2}{4} \mathrm{Tr}(\bm{\Gamma}) \beta \right )
\sqrt{\det \left ( 2 \left [ \bm{1} - \exp (-\hbar^2 \bm{\Gamma} \beta)
\right ]^{-1} \right )} \\ \times
\int_{-\infty}^\infty \frac{\mathrm{d}\bm{q}^{3N}}{(2\pi)^{3N}} \exp \biggl (-2
\int_0^{\beta/2} \mathrm{d}\tau \langle V(\bm{q}(\tau)) \rangle \biggr )
\int_{-\infty}^\infty \mathrm{d}\bm{x}^{3N} \exp \biggl (- [\bm{x}- \bm{q}
(\beta/2)]^\mathrm{T} \bm{\Gamma} [\bm{x}-\bm{q}(\beta/2)] \biggr )
| \bm{x}_i - \bm{x}_j | ,
\label{eq:distance_full_term}
\end{gathered}$$
in which an explicit integration over the $3N$ position variables $\bm{x}$ remains in addition to the evaluation of the partition function . As mentioned previously [@Cartarius12a] it is very important to reduce the numerical effort as much as possible for many-particle systems. In particular, the position space integrations require an expensive Monte Carlo sampling in a high-dimensional configuration space.
For usual applications a numerical evaluation of the $\bm{x}$ integration can be avoided in a reasonable approximation. This can be seen with the variable $\bm{y} = \bm{x} - \bm{q}(\beta/2)$, which transforms the $\bm{x}$ integral in to $$\begin{gathered}
I = \int_{-\infty}^\infty \mathrm{d}\bm{y}^{3N} \exp \biggl (- \bm{y}^\mathrm{T}
\bm{\Gamma} \bm{y} \biggr ) \\ \quad \times
| \bm{y}_i - \bm{y}_j + \bm{q}_i(\beta/2) - \bm{q}_j(\beta/2)| .
\label{eq:distance_y}\end{gathered}$$ The widths of the atom’s wave functions contribute only at low temperatures significantly to the distance. As will be seen, in practical applications a very narrow Gaussian centers all values $\bm{y}_i$ strongly around zero, i.e. $\bm{x}_i$ is almost identical with $\bm{q}_i(\beta/2)$ for a nonvanishing Gaussian weight. Thus, the integral is calculated for the case $| \bm{y}_i - \bm{y}_j | \ll | \bm{q}_i
- \bm{q}_j|$. With the expansion $$\begin{gathered}
| \bm{y}_i - \bm{y}_j + \bm{q}_i - \bm{q}_j| \approx | \bm{q}_i - \bm{q}_j |
- (\bm{y}_i - \bm{y}_j)\cdot \frac{\bm{q}_i - \bm{q}_j}{|\bm{q}_i
- \bm{q}_j|} \\
+ \frac{1}{2} (\bm{y}_i - \bm{y}_j)^2 - \frac{1}{2} \frac{\left [ (\bm{y}_i
- \bm{y}_j) \cdot (\bm{q}_i - \bm{q}_j ) \right ]^2}{|\bm{q}_i
- \bm{q}_j|^3}
\label{eq:approximation_dist}\end{gathered}$$ the integral evaluates to $$\begin{gathered}
I = \sqrt{\frac{\pi^{3N}}{\det (\bm{\Gamma})}} \bigg [ |\bm{q}_i(\beta/2)
- \bm{q}_j(\beta/2)| \\
+ \frac{\mathrm{Tr} \big ( \bm{\Gamma}_{ii}^{-1} + \bm{\Gamma}_{jj}^{-1}
- \bm{\Gamma}_{ij}^{-1} - \bm{\Gamma}_{ji}^{-1} \big )}{6 |\bm{q}_i(\beta/2
)
- \bm{q}_j(\beta/2)|} \bigg ] ,\end{gathered}$$ where $\bm{\Gamma}_{ij}$ is the $3\times 3$ submatrix of $\bm{\Gamma}$ at the rows and columns representing particles $i$ and $j$. In total
$$\begin{gathered}
\bar{d}_{ij}^\mathrm{(FG)}(\beta) = \frac{1}{Z_\mathrm{FG}(\beta)}
\mathrm{Tr}(K_\mathrm{FG}(\beta) \, | \bm{x}_i - \bm{x}_j |)
\approx \frac{1}{Z_\mathrm{FG}(\beta)} \sqrt{\det(\bm{\Gamma})}
\exp \left ( -\frac{\hbar^2}{4} \mathrm{Tr}(\bm{\Gamma}) \beta \right )
\sqrt{\det \left ( \left [ \bm{1} - \exp (-\hbar^2 \bm{\Gamma} \beta)
\right ]^{-1} \right )} \\ \times
\int_{-\infty}^\infty \frac{\mathrm{d}\bm{q}^{3N}}{(2\pi)^{3N/2}}
\exp \biggl (-2 \int_0^{\beta/2} \mathrm{d}\tau \langle V(\bm{q}(\tau))
\rangle \biggr ) \bigg [ |\bm{q}_i(\beta/2) - \bm{q}_j(\beta/2)|
+ \frac{\mathrm{Tr} \big ( \bm{\Gamma}_{ii}^{-1} + \bm{\Gamma}_{jj}^{-1}
- \bm{\Gamma}_{ij}^{-1} - \bm{\Gamma}_{ji}^{-1} \big )}{6 |\bm{q}_i(\beta/2)
- \bm{q}_j(\beta/2)|} \bigg ]
\label{eq:distance_full_approximated}
\end{gathered}$$
is obtained.
The first term in equation , $\propto
|\bm{q}_i - \bm{q}_j|$, reflects the core of the semiclassical approximation, in which the positions of the atoms are given by the centers $\bm{q}_i$ of the Gaussian wave packets . It corresponds to $$O = | \bm{q}_i - \bm{q}_j | .$$ The second term contains a correction due to the finite width of an atom’s wave packet. It is completely sufficient to include this lowest-order term, of which the $\bm{x}$ integration could be done analytically with a simple result, thus reducing the numerical effort drastically. For the frozen Gaussian method any higher terms beyond those included in the approximation for the mean distances are of lower interest. From the physical point of view it is expected that the width of the atom’s wave function only plays a role at very low temperatures at which the structural configuration is unambiguously in a highly symmetric ground state configuration. Indeed, as will be seen in the results already the correction term in the approximation is very small.
To estimate the quality and validity of these mean values additionally the variances of the distance distributions are calculated. With the operator $$O = v_{ij} = (\bm{x}_i - \bm{x}_j)^2 - {\bar{d}_{ij}}^2$$ and the integral
$$\begin{gathered}
\frac{1}{Z_\mathrm{FG}(\beta)}
\mathrm{Tr} \left ( K_\mathrm{FG}(\beta) \, [ \bm{x}_i - \bm{x}_j ]^2
\right ) = \frac{1}{Z_\mathrm{FG}(\beta)} \sqrt{\det(\bm{\Gamma})}
\exp \left ( -\frac{\hbar^2}{4} \mathrm{Tr}(\bm{\Gamma}) \beta \right )
\sqrt{\det \left ( \left [ \bm{1} - \exp (-\hbar^2 \bm{\Gamma} \beta)
\right ]^{-1} \right )} \\ \times
\int_{-\infty}^\infty \frac{\mathrm{d}\bm{q}^{3N}}{(2\pi)^{3N/2}} \exp \biggl
(-2 \int_0^{\beta/2} \mathrm{d}\tau \langle V(\bm{q}(\tau)) \rangle \biggr )
\bigg [ (\bm{q}_i(\beta/2) - \bm{q}_j(\beta/2))^2 + \frac{1}{2}
\mathrm{Tr} \big ( \bm{\Gamma}_{ii}^{-1} + \bm{\Gamma}_{jj}^{-1}
- \bm{\Gamma}_{ij}^{-1} - \bm{\Gamma}_{ji}^{-1} \big )\bigg ]
\label{eq:distvariances}
\end{gathered}$$
the standard deviations $\sigma_{ij} = \sqrt{\bar{v}_{ij}^\mathrm{(FG)}}$ of the distances $\bar{d}_{ij}^\mathrm{(FG)}$ are obtained.
Sorted distances {#sec:sizeorder}
----------------
For the $\mathrm{Ar}_6$ cluster there are 15 possible combinations $i$ and $j$, and thus 15 distances. The clusters are oriented arbitrarily in the simulation. The numbers $i$ and $j$ of the atoms have no meaning for the true configuration, and thus are not appropriate quantities to define the pairwise distances. The average of all calculations simply results in identical values for all $\bar{d}_{ij}$, which correspond to the mean value of all 15 atom-atom distances in a certain configuration. To obtain a meaningful quantity the distances are sorted according to their size, $$d_1 < d_2 < \hdots < d_{15} ,$$ and the thermal average of these size-ordered distances is determined, i.e.the thermal mean values of the smallest distance, the second smallest, and so forth are obtained. These values can be compared with the expectations of geometrical configurations. In an experiment the single distances are accessible [@Kwon1996a] and can in a given sample be sorted the same way. Alternatively, results from this calculation can be used to determine the distance of the atoms with a well-grounded assumption about the configuration [@Ulrich2011a].
Comparison with the thawed Gaussian propagator
----------------------------------------------
The frozen Gaussian method has proved to provide good results for thermodynamic quantities. We want to know whether or not this is also true for the widths calculated in this article. Thus, the structural information of the frozen Gaussian method is compared with that of a more flexible thawed Gaussian ansatz. It is based on a time-dependent width matrix $\bm{G}(\tau)$, which adapts itself to the given temperature. With the restriction to Gaussian wave packets the thawed Gaussian variant is usually the most accurate approximation. The variable width matrix adds an additional freedom in the parameters. This is reflected in the quality of the results as has clearly been demonstrated for a double well potential [@Conte10a]. For a large number of degrees of freedom it suffers, however, from the higher numerical costs. The single-particle ansatz of Frantsuzov et al. [@Frantsuzov04a] avoids these difficulties by reducing the matrix $\bm{G}(\tau)$ to a block-diagonal structure, where $3\times 3$ matrices representing the three spacial coordinates of one particle are the only non-vanishing matrix elements. In the case of six atoms this reduction is not required and there is no need to ignore the inter-particle correlations.
The thawed Gaussian propagator used for comparison with the frozen Gaussian method is the time evolved Gaussian approximation (TEGA) suggested by Frantsuzov et al. [@Frantsuzov03a; @Frantsuzov04a] with a full width matrix $\bm{G}$. It is based on the solution of the Bloch equation with the coherent state $$\begin{gathered}
\langle \bm{x} | g(\bm{q}(\tau),\bm{G}(\tau)) \rangle
= \left (\pi^{3N} |\det \bm{G}(\tau)|\right )^{-1/4} \\ \times
\exp \left ( -\frac{1}{2} [\bm{x}-\bm{q}(\tau)]^\mathrm{T} \bm{G}(\tau)^{-1}
[\bm{x}-\bm{q}(\tau)] \right ) .
\label{eq:tega_gaussian}\end{gathered}$$ The resulting symmetrized propagator reads $$\begin{gathered}
\langle \bm{x} | K_\mathrm{TG}(\tau) | \bm{x}' \rangle
= \int \frac{\mathrm{d}\bm{q}^{3N}}{(2\pi)^{3N}} \frac{\exp[2\gamma(\tau/2)]}
{\det[\bm{G}(\tau/2)]} \\
\times \exp \left ( -\frac{1}{2} [\bm{x}-\bm{q}(\tau/2)
]^\mathrm{T} \bm{G}(\tau/2)^{-1} [\bm{x}-\bm{q}(\tau/2)] \right ) \\ \times
\exp \left ( -\frac{1}{2} [\bm{x}'-\bm{q}(\tau/2)]^\mathrm{T}
\bm{G}(\tau/2)^{-1} [\bm{x}'-\bm{q}(\tau/2)] \right )
\label{eq:prop_TG}\end{gathered}$$ with the time-dependent width matrix $\bm{G}(\tau)$. In imaginary time $\tau$ the equations of motion for the Gaussian parameters $\bm{G}$, $\bm{q}$, and $\gamma$ are
$$\begin{aligned}
\frac{d}{d\tau} \bm{G}(\tau) &= -\bm{G}(\tau) \langle \nabla
\nabla^\mathrm{T} V(\bm{q}(\tau)) \rangle \bm{G}(\tau) + \hbar^2 \bm{1},
\label{eq:tg_eqs_motion_1} \\
\frac{d}{d\tau} \bm{q}(\tau) &= -\bm{G}(\tau) \langle \nabla V(\bm{q}
(\tau)) \rangle, \\
\frac{d}{d\tau} \gamma(\tau) &= -\frac{1}{4} \mathrm{Tr} \left [ \langle
\nabla\nabla^\mathrm{T} V(\bm{q}(\tau)) \rangle \bm{G}(\tau) \right ]
- \langle V(\bm{q}(\tau)) \rangle ,
\label{eq:tg_eqs_motion_3}
\end{aligned}$$
which have to be integrated from $\tau = 0$ to larger times with the initial conditions
$$\begin{aligned}
\bm{q}(\tau \approx 0) &= \bm{q}_0 , \\
G(\tau \approx 0) &= \hbar^2 \bm{1} \tau , \label{eq:initial_matrix} \\
\gamma(\tau \approx 0) &= - V(\bm{q}_0) \tau .
\end{aligned}$$
In all expressions $\langle \dots \rangle$ represents Gaussian averaged quantities of the form with the wave packet , and $\bm{1}$ is the $3N \times 3N$-dimensional identity matrix. The relevant quantities are the partition function $$Z_\mathrm{TG} = \int \frac{\mathrm{d}\bm{q}^{3N}}{(2\sqrt{\pi})^{3N}}
\frac{\exp[2\gamma(\tau/2)]}{\sqrt{\det[\bm{G}(\tau/2)]}}
\label{eq:pf_TG}$$ and the mean value of the distances in the same approximation as for the frozen Gaussian method, $$\begin{gathered}
\bar{d}_{ij}^\mathrm{(TG)}(\beta) \approx \frac{1}{Z_\mathrm{TG}(\beta)}
\int \frac{\mathrm{d}\bm{q}^{3N}}{(2 \sqrt{\pi})^{3N}}
\frac{\exp[2\gamma(\beta/2)]}{\sqrt{\det[\bm{G}(\beta/2)]}} \\
\times \bigg [ |\bm{q}_i(\beta/2) - \bm{q}_j(\beta/2)| \\+ \frac{\mathrm{Tr}
\big ( \bm{G}_{ii}(\beta/2) + \bm{G}_{jj}(\beta/2) - \bm{G}_{ij}(\beta/2)
- \bm{G}_{ji}(\beta/2) \big )}{6 |\bm{q}_i(\beta/2)
- \bm{q}_j(\beta/2)|} \bigg ] .\end{gathered}$$
The thawed Gaussian approximation allows for an additional important information. Its temperature-dependent width matrix $\bm{G}(\tau)$ provides easier access to the width of the wave function, which influences the variances of the distances. The quantum mechanical part of the variances, i.e. that originating from the spread of the wave function, is expected to increase at lower temperatures. For a frozen Gaussian this can be described correctly if the constant matrix $\bm{\Gamma}$ is optimized for every single temperature. In the thawed Gaussian case the variances read $$\begin{gathered}
\bar{v}_{ij}^\mathrm{(TG)}(\beta) \approx \frac{1}{Z_\mathrm{TG}(\beta)}
\int \frac{\mathrm{d}\bm{q}^{3N}}{(2 \sqrt{\pi})^{3N}}
\frac{\exp[2\gamma(\beta/2)]}{\sqrt{\det[\bm{G}(\beta/2)]}} \\
\times \bigg [ (\bm{q}_i(\beta/2) - \bm{q}_j(\beta/2))^2
+ \frac{1}{2} \mathrm{Tr} \big ( \bm{G}_{ii}(\beta/2) + \bm{G}_{jj}(\beta/2)
\\ - \bm{G}_{ij}(\beta/2) - \bm{G}_{ji}(\beta/2) \big )\bigg ]
- \left ( \bar{d}_{ij}^\mathrm{(TG)} \right )^2
\label{eq:var_fctg}\end{gathered}$$ and follow directly from the imaginary time evolution of $\bm{q}(\tau)$ *and* $\bm{G}(\tau)$. We are mainly interested in the quantum mechanical part of the variances, viz.$$\begin{gathered}
\bar{v}_{ij}^\mathrm{(TG,qm)}(\beta) \approx \frac{1}{Z_\mathrm{TG}(\beta)}
\int \frac{\mathrm{d}\bm{q}^{3N}}{(2 \sqrt{\pi})^{3N}}
\frac{\exp[2\gamma(\beta/2)]}{\sqrt{\det[\bm{G}(\beta/2)]}} \\
\times \frac{1}{2} \mathrm{Tr} \big ( \bm{G}_{ii}(\beta/2)
+ \bm{G}_{jj}(\beta/2) - \bm{G}_{ij}(\beta/2) - \bm{G}_{ji}(\beta/2) \big ) .
\label{eq:qm_var_fctg}\end{gathered}$$
Structural information about the $\mathrm{Ar}_6$ cluster {#sec:ar6}
========================================================
Representation of the system {#sec:system}
----------------------------
The argon cluster consists of 6 atoms, where the Hamiltonian in mass scaled coordinates reads $$H = -\frac{\hbar^2}{2} \sum_{i=1}^6 \Delta_i
+ \sum_{j<i} V(r_{ij})
\label{eq:Hamiltonian}$$ with the Laplacian $\Delta_i$ of particle $i$. The two-body potential $V(r_{ij})$ of Argon is still a very challenging task. One of the best analytic expressions at hand is a fit to experimental results by Aziz and Slaman [@Aziz86a] of which an adoption to a Morse potential [@Gonzales-Lezana99a] is used, $$V(r_{ij}) = D \left ( \exp \left [ -2\alpha (r_{ij}-R_\mathrm{e}) \right ]
- 2 \exp \left [ -\alpha (r_{ij}-R_\mathrm{e}) \right ] \right)
\label{eq:Morse_potential}$$ with the parameters $D = 99.00\,\mathrm{cm}^{-1}$, $\alpha = 1.717\,
\text{\r{A}}$, and $R_\mathrm{e} = 3.757\,\text{\r{A}}$ in consistence with previous studies of the Argon trimer [@Perez10a; @Cartarius11a; @Cartarius12a].
The numerical efficiency of the frozen Gaussian method is increased with an expansion of the potential in terms of Gaussians, viz.$$V(|\bm{r}_i - \bm{r}_j|) = \sum_{p} c_p e^{-\alpha_p r_{ij}^2} ,
\qquad r_{ij} = |\bm{r}_i - \bm{r}_j| .
\label{eq:Gaussian_fit}$$ This procedure was suggested by Frantsuzov et al. [@Frantsuzov04a] and has successfully been applied [@Frantsuzov04a; @Cartarius11a; @Cartarius12a]. In the form Gaussian integrals of the potential or its derivatives can be done analytically. The required parameters for a fit to three Gaussians are listed in Table \[tab:Gaussian\_parameters\]
[lD[.]{}[.]{}[8]{}D[.]{}[.]{}[5]{}]{} $p$ & &\
1 & 3.29610\^[5]{} & 0.6551\
2 & -1.27910\^[3]{} & 0.1616\
3 & -9.94610\^[3]{} & 6.0600\
and were previously obtained in Ref. [@Cartarius11a].
Confining sphere {#sec:sphere}
----------------
An additional potential is usually introduced to converge the numerical $\bm{q}$ integration. All particles are confined within a sphere around the center of mass $\bm{R}_\mathrm{cm}$ by the condition $|\bm{q}-\bm{R}_\mathrm{cm}|
< R_c$, where $R_c$ is the confining radius. This can be achieved with the steep potential $$V_\mathrm{c}(\bm{r}) \propto \sum_{i=1}^{N} \left ( \frac{\bm{r}_i
- \bm{R}_\mathrm{cm}}{R_\mathrm{c}} \right )^{20}
\label{eq:confinement}$$ added to the Hamiltonian or, as in our study, by a restriction of the volume for the $\bm{q}$ integration.
Of course, an additional potential influences the results and can crucially change the behavior of the cluster [@Predescu03a; @Etters75a]. If only bound configurations are investigated $R_\mathrm{c}$ is usually chosen such that the bound configurations are not affected, i.e. $R_\mathrm{c}$ is larger than the extension of the bound cluster. However, we are interested also in the dissociation process for which the choice of $R_\mathrm{c}$ is nontrivial [@Cartarius11a; @Cartarius12a]. A larger radius $R_\mathrm{c}$ always allows for a dissociation at lower temperatures. In principle it has to be adopted to the physical conditions as, e.g. the pressure. We are interested in the qualitative behavior at the dissociation and it was checked carefully that the choice $R_\mathrm{c} = 35\,\text{\r{A}}$ does not influence the qualitative change of the relevant observables, i.e. the mean energy, the specific heat, the mean values of the inter-atomic distances and their variances. In particular, it was assured that the case of a completely dissociated cluster is present for temperatures above $40\,\mathrm{K}$ and the form of the dissociation process is not altered. The value of the confining radius $R_\mathrm{c}$, i.e. the pressure in physical terms, affects the temperature at which the dissociation occurs.
Choice of the width matrix {#sec:width_matrix}
--------------------------
While in a thawed Gaussian calculation the initial condition for the width matrix is defined, the constant matrix $\bm{\Gamma}$ of its frozen Gaussian counterpart has to be chosen carefully. It is a free parameter of the system. It is not trivial to find a good choice of $\bm{\Gamma}$. However, as was mentioned above, the structure with the $3\times 3$ submatrices is well suited. Thus, only the two parameters $D_1$ and $D_2$ need to be chosen.
A detailed investigation of the $\mathrm{Ar}_3$ cluster revealed that there is a reliable and simple method to find the best choice for the inter-particle width parameter $D_2$ [@Cartarius11a]. Propagating the partition function to long imaginary times $\beta \to \infty$, i.e. $T \to 0$, one can extract the thermodynamic mean energy to correspond to the ground state energy $E_0$. The parameter $D_2$ providing the lowest value for $E_0$ has shown to lead to the best agreement with numerically exact calculations and the more flexible thawed Gaussian approximation. This result is almost independent of the temperature at which the partition function, mean energy or specific heat of all methods are compared. Furthermore, calculating the first-order correction to the frozen Gaussian propagator showed that this choice also requires the smallest correction. Thus, the simple minimization of the ground state energy gives us a reliable way of determining $D_2$. For the $\mathrm{Ar}_6$ cluster in this article it was found that $D_2 = 32\,\text{\r{A}}^{-2}$ is the best choice.
The center of mass is free and it can exactly be described by a Gaussian in the limit $D_1 \to 0$. This means the value should be as small as possible. For the numerical evaluation one needs a finite value. It is known that $D_1 = 0.1\,\text{\r{A}}^{-2}$ is small enough [@Cartarius11a]. The results cannot be distinguished from those of even lower values for $D_1$.
Mean energy and specific heat {#sec:energy}
-----------------------------
To compare the $\mathrm{Ar}_6$ cluster with the trimer considered in [@Cartarius11a; @Cartarius12a] the mean energy and the specific heat are studied first. They are shown in Fig. \[fig:energy\]
 (a) Mean energies of the $\mathrm{Ar}_6$ cluster calculated with the two-parameter frozen Gaussian propagator (2P-FG) and its classical counterpart. For the high-temperature limit both results agree well. At low temperatures the classical mean energy tends to the potential minimum and the frozen Gaussian propagator approximates the quantum mechanical ground state. The inset shows a comparison of the 2P-FG method with the fully coupled thawed Gaussian method (FC-TG). The energies differ by a few percent. (b) Specific heat around the dissociation, which seems to happen in one step.](figure_01){width="\columnwidth"}
for the two-parameter ($D_1$ and $D_2$) frozen Gaussian propagator and the derivatives of the classical partition function $$Z_\mathrm{cl} = \left ( \frac{\mathrm{k} T}{2\pi \hbar^2} \right )^{3/2 N}
\int \mathrm{e}^{-\beta V(\bm{q})} \, \mathrm{d}\bm{q}^{3N} .
\label{eq:pf_classical}$$ The observations are very similar to those obtained for trimer. At very low temperatures the classical calculation tends to the potential minimum. At $T = 1\,\mathrm{K}$ a mean energy of $E = -1216\,\mathrm{cm}^{-1}$ is found. The frozen Gaussian results are at this temperature already in a very flat regime, in which the mean energy is almost independent of the temperature and approximates the quantum mechanical ground state energy. The method leads to $E_0 \approx -1015\, \mathrm{cm}^{-1}$. In the inset of Fig. \[fig:energy\] (a) a comparison with the more flexible fully-coupled thawed Gaussian propagator is shown. It leads to a value of $E_0 \approx -1040\,
\mathrm{cm}^{-1}$, i.e. the difference of the ground state’s binding energy is only $2.4\%$. Thus, one may conclude that also for the larger $\mathrm{Ar}_6$ cluster the quality of the frozen Gaussian propagator is acceptable in comparison with the numerically more expensive thawed Gaussian variant even for the low-temperature limit.
The dissociation appears in the mean energy as a step. The energy raises almost directly to that of six free particles. This indicates a dissociation of all atoms at once as was observed for the trimer. The same information can be gained from the specific heat, which is shown around the dissociation in Fig. \[fig:energy\] (b). One broad peak confirms that the dissociation occurs in one step. The classical calculation shows a transition at a slightly lower temperature, and the difference between the two maxima in the specific heat is approximately $0.5\,\mathrm{K}$, which is lower than for the trimer, where a difference of $1.5\,\mathrm{K}$ was observed. Certainly the difference can again be related to the zero point energy in quantum mechanics. This is larger for six atoms than for three and one could expect that also the temperature difference is larger. However, one has to keep in mind that this energy has to be distributed among a larger number of atoms during the dissociation. Since aside from the small shift in the temperature the dissociation process is almost identical in the classical and the frozen Gaussian calculation one may conclude that it is a purely classical phenomenon.
Structural information for low temperatures and for the dissociation {#sec:distances}
--------------------------------------------------------------------
In addition to the information of the simple derivatives of the partition function the mean distances are studied. Since so far the dissociation seems to be purely classical it is interesting to also compare the structural information with the classical one. To do so, the classical mean distances $$\bar{d}_{ij}^\mathrm{(classical)} = \frac{1}{Z_\mathrm{cl}}
\left ( \frac{\mathrm{k} T}{2\pi \hbar^2} \right )^{3/2 N}
\int \mathrm{e}^{-\beta V(\bm{q})} | \bm{q}_i - \bm{q}_j| \,
\mathrm{d}\bm{q}^{3N}
\label{eq:mean_dist_classical}$$ are added in the calculations below.
### Structure at low temperatures
The mean distances obtained for temperatures below $20\,\mathrm{K}$, i.e.significantly below the dissociation process, are shown in Fig.\[fig:dist\_low\],
 (a) Comparison of the mean values of all 15 distances calculated with the frozen Gaussian method (2P-FG) and a classical calculation at temperatures $T \leq 20 $. The distances appear in groups. Three distances converge for $T\to 0$ to values above $5\,\text{\r{A}}$, whereas the remaining 12 are below $4\,\text{\r{A}}$. (b) A comparison of the frozen Gaussian and thawed Gaussian method (FC-TG) shows that the mean distances agree very well.](figure_02){width="\columnwidth"}
where first the frozen Gaussian method (2P-FG) is compared with the classical calculation in Fig. \[fig:dist\_low\] (a), and then the fully-coupled thawed Gaussian approximation is added in Fig. \[fig:dist\_low\] (b). The most striking observation is that the distances appear in the low-temperature limit in two groups. A group of three “long” distances, of which the values are always above $5\,\text{\r{A}}$ for $T\to 0$, and a second group of the 12 remaining “short” distances, which converge to a value below $4\,\text{\r{A}}$, exist. This already gives a clear answer to the question about the ground state configuration of $\mathrm{Ar}_6$. It is consistent with the distances in an octahedron, or in other words, the atoms are located at the centers of the surfaces of its dual polyhedron, viz.the cube. Twelve short distances $d_s$ from the atoms on neighboring surfaces and three longer distances $d_l$ between the atoms on opposite surfaces are expected. The ratio of the distances is supposed to be $d_l = \sqrt{2} d_s$, which is fulfilled excellently in both the classical and semiclassical calculations.
In the classical case the atoms seek directly the potential minima, whereas in the quantum case always a wave function with a finite width leading automatically to larger mean distances is present. Furthermore, in the classical case the fixed octahedron configuration is only observable for $T\to 0$. This is a consequence of the fact that classically every nonvanishing energy allows for a thermal excitation. In contrast to this there should be no excitation possible if $\mathrm{k}T$ is clearly below the energy difference between the ground state and the first excited state in the quantum mechanical case. This is also reflected in the mean distances. For temperatures below $T\approx 3\,\mathrm{K}$ no differences between the distances in one group are observed, and the distances do not change for even lower temperatures. This indicates that the cluster is already in the ground state configuration.
A comparison of the frozen Gaussian method with the fully-coupled thawed Gaussian propagator reveals that the distances agree very well. This is in particular true for all larger distances. Also the low temperature limit shows an excellent agreement. The edge length of the cube containing the octahedron is $d_l = 5.39\,\text{\r{A}}$ in the frozen Gaussian calculation and $d_l = 5.35\,\text{\r{A}}$ in the thawed Gaussian approximation. The difference is below $1\,\%$, and thus even smaller than that of the mean energy. Obviously the structural information of the frozen Gaussian method is less affected by the constant Gaussian width approximation.
With the data of Fig. \[fig:dist\_low\] we are also able to estimate the quality of the approximation , in which the power series expansion of the distances was introduced. The first-order term retained in the expansion is of the size $$\frac{\mathrm{Tr} \big ( \bm{\Gamma}_{ii}^{-1} + \bm{\Gamma}_{jj}^{-1}
- \bm{\Gamma}_{ij}^{-1} - \bm{\Gamma}_{ji}^{-1} \big )}{6 |\bm{q}_i
- \bm{q}_j|} ,$$ where for the width matrix $\mathrm{Tr} \big ( \bm{\Gamma}_{ii}^{-1} + \bm{\Gamma}_{jj}^{-1}
- \bm{\Gamma}_{ij}^{-1} - \bm{\Gamma}_{ji}^{-1} \big )/6 = D_2^{-1}
= (32\,\text{\r{A}}^{-2})^{-1} = 0.031\,\text{\r{A}}^{2}$ is obtained. With the knowledge that the typical distances $|\bm{q}_i - \bm{q}_j|$ are even in the bound phase of the order of a few ngströms this correction can be estimated to be always less than $1\,\%$ of the leading order. Hence, it has at most the same size as the difference between the two semiclassical propagators. Higher orders in the series expansion would lead to even smaller corrections, which do not need to be taken into account since they are below the error of the semiclassical approximation.
### Dissociation to six free atoms
The distances around the dissociation are shown in Fig. \[fig:dist\_high\],
 (a) Mean values of all 15 distances calculated with the frozen Gaussian method (2P-FG) around the dissociation. Two of the smaller distances join the group of the three larger distances starting at $T \approx 23\,\mathrm{K}$, but then the dissociation occurs at once. (b) The same behavior is observed for the classical calculation.](figure_03.eps){width="\columnwidth"}
where the classical and 2P-FG results are compared. Since the frozen Gaussian approximation is known to provide good results at these temperatures (cf.Ref. [@Cartarius12a]), a comparison with the thawed Gaussian propagator does not give any new information. Figure \[fig:dist\_high\] confirms the finding of the consideration of the mean energy and the specific heat in Fig. \[fig:energy\]. The dissociation effect is classical. The semiclassical approximation of the quantum mechanical propagator and the purely classical calculation lead to the same behavior. Apart from a small shift in temperature both diagrams agree very well.
The calculation of the distances gives additional insight into the dissociation process. For temperatures $T \lessapprox 23\,\mathrm{K}$ the groups of short and long distances are unchanged. Above this temperature two of the twelve distances $d_s$ are separating from the others and join the three longer distances $d_l$. Two new groups with five and ten distances start to from. For $\text{Ar}_6$ a second isomer in the form of a tri-tetrahedron is known to contribute at increasing temperatures [@Franke1993a]. This would exactly agree with a grouping of five longer and ten shorter distances and is also visible in Fig. \[fig:cmdist\], in which the distances of all six atoms from the center of mass are shown for a classical calculation.
 Distances of all six atoms from the center of mass in a classical calculation for $R_\mathrm{c} = 35\,
\text{\r{A}}$. Shown is a thermal average of size-ordered distances as explained in Sec. \[sec:sizeorder\]. With increasing temperature one distance becomes larger than the others before all raise drastically.](figure_04.eps){width="\columnwidth"}
At low temperatures all distances have almost the same size, which agrees with a pure octahedron configuration. With increasing temperature one distance grows and indicates a coexistence of octahedron and tri-tetrahedron configurations. Signatures of further arrangements of the atoms are not found.
However, this rearrangement of the atoms remains in its beginnings. A new structure cannot completely arise since the whole process does not finish before all distances raise drastically and indicate with this increase the dissociation of the cluster. Comparisons with calculations, in which due to a smaller value of $R_c$ no dissociation is allowed, show that this effect only appears in connection with the dissociation. Thus, the separation of the two distances is more a precursor of the total destruction of the cluster. The dissociation happens then at once. After the dissociation the distances obtain new almost constant values which correspond to the distribution of atoms moving freely within the confining sphere.
### Variances of the distances
To learn more about the actual distribution of the distances their standard deviations are shown in Figs. \[fig:variances\] (a) and (b)
 (a) Standard deviations $\sigma_{ij}$ of the distances in the frozen Gaussian approximation (2P-FG) and (b) of the classical calculation. They increase drastically around the dissociation. (c) Quantum mechanical part of the variances for the fully coupled thawed Gaussian propagator. For low temperatures it increases due to an increasing width of the Gaussians.](figure_05.eps){width="\columnwidth"}
for the classical and the 2P-FG method. The most significant feature is the drastic increase of the standard deviations around the transition. At this temperature range parts of the simulated clusters are still in a bound configuration whereas others are already dissociated. The effect is similar and of the same size for the classical and the semiclassical calculation. For temperatures above the dissociation the standard deviations are almost the same for all distances, which is also expected for six free atoms.
Of more interest is the behavior of the standard deviations below the dissociation. The longer distances are expected to show more fluctuations. Additionally two of the shorter distances join the group of the three longer distances for increasing temperatures as a consequence of contributions from two isomers. It can also be expected that these two show higher standard deviations than the short distances since the separation of the two distances does not happen abruptly at one temperature as can be seen in Fig.\[fig:dist\_high\]. Consequently, below the dissociation the standard deviations are found to form two groups. One group combines the standard deviations of five distances, i.e. the three longer ones and the two joining them. The other group consists of the standard deviations of the 10 short distances which stay together up to the temperature of the dissociation.
In Figs. \[fig:variances\] (a) and (b) it seems that the standard deviations always decrease for lower temperatures. This is definitely expected for the classical calculation. The frozen Gaussian approximation can, since the width of the wave function is determined by the constant values of $\bm{\Gamma}$, not reflect the quantum mechanical expectation that the wave function stretches at lower temperatures. To cover also this effect the quantum mechanical part according to equation is plotted in Fig. \[fig:variances\] (c). Only the variance of one of the distances is shown since this part is almost identical for all of them in the highly symmetric situation of a monoatomic cluster. At $T \approx
20\,\mathrm{K}$ the variance has approximately the same size as that following from the frozen Gaussian method $v_{ij}^\mathrm{(FG,qm)} = \mathrm{Tr}
\big ( \bm{\Gamma}_{ii}^{-1} + \bm{\Gamma}_{jj}^{-1} - \bm{\Gamma}_{ij}^{-1}
- \bm{\Gamma}_{ji}^{-1} \big )/2 = 3 D_2^{-1} = 0.093\,\text{\r{A}}^2$. In particular, these quantum mechanical parts of the variances are considerably lower than other contributions in equations and . Figure \[fig:variances\] (a) would not change with the thawed Gaussian propagator. Only for temperatures $T \lessapprox 10\,
\mathrm{K}$ the extension of the wave function becomes important for a measurement of the distances. For temperatures in the range of the dissociation the information from the frozen Gaussian method is completely sufficient.
All calculations shown in this section could be implemented and performed easily on a NVIDIA Tesla C2070 GPU. On this architecture a converged result for low temperatures is obtained in less than 20 hours. The most critical part is that around the dissociation, where a very detailed sampling for bound configurations has to be done alongside an inclusion of large distances allowing for an unbound cluster. This can require an increase of the sampling points by a factor of 10.
Summary and outlook {#sec:discussion}
===================
In this article it was shown that structural information about a cluster of atoms can be obtained with the frozen Gaussian semiclassical method in a numerically cheap way. The evaluation of the corresponding integrals can be done in parallel to that of the partition function. With this method the full information about all the distances of all combinations of the atoms can be obtained. A comparison with the more flexible thawed Gaussian propagator revealed that the quality of the distances is on the same level as that of the mean energy or the specific heat, or even better. To avoid inefficient numerical computations of a position space integral an approximation for the distances was introduced. It was possible, however, to show that this approximation does not reduce the quality of the results below that obtained in the semiclassical approximation of the propagator.
On the physical side it was found that with increasing temperatures the $\mathrm{Ar}_6$ cluster undergoes an almost direct transition to six free atoms. However, it shows precursors in the distances. At temperatures slightly below the dissociation a reordering of the atoms starts, in which contributions from a second isomer, viz. a tri-tetrahedron [@Franke1993a], appear, but then vanish in the increasing distances at the dissociation. The dissociation is a purely classical effect. The semiclassical approximation shows exactly the same behavior with just a small shift in the temperature of $0.5\,\mathrm{K}$. Around the dissociation the standard deviations of the distances are almost completely determined by classical contributions. Only for lower temperatures the extension of the wave functions becomes important as was seen in a thawed Gaussian approximation.
At low temperatures the cluster assumes the shape of an octahedron, where the longer distance between the atoms is $d_l = 5.4\,\text{\r{A}}$, and the shorter has the value $d_s = d_l/\sqrt{2} = 3.8\,\text{\r{A}}$. Classically the fixed configuration is only obtained in the limit $T \to 0$ whereas in the quantum mechanical case the ground state configuration is present for all temperatures $T\lessapprox 3\,\mathrm{K}$.
The frozen Gaussian method has proved to provide reliable results for quantum mechanical calculations. There is a large number of investigations which can be done with it. In particular, the results for $\mathrm{Ar}_3$ in [@Cartarius11a] and for $\mathrm{Ar}_6$ in this work indicate that the confinement to very small spheres usually applied in the calculation of the partition function and values deduced from it [@Predescu03a; @Frantsuzov04a; @Predescu05a; @Frantsuzov08a] only sample bound cluster configurations. This is physically realized at high pressures. If one is interested in lower pressures, at which a dissociation is allowed, this is too restrictive to fully understand the low-temperature behavior of the clusters. The dissociation can set in before structural changes or a melting can be observed. To take this into account it is necessary to advance the investigations done here to clusters with higher numbers of atoms. In particular, the cases of $\mathrm{Ar}_{13}$ , $\mathrm{Ne}_{13}$ [@Frantsuzov04a] or $\mathrm{Ne}_{38}$ [@Predescu05a] are of special interest since they showed interesting structural transformations in the non-dissociated cases. Whereas the stronger quantum effects in the completely bound case are well covered by a variable width matrix the numerically cheaper frozen Gaussian method has advantages in the numerically more challenging case of the dissociation requiring a sampling of bound and unbound configurations of the atoms. The exactness of both methods can then be monitored and (if necessary) improved with the the series expansion of the imaginary time propagator [@Shao06a; @Zhang09a; @Cartarius12a].
The importance of the series expansion is not restricted to the dissociation. Most effects in rare gas clusters such as structural transformations or dissociations appear at such low temperatures that it is necessary to analyze whether the semiclassical approximations used in the calculations correctly reproduce the true quantum mechanical behavior. An important example will be $\mathrm{Ne}_{38}$, for which strong differences are found between the approximate quantum computations and a purely classical theory [@Frantsuzov06a].
H.C. is grateful for a Minerva fellowship. He thanks Eli Pollak for valuable comments and kind hospitality at the Weizmann Institute of Science, where this work has been started.
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|
---
abstract: 'We provide a comprehensive and pedagogical introduction to the [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{} framework, with a particular focus on its usage for reinterpretation studies. To this end, we first review the main features of the normal mode of the program and how a detector simulation can be handled. We then detail, step-by-step, how to implement and validate an existing LHC analysis in the [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{} framework and how to use this reimplementation, possibly together with other recast codes available from the [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{} Public Analysis Database, for reinterpreting ATLAS and CMS searches in the context of a new model. Each of these points is illustrated by concrete examples. Moreover, complete reference cards for the normal and expert modes of [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{} are provided in two technical appendices.'
address:
- ' Institut Pluridisciplinaire Hubert Curien/Département Recherches Subatomiques, Université de Strasbourg/CNRS-IN2P3, 23 Rue du Loess, F-67037 Strasbourg, France'
- |
Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Énergies, LPTHE, F-75005 Paris, France\
Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France
author:
- Eric Conte
- Benjamin Fuks
bibliography:
- 'recasting.bib'
title: 'Confronting new physics theories to LHC data with [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{}'
---
Introduction
============
[<span style="font-variant:small-caps;">MadAnalysis</span> 5]{} in a nutshell
=============================================================================
Fast detector simulation with [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{}
=============================================================================================
Reimplementing an LHC analysis with [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{} {#sec:implementation}
===================================================================================================
Using [<span style="font-variant:small-caps;">MadAnalysis</span> 5]{} for reinterpreting LHC results
====================================================================================================
Conclusion
==========
Acknowledgements {#acknowledgements .unnumbered}
================
We thank both ATLAS and CMS for providing plentiful information on their searches to make them recastable by people outside the collaborations, this effort being crucial for the legacy of the LHC. This indeed allows the whole high-energy physics community to exploit the LHC experimental results for new physics studies in the best possible manner.
We are grateful to our CMS experimental colleagues from the exotica working group, and in particular to Andreas Albert, Olivier Buchmüller and Viatcheslav Valuev, for their help in the validation of the reimplementation of the CMS-EXO-16-010 analysis. We also heartfully thank Guillaume Chalons, Sabine Kraml and Dipan Sengupta for valuable comments on the manuscript and all lively discussions on LHC recasting of the last 5 years.
BF has been supported in part by French state funds managed by the Agence Nationale de la Recherche (ANR), in the context of the LABEX ILP (ANR-11-IDEX-0004-02, ANR-10-LABX-63).
Reference card for the normal mode {#app:normal}
==================================
Reference card for the expert mode {#app:expert}
==================================
|
---
abstract: 'We compute derived quantities for various values of the model parameter of the Cornell potential model for the $S$-wave heavy quarkonia with radial quantum numbers $n=1$, $2$, and $3$. Our results can be used to determine leading and relative-order-$v^2$ nonrelativistic quantum chromodynamics matrix elements for $S$-wave charmonia and bottomonia such as $\psi(2S)$, $\eta_c(2S)$, and $\Upsilon(nS)$ for $n=1$, $2$, and $3$. These matrix elements will be essential ingredients for resumming relativistic corrections to processes involving those $S$-wave heavy quarkonium states.'
author:
- Hee Sok Chung
- Jungil Lee
- Daekyoung Kang
title: 'Cornell Potential Parameters for $\bm{S}$-wave Heavy Quarkonia '
---
Introduction\[intro\]
=====================
One of the most interesting recent developments in heavy-quarkonium phenomenology is the introduction of a new technique for resumming relativistic corrections to $S$-wave quarkonium production and decay rates. The technique resums corrections to all orders in the heavy-quark velocity $v$ in the heavy-quark-antiquark ($Q\bar{Q}$) rest frame [@Bodwin:2006dn; @Bodwin:2007fz] within the color-singlet mechanism of the nonrelativistic quantum chromodynamics (NRQCD) factorization approach [@Bodwin:1994jh]. With the method, the relative-order-$v^2$ NRQCD matrix element for the $S$-wave charmonium, which has a power-ultraviolet divergence and needs subtraction, has been evaluated with improved accuracy [@Bodwin:2006dn; @Bodwin:2007fz] compared with lattice calculations [@bks] and with the determination [@Braaten:2002fi] obtained by using the Gremm-Kapustin relation [@Gremm:1997dq]. In Ref. [@Bodwin:2006dn], the generalized version of the Gremm-Kapustin relation within the Cornell potential model [@Eichten:1978tg] was derived. The generalized Gremm-Kapustin relation allows one to resum a class of relativistic corrections in a potential-model color-singlet $Q\bar{Q}$ wave function. The resummation method has also been applied to determine leading-order NRQCD matrix elements for the $1S$ charmonium states. The resultant values for the matrix elements are significantly greater than those known previously [@Bodwin:2006dn; @Bodwin:2007fz]. In addition, the method has provided a reasonable solution [@Bodwin:2007ga] to the long-standing puzzle of the cross section for $e^+e^-\to J/\psi+\eta_c$ measured at the $B$-factories [@puzzle], which has been one of the greatest discrepancies between theory and experiment within the Standard Model.
Therefore, it is worthwhile to extend this method to study radially excited $S$-wave charmonia, $\psi(2S)$ and $\eta_c(2S)$, and spin-triplet $S$-wave bottomonia, $\Upsilon(nS)$, with radial quantum numbers $n=1$, $2$, and $3$. Unfortunately, the potential-model parameters and derived quantities reported in Ref. [@Bodwin:2006dn] are only for the $1S$ and the $2S$ states. The tabulation of the parameters in Ref. [@Bodwin:2006dn] is not convenient to use in combination with the improved version of the resummation method reported in Ref. [@Bodwin:2007fz], which considered only the $1S$ charmonium states.
In this paper, we compute derived quantities for various values of the model parameter of the Cornell potential model for $S$-wave heavy quarkonia with radial quantum numbers $n=1$, $2$, and $3$. The Cornell potential is a linear combination of the Coulomb and linear potentials. Following Ref. [@Eichten:1978tg], we rescale the Schrödinger equation and solve the equation numerically for various values of the model parameter $\lambda$, which determines the strength of the Coulomb potential relative to the linear potential. To achieve improved accuracies in calculating energy eigenvalues, we use the numerical method[^1] given in Refs. [@Kang:2006jd; @Aichinger:2005]. The potential-model parameters and derived quantities are listed as functions of $\lambda$. These parameters are ready for use to determine leading and relative-order-$v^2$ color-singlet NRQCD matrix elements for both $S$-wave charmonia and bottomonia.
The remainder of this paper is organized as follows. In Sec. \[sec:model\], we give a brief description of the Cornell potential model. Our numerical results are given in Sec. \[sec:result\], which is followed by a discussion in Sec. \[sec:discussion\].
Potential Model\[sec:model\]
============================
In order to compute leading and higher-order color-singlet NRQCD matrix elements for an $S$-wave heavy quarkonium with the radial quantum number $n=1$, $2$, or $3$, we need to compute the binding energy $\epsilon_{nS}$ of the $nS$ state that appears in the generalized Gremm-Kapustin relation [@Bodwin:2006dn; @Gremm:1997dq]. In this section, we describe briefly the Cornell potential model [@Eichten:1978tg] that we use to compute $\epsilon_{nS}$. We refer the reader to Refs. [@Bodwin:2006dn; @Bodwin:2007fz] which contain more complete descriptions of the model in conjunction with the NRQCD factorization formalism and the resummation technique for relativistic corrections to quarkonium processes.
We employ the Cornell potential [@Eichten:1978tg], which parametrizes the $Q\bar Q$ potential $V(r)$ as a linear combination of the Coulomb and linear potentials: $$V(r)=-\frac{\kappa}{r}+\sigma r,
\label{model-V}%$$ where $\kappa$ is a model parameter for the Coulomb strength and $\sigma$ is the string tension. The relation between the string tension $\sigma$ and the corresponding parameter $a$ in the original formulation of the Cornell potential model [@Eichten:1978tg] is $a=1/\sqrt{\sigma}$. By varying the parameters in the Cornell potential, one can obtain good fits to lattice measurements of the $Q\bar Q$ static potential [@Bali:2000gf]. Therefore, we assume that the use of the Cornell parametrization of the $Q\bar Q$ potential should result in errors that are much less than the order-$v^2$ errors (about 30% for a charmonium and about 10% for a bottomonium) that are inherent in the leading-potential approximation to NRQCD [@Bodwin:2006dn; @Bodwin:2007fz].
The Schrödinger equation for the $S$-wave radial wave function $R_{nS}(r)$ with the radial quantum number $n$ is $$\left[
-\frac{1}{mr^2} \frac{d}{dr}\left( r^2 \frac{d}{dr}\right)
+V(r)
\right]R_{nS}(r)=\epsilon_{nS} R_{nS}(r),
\label{radial}%$$ where $m$ is the quark mass and $\epsilon_{nS}$ is the binding energy for the $nS$ state. The model parameter $m$ is distinguished from the heavy-quark mass $m_Q$ that appears in the short-distance coefficients of NRQCD factorization formulas. For an $S$-wave state, the wave function is $\psi_{nS}(r)=R_{nS}(r)/\sqrt{4\pi}$.
The Schrödinger equation in Eq. (\[radial\]) depends on the model parameters $m$ and $\kappa$, where we assume that the string tension $\sigma$ is common to both charmonium and bottomonium states and that the value for $\sigma$ can be determined from lattice measurements of the $Q\bar{Q}$ static potential. Then, the dependence on the flavor appears through $m$ and $\kappa$. Introducing the scaled radius $\rho$ and scaled coupling $\lambda$ [@Eichten:1978tg],
$$\begin{aligned}
\rho&=&(\sigma m)^{1/3}\, r,
\label{r-rho}%
\\
\lambda&=&\frac{\kappa}{(\sigma/m^2)^{1/3}},
\label{kappa-lambda}%\end{aligned}$$
\[dimensionless\]
which are dimensionless, one can rewrite the radial equation in Eq. (\[radial\]) as [@Eichten:1978tg] $$\left[
\frac{d^2}{d\rho^2}
+\frac{\lambda}{\rho}-\rho+\zeta_{nS}
\right]u_{nS}(\rho)=0,
\label{eq:u}%$$ where $u_{nS}(\rho)$ and $\zeta_{nS}$ are the dimensionless radial wave function and the dimensionless energy eigenvalue of the $nS$ state, respectively. The relation between $R_{nS}(r)$ and $u_{nS}(\rho)$ is $$\label{eq:psir}%
R_{nS}(r)=
\sqrt{\sigma m}
\, \frac{u_{nS}(\rho)}{\rho},$$ where the wave functions are normalized according to $$\int_0^\infty |u_{nS}(\rho)|^2 d\rho
=\int_0^\infty |R_{nS}(r)|^2 r^2dr=1.$$ The binding energy is related to the dimensionless eigenvalue $\zeta_{nS}$ as $$\epsilon_{nS}=
[\sigma^2/m]^{1/3}\zeta_{nS}(\lambda).
\label{eq:ezeta}%$$ Note that Eq. (\[eq:u\]) depends only on $\lambda$. Therefore, the scaled equation can be solved for a given $\lambda$ to get the wave function $u_{nS}(\rho)$ and the eigenvalue $\zeta_{nS}$. The flavor dependence appears when we invert them to get the radial wave function $R_{nS}(r)$ and the energy eigenvalue $\epsilon_{nS}$. In this step, $m$ looks independent of the model parameter $\lambda$. However, we can express $m$ in terms of $\sigma$, $\lambda$, and the $1S$-$2S$ mass splitting [@Eichten:1978tg; @Bodwin:2006dn; @Bodwin:2007fz]: $$m (\lambda)=
\sigma^2 \left[\frac{\zeta_{2S}(\lambda)-\zeta_{1S}(\lambda)}
{m_{2S}-m_{1S}}\right]^3.
\label{m-lam}%$$ For $S$-wave states, the wave function at the origin $\psi_{nS}(0)=R_{nS}(0)/\sqrt{4\pi}$ can be expressed as [@Eichten:1978tg; @Bodwin:2006dn; @Bodwin:2007fz] $$|\psi_{nS}(0)|^2
=\frac{m}{4\pi}\int d^3r
|\psi_{nS}(r)|^2
\frac{\partial V(r)}{\partial\, r}
=\frac{\sigma\, m(\lambda)}{4\pi}
\left[
1+\lambda F_{nS}(\lambda)
\right]
,
\label{eq:psi0rhom2}%$$ where $F_{nS}(\lambda)$ is the expectation value of $1/\rho^2$ for the $nS$ state: $$F_{nS}(\lambda)=
\int_0^\infty \frac{d\rho}{\rho^2} \,\left|u_{nS}(\rho)\right|^2.
\label{eq:rhom2}%$$ For purposes of computation of the NRQCD matrix elements, it is convenient to express those matrix elements in terms of the potential-model parameters listed above. A convenient parametrization can be found in Ref. [@Bodwin:2007fz]. According to Ref. [@Bodwin:2007fz], the leading-order NRQCD matrix element for the $S$-wave heavy quarkonium depends on $\lambda$, $m(\lambda)$, and $F_{nS}(\lambda)$. The ratio of the relative-order-$v^2$ NRQCD matrix element to the leading-order one is proportional to $\zeta_{nS}(\lambda)$. The electromagnetic decay rate for the quarkonium, in which relativistic corrections to all orders in $v$ are resummed, is expressed in terms of these NRQCD matrix elements [@Bodwin:2007fz]. Because $m(\lambda)$, $F_{nS}(\lambda)$, and $\zeta_{nS}(\lambda)$ depend on $\lambda$, the decay rate is completely determined by $\lambda$. Determination of the best value for $\lambda$ can be made by imposing a requirement that the resummed formula for the decay rate should reproduce the measured rate. Therefore, the leading-order and the relative-order-$v^2$ NRQCD matrix elements are completely determined [@Bodwin:2007fz].
Numerical Results \[sec:result\]
================================
$\lambda$ $\zeta_{1S}$ $F_{1S}$ $\zeta_{2S}$ $F_{2S}$ $\zeta_{3S}$ $F_{3S}$
---------------- -------------- ---------- -------------- ---------- -------------- ----------
0.0 2.338107 1.1248 4.087949 0.8237 5.520560 0.6983
\[-0.8ex\] 0.1 2.253678 1.1869 4.029425 0.8525 5.473169 0.7178
\[-0.8ex\] 0.2 2.167316 1.2532 3.970286 0.8821 5.425462 0.7375
\[-0.8ex\] 0.3 2.078949 1.3237 3.910531 0.9125 5.377441 0.7576
\[-0.8ex\] 0.4 1.988504 1.3989 3.850160 0.9437 5.329112 0.7780
\[-0.8ex\] 0.5 1.895904 1.4789 3.789174 0.9756 5.280478 0.7987
\[-0.8ex\] 0.6 1.801074 1.5641 3.727575 1.0083 5.231545 0.8196
\[-0.8ex\] 0.7 1.703935 1.6546 3.665364 1.0417 5.182316 0.8408
\[-0.8ex\] 0.8 1.604409 1.7507 3.602543 1.0758 5.132798 0.8622
\[-0.8ex\] 0.9 1.502415 1.8527 3.539116 1.1105 5.082996 0.8837
\[-0.8ex\] 1.0 1.397876 1.9608 3.475087 1.1459 5.032914 0.9054
\[-0.8ex\] 1.1 1.290709 2.0753 3.410458 1.1818 4.982560 0.9272
\[-0.8ex\] 1.2 1.180834 2.1965 3.345233 1.2183 4.931938 0.9492
\[-0.8ex\] 1.3 1.068171 2.3246 3.279418 1.2552 4.881053 0.9712
\[-0.8ex\] 1.4 0.952640 2.4599 3.213016 1.2927 4.829913 0.9933
\[-0.8ex\] 1.5 0.834162 2.6026 3.146031 1.3306 4.778522 1.0155
\[-0.8ex\] 1.6 0.712658 2.7529 3.078468 1.3689 4.726886 1.0377
\[-0.8ex\] 1.7 0.588049 2.9110 3.010330 1.4077 4.675010 1.0598
\[-0.8ex\] 1.8 0.460260 3.0773 2.941621 1.4468 4.622899 1.0820
\[-0.8ex\] 1.9 0.329215 3.2518 2.872344 1.4862 4.570560 1.1041
\[-0.8ex\] 2.0 0.194841 3.4348 2.802503 1.5260 4.517996 1.1262
\[-0.8ex\] 2.1 0.057065 3.6264 2.732099 1.5662 4.465212 1.1482
\[-0.8ex\] 2.2 $-$0.084182 3.8269 2.661134 1.6066 4.412212 1.1702
\[-0.8ex\] 2.3 $-$0.228969 4.0364 2.589611 1.6474 4.359001 1.1921
\[-0.8ex\] 2.4 $-$0.377362 4.2550 2.517529 1.6885 4.305582 1.2139
\[-0.8ex\] 2.5 $-$0.529425 4.4829 2.444888 1.7299 4.251959 1.2356
\[-0.8ex\] 2.6 $-$0.685221 4.7202 2.371688 1.7716 4.198135 1.2573
\[-0.8ex\] 2.7 $-$0.844808 4.9670 2.297928 1.8137 4.144112 1.2788
\[-0.8ex\] 2.8 $-$1.008244 5.2235 2.223605 1.8561 4.089893 1.3002
\[-0.8ex\] 2.9 $-$1.175584 5.4897 2.148718 1.8989 4.035481 1.3216
\[-0.8ex\] 3.0 $-$1.346882 5.7657 2.073261 1.9421 3.980877 1.3428
: \[table:param\] Scaled energy eigenvalues $\zeta_{nS}$ and $F_{nS}$ of the $S$-wave heavy quarkonium with radial quantum numbers $n=1$, $2$, and $3$ as functions of the Coulomb strength parameter $\lambda$.
In this section, we list our numerical values for the parameters and derived quantities for the $S$-wave heavy quarkonia with radial quantum numbers $n=1$, $2$, and $3$. In order to provide a set of parameters that can be used for both charmonia and bottomonia, we list the values for $\zeta_{nS}$ and $F_{nS}$ that are common to both cases. In Table \[table:param\], we tabulate those values as functions of the model parameter $\lambda$, whose range has been chosen so that the parameters can be used to determine the NRQCD matrix elements for both charmonia and bottomonia. The dependence of the eigenvalues $\zeta_{nS}$ and $F_{nS}$ on $\lambda$ are shown in Figs. \[figure1\] and \[figure2\], respectively.
In Ref. [@Kang:2006jd], the authors solved the Schrödinger equation in Eq. (\[radial\]) numerically by using the inverse iteration method introduced in Ref. [@Aichinger:2005]. The method uses a trial wave packet which is a linear combination of the eigenfunctions for a given system and the amplification operator of the form $(H-\zeta_{\rm trial})^{-1}$, where $H$ is the Hamiltonian of the system and $\zeta_{\rm trial}$ is a trial eigenvalue. If this operator acts on an eigenfunction of the Hamiltonian, then the operator yields a multiplicative factor $(\zeta-\zeta_{\rm trial})^{-1}$, where $\zeta$ is the eigenvalue of that eigenfunction. As $\zeta_{\rm trial}$ gets closer to $\zeta$, the multiplicative factor blows up. Therefore, if this operator acts on a wave packet, then an eigenfunction, whose eigenvalue is the nearest to $\zeta_{\rm trial}$, is selectively amplified. Therefore, by applying the operator repeatedly, one could obtain the eigenfunction and the corresponding eigenvalue from the expectation value of the Hamiltonian. By varying $\zeta_{\rm trial}$, one could also obtain the energy spectrum. The details of the numerical method can be found in Refs. [@Kang:2006jd; @Aichinger:2005]. Here, we use this method to generate Table \[table:param\].
![Eigenvalues $\zeta_{nS}$ as functions of $\lambda$. []{data-label="figure1"}](fig1.eps){width="11.0cm"}
![$F_{nS}$ as functions of $\lambda$. []{data-label="figure2"}](fig2.eps){width="11.0cm"}
If one substitutes $\zeta_{nS}(\lambda)$ listed in Table \[table:param\] and the measured $1S$-$2S$ mass splitting for either the charmonium or the bottomonium into Eq. (\[m-lam\]), one can obtain the mass parameter $m(\lambda)$ as a function of $\lambda$. $\epsilon_{nS}$ and $\psi_{nS}(0)$ are calculable by substituting the parameters $\zeta_{nS}(\lambda)$, $m(\lambda)$, and $F_{nS}(\lambda)$ into Eqs. (\[eq:ezeta\]) and (\[eq:psi0rhom2\]), respectively. A convenient way to determine the optimal value for the parameter $\lambda$ is given in Refs. [@Bodwin:2006dn; @Bodwin:2007fz].
Discussion \[sec:discussion\]
=============================
We have computed derived quantities for various values of the model parameter of the Cornell potential model for the $S$-wave heavy quarkonia with radial quantum numbers $n=1$, $2$, and $3$. The scaled Schrödinger equation is solved numerically for various values for the model parameter $\lambda$ which determines the strength of the Coulomb potential relative to the linear potential. The scaled energy eigenvalue $\zeta_{nS}$ and a derived potential quantity $F_{nS}$ are computed as functions of $\lambda$. These numbers are useful in determining the leading and relative-order-$v^2$ NRQCD matrix elements for both $S$-wave charmonia and bottomonia with radial quantum numbers $n=1$, $2$, and $3$. As an application, the leading-order NRQCD matrix elements for the $\Upsilon(nS)$ have already been calculated and used to determine the branching fractions and charm-momentum distributions for the inclusive charm production in $\Upsilon(nS)$ decays, which are being analyzed by the CLEO Collaboration [@Kang:2007uv]. The result listed in this paper can also be used to determine the NRQCD matrix elements for the $2S$ charmonium states. Once recent studies [@Chung:2007ke; @Lee:2007kf; @Lee:2007kg] to compute relativistic corrections to the leptonic width of the $S$-wave spin-triplet quarkonium are extended to complete the resummation of relativistic corrections to the process at next-to-leading order in $\alpha_s$, our results can also be used to determine the leading-order and relative-order-$v^2$ NRQCD matrix elements for the $S$-wave quarkonium with accuracies better than the best available values in Ref. [@Bodwin:2007fz].
We wish to thank Geoff Bodwin for checking the values of the parameters for the $1S$ state, which were published in Refs., and useful discussions. We also express our gratitude to Eunil Won for a useful discussion on the numerical method in Ref. [@Kang:2006jd]. Critical reading of the manuscript by Chaehyun Yu is also acknowledged. The work of HSC was supported by the Korea Research Foundation under MOEHRD Basic Research Promotion grant KRF-2006-311-C00020. The work of DK was supported by the Basic Research Program of the Korea Science and Engineering Foundation (KOSEF) under Grant No. R01-2005-000-10089-0. The work of JL was supported by the Korea Research Foundation under Grant No. KRF-2004-015-C00092.
G. T. Bodwin, D. Kang, and J. Lee, Phys. Rev. D [**74**]{}, 014014 (2006) \[arXiv:hep-ph/0603186\]. G. T. Bodwin, H. S. Chung, D. Kang, J. Lee, and C. Yu, arXiv:0710.0994 \[hep-ph\]. G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D [**51**]{}, 1125 (1995); [**55**]{}, 5853(E) (1997) \[arXiv:hep-ph/9407339\].
G. T. Bodwin, S. Kim, and D. K. Sinclair, Nucl. Phys. B, Proc. Suppl. [**34**]{}, 434 (1994); [**42**]{}, 306 (1995) \[arXiv:hep-lat/9412011\]; G. T. Bodwin, D. K. Sinclair, and S. Kim, Phys. Rev. Lett. [**77**]{}, 2376 (1996) \[arXiv:hep-lat/9605023\]; Int. J. Mod. Phys. A [**12**]{}, 4019 (1997) \[arXiv:hep-ph/9609371\]; Phys. Rev. D [**65**]{}, 054504 (2002) \[arXiv:hep-lat/0107011\]. E. Braaten and J. Lee, Phys. Rev. D [**67**]{}, 054007 (2003); [**72**]{}, 099901(E) (2005) \[arXiv:hep-ph/0211085\].
M. Gremm and A. Kapustin, Phys. Lett. B [**407**]{}, 323 (1997) \[arXiv:hep-ph/9701353\]. E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T. M. Yan, Phys. Rev. D [**17**]{}, 3090 (1978); [**21**]{}, 313(E) (1980). G. T. Bodwin, J. Lee, and C. Yu, arXiv:0710.0995 \[hep-ph\]. See, for example, G. T. Bodwin, J. Korean Phys. Soc. [**45**]{}, S306 (2004) \[arXiv:hep-ph/0312173\]; J. Lee, J. Korean Phys. Soc. [**45**]{}, S354 (2004) \[arXiv:hep-ph/0312251\] and references therein. G. T. Bodwin, D. Kang, and J. Lee, Phys. Rev. D [**74**]{}, 114028 (2006) \[arXiv:hep-ph/0603185\]. D. Kang and E. Won, arXiv:physics/0609176. M. Aichingera and E. Krotscheck, Comput. Mater. Sci. [**34**]{}, 188 (2005).
G. S. Bali, Phys. Rep. [**343**]{}, 1 (2001) \[arXiv:hep-ph/0001312\].
D. Kang, T. Kim, J. Lee, and C. Yu, Phys. Rev. D [**76**]{}, 114018 (2007) \[arXiv:0707.4056 \[hep-ph\]\]. H. S. Chung, J. Lee, and C. H. Yu, J. Korean Phys. Soc. [**50**]{}, L357 (2007). J. Lee, H. K. Noh, and C. H. Yu, J. Korean Phys. Soc. [**50**]{}, 398 (2007). J. Lee, H. K. Noh, and C. H. Yu, J. Korean Phys. Soc. [**50**]{}, 403 (2007).
[^1]: In previous calculations in Refs. [@Bodwin:2006dm; @Bodwin:2006dn], the authors quoted the energy eigenvalues determined in Ref. [@Eichten:1978tg]. A recently improved analysis given in Ref. [@Bodwin:2007fz] employed the numerical method used in the present work.
|
---
abstract: 'We have studied the magnetic excitations of electron-doped Sr$_{2-x}$La$_x$IrO$_4$ ($0 \leq x \leq 0.10$) using resonant inelastic x-ray scattering (RIXS) at the Ir L$_3$-edge. The long range magnetic order is rapidly lost with increasing $x$, but two-dimensional short-range order (SRO) and dispersive magnon excitations with nearly undiminished spectral weight persist well into the metallic part of the phase diagram. The magnons in the SRO phase are heavily damped and exhibit anisotropic softening. Their dispersions are well described by a pseudospin-1/2 Heisenberg model with exchange interactions whose spatial range increases with doping. We also find a doping-independent high-energy magnetic continuum, which is not described by this model. The spin-orbit excitons arising from the pseudospin-3/2 manifold of the Ir ions broaden substantially in the SRO phase, but remain largely separated from the low-energy magnons. Pseudospin-1/2 models are therefore a good starting point for the theoretical description of the low-energy magnetic dynamics of doped iridates.'
author:
- 'H. Gretarsson'
- 'N. H. Sung'
- 'J. Porras'
- 'J. Bertinshaw'
- 'C. Dietl'
- 'Jan A. N. Bruin'
- 'A. F. Bangura'
- 'Y. K. Kim'
- 'R. Dinnebier'
- Jungho Kim
- 'A. Al-Zein'
- 'M. Moretti Sala'
- 'M. Krisch'
- 'M. Le Tacon'
- 'B. Keimer'
- 'B. J. Kim'
title: 'Persistent paramagnons deep in the metallic phase of Sr$_{2-x}$La$_x$IrO$_4$'
---
The proximity of two-dimensional antiferromagnetism and high-temperature superconductivity in copper oxides [@Keimer_Nature] and iron pnictides [@Paglione_Nature] suggest that both phases are intimately related. Neutron scattering [@Fujita_review_2012; @Tranquada_review_2014] and resonant inelastic x-ray scattering (RIXS) [@Ament_RMP_RIXS_2011] experiments on both sets of materials [@LeTacon_NatPhys_2011; @MPDean_NatMat_2013; @Minola_PRL_2015; @KIshii_NatComm_2013; @WSLee_NatPhys_2014; @KeJin_NatComm_2012] have indeed revealed damped spin excitations in the superconducting regimes of the phase diagrams. In RIXS experiments, their dispersions and spectral weights are closely similar to those of magnons in the magnetically ordered parent compounds.
The notion of magnetically mediated high-temperature superconductivity has motivated an extensive search for new materials that realize two-dimensional quantum antiferromagnets akin to those in the undoped cuprates and iron pnictides. The antiferromagnetic Mott insulator Sr$_2$IrO$_4$ has emerged as a particularly promising candidate, because it is isostructural to La$_2$CuO$_4$, the progenitor of one of the most prominent families of superconducting cuprates, and it exhibits a closely analogous electronic structure. The Mott-insulating state in Sr$_2$IrO$_4$ is driven by the combination of intra-atomic spin-orbit coupling and electronic correlations (“spin-orbit Mott insulator”). Its magnetic ground state and low-energy magnetic excitations are well described by the pseudospin $j_{\rm{eff}}$=1/2 states arising from the spin-orbit coupled spin and orbital angular momenta of the iridium ions [@BJKim_PRL_2008; @BJKim_Science_2009]. The pseudospins decorate a square lattice, and their interactions are well described by a Heisenberg model [@Jungho_PRL_2012] akin to the spin-1/2 Hamiltonian describing the magnetic dynamics of La$_2$CuO$_4$ [@Coldea_PRL_2001].
These similarities are driving theoretical work predicting $d$-wave superconductivity in electron-doped Sr$_2$IrO$_4$ [@FaWang_PRL2011; @Watanabe_PRL_2013; @Yang_PRB2014; @Kee_PRL_2014]. Evidence supporting this prediction has emerged from angle-resolved photoemission spectroscopy (ARPES) measurements on electron-doped Sr$_2$IrO$_4$ surfaces, doped by charge transfer from a monolayer of adatoms. The Fermi surface of the surface electron system is split up into disjointed segments (“Fermi arcs”) [@Kim_Science2014] and exhibits a gap with $d$-wave symmetry at low temperature [@Kim_dWave; @Feng_dWave] – features that are hallmarks of $d$-wave superconductivity in the hole-doped cuprates [@Keimer_Nature]. These results have also been partially reproduced in [@Baumberger_PRL_2015], although impurity-driven disorder [@Wilson_LaDoped_PRB_2015] and oxygen vacancies [@Nakheon_arXiv_2015] have been an impediment to the progress in bulk, chemically-doped counterpart.
Despite these encouraging results, it is still unclear whether the magnetic correlations are as robust against carrier doping in Sr$_2$IrO$_4$ as they are in the cuprates. In particular, the on-site Coulomb repulsion is relatively weak in Sr$_2$IrO$_4$, resulting in a small charge gap ($\sim 0.4$ eV [@Moon_PRB_2009]) which is comparable to the magnon bandwidth [@Jungho_PRL_2012]. Recent Raman experiments have also revealed a strong pseudospin-lattice coupling in Sr$_2$IrO$_4$, indicating unquenched orbital dynamics that is quite unusual for a Mott insulator [@Gretarsson_TwoMagnon_2015]. These observations indicate a complex interplay between pseudospin, charge, and lattice degrees of freedom. All of these variables are expected to be sensitive to chemical doping, highlighting the need for a systematic study on how the magnetic correlations evolve in electron-doped Sr$_2$IrO$_4$.
 (a) Schematic phase diagram of Sr$_{2-x}$La$_x$IrO$_4$. Data points for the Neel temperature are taken from the magnetization measurements in Ref. [@sup]. (b) $HH$-scans through the magnetic Bragg peak Q$_{\rm M}=$ (0.5 0.5 $L_{even})$ at $T = 20$ K (includes x=0.10 at $T=300$ K). The maximum intensity of each scan was normalized to unity. ](fig01){width="0.975\columnwidth"}
Here we report a systematic investigation of the doping evolution of the magnetic structure and magnetic excitations in Sr$_{2-x}$La$_x$IrO$_4$ (x=0, 0.015, 0.04, and 0.10) by Ir L$_3$-edge RIXS. Our results show that antiferromagnetic long-range order (LRO) is rapidly lost upon electron doping ($0.015\!\!<\!\!x_c\!\!\leq$0.04), followed by persistent two-dimensional short-range order (SRO) deep in the metallic phase of . We report a detailed description of the low-energy magnon and high-energy spin-orbit exciton in the SRO phase, which provides an excellent basis for the theoretical description of the electronic properties of the doped iridates.
The RIXS experiments were carried out at the European Synchrotron Radiation Facility using the ID20 beamline. A spherical (2 m radius) diced Si(844) analyzer with a 60 mm mask and a Si(844) secondary monochromator were used to obtain an overall energy resolution of $\sim 25$ meV (full width at half maximum) and an in-plane momentum resolution of $\sim 0.035$ reciprocal lattice units. (The momentum transfer Q$=(H,K,L)$ is quoted in terms of reciprocal vector $\pi/(a,b,c)$ where $a\approx b \approx 3.88$ ${\rm \AA}$ (undistorted unit cell) and $c \approx $ 25.8 ${\rm \AA}$ are the lattice parameters. In-plane crystal momenta q are quoted in square-lattice notation with unit lattice constant.) ARPES measurements were performed at Beamline 4.0.3 at the Advanced Light Source. The energy of the incident light was $h\nu$=90 eV and overall energy resolution of $\sim 25$ meV was achieved. Single crystals of La-doped were grown by the flux method, previously described in detail [@Nakheon_arXiv_2015]. The La concentrations were checked via electron probe micro-analysis.
{width="1.9\columnwidth"}
The physical properties of Sr$_2$IrO$_4$ are very sensitive to the doping level and to the crystal growth conditions [@Cao_LaDoped_PRB_2011; @Wilson_LaDoped_PRB_2015; @Nakheon_arXiv_2015]. We have therefore carefully characterized all samples using susceptibility and resonant x-ray diffraction measurements. Fig. \[fig01\](a) shows the phase diagram derived from our magnetization measurements (see Supplemental Material (SM) in Ref. [@sup]). In crystals with $x=0$ and 0.015, the long range magnetic order (LRO) [@BJKim_Science_2009] sets in around T$_N$ = 240 K and 230 K, respectively, while no signature of such order is found in both $x=0.04$ and 0.10, indicating a doping-induced transition into a paramagnetic state. This observation is confirmed by monitoring the antiferromagnetic Bragg peaks, which are visible in the elastic scattering channel of the RIXS spectra for Q$_{\rm M}=$ (0.5 0.5 $L_{even})$ (Fig. \[fig01\] (b)). Whereas in the $x=0$ and 0.015 crystals the magnetic Bragg peaks are sharp and narrow along the $HH$-directions, the elastic magnetic response of the $x=0.04$ and $x=0.10$ samples becomes extremely broad along the $HH$-direction and no correlation are observed along the $L$-direction (see SM in Ref. [@sup]), implying two-dimensional (2D) magnetic short-range order. By fitting the profiles to 2D Lorentzians, we determined the in-plane correlation lengths $\xi = 10$ ${\rm\AA}$ and 8 ${\rm\AA}$ for $x=0.04$ and $x=0.10$, respectively [@note]. We also note that the 2D magnetic correlations persist to high temperature ($\xi \sim 5$ ${\rm\AA}$ for $x = 0.10$ at $T = 300$ K), consistent with the large in-plane exchange coupling determined by RIXS and Raman scattering [@Jungho_PRL_2012; @Gretarsson_TwoMagnon_2015].
The rapid suppression of magnetic LRO in electron-doped Sr$_{2-x}$La$_x$IrO$_4$ (see Figure \[fig01\](a)) agrees with neutron diffraction results from Ref. and parallels the behavior in hole-doped cuprates (such as La$_{2-x}$Sr$_{x}$CuO$_4$), where the LRO is quenched for $x \sim0.02$ [@Keimer_PRB_LCO_1992]. In the electron-doped cuprates, on the other hand, LRO survives well above 0.1 electrons per Cu [@Armitage_RMP_2010]. This supports the analogy between electron doping in the iridates and hole doping in the cuprates proposed on the basis of the opposite signs of the next-nearest-neighbor hopping amplitudes in the two sets of materials [@FaWang_PRL2011].
Having studied the static magnetic properties, we now turn to the evolution of the magnetic excitations in across the transition between LRO and SRO phases. Figure \[fig02\] (a,b) displays color maps of RIXS spectra (x=0 and x=0.10) taken at $T = 20$ K along high-symmetry directions of reciprocal space. In agreement with prior work on the undoped compound [@Jungho_PRL_2012], the data show a highly dispersive magnon excitation emanating from the antiferromagnetic ordering vector and extending up to about 0.2 eV. Our new data on the highest doped compound shows that magnon excitations (paramagnons) with closely similar dispersion and nearly undiminished spectral weight persist deep into the phase diagram of . The magnetic dynamics in the doped iridates are thus strongly reminiscent of those in the hole-doped layered cuprates, where paramagnon excitations akin to the spin waves of antiferromagnetic La$_2$CuO$_4$ were found deeply in the metallic and superconducting regimes of the phase diagram [@LeTacon_NatPhys_2011; @MPDean_NatMat_2013].
To emphasize the importance of our findings, we show in Fig. \[fig03\] (a) an ARPES intensity map in the x=0.10 sample, taken at $T = 10$ K and at the Fermi level energy (E$_F$). Despite the detection of well-defined paramagnons in the x=0.10 sample a clear Fermi surface (FS) is apparent, indicating that the sample is deep in the metallic phase [@note2]. The dispersive charge excitation, crossing E$_F$, can be seen in more detail in \[fig03\] (b) where we plot the binding energy as a function of momentum (taken along the dashed horizontal line in (a)). In a previous report [@Baumberger_PRL_2015], it has been shown that this band is gapped near ($\pi,0$), making it a “Fermi arc” instead of a usual FS. Investigating how the magnetic correlations can coexist with a FS (i.e. system with Fermi-Dirac statistics), similar to the case in optimally hole-doped cuprates, is a question that should be addressed. One possible explanation for this discrepancy could be electronic phase separation on a macroscopic scale. This would give rise to a RIXS spectrum that looks like a superposition of metallic and insulating parts which is not supported by our data (e.g. Fig. \[fig02\](f) shows no hint of parent phase in the x=0.10 sample). Microscopic phase separation, as seen in STM measurements on metallic [@Wilson_LaDoped_PRB_2015], can however not be excluded.
To gain more insight into the magnetic excitations it is important to look systematically at how the RIXS spectra evolve with increasing doping. Fig. \[fig02\] (c) presents two salient features of the magnon (A) in the constant-momentum cuts plots (x=0, 0.015, 0.04 and x=0.10). First, the magnon energy at q=($\pi,0$) remains unchanged, although the magnon becomes severely damped. Second, at q=($\pi/2,\pi/2$), the magnon shows clear softening in addition to becoming broader. The observed anisotropic magnetic softening along $(0,0)\rightarrow(\pi,\pi)$ direction is in stark contrast to the hardening of the magnon in electron-doped cuprates but resembles results seen in the metallic Bi-based cuprates [@Grioni_NatComm_2014; @MPDean_PRB_2014]. This strengthens the analogy between electron-doped iridates and hole-doped cuprates already pointed out above.
To quantify the doping dependence of the magnon (or paramagnon) dispersion relations, we have tracked the peak positions in the RIXS spectra as a function of momentum (e.g. red triangles in Fig. \[fig02\](d-h)). The resulting dispersions are plotted in Fig. \[fig04\]. The error bars were determined by selecting the energy interval in which the measured intensity is within 95$\%$ of the maximum intensity, following a recent RIXS study of a doped cuprate [@DJHuang_RIXS_2015]. Whereas elastic and inelastic contributions could be reliably separated in most of the spectra, the magnon peak positions for q=($0,0$) and q=($\pi,\pi$) could not be accurately determined due to the low intensity of the magnetic signal and the strong elastic line, respectively. These q-points were therefore not included in the dispersions in Fig. \[fig04\].
To obtain the doping dependent magnetic interactions, we have fitted the magnon dispersions obtained in this way by a Heisenberg model with exchange interactions between first, second, and third nearest neighbors in the IrO$_2$ planes, termed $J$, $J^{\prime}$, $J^{\prime\prime}$. For the $x=0$ and $x=0.015$ samples, we obtain $J=60$ meV, $J^\prime=-20$ meV, and $J^{\prime\prime}=15$ meV, in excellent agreement with prior work on undoped (solid lines in Fig. \[fig04\] ) [@Jungho_PRL_2012]. To describe the anisotropic softening of the magnon dispersion in the $x=0.04$ and 0.10 samples noted above, the fit parameters were altered to $J=48$ meV, $J^\prime=-27$ meV, and $J^{\prime\prime}=20$ meV (dashed lines in Fig. \[fig04\]); here, the ratio between $J^{\prime}$ and $J^{\prime\prime}$ was fixed in order to reduce the free parameters. Doping $\rm Sr_2IrO_4$ with electrons thus reduces the nearest-neighbor exchange interaction by $\sim20\%$, while enhancing the longer-range interactions by $\sim30\%$. This finding is in general accord with expectations for magnetic interactions in itinerant-electron systems, and it will guide future theoretical work on electronic correlations in doped iridates.
The intensity maps in Fig. \[fig02\](a,b) and constant-momentum cuts in Fig. \[fig02\](d-h) highlight yet another intriguing feature of this 2D quantum magnet. By looking at the doping dependence at $(\pi/2,\pi/2)$, it becomes clear that the magnetic excitation has two components: the sharp mode discussed above which reacts strongly to electron-doping, and a high-energy tail which retains its intensity upon doping (e.g. Fig. \[fig02\](c)). This momentum dependent high-energy tail is bounded at low energies by the magnon dispersion, and at high energies by the dashed black line in Fig. \[fig02\](a,b). The same behavior is also evident along the Brillouin zone border (e.g. Fig. \[fig02\](e)). One explanation for this marked departure in doping behavior between the single-magnon and the high-energy tail is that the latter arises from spinon-like excitations not directly associated with the Néel-ordered state. Indeed, in the cuprates, the spinon-picture [@Anderson_PRL_2001] has been evoked to explain the strong damping of the magnon at $(\pi,0)$ [@Perring_PRL_2010] and the pronounced asymmetrical line shape of the two-magnon Raman scattering [@Suga_PRB_1990; @Kampf_PRB_2003]. The latter was recently also observed in $\rm Sr_2IrO_4$ [@Gretarsson_TwoMagnon_2015]. Although this analogy and explanation are intriguing, further measurements, targeting the polarization dependence of the signal [@Ronnow_NatPhys2015] are required to determine the origin of the high-energy magnetic tail in $\rm Sr_2IrO_4$.
For excitation energies above the optical gap ($> 0.4$ eV), we find a dispersive spin-orbit exciton mode arising from the =3/2 pseudospin manifold [@Moon_PRB_2009; @Jungho_PRL_2012]. In the parent compound, this feature comprises a sharp excitation near the optical gap (marked B in Fig. \[fig02\](c)) followed by incoherent higher-energy excitations (marked C). As pointed out in Ref. , the broadening of feature C may be due to overlap with the particle-hole continuum. Upon La-doping, the dispersion of the spin-orbit excitons (B+C) becomes less pronounced, and the sharp feature B disappears. At ($\pi/2,\pi/2$), in particular, feature B is drastically suppressed upon doping, while C loses intensity. The strong damping of the spin-orbit exciton can be attributed to a lowering of the particle–hole continuum threshold as the system becomes metallic.
In summary, our systematic RIXS measurements of electron-doped Sr$_{2-x}$La$_x$IrO$_4$ have uncovered persistent short-range magnetic order deep in the metallic phase of accompanied by dispersive magnetic excitations (paramagnons) of nearly undiminished spectral weight. Both the persistence of magnon-like excitations upon doping and their anisotropic softening indicate an intriguing analogy between electron-doped iridates and hole-doped cuprates. The low-energy magnon dispersions are well described by a $j_{\rm{eff}}$=1/2 Heisenberg model with exchange interactions whose spatial range increases with increasing doping. A doping independent high-energy magnetic continuum has also been found, suggesting that the ground state of contains additional short-range correlations not captured by the Heisenberg model. Finally, the $j_{\rm{eff}}$=3/2 spin-orbit exciton broadens and becomes less dispersive with increasing doping, but the overlap with the low-energy paramagnon branch remains small. Models based on $j_{\rm{eff}}$=1/2 pseudospins therefore appear to be a good starting point for the theoretical description of the electronic structure and for a realistic assessment of the prospects for unconventional superconductivity in doped iridates.
*Note added*: Damped magnon and anisotropic softening have recently also been observed by Xuerong Liu and collaborators in La-doped [@Dean_arXiv_2016].
[33]{}
B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, [Nature [**518**]{}, 179 (2015)](http://www.nature.com/nature/journal/v518/n7538/full/nature14165.html).
J. Paglione and R. L. Greene, [Nature Physics [**6**]{}, 645 (2010)](http://www.nature.com/nphys/journal/v6/n9/abs/nphys1759.html).
M. Fujita, H. Hiraka, M. Matsuda, M. Matsuura, J. M. Tranquada, S. Wakimoto, G. Xu, K. Yamada, [Journal of the Physical Society of Japan [**81**]{}, 011007 (2012)]( http://dx.doi.org/10.1143/JPSJ.81.011007).
J. M. Tranquada, G. Xu, I. A. Zaliznyak, [Journal of Magnetism and Magnetic Materials [**350**]{}, 148 (2014)](http://dx.doi.org/10.1016/j.jmmm.2013.09.029).
L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, [Rev. Mod. Phys. [**83**]{}, 705 (2011)](http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.83.705).
M. Le Tacon, G. Ghiringhelli, J. Chaloupka, M. Moretti Sala, V. Hinkov, M. W. Haverkort, M. Minola, M. Bakr, K. J. Zhou, S. Blanco-Canosa, C. Monney, Y. T. Song, G. L. Sun, C. T. Lin, G. M. De Luca, M. Salluzzo, G. Khaliullin, T. Schmitt, L. Braicovich, and B. Keimer, [Nature Physics [**7**]{}, 725-730 (2011)](http://www.nature.com/nphys/journal/v7/n9/full/nphys2041.html).
M. P. M. Dean, G. Dellea, R. S. Springell, F. Yakhou-Harris, K. Kummer, N. B. Brookes, X. Liu, Y-J. Sun, J. Strle, T. Schmitt, L. Braicovich, G. Ghiringhelli, I. Božović, and J. P. Hill, [Nature Materials [**12**]{}, 1019-1023 (2013)](http://www.nature.com/nmat/journal/v12/n11/full/nmat3723.html).
M. Minola, G. Dellea, H. Gretarsson, Y. Y. Peng, Y. Lu, J. Porras, T. Loew, F. Yakhou, N. B. Brookes, Y. B. Huang, J. Pelliciari, T. Schmitt, G. Ghiringhelli, B. Keimer, L. Braicovich, and M. Le Tacon, [Phys. Rev. Lett. [**114**]{}, 217003 (2015)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.217003).
K. Ishii, M. Fujita, T. Sasaki, M. Minola, G. Dellea, C. Mazzoli, K. Kummer, G. Ghiringhelli, L. Braicovich, T. Tohyama, K. Tsutsumi, K. Sato, R. Kajimoto, K. Ikeuchi, K. Yamada, M. Yoshida, M. Kurooka, and J. Mizuki, [Nature Communications [**5**]{}, 3714 (2014)](http://www.nature.com/ncomms/2014/140425/ncomms4714/full/ncomms4714.html).
W. S. Lee, J. J. Lee, E. A. Nowadnick, S. Gerber, W. Tabis, S. W. Huang, V. N. Strocov, E. M. Motoyama, G. Yu, B. Moritz, H. Y. Huang, R. P. Wang, Y. B. Huang, W. B. Wu, C. T. Chen, D. J. Huang, M. Greven, T. Schmitt, Z. X. Shen, and T. P. Devereaux, [Nature Physics [**10**]{}, 883-889 (2014)](http://www.nature.com/nphys/journal/v10/n11/full/nphys3117.html).
Ke-Jin Zhou, Yao-Bo Huang, C. Monney, Xi Dai, V. N. Strocov, Nan-Lin Wang, Zhi-Guo Chen, Chenglin Zhang, Pengcheng Dai, Luc Patthey, J. van den Brink, H. Ding, and T. Schmitt, [Nature Communications [**4**]{}, 1470 (2013)](http://www.nature.com/ncomms/journal/v4/n2/full/ncomms2428.html).
B. J. Kim, Hosub Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, Jaejun Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, [Phys. Rev. Lett. [**101**]{}, 076402 (2008)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.076402).
B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, [Science [**323**]{}, 1329 (2009)](http://www.sciencemag.org/content/323/5919/1329).
Jungho Kim, D. Casa, M. H. Upton, T. Gog, Young-June Kim, J. F. Mitchell, M. van Veenendaal, M. Daghofer, J. van den Brink, G. Khaliullin, and B. J. Kim, [Phys. Rev. Lett. [**108**]{}, 177003 (2012)](http://link.aps.org/doi/10.1103/PhysRevLett.108.177003).
R. Coldea, S. M. Hayden, G. Aeppli, T. G. Perring, C. D. Frost, T. E. Mason, S.-W. Cheong, and Z. Fisk, [Phys. Rev. Lett. [**86**]{}, 5377 (2001)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.86.5377).
F. Wang and T. Senthil, [Phys. Rev. Lett. [**106**]{}, 136402 (2011)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.136402).
Y. Yang, W.-S. Wang, J.-G. Liu, H. Chen, J.-H. Dai, and Q.-H. Wang, [Phys. Rev. B [**89**]{}, 094518 (2014)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.094518).
Hiroshi Watanabe, Tomonori Shirakawa, and Seiji Yunoki, [Phys. Rev. Lett. [**110**]{}, 027002 (2013)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.027002).
Zi Yang Meng, Yong Baek Kim, and Hae-Young Kee, [Phys. Rev. Lett. [**113**]{}, 177003 (2014)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.177003).
Y. K. Kim, O. Krupin, J. D. Denlinger, A. Bostwick, E. Rotenberg, Q. Zhao, J. F. Mitchell, J. W. Allen, and B. J. Kim, [Science [**345**]{}, 187 (2014)](http://www.sciencemag.org/content/345/6193/187.abstract).
Y. K. Kim, N. H. Sung, J. D. Denlinger, and B. J. Kim, [Nature Physics [**12**]{}, 37 (2016)](http://www.nature.com/nphys/journal/v12/n1/full/nphys3503.html).
Y. J. Yan, M. Q. Ren, H. C. Xu, B. P. Xie, R. Tao, H. Y. Choi, N. Lee, Y. J. Choi, T. Zhang, and D. L. Feng, [Phys. Rev. X [**5**]{}, 041018 (2015)](http://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.041018).
Xiang Chen, Tom Hogan, D. Walkup, Wenwen Zhou, M. Pokharel, Mengliang Yao, Wei Tian, Thomas Z. Ward, Y. Zhao, D. Parshall, C. Opeil, J. W. Lynn, Vidya Madhavan, and Stephen D. Wilson, [Phys. Rev. B [**92**]{}, 075125 (2015)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.075125).
N. H. Sung, H. Gretarsson, D. Pröpper, J. Porras, M. Le Tacon, A. V. Boris, B. Keimer, and B. J. Kim, [Phil. Mag. [**96**]{}, 413 (2016)](http://www.tandfonline.com/doi/abs/10.1080/14786435.2015.1134835).
A. de la Torre, S. McKeown Walker, F. Y. Bruno, S. Ricco, Z. Wang, I. Gutierrez Lezama, G. Scheerer, G. Giriat, D. Jaccard, C. Berthod, T. K. Kim, M. Hoesch, E. C. Hunter, R. S. Perry, A. Tamai, and F. Baumberger, [Phys. Rev. Lett. [**115**]{}, 176402 (2015)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.115.176402).
S. J. Moon, Hosub Jin, W. S. Choi, J. S. Lee, S. S. A. Seo, J. Yu, G. Cao, T. W. Noh, and Y. S. Lee, [Phys. Rev. B [**80**]{}, 195110 (2009)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.195110).
H. Gretarsson, N. H. Sung, M. Höppner, B. J. Kim, B. Keimer, and M. Le Tacon, [arXiv:1509.03396 (2015)](http://arxiv.org/abs/1509.03396).
M. Ge, T. F. Qi, O. B. Korneta, D. E. De Long, P. Schlottmann, W. P. Crummett, and G. Cao, [Phys. Rev. B [**84**]{}, 100402(R) (2011)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.84.100402).
We cannot rule out that the broadening of the Bragg reflections is partially due to unresolved incommensurability of the magnetic correlations, in analogy to the hole-doped cuprates. The values of the correlation length we quote should therefore be regarded as lower bounds.
A weakly metallic behavior is seen in transport measurement on the x=0.04 sample [@sup], indicating that the insulator to metal phase boundary occurs close to that doping level.
B. Keimer, N. Belk, R. J. Birgeneau, A. Cassanho, C. Y. Chen, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh, R. W. Erwin, and G. Shirane, [Phys. Rev. B [**46**]{}, 14034 (1992)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.46.14034).
N. P. Armitage, P. Fournier, and R. L. Greene, [Rev. Mod. Phys. [**82**]{}, 2421 (2010)](http://link.aps.org/doi/10.1103/RevModPhys.82.2421).
M. Guarise, B. Dalla Piazza, H. Berger, E. Giannini, T. Schmitt, H. M. Rønnow, G.A. Sawatzky, J. van den Brink, D. Altenfeld, I. Eremin, and M. Grioni, [Nat. Comm. [**5**]{}, 5760 (2014)](http://www.nature.com/ncomms/2014/141218/ncomms6760/full/ncomms6760.html).
M. P. M. Dean, A. J. A. James, A. C. Walters, V. Bisogni, I. Jarrige, M. Hücker, E. Giannini, M. Fujita, J. Pelliciari, Y. B. Huang, R. M. Konik, T. Schmitt, and J. P. Hill, [Phys. Rev. B [**90**]{}, 220506 (2014)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.220506).
H. Y. Huang, C. J. Jia, Z. Y. Chen, K. Wohlfeld, B. Moritz, T. P. Devereaux, W. B. Wu, J. Okamoto, W. S. Lee, M. Hashimoto, Y. He, Z. X. Shen, Y. Yoshida, H. Eisaki, C. Y. Mou, C. T. Chen, and D. J. Huang, [Scientific Reports [**6**]{}, 19657 (2016)](http://dx.doi.org/10.1038/srep19657).
Chang-Ming Ho, V. N. Muthukumar, Masao Ogata, and P. W. Anderson, [Phys. Rev. Lett. [**86**]{}, 1626 (2001)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.86.1626).
N. S. Headings, S. M. Hayden, R. Coldea, and T. G. Perring, [Phys. Rev. Lett. [**105**]{}, 247001 (2010)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.247001).
S. Sugai, M. Sato, T. Kobayashi, J. Akimitsu, T. Ito, H. Takagi, S. Uchida, S. Hosoya, T. Kajitani, and T. Fukuda, [Phys. Rev. B [**42**]{}, 1045 (1990)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.42.1045).
A. A. Katanin and A. P. Kampf, [Phys. Rev. B [**67**]{}, 100404 (2003)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.67.100404).
B. Dalla Piazza, M. Mourigal, N. B. Christensen, G. J. Nilsen, P. Tregenna-Piggott, T. G. Perring, M. Enderle, D. F. McMorrow, D. A. Ivanov, and H. M. Rønnow, [Nature Physics [**11**]{}, 62 (2015)](http://www.nature.com/nphys/journal/v11/n1/full/nphys3172.html).
Jungho Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Khaliullin, and B. J. Kim, [Nature Communications [**5**]{}, 4453 (2014)](http://www.nature.com/ncomms/2014/140717/ncomms5453/full/ncomms5453.html).
See Supplemental Material at http://link.aps.org/supplemental/ for additional information.
Xuerong Liu, M. P. M. Dean, Z. Y. Meng, M. H. Upton, T. Qi, T. Gog, H. Ding, G. Cao, and J. P. Hill, [Phys. Rev. B [**93**]{}, 241102(R) (2016)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.241102).
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---
abstract: 'Sum-of-squares objective functions are very popular in computer vision algorithms. However, these objective functions are not always easy to optimize. The underlying assumptions made by solvers are often not satisfied and many problems are inherently ill-posed. In this paper, we propose LS-Net, a neural nonlinear least squares optimization algorithm which learns to effectively optimize these cost functions even in the presence of adversities. Unlike traditional approaches, the proposed solver requires no hand-crafted regularizers or priors as these are implicitly learned from the data. We apply our method to the problem of motion stereo ie. jointly estimating the motion and scene geometry from pairs of images of a monocular sequence. We show that our learned optimizer is able to efficiently and effectively solve this challenging optimization problem.'
author:
- Ronald Clark
- Michael Bloesch
- Jan Czarnowski
- Stefan Leutenegger
- 'Andrew J. Davison'
title: |
LS-Net: Learning to Solve Nonlinear Least Squares\
for Monocular Stereo
---
Introduction
============
Most algorithms in computer vision use some form of optimization to obtain a solution that best satisfies some objective function for the problem at hand. The optimization method itself can be seen as simply an intelligent means of searching the solution space for the answer, possibly exploiting the specific structure of the objective function to guide the search.
One particularly interesting form of objective function is one that is composed of a sum of many squared residual terms. $$E = \frac{1}{2}\sum_j r_j^2(\mathbf{x})$$ where $r_j$ is the j-th residual term and $E$ is the optimization objective.
In most cases the residual terms are a nonlinear function of the optimization variables and problems with this type of objective function are called nonlinear least square (NLLS) problems (NLSPs). NLSPs can be efficiently solved using second-order methods [@kelley1999iterative].
However, the success in finding a good solution also depends on the characteristics of the problem itself. The set of residual functions can be likened to a system of equations with their solution at zero, $r_j(\mathbf{x}) = 0$. If the number of variables in this system is larger than the number of equations then the system is underdetermined, if they are equal then it is well-determined and if there are more equations than variables then it is overdetermined. Well-posed problems need to satisfy three conditions: 1) a solution must exist 2) there must be a unique solution and 3) the solution must be continuous as a function of its parameters [@Tikhonov:Arsenin:Solutions1977].
Undetermined problems are ill-posed as they have infinitely many solutions and therefore no unique solution exists. To cope with this, traditional optimizers use hand-crafted regularizers and priors to make the ill-posed problem well-posed.
In this paper we aim to utilize strong and well-developed ideas from traditional nonlinear least squares solvers and integrate these with the promising new learning-based approaches. In doing so, we seek to capitalize on the ability of neural network-based methods to learn robust data-driven priors, and a traditional optimization-based approach to obtain refined solutions of high-precision. In particular, we propose to learn how to compute the update based on the current residual and Jacobian (and some extra parameters) to make the NLLS optimization algorithm more efficient and more robust to high noise.
We apply our optimizer to the problem of estimating the pose and depths of pairs of frames from a monocular image sequence known as monocular stereo as illustrated in Fig. \[fig:teaser\].
To summarise, the contributions of our paper are the following:
1. We propose an end-to-end trainable optimization method that builds upon the powerful approximate Hessian-based optimization approaches to NLLS problems.
2. The implicit learning of priors and regularizers for least squares problems directly from data.
3. The first approach to use a learned optimizer for efficiently minimizing photometric residuals for monocular stereo reconstruction.
Compared to existing learning-based approaches, our method is designed to produce predictions that are accurate and photometrically consistent.
The rest of the paper is structured as follows. First we outline related work on dense reconstruction using traditional and learning-based approaches. We then visit some preliminaries such as the structure of traditional Gauss-Newton optimizers for nonlinear least square problems. We then introduce our proposed system and finally carry out an evaluation of our method in terms of structure and motion accuracy on a number of sequences from publicly available datasets.
![Overview of our system for jointly optimizing a nonlinear least squares objective[]{data-label="fig:teaser"}](figs/teaser.pdf){width="\textwidth"}
Related Work
============
**Optimization for SLAM** In visual SLAM we are faced with the problem of estimating both the geometry of the scene and the motion of the camera. This is most often formulated as an optimization over the pixel depths and transformation parameters between pairs of frames. The cost function comprises some form of reprojection error which may be formulated either in terms of geometric or photometric residuals. Geometric residuals require the correspondence of points to be known and thus are only feasible for sparse recostructions. Photometric residuals are formulated in terms of intensity differences and can be computed across the entire image. However, this photometric optimization is difficult as the photometric residuals have high noise levels and various strategies have been proposed to cope with this. In DTAM [@Newcombe:etal:ISMAR2011], for example, this is accomplished by formulating a cost volume and integrating the residuals from multiple frames before performing the optimization. Even then, the residuals need to be combined with a TV-L1 regularization term to ensure noise does not dominate the reconstruction. Other approaches, such as LSD-SLAM [@Engel:etal:ECCV2014], operate only on high-gradient pixels where the signal-to-noise ratio of the photometric residual is high. Even so, none of these systems are able to estimate the geometry and motion in a single joint optimization. Rather, they resort to an approach which swithches between independently optimizing the motion parameters and then the depths in an alternating fashion. CodeSLAM [@Bloesch:etal:CVPR2018] overcomes this problem by using an autoencoder to compress the scene geometry into a small optimizable code, allowing for the joint optimization of both the geometry and motion.
**Learning for Monocular Stereo** There has been much interest recently in using end-to-end learning to estimate the motion of a camera [@Clark:etal:AAAI2017; @Clark:etal:CVPR2017; @Wang:etal:ICRA2017] and reconstruct scenes from monocular images [@Eigen:etal:ICCV2015]. Most of these [@Eigen:etal:ICCV2015; @Zhou:etal:CVPR2017] are based on feed-forward inference networks. The training signal for these networks can be obtained in many ways. The first approaches were based on a fully-supervised learning signal where labelled depth and pose information were used. Subsequent works have shown that the networks can be learned in a self-supervised manner using a learning signal derived, for example, from photometric error of pixel-wise reprojection [@Zhou:etal:CVPR2017], from the consistency of rays projected into a common volume [@Tulsiani:CVPR2017] or even using an adversarial signal by modelling the image formation process in a GAN framework [@choy20163d]. Even so, these approaches only utilize the photometric consistency in an offline manner, i.e. during training, and do not attempt to optimize it online as is common in traditional dense reconstruction methods.
To this extent, some works such as [@Ummenhofer:Brox:CVPR2017], have demonstrated that it is beneficial to include multiple views and a recurrent refinement procedure in the reconstruction process. Their network, comprising three stages, is closely related to the structure which we build on in this work. The first stage consists of a *bootstrap network* which produces a rough low-resolution prediction; the second stage consists of an *iterative network* which iteratively refines the bootstrap prediction; and finally a *refinement network* which computes a refined and upscaled depth map.
In this paper, we adopt the same structure but formalize the iterative network as an optimization designed to enforce multiview photometric consistency where the bootstrap network acts as an initialization of the optimization and the refinement acts as an upscaling. In essence, our reconstruction is based on an optimization procedure that is itself optimized using data. This is commonly referred to in the machine-learning literature as a *meta-learned optimizer*.
**Meta-learning and Learning to Optimize** A popular and very promising avenue of research which has been receiving increasing attention is that of meta-learned optimizers. Such approaches have shown great utility in performing few-shot learning without overfitting [@Ravi:ICLR2016; @clark:etal:metalearn], for optimizing GANS which are traditionally very difficult to train [@Metz:2016], for optimizing general black box functions [@Chen:ICML2017] and even for solving difficult combinatorial problems [@Dai:2017]. Perhaps the most important advantage is to learn data-driven regularization as demonstrated in [@Oktem:IOP2017] where the authors use a partially learned optimization approach for solving ill-posed inverse problems. In [@lin2016inverse], the authors train through a multi-step inverse compositional Lukas Kanade algorithm for aligning 2D images. In our method, we utilize a learned multi-step optimization model by using a recurrent network to compute the update steps for the optimization variables. While most approaches that attempt to learn optimization updates use either fully learned update steps [@Xiong:etal:CVPR2013; @Li:ICLR2017] about the objective and first-order gradient information [@Chen:ICML2017], we exploit the least-square structure of our problem and forward the full Jacobian matrix to provide the network with richer information. Our approach is – to the best of our knowledge – the first to use second-order approximations of the objective to learn optimization updates.
Preliminaries
=============
Nonlinear Least Squares Solvers
-------------------------------
Many optimization problems have an objective that takes the form of a sum of squared residual terms, $E = \frac{1}{2}\sum_j r_j^2(\mathbf{x})$ where $r_j$ is the j-th residual term and $E$ is the optimization objective. As such, much research has been devoted to finding efficient solvers for problems of this form. Two of the most successful and widely used approaches are the Gauss-Newton (GN) and Levenberg-Marquadt (LM) methods. Both of these are second-order, iterative optimization methods. However, instead of computing the true Hessian, they exploit the least-squares structure of the objective to compute an approximate Hessian that is used in the updates. Given an initial estimate of the variables, $\mathbf{x}_0$, these approaches compute updates to the optimization variable in the attempt to find a better solution, $\mathbf{x}_i$, at each step $i$. The incremental update, $\Delta \mathbf{x}_i$ is computed by solving a linear least squares problem which is formed by linearising the residual at the current estimate $\mathbf{r}(\mathbf{x}_i + \Delta \mathbf{x}_i) \approx \mathbf{r}_i + \mathbf{J}_i \Delta \mathbf{x}_i$ [@kelley1999iterative], with the abbreviations: $$\mathbf{r}_i = \mathbf{r}(\mathbf{x}_i), \ \ \ \ \mathbf{J}_i = \left.\frac{d\mathbf{r}}{d\mathbf{x}} \right\vert_{\mathbf{x} = \mathbf{x}_i}.$$
Using the linearized residual, the optimal update can be found as the solution to the quadratic problem [@kelley1999iterative] $$\Delta \mathbf{x}_i = \underset{ \Delta \mathbf{x}_i}{\arg\min} \frac{1}{2}|| \mathbf{r}_i + \mathbf{J}_i \Delta \mathbf{x}_i ||^2.$$ The well known Normal equations to this can be computed analytically by differentiating the problem and equating to zero. The update step used in GN is then given by solving: $$\mathbf{J}_i^T \mathbf{J}_i \Delta \mathbf{x}_i = - \mathbf{J}_i^T \mathbf{r}_i$$ By comparing this to Newton’s method which requires the computation of the true Hessian $\mathbf{H}(\mathbf{x}_i)$ for finding updates [@fletcher2013practical], we see that the GN method effectively approximates $\mathbf{H}(\mathbf{x}_i)$ using $ \mathbf{J}_i^T \mathbf{J}_i$, which is usually more efficient to compute. LM extends GN by adding a damping factor $\lambda$ to the update $\Delta \mathbf{x}_i = - (\mathbf{J}_i^T \mathbf{J}_i+\lambda \, \mathrm{diag}(\mathbf{J}_i^T \mathbf{J}_i))^{-1} \mathbf{J}_i^T \mathbf{r}_i$ to better condition the updates and make the optimization more robust [@fletcher2013practical].
In our proposed approach, we build on the GN method by not restricting the updates to be a static function of $\mathbf{J}_i$. Compared to LM which adaptively sets a single parameter, $\lambda$, we compute the entire update step by using a neural network which has as its input the full Jacobian $\mathbf{J}_i$. The details of this are described in Section \[sec:optimization\].
Warping and Photometric Cost Function
-------------------------------------
The warping function we use for the least squares cost function is similar to the loss used in the usupervised training in [@Zhou:etal:CVPR2017]. The warping is based on a spatial transformer which first transforms the coordinates of points in the target view to points in the source view and then samples the source view. The 4x4 transformation matrix, $\hat{T}_{t\rightarrow s}$ is obtained by applying an exponential map to the output of the network, i.e. $\hat{T}_{t\rightarrow s}= \exp{(\mathbf{p}^\times)}$ where $\mathbf{p}$ (bold face) is the relative pose represented as a six-vector and $p_s$ (non-bold face) is the pixel location in the source image and $p_t$ (non-bold face) is a pixel location in the target image (consistent with the notation in the paper)
$$p_s \sim K \hat{T}_{t\rightarrow s}\hat{D}_{t}(p_t)K^{-1}p_t
\label{eq:projection}$$
Using these warped coordinates, a synthesized image $\hat{I}_s(p)$ is obtained through bilinear sampling of the the source view at the locations $p_s$ computed in Eqn. \[eq:projection\]. The least squares loss function from which we derive $\mathbf{J}$ is then,
$$L = \sum_p || I_t (p) - \hat{I}_s(p) ||_2~,
\label{eq:objective}$$
where $I_t$ and $I_s$ are the source and target intensity images and the residual corresponding to each pixel is $\mathbf{r}_p = I_t (p) - \hat{I}_s(p)$. The elements of the Jcaobian of the warping function, $\mathbf{J}$, can be easily computed using autodiff (in Tensorflow simply [tf.gradients(res\[i\],x))]{} for each residual. However, to speed up our implementation we anylytically compute the elements of the Jacobian in our computation graph.
Model
=====
The model is built around the optimization of the photometric consistency of the depth and motion predictions for a short sequence of input images. Each sequence of images has a single “target" keyframe (which we choose as the first frame) for which we optimize the depth values. In all cases, we operate on inverse depths, $z = \frac{1}{d}$ for better handling of large depths values. Our model additionally seeks to optimize for the relative transformations between each source frame $s$ in the sequence and the target keyframe $t$, $\mathbf{p}_{t\rightarrow s}$. The full model consists of three stages. All iterative optimization procedures require an initial starting point and thus the initialization stage serves the purpose of predicting a good initial estimate. The optimization stage consists of a learned optimizer which benefits from explicitly computed residuals and Jacobians. To make the optimization computationally tractable, the optimization network operates on a down-sampled version of the input and exploits the sparsity of the problem. The final stage of the network upsamples the prediction to the original resolution. The networks (including those of the optimizer) are trained using a supervised loss. We now describe each of the three network components in detail.
Residual function $\mathbf{r}(\mathbf{x})$, image sequence $\mathbf I_1,\mathbf I_2, \ldots $ $\mathbf{x}_{0} \gets f_{\theta_0} (\mathbf I_1,\mathbf I_2,\ldots )$ $\Delta \mathbf{x}_i, \mathbf{h}_{i+1} \gets f_\theta \left(\Phi(\mathbf{J}_i, \mathbf{r}_i), \mathbf{h}_{i} \right)$ $\mathbf{x}_{i+1} \gets \mathbf{x}_{i} + \Delta \mathbf{x}_i$
Initialization Network
----------------------
The purpose of the initialization network is to predict a suitable starting point for the optimization stage. We provide the initialization network with both RGB images and thereby allow it to leverage stereopsis. The architecture of this stage is a simple convolutional network. For this stage we use 3 convolutions with stride 2, one convolution with stride 1 and one upsamplings + convolutional layers. This results in the output of the network being downscaled by a factor of 4 for feeding into the optimization stage. The network also produces an initial pose using a fully connected layer branched from the central layers of the network. Thus the output of the initialization stage consists of an initial depth image and pose.
Optimizing Nonlinear Least Squares with LS-Net {#sec:optimization}
----------------------------------------------
The learnt optimization procedure is outlined in Algorithm \[alg:opt\_alg\_structure\]. The optimization network attempts to optimize the photometric objective $E(\mathbf{x})$ where $\mathbf{x} = (\mathbf{z},\mathbf{p})$ are the optimization variables (inverse depths $\mathbf{z}$ and pose $\mathbf{p}$). The objective $E(\mathbf{x})$ is a nonlinear least squares expression defined in terms of the photometric residual vector $\mathbf{r}(\mathbf{x})$ $$\begin{aligned}
E(\mathbf{x}) = \frac{1}{2} ||\mathbf{r}(\mathbf{x})||^2.\end{aligned}$$
The updates of the parameters to be optimized, $\mathbf{x}$, follow a standard iterative optimization scheme, i.e. $${\mathbf{x}}_{i+1} = {\mathbf{x}}_i + \Delta \mathbf{x}_i.
\label{eqn:update}$$ In our case, the updates $\Delta \mathbf{x}_i$ are predicted using a Long Short Term Memory Recurrent Neural Network (LSTM-RNN) [@Hochreiter:2001]. In order to compute the Jacobian we use automatic differentiation available in the Tensorflow library [@abadi2016tensorflow]. Using the automatic differentiation operation, we add operations to the Tensorflow computation graph [@abadi2016tensorflow] which compute the Jacobian of our residual vector with respect to the dense depth and motion. As the structure of the Jacobian often exhibits problem specific properties, we apply a transformation to the Jacobian, $\Phi(\mathbf{J}_i, \mathbf{r}_i)$ before feeding this Jacobian into our network. The operation $\Phi$ may involve element-wise matrix operations such as gather or other operations which simplify the Jacobian input. The operations we use for the problems addressed in this paper are detailed in Section \[sec:sparsity\].
To allow for the computation of parameter updates which are not restricted to those derived from the approximate Hessian, we turn to the powerful function approximation ability of the LSTM-RNN [@Hochreiter:2001] to learn the final parameter update operation from data. As the number of coordinates are likely to be very large for most optimization problems, [@Chen:ICML2017] propose to use one LSTM-RNN for each coordinate. For our problem, we have Jacobians with high spatial correlations and thus we replace the coordinate-wise LSTM with a convolutional LSTM. The per-iteration updates, $\Delta \mathbf{x}_i$ are predicted by a network which in this case is an LSTM-RNN, $$\begin{bmatrix}
\Delta x_i \\ h_{i+1}
\end{bmatrix}
= \mbox{LSTM}_{cell}\left(\Phi(\mathbf{J}_i, \mathbf{r}_i), h_i, \mathbf{x}_i; \mathbf{\theta}\right),$$ where $\mathbf{\theta}$ are the parameters of the networks and $\mbox{LSTM}_{cell}$ is a standard LSTM cell update function with hidden layer $h_i$.
The Jacobian input structure {#sec:sparsity}
----------------------------
Each type of least squares cost function gives rise to a special Jacobian structure. The input function, $\Phi(\mathbf{J}, \mathbf{r})$, to our network serves two purposes; one functional and the other structural. Firstly, $\Phi$ serves to compute the *approximate Hessian* as is done with the classical Gauss-Newton optimization method: $$\Phi(\mathbf{J}, \mathbf{r}) = [\mathbf{J^T} \mathbf{J},\mathbf{r}].$$ The structure of $\Phi(\mathbf{J}, \mathbf{r})$ is shown in Figure \[fig:sparsity\]. We note that we choose not to compute the full $(\mathbf{J}^T \mathbf{J})^{-1}\mathbf{J}$ as this adds additional computational complexity to the operation which is repeated many times during training. We also compress the sparse $\mathbf{J}^T \mathbf{J}$ into a compact form as illustrated in Figure \[fig:sparsity\]. The output of this restructuring yields the same image shape as the image. The compressed structure allows efficient processing of the matrix.
![The block-sparsity structure of $\mathbf{J}$ and $\mathbf{J}^T\mathbf{J}$ for the depth and egomotion estimation problem.[]{data-label="fig:sparsity"}](figs/sparsity_pose_depth.pdf){width="\columnwidth"}
Upscaling Network
-----------------
As the optimization network operates on low-resolution predictions, an upscaling network is used to produce outputs of the desired size. The upscaling network consists of a series of bilinear upsampling layers concatenated with convolutions and acts as a super-resolution network. The input to the upscaling network consists of the low-resolution depth prediction and the RGB image.
Loss Function {#sec:losses}
=============
In this section we describe the loss function which we use to train the network weights of all three stages of our model.
The current state-of-the-art depth and motion prediction networks still rely on labelled images to provide a strong learning signal. We include a loss term based on labelled ground truth inverse depth images $\tilde{\mathbf{z}}$,
$$L_{depth}(\mathbf{x}) = \frac{1}{wh} \| \mathbf{z} - \tilde{\mathbf{z}} \|_1$$
with image width $w$ and height $h$, and where $\mathbf{z}$ is the predicted inverse depth image.
We also use a loss term based on the relative pose between the source (s) and target (t) frame, $\tilde{\mathbf{p}} = (\tilde{\mathbf{t}}_{t\rightarrow s}, \tilde{\boldsymbol{\alpha}}_{t\rightarrow s}) $ with translation $\tilde{\mathbf{t}}_{t\rightarrow s}$ and rotation vector $\tilde{\boldsymbol{\alpha}}_{t\rightarrow s}$ from ground-truth data, $$L_{pose}(\mathbf{x}) = \sum_{s} \| \boldsymbol{\alpha}_{t\rightarrow s} - \tilde{\boldsymbol{\alpha}}_{t\rightarrow s}\|_1 + \| \mathbf{t}_{t\rightarrow s} - \tilde{\mathbf{t}}_{t\rightarrow s}\|_1$$ Note that this loss function need not be a sum of squares and can be computed using any other form using eg. L1 etc. The final loss function consists of a weighted combination of the individual loss terms:
$$\begin{aligned}
L_{tot}(\mathbf{\theta}) = \sum_{i} w_{pose} L_{pose}(\mathbf{x}_i(\mathbf{\theta}))+ w_{depth} L_{depth}(\mathbf{x}_i(\mathbf{\theta})). \label{eq:ltot}\end{aligned}$$
Note that our objective here includes the ground-truth inverse depth which we do not have access to when computing the residuals $\mathbf{r}$ (and then the Jacobian $\mathbf{J}$) in the recurrent optimization network in Section \[sec:optimization\].
The optimization network is never directly privy to the ground truth depth and poses, it only benefits from these by what is learned in the network parameters during training. In this manner, we have a system which is trained offline to best minimize our objective online. During the offline training phase, our system learns robust priors for the optimization by using the large amounts of labelled data. During the online phase our system optimizes for photometric consistency only but is able to utilize the knowledge it has learned during the offline training to better condition the optimization process.
Training
========
During the training, we unroll our iterative optimization network for a set number of steps and backpropogate the loss through the network weights, $\theta$. In order to find the parameters of the optimizer network, the meta-loss, $L_{tot}(\mathbf{\theta})$, is minimized using the ADAM optimizer where the total meta-loss is computed as the loss summed over the $N$ iterations of the learned optimization (see Eq. \[eq:ltot\]). For each step $i$ in the optimization process we update the state $\mathbf{x}_i$ of the optimization network according to Eqn. \[eqn:update\].
As our loss depends on variables which are updated recurrently over a number of timesteps, we use backpropogation through time to train the network. Backpropogation through time unrolls each step and updates the parameters by computing the gradients through the unrolled network. In our experiments we unroll our optimization for 15 steps.
We find that training the whole network at once is difficult and thus train the initialization network first before adding the optimization stage.
Evaluation
==========
In this section we evaluate the proposed method on both synthetic and real datasets. We aim to determine the efficiency of our approach i.e. how quickly it converges to an optimum and how it compares to a network which does not explicitly incorporate the problem structure in its iterations.
Synthetic data experiments
--------------------------
In this section we evaluate the performance of our proposed method on a number of least squares curve fitting problems. We experiment on curves parameterized by two variables, $\mathbf{x} = (a,b)$. We chose a set of four functions to use for our experiment as follows $$y = x \exp(a t) + x\exp(b t) + \epsilon,
\label{eqn:eqn1}$$ $$y = \sin(a t + b) + \epsilon,$$ $$y = \mathrm{sinc}(a t + b) + \epsilon,$$ $$y = \mathcal{N}(t|\mu=a,\sigma=b) \hspace{5pt} \mbox{(fitting a Gaussian)}
\label{eqn:eqn4}$$
For these experiments we generate the data by randomly sampling one of four parametric functions (Eqn. \[eqn:eqn1\] to Eqn. \[eqn:eqn4\]) as well as the two parameters $a$ and $b$. For the training data we add noise $\epsilon \sim \mathcal{N}(0,0.1)$ to the true function values. In Figure \[fig:syn\_results\] we show the results on a test set of sampled functions. Figure \[fig:syn\_results\] a) shows the fitted function after 5 iterations (of a total of 15 iterations) for our method and standard LM. The learned approach clearly outperforms LM in terms of speed of convergence. In Figure \[fig:syn\_results\] b) we see the learned errors vs LM for all steps in the optimization, where again, the learned method clearly outperforms LM.
![Comparison between our method and standard least squares for fitting parametric functions to noisy data with a least-squares objective. In a) the fitted functions limited to 5 iterations is shown, in b) the error as a function of iteration no. is shown for 10 test functions and in c) the LM error is plotted against the error of the proposed method for all iterations.[]{data-label="fig:syn_results"}](figs/function.pdf "fig:"){width="0.32\linewidth"} ![Comparison between our method and standard least squares for fitting parametric functions to noisy data with a least-squares objective. In a) the fitted functions limited to 5 iterations is shown, in b) the error as a function of iteration no. is shown for 10 test functions and in c) the LM error is plotted against the error of the proposed method for all iterations.[]{data-label="fig:syn_results"}](figs/loss_iterations.pdf "fig:"){width="0.34\linewidth"} ![Comparison between our method and standard least squares for fitting parametric functions to noisy data with a least-squares objective. In a) the fitted functions limited to 5 iterations is shown, in b) the error as a function of iteration no. is shown for 10 test functions and in c) the LM error is plotted against the error of the proposed method for all iterations.[]{data-label="fig:syn_results"}](figs/loss_vs_gn.pdf "fig:"){width="0.30\linewidth"}
Real-world test: depth and pose estimation
------------------------------------------
In this section we test the ability of our proposed method on estimating the depth and egomotion of a moving camera. To provide a fair evaluation of the proposed approach, we use the same evaluation procedure as in [@Ummenhofer:Brox:CVPR2017] and report the same baselines, where oracle uses MVS with known poses, *SIFT* uses sparse-feature for correspondences, *FF* uses optical flow, *Matlab* uses the KLT tracker in Matlab as the basis of a bundle-adjusted reconstruction.
{width="\textwidth"}
Metrics
-------
We evaluate the performance of our approach on the depth as well as the motion prediction performance. For depth prediction we use the absolute, scale-invariant and relative performance metrics.
Datasets
--------
The datasets which we use to evaluate the network consist of both indoor and outdoor scenes. For all the datasets, the camera undergoes free 6-DoF motion. To train our network we use images from all the datasets partitioned into testing and training sets.
**MVS** The multiview stereo dataset consists of a collection of scenes obtained using struction from motion software followed by dense multi-view stereo reconstruction. We use the same training/test split as in [@Ummenhofer:Brox:CVPR2017]. The training set of images used included “Citywall", “Achteckturm" and “Breisach" scenes with “Person-Hall", “Graham-Hall", and “South-Building" for testing.
**TUM** The TUM RGB-D dataset consists of Kinect-captured RGB-D image sequences with ground truth poses obtained from a Vicon system. It comprises a total of 19 sequences with 45356 images. We use the same test / train split as in [@Ummenhofer:Brox:CVPR2017] with 80 held-out images for test.
**Sun3D** The SUN3D dataset consists of scenes reconstructed using RGB-D structure-from-motion. The dataset has a variety of indoor scenes, with absolute scale and consists of 10,000 individual images. The poses are less accurate than the TUM dataset as they were obtained using an RGB-D reconstruction.
![Qualitative results on the NYU dataset. Compared to DeMoN our network has fewer “hallucinations” of structures which do not exist in the scene. []{data-label="fig:qualitative_comparison"}](figs/extended.pdf){width="1.0\columnwidth"}
A qualitative evaluation of our method compared to standard multiview stereo and DeMoN [@Ummenhofer:Brox:CVPR2017] is shown in Figure \[fig:qualitative\_comparison\]. Our method produces depth maps with sharper structures compared to DeMoN, even with a lower output resolution. Compared to COLMAP [@schoenberger2016mvs] our reconstruction is more dense and does not include as many outlier pixels. Numerical results on the testing data-sets are shown in Table \[tbl:results1\]. As is evident from the Table, our learned optimization approach outperforms most of the traditional baseline approaches, and performs better or on par with DeMoN on most cases. This may be due to our architectural choice as we do not include any alternating flow and depth predictions.
-- ---------- -------- -------- -------- ---------- -------------
Method L1-inv sc-inv L1-rel Rotation Translation
MVS 0.019 0.197 0.105 0 0
SIFT 0.056 0.309 0.361 21.180 60.516
FF 0.055 0.308 0.322 4.834 17.252
Matlab - - - 10.843 32.736
DeMoN 0.047 0.202 0.305 5.156 14.447
Proposed 0.051 0.221 0.311 4.653 11.221
Oracle 0.023 0.618 0.349 0 0
SIFT 0.051 0.900 1.027 6.179 56.650
FF 0.038 0.793 0.776 1.309 19.425
Matlab - - - 0.917 14.639
DeMoN 0.019 0.315 0.248 0.809 8.918
Proposed 0.010 0.410 0.210 0.910 8.21
Oracle 0.026 0.398 0.336 0 0
SIFT 0.050 0.577 0.703 12.010 56.021
FF 0.045 0.548 0.613 4.709 46.058
Matlab - - - 12.831 49.612
DeMoN 0.028 0.130 0.212 2.641 20.585
Proposed 0.019 0.09 0.301 1.01 22.1
oracle 0.020 0.241 0.220 0 0
SIFT 0.029 0.290 0.286 7.702 41.825
FF 0.029 0.284 0.297 3.681 33.301
Matlab - - - 5.920 32.298
DeMoN 0.019 0.114 0.172 1.801 18.811
Proposed 0.015 0.189 0.650 1.521 14.347
-- ---------- -------- -------- -------- ---------- -------------
: Quantitative results on the evaluation datasets. Green highlights the best performing method for a particular task.[]{data-label="tbl:results1"}
Discussion
==========
In the context of optimisation, our network-based updates accomplish something which a classical optimisation approach cannot in that it is able to reliably optimise a large under-determined system with implicitly learned priors.
[l]{}[0.5]{} \[tbl:comparison\]
---------- ------- -------- -------
Small Medium Large
Accuracy Ours Ours Ours
Memory Tie Ours Ours
Speed Tie Ours Ours
---------- ------- -------- -------
For a large under-determined problem like in the depth and motion case, standard Levenberg-Marquadt (LM) fails to improve the objective and the required sparse matrix inversion for a $J^TJ$ with $\approx91K$ non-zero elements (128 $\times$ 96 size image) takes 532ms, compared to our network forward pass which takes 25ms. For small, overdetermined problems LM does work and for this reason, in Section 7.1, we have compared our approach to LM on a small curve fitting problem and found that our approach significantly outperforms it in terms of accuracy and convergence rate. For the small problem, the matrix inversion in the standard approach (LM) is very quick but we are also able to use a smaller network so our time per-iteration is tie with LM. This is summarised in Table \[tbl:comparison\].
Ablation study
--------------
We conduct an experiment to verify the efficacy of the learned optimization procedure. The first part of our ablation study considers the effect of increasing the number of optimization iterations. These results are shown in Table \[tbl:ablation\_1\] and a qualitative overview of the operation of our network is shown in Figure \[fig:results\] which visualizes the learned optimization process. The second part of our ablation study evaluates the efficacy of the learned optimizer compared to DeMoN’s iterative network. This is show in Figure \[fig:ablation\_graph\].
![Comparison between our learned optimizer and the (larger) RNN refinement network from DeMon.[]{data-label="fig:ablation_graph"}](figs/ablation.pdf){width="0.5\columnwidth"}
Number of parameters and inference speed
----------------------------------------
An advantage of our approach is its parameter efficiency. Compared to DeMoN, our model has significantly fewer parameters. The DeMoN network contains 45,753,883 wheres ours has only 11,438,470 – making it over $3\times$ more parameter efficient. Ours also has an advantage in terms of inference speed, as although we have to compute the large Jacobian, it still runs around $1.5\times$ faster during inference compared to DeMoN.
Conclusion
==========
In this paper we have presented an approach for robustly solving nonlinear least squares optimization problems by integrating deep neural models with traditional knowledge of the optimization structure. Our method is based on a novel nonlinear least squares optimizer which is trained to robustly optimize the residuals. Although it is generally applicable to any least squares problem, we have demonstrated the proposed method on the real-world problem of computing depth and egomotion for frames of a monocular video sequence. Our method can cope with images captured from a wide baseline. In future work we plan to investigate means of increasing the number of residuals that are optimized and thereby achieve an even more detailed prediction. We also plan to further study the interplay between the recurrent neural network and optimization structure and want to investigate the use of predicted confidence estimates in the learned optimization.
[10]{} \[1\][`#1`]{} \[1\][https://doi.org/\#1]{}
Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., et al.: Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467 (2016)
Bloesch, M., Czarnowski, J., Clark, R., Leutenegger, S., Davison, A.J.: [CodeSLAM]{} — learning a compact, optimisable representation for dense visual [SLAM]{}. In: [Proceedings of the [IEEE]{} Conference on Computer Vision and Pattern Recognition ([CVPR]{})]{} (2018)
Chen, Y., Hoffman, M.W., Colmenarejo, S.G., Denil, M., Lillicrap, T.P., Botvinick, M., de Freitas, N.: Learning to learn without gradient descent by gradient descent. In: Proceedings of the 34th International Conference on Machine Learning. vol. 70, pp. 748–756. PMLR (06–11 Aug 2017)
Choy, C.B., Xu, D., Gwak, J., Chen, K., Savarese, S.: 3d-r2n2: A unified approach for single and multi-view 3d object reconstruction. In: Proceedings of the European Conference on Computer Vision ([ECCV]{}) (2016)
Clark, R., McCormac, J., Leutenegger, S., Davison, A.: [Meta-Learning for Instance-Level Data Association]{}. In: [Neural Information Processing Systems ([NIPS]{})]{} (2017)
Clark, R., Wang, S., Wen, H., Markham, A., Trigoni, N.: [VidLoc]{}: A deep spatio-temporal model for 6-dof video-clip relocalization. In: [Proceedings of the [IEEE]{} Conference on Computer Vision and Pattern Recognition ([CVPR]{})]{} (2017)
Clark, R., Wang, S., Wen, H., Markham, A., Trigoni, N.: [VINet]{}: Visual-inertial odometry as a sequence-to-sequence learning problem. In: [Proceedings of the National Conference on Artificial Intelligence ([AAAI]{})]{} (2017)
Dai, H., Khalil, E.B., Zhang, Y., Dilkina, B., Song, L.: Learning combinatorial optimization algorithms over graphs. arXiv preprint arXiv:1704.01665 (2017)
Eigen, D., Fergus, R.: [Predicting Depth, Surface Normals and Semantic Labels with a Common Multi-Scale Convolutional Architecture]{}. In: [Proceedings of the International Conference on Computer Vision ([ICCV]{})]{} (2015)
Engel, J., Schoeps, T., Cremers, D.: [LSD-SLAM]{}: Large-scale direct monocular [SLAM]{}. In: [Proceedings of the European Conference on Computer Vision ([ECCV]{})]{} (2014)
Fletcher, R.: Practical methods of optimization. John Wiley & Sons (2013)
Hochreiter, S., Younger, A.S., Conwell, P.R.: Learning to Learn Using Gradient Descent, pp. 87–94 (2001)
Kelley, C.T.: Iterative methods for optimization, vol. 18. Siam (1999)
Li, K., Malik, J.: Learning to optimize (2016)
Lin, C.H., Lucey, S.: Inverse compositional spatial transformer networks. arXiv preprint arXiv:1612.03897 (2016)
Metz, L., Poole, B., Pfau, D., Sohl-Dickstein, J.: Unrolled generative adversarial networks. arXiv preprint arXiv:1611.02163 (2016)
Newcombe, R.A., Izadi, S., Hilliges, O., Molyneaux, D., Kim, D., Davison, A.J., Kohli, P., Shotton, J., Hodges, S., Fitzgibbon, A.: [[KinectFusion]{}: Real-Time Dense Surface Mapping and Tracking]{}. In: [Proceedings of the International Symposium on Mixed and Augmented Reality ([ISMAR]{})]{} (2011)
ktem, O., Adler, J.: Solving ill-posed inverse problems using iterative deep neural networks. Inverse Problems (2017)
Ravi, S., Larochelle, H.: Optimization as a model for few-shot learning. International Conference on Learning Representations (ICLR) (2016)
Schönberger, J.L., Zheng, E., Pollefeys, M., Frahm, J.M.: Pixelwise view selection for unstructured multi-view stereo. In: European Conference on Computer Vision (ECCV) (2016)
Tikhonov, A., Arsenin, V.: [Solutions of ill-posed problems]{}. Winston, Washington,DC (1977)
Tulsiani, S., Zhou, T., Efros, A.A., Malik, J.: Multi-view supervision for single-view reconstruction via differentiable ray consistency. In: [Proceedings of the [IEEE]{} Conference on Computer Vision and Pattern Recognition ([CVPR]{})]{} (2017)
Ummenhofer, B., Zhou, H., Uhrig, J., Mayer, N., Ilg, E., Dosovitskiy, A., Brox, T.: Demon: Depth and motion network for learning monocular stereo. In: [Proceedings of the [IEEE]{} Conference on Computer Vision and Pattern Recognition ([CVPR]{})]{}
Wang, S., Clark, R., Wen, H., Trigoni, N.: [DeepVO]{}: Towards end to end visual odometry with deep recurrent convolutional neural networks. In: [Proceedings of the [IEEE]{} International Conference on Robotics and Automation ([ICRA]{})]{} (2017)
Xiong, X., De la Torre, F.: Supervised descent method and its applications to face alignment. In: [Proceedings of the [IEEE]{} Conference on Computer Vision and Pattern Recognition ([CVPR]{})]{}. pp. 532–539 (2013)
Zhou, T., Brown, M., Snavely, N., Lowe, D.G.: Unsupervised learning of depth and ego-motion from video. In: [Proceedings of the [IEEE]{} Conference on Computer Vision and Pattern Recognition ([CVPR]{})]{} (2017)
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---
abstract: |
This paper studies a class of constrained restless multi-armed bandits. The constraints are in the form of time varying availability of arms. This variation can be either stochastic or semi-deterministic. A fixed number of arms can be chosen to be played in each decision interval. The play of each arm yields a state dependent reward. The current states of arms are partially observable through binary feedback signals from arms that are played. The current availability of arms is fully observable. The objective is to maximize long term cumulative reward. The uncertainty about future availability of arms along with partial state information makes this objective challenging.
This optimization problem is analyzed using Whittle’s index policy. To this end, a constrained restless single-armed bandit is studied. It is shown to admit a threshold-type optimal policy, and is also *indexable*. An algorithm to compute Whittle’s index is presented. Further, upper bounds on the value function are derived in order to estimate the degree of sub-optimality of various solutions. The simulation study compares the performance of Whittle’s index, modified Whittle’s index and myopic policies.
bibliography:
- 'limfee.bib'
title: Sequential Decision Making under Uncertainty with Dynamic Resource Constraints
---
Introduction
============
Motivation
----------
Restless multi-arm bandits (RMAB) have been extensively studied for applications in opportunistic communication systems, scheduling in queuing systems, recommendation systems, [@LiuZhao10; @Gittins11; @Ansell03; @Nino-Mora01; @Nino-Mora07; @Avrachenkov18; @Meshram17]. A restless multi-armed bandit is a class of sequential decision problems. It is used to solve policy planning problems under uncertain system environments. A restless multi-arm bandit model is described as follows. There is a decision maker or source that has $N$ independent arms. Each arm can be in one of a finite set of states and the state evolves according to Markovian law. The play of an arm yields a state dependent reward. It is assumed that the decision maker knows the statistical characteristics of state evolution for each arm. The system is time-slotted and it is discretized. The decision maker plays $M$ out of $N$ arms in each slot. The goal is to determine the sequence of plays of the arms that maximizes the long term cumulative reward. These planning problems are non-trivial because there is a trade-off between the immediate, and the future rewards. The choice that yields low immediate reward may yield better future reward.
[ A typical restless multi-armed bandit model assumes that the arms are always available and the objective is to determine the optimal subset of arms to play in a given state. However, there may be scenarios where the availability of arms is intermittent and time varying. We refer to such RMABs as *constrained restless multi-armed bandits (CRMAB)*. This is an important generalization of restless multi-armed bandits we consider in this paper. Also, the availability of arms may vary across applications. We consider stochastic and semi-deterministic availability models. ]{}
[ A couple of motivating applications for these constrained restless bandits can be found in the domains of cyber-security and wireless relay networks. These application scenarios are described below. ]{}
Honeypots are physical or virtual machines which are deployed as a cyber security mechanism for intrusion detection and prevention. Honeypots attract the attention of attackers by appearing as genuine servers with a security vulnerability. When an attack happens on a honeypot, information about the attack is logged and the attack is contained from spreading. An intelligent attacker might realize after some delay, that it has been trapped by a honeypot. Now, if the locations of honeypots in a network are fixed, the attacker can learn about them through repeated interaction, and avoid them in future. Hence, the idea of roaming honeypots has been proposed in the literature, where locations of honeypots are shifted in order minimize the chance of a denial of service attack [@Khattab04; @Tapaswi14]. Consider a set of $N$ servers. Each server may either be in a high interest or low interest state. High interest state would mean a higher probability of attack. The change of interest states of servers can be for various reasons. These include, current utility or purpose of the server (email, ftp, etc.), specific application running on the server, or the strategy of the attacker. These states are continuously evolving with time. For every interval of length $\tau,$ $M$ out of $N$ servers are chosen to act as honeypots. Sometimes it may happen that only $N'$ of the $N$ servers are available for deployment as honeypots. In the current work we will assume that $M\leqslant N'$ with unit probability. If a honeypot is attacked, the intrusion is detected; hence it can be considered as a positive reward event by the decision maker. The goal of the decision maker is to schedule honeypots such that the long term cumulative reward (total intrusion detection) is maximized. It can be modeled as a CRMAB with it’s arms as servers whose states are evolving with time. In this paper, we model the evolution of interest states as a Markov chain whose characteristics are known to the decision maker. It can later be extended to scenarios which might require estimation or learning of these characteristics.
[ Let us now look at another motivating application. Consider a wireless relay network with a source, destination and a set of $N$ relay nodes. The availability of each relay is time varying and may be due to reasons such as power constraints, outage or mobility. Further, the channel conditions of links are evolving with time. The objective here is to schedule a relay in each time slot under such constrained availability to maximize cumulative long term reward, i.e., throughput. In this network each link can be represented as the arm of the bandit, and each link evolves independent of transmission in every time slot. Thus, it is can be modeled as a CRMAB. ]{}
[ ]{}
[ ]{}
[ ]{}
Related work
------------
The literature on restless bandits is vast and includes different variations on bandits and their applications. We mention a few of them that are relevant to our work.
The resltess multi-armed bandit problem was first proposed in [@Whittle88]. It was inspired from the work on rested bandits [@Gittins79]. In [@Gittins79], index policies were introduced for rested multi-armed bandits, where states of arms do not change when they are not played. This index policy is now known as Gittins index policy. Later, [@Whittle88] studied restless bandits and introduced an index policy which is now referred to as Whittle’s index policy. The popularity of Whittle’s index policy is due to its asymptotic optimality in some examples and its near optimal performance in some others (see [@Whittle88; @Weber90; @Ouyang12]). The Whittle’s index policy for other applications such as machine repair problems are given in [@Gittins11].
In classical restless bandit literature, current states of all the arms are observable in every time slot [@Nino-Mora01; @Nino-Mora07; @Whittle88]. Later, this assumption was relaxed and restless bandit models with partially observable states were studied, where states are observable only for those arms that are played [@LiuZhao10; @Ouyang12]. Recent work on restless bandits further generalized this model to the case where states of all arms are partially observable. This is referred to as the *hidden restless bandit* [@Borkar17; @Meshram18]. In [@Kaza19], further generalization is considered where multiple state transitions are allowed in a single decision interval.
The Whittle index policy for RMABs was studied for job scheduling and dynamic routing on servers in [@Martin05; @Glazebrook07], where authors considered scenario of servers being available intermittently.
Earlier, a variant of the restless multi-armed bandit with availability constraints was proposed in [@Dayanik02]. It was applied to the machine repair problem where machine availability is time varying. This model was further generalized in [@Varun18] by considering partially observable states. In [@Varun18], the authors consider a penalty for playing an unavailable arm. That is, arms can be played both when they are available and unavailable. Whittle’s index policy and myopic policy are analyzed. [There are several subtle, but important differences between the current model and the model in [@Varun18] . The CRMABs considered in this paper do not allow the play of unavailable arms. Our proposed model differentiates between the actions “don’t play” and “can’t play”. That is, the belief update rules are different for the case where an arm is available and is not played and the case where the arm is unavailable and cannot be played. In this work, we provide an upper bound on optimal value function which can be used as a reference to measure sub-optimality gap of the Whittle’s index policy. ]{}
[ The literature on POMDPs, RMABs makes use of certain common techniques and procedures. These include defining action value functions and using induction principle to derive their structural properties. Another common aspect is proving sub-modularity of the value function, which will lead to a threshold structure of optimal policy (see [@Lovejoy87; @Puterman14]). One must note the differences in modeling that require redoing or following similar procedures, as it is not obvious that the same results hold. ]{}
[ ]{}
Contributions
-------------
We propose a novel methodology to solve the problem of sequential decision making under dynamic resource constraints. It is modeled as a constrained restless multi-armed bandit problem. The resource constraints occur in the form of time-varying availability of arms. We consider two availability models, namely, stochastic and semi-deterministic availability. The analysis of the constrained restless single armed bandit (CRSAB) forms the basis of the solution methodology. They are shown to admit a threshold type policy in belief space. This holds for both stochastic and semi-deterministic availability models. Indexability of CRSABs is claimed by imposing certain conditions on the parameters. An algorithm for computation of Whittle’s index is also presented. An upper bound on the optimal value function is derived. The Lagrangian relaxation of the original problem provides an upper bound on the value function. The relationship between the Lagrangian bound for CRMAB and unconstrained RMAB is studied. It is shown that under certain conditions the former gives a tighter bound than the later. An extensive simulation study is presented with performance comparison of various solution schemes such as Whittle’s index policy, modified Whittle’s index policy and myopic policy. The rest of this document is organized as follows. The system model is explained in Section \[sec:model\] and the constrained restless single armed bandit is analyzed in Section \[sec:crsab\]. Bounds on value functions are derived in Section \[sec:bounds\]. Numerical simulations are presented in Section \[sec:simulation\], and concluding remarks in Section \[sec:conc\].
Model Description and Preliminaries {#sec:model}
===================================
Consider a restless multi-armed bandit with $N$ independent arms. Each arm can be one of two states, say, state $0$ or state $1$. The state of each arm evolves according to a discrete time Markov chain. Some times arms might become unavailable. Hence, the evolution of states also depends on availability of arms. Also, the availability of arms is time varying. Let us introduce some notation to formalize the model. Assume that the system is time-slotted and time is indexed by $t.$ Let $X_n(t) \in \{0,1\}$ denote the state of arm $n$ at the beginning of time slot $t.$
Let $Y_n(t) \in \{0,1\}$ denote the availability of arm $n$ at the beginning of time slot $t$ and [[ $$\begin{aligned}
Y_n(t) =
\begin{cases}
1 & \mbox{if arm $n$ is available,} \\
0 & \mbox{if arm $n$ is not available.}
\end{cases}\end{aligned}$$ ]{}]{} Each arm has two actions associated with it when it is available, either ‘play’ or ‘don’t play’. When it is unavailable it cannot be played. However, it’s state still evolves. Let $a_n^{1}(t) \in \{0,1 \}$ be an action corresponding to arm $n$ when it is available and it is described as follows. [[ $$\begin{aligned}
a_n^{1}(t) = \begin{cases}
1 & \mbox{if arm $n$ is available and played,}\\
0 & \mbox{if arm $n$ is available and not played.}
\end{cases}\end{aligned}$$ ]{}]{} Let $a_n^{0}(t)$ be the action corresponding to arm $n$ when it is not available. As it cannot be played, $a_n^{0}(t) \coloneqq 0.$
The state of arm $n$ changes at beginning of time slot $(t+1)$ from state $i$ to $j$ according to transition probabilities $p_{ij}^n.$ These are defined as follows. [[ $$\begin{aligned}
p_{ij}^n \coloneqq \Pr \{ {X_n}(t + 1) = j~|~{X_n}(t) = i\}\end{aligned}$$ ]{}]{} When arm $n$ is played, the result is either success or failure. A binary signal is observed at the end of each slot that describes the event of success or failure (ACK or NACK in communication parlance). Let $Z_n(t)$ be the binary signal that is received by the source at the end of slot $t.$ It is given as
[[ $$\begin{aligned}
Z_n(t) =
\begin{cases}
1 & \mbox{if arm $n$ is played and that resulted success,} \\
0 & \mbox{If arm $n$ is played and no success.}
\end{cases}
\end{aligned}$$ ]{}]{} When arm $n$ is not played, no signal is observed from that arm. Let $\rho_n(i)$ be the probability of success from playing arm $n.$ for $i = 0,1.$ It is the probability that signal $Z_n(t)=1$ is observed given that arm $n$ is in state $i$ and action $a_n(t) =1.$ We will assume $ \rho_{n,0} <
\rho_{n,1},$ i.e., the probability of success is higher from state $1$ than from state $0.$
The play of arm $n$ yields a state dependent reward. Let $\eta_{n,i}$ be the reward obtained by playing arm $n$ given that $X_n(t) = i.$ When arm $n$ is not played, no reward is obtained. Further, we suppose that $0 \leq \eta_{n,0} < \eta_{n,1} \leq 1$ for all $n.$
The decision maker or source cannot exactly observe the state vector at any arbitrary time $t$. However, the source can exactly observe the current availability vector at the beginning of each time slot. That is, ${\boldsymbol{Y}}(t) = [Y_1(t),...,Y_n(t)]$ is known at beginning of slot $t.$ Since the source does not know the exact states of arms, it maintains a ‘belief’ about each of them. Let $\pi_n(t)$ be the belief about arm $n.$ It is the probability of being in state $0,$ given the history $H_t$ upto time $t.$ The history upto time $t$ is given as [[ $$H_t := \left(Y_n(s),a_{n}(s) ,Z_{n}(s) \right)_{0 \leq n \leq N, 1 \leq s < t}.$$ ]{}]{} The belief vector is given as $\boldsymbol{\pi}(t) =[\pi_1(t),...,\pi_n(t)],$ with $$\pi_n(t) = \Pr{\left( X_n(t) = 0~|~ H_t \right)}.$$
Availability models
-------------------
We consider two availability models, namely, stochastic and semi-deterministic. In the stochastic model, future availability depends on a probability law conditioned on current availability. In the semi-deterministic model, the future availability is deterministic when an arm goes unavailable. This model is useful in applications in which some sub-systems are occasionally down for a fixed maintenance time.
### Stochastic
Here, future availability, $Y_n(t+1)$ is based on current availability $Y_n(t)=y,$ action $a_n(t)=a$ and belief $\pi_n(t)=\pi$ according to the following probability law. Let $$Y_n(t+1) =
\begin{cases}
& 1, \texttt{ } w.p. \texttt{ } \theta_n^a(\pi,y)\\
& 0, \texttt{ } w.p. \texttt{ } 1-\theta_n^a(\pi,y).
\end{cases}$$ The source knows the probability of availability $\theta_n^a(\pi,y).$ Notice that this model satisfies Markov property.
In general $\theta_n^a(i,y)$ depends on the state of arm $n,$ the current availability $y$ and action of that arm $a.$ For simplicity we assume that it is independent of state, *i.e.,* $\theta_n^a(i,y) = \theta_n^a(y).$
### Semi-deterministic
Here, future availability for unavailable arms has a deterministic model. When available arms turn unavailable, they remain unavailable for exactly $T_0$ slots and then become available. That is,\
if $Y_n(t)=0,$ then $$\begin{aligned}
Y_n(t+t') = \begin{cases}
& 0, \texttt{ } for \texttt{ } t' = 1,...,T_0-1 \nonumber\\
& 1, \texttt{ } for \texttt{ } t' = T_0,
\end{cases}\end{aligned}$$ if $Y_n(t)=1,$ then $$Y_n(t+1) =
\begin{cases}
& 1, \texttt{ } w.p. \texttt{ } \theta_n^a(\pi,1)\\
& 0, \texttt{ } w.p. \texttt{ } 1-\theta_n^a(\pi,1).
\end{cases}$$\
Problem formulation
-------------------
Let us describe the state in terms of the belief and availability. Consider the perceived state $S_n(t) = (\pi_n(t), Y_n(t)) \in [0,1]
\times \{0,1\}$ in beginning of time slot $t.$ Using the belief $\pi_n(t),$ we compute the expected reward from play of arm $n$ at time $t$ as follows. $$\eta(\pi_n(t), y=1) \coloneqq \pi_n(t) \eta_{n,0} + (1-\pi_n(t)) \eta_{n,1}$$ and $\eta(\pi_n(t), y=0)\coloneqq 0.$
We next define the optimization problem as reward maximization. Let $\phi(t)$ be the policy of the source such that $\phi(t): H_t
\rightarrow \{1, \cdots,N\}$ maps the history to $M$ arms in slot $t.$ Let $$\begin{aligned}
a_n^{\phi}(t) =
\begin{cases}
1 & \mbox{if $n \in \phi(t),$ } \\
0 & \mbox{if $n \notin \phi(t).$}
\end{cases}\end{aligned}$$ The infinite horizon discounted cumulative reward under strategy $\phi$ for initial state information $({\boldsymbol{\pi}},
{\boldsymbol{y}}),$ ${\boldsymbol{\pi}}=(\pi_1(1), \cdots, \pi_N(1))$ and ${\boldsymbol{y}} = (y_1(1), \cdots, y_N(1))$ is given by $$V_{\phi}({\boldsymbol{\pi}}, {\boldsymbol{y}}) =
\mathrm{E}^{\phi}\left({\sum_{t=1}^{\infty} \beta^{t-1}
\left[ \sum_{n=1}^{N}
a_n^{\phi}(t) \eta(\pi_n(t), Y_n(t) ),
\right]
}\right).
\label{eqn:opt1}$$ And, in each time slot $M$ arms are played; hence, $\sum_{n=1}^{N}
a_n^{\phi}(t) = M.$ Here, $\beta \in (0,1)$ is the discount parameter. The objective is to find a policy $\phi$ that maximizes $V_{\phi}({\boldsymbol{\pi}}, {\boldsymbol{y}})$ for all $ {\boldsymbol{\pi}}
\in [0,1]^N,$ ${\boldsymbol{y}} \in \{0,1\}^N.$ The problem is a constrained hidden Markov restless multi-armed bandit.
The analysis of restless multi-armed bandits proceeds in the following manner. 1) Consider the restless single armed bandit. Write down Bellman optimality equations along with action value functions, 2) Study the properties of action value functions, 3) Use these properties to prove the optimal threshold policy result, 4) Use this result to prove indexability and compute index.
Constrained restless single armed bandit {#sec:crsab}
========================================
We will now study the constrained restless single armed bandit. The problem of the decision maker here is to decide in each time slot whether or not to play the arm. As there is only one arm, we drop the subscript $n$, the sequence number of the arms; so, $\rho_{n,i} \equiv \rho_i, $ $\eta_{n,i} \equiv \eta_i,$ $\theta_n^{a}(y) \equiv \theta^{a}_{y},$ and so on. The analysis of the single arm problem proceeds by assigning a subsidy $w$ for not playing the arm, see [@Gittins11; @Whittle88]. Recall that the source maintains and updates its belief about state of the arm at the end of every time slot. The update rules are based on previous actions and observations as follows.
1. If $a(t)=1, Y(t)=1, Z(t)=1,$ the new belief $\pi(t+1) =
\Gamma_{1}(\pi(t)).$ $$\Gamma_{1}(\pi) = \frac{\pi \rho_0 p_{00} + (1-\pi) \rho_1
p_{10}}{\pi \rho_0 + (1-\pi)\rho_1}$$ In this case the arm is available, played and a success observed. Then, the belief update is according to the Bayes rule.
2. If $a(t)=1, Y(t)=1, Z(t)=0,$ the new belief $\pi(t+1) =
\Gamma_{0}(\pi(t)).$ $$\Gamma_{0}(\pi) = \frac{ \pi (1-\rho_0)p_{00} +
(1-\pi)(1-\rho_1)p_{10} }{ \pi(1-\rho_0) + (1-\pi)(1-\rho_1) }$$ In this case also, the arm is available, played and a failure observed. Again the belief update makes use of Bayes rule.
3. If $a(t)=0, Y(t)=1$ the new belief $\pi(t+1) =
\gamma^{0}_{1}(\pi(t)).$ $$\gamma^0_1(\pi) = \pi p_{00} + (1-\pi)p_{10}$$ This is the case where the arm is available but not played and no observation is obtained. So, the Markov chain evolves naturally from its current state.
4. If $a(t)=0, Y(t)=0$ the new belief $\pi(t+1) =
\gamma^{0}_{0}(\pi(t)).$ $$\gamma^0_0(\pi) = \frac{p_{10}}{p_{01}+p_{10}} \texttt{ }or\texttt{ } \pi p_{00} + (1-\pi)p_{10}$$ Here, the arm is not available and cannot be played. Again, no observation is obtained. The updated belief in this case is either taken to be the stationary probability or the natural evolution of the Markov chain.
Value functions
---------------
We now define the values of different actions depending on the belief and availability. Value function for action $a,$ belief $\pi$ and availability $y$ is denoted as $\mathcal{L}^aV(\pi,y), a\in\mathcal{A}_y.$ The set $\mathcal{A}_y$ is the set of possible actions for availability $y.$ For the current system $\mathcal{A}_{y=1} = \{0,1\}$ and $\mathcal{A}_{y=0} = \{0\}.$ This means that when arm is unavailable ($y=0$), it cannot be played.
These value functions for different availability models can be defined recursively as follows.
### Stochastic availability
- Action $a=1,$ availability $y=1,$\
- Action $a=0,$ availability $y=1,$
- Action $a=0,$ availability $y=0,$
### Semi-deterministic availability
- Action $a=1,$ availability $y=1,$\
- Action $a=0,$ availability $y=1,$
- Action $a=0,$ availability $y=0,$ $$\begin{aligned}
\mathcal{ L}^{0}V(\pi ,0) = w\frac{1-\beta^{T_0}}{1-\beta} + \beta^{T_0}V\left( (\gamma^0_0)^{T_0}(\pi),1 \right)\end{aligned}$$ Here, $(\gamma^0_0)^{T_0}(\pi)=q,$ with $q$ being the stationary probability, or $(\gamma^0_0)^{T_0}(\pi) = (\gamma^0_1)^{T_0}(\pi)$ .
Notice that a subsidy $w$ is being provided for not playing ($a=0$). Also, not playing is the only possible action when the arm is unavailable. Although there is no apparent choice of action for $y=0,$ the value function $\mathcal{L}^0V(\pi,0)$ is important as it impacts other value functions.
The optimality equations are written as follows. Given availability $y$ and belief $\pi,$ the maximum value $V(\pi,y)$ that can be achieved is given by $$\begin{aligned}
V(\pi,y) = \max_{a\in \mathcal{A}_y} \mathcal{L}^{a}V(\pi ,y) \\
\mathcal{A}_{y=1} = \{0,1\} \texttt{ }
\mathcal{A}_{y=0} = \{ 0 \} \nonumber\end{aligned}$$ $\mathcal{A}_y$ is the feasible set of actions, given availability $y.$ **Note**: We will sometimes use the notation $\mathcal{L}_w^aV(\pi,y)$ in place of $\mathcal{L}^aV(\pi,y)$ when it is required to emphasize the dependence on $w.$ We assume $\eta_0 = \rho_0,$ $\eta_1 = \rho_1$ while proving the results.
In the following discussion we present results for the stochastic availability model. These results also hold true for the semi-deterministic availability model.
Threshold type policy
---------------------
(Threshold type policy) A policy is said to be of threshold type if one of the following is true.
1. For any $\pi \in [0,1],$ $\mathcal{L}^1V(\pi,1) > \mathcal{L}^0V(\pi,1)$; in this case playing the arm is always optimal.
2. For any $\pi \in [0,1],$ $\mathcal{L}^1V(\pi,1) <
\mathcal{L}^0V(\pi,1)$; in this case not playing the arm is always optimal.
3. There exists a $\pi_T \in (0,1),$ such that, $\mathcal{L}^1V(\pi,1) > \mathcal{L}^0V(\pi,1)$ for $\pi<\pi_T$ and $\mathcal{L}^1V(\pi,1) <\mathcal{L}^0V(\pi,1)$ for $\pi>\pi_T $; there exist a unique $\pi_T$ at which optimal action changes from playing to not playing .
\[def:threhold\]
One major claim in this work is the existence of an optimal threshold type policy. To prove this we will make use of some structural properties of the value functions.
1. Value functions $V(\pi,y),\mathcal{L}^{a}V(\pi,y), \pi\in[0,1], a\in \mathcal{A}_y, y\in\{0,1\}$ are convex in belief $\pi.$
2. Value functions $V(\pi,y),\mathcal{L}_{w}^{a}V(\pi,y), \pi\in[0,1], a\in \mathcal{A}_y, y\in\{0,1\}, w\in \mathbb{R}$ are convex in subsidy $w.$
\[lemma:valfuncprops\]
Lemma \[lemma:valfuncprops\] can be claimed by the principle of induction and the convergence of value iteration (Bellman operation).
For any $\pi\in[0,1],w\in \mathbb{R},$ $\left|\frac{\partial \mathcal{L}^0 V(\pi,1)}{\partial \pi} \right|,$ $\left|\frac{\partial \mathcal{L}^1 V(\pi,1)}{\partial \pi}\right|$ and $\left|\frac{\partial V(\pi,1)}{\partial \pi}\right|$ are bounded by $\kappa |\rho_1 - \rho_0|, \kappa = \frac{1}{1-\beta|p_{00} - p_{10}|} $ under either of the following conditions
1. ${0<p_{00}-p_{10}<1/2},$
2. $0<p_{10}-p_{00}<1.$
\[lemma:derivativewrt\_pi\_bound\]
*Proof sketch:* The steps are as follows. (1) Let $V_1(\pi,1) = \max \{\rho(\pi) , w \},$ where $\rho(\pi) = \pi(\rho_0 - \rho_1) + \rho_1$ and $\rho_0 < \rho_1 .$ Hence, the absolute value of slope of $V_1(\pi,1)$ is bounded by $\rho_1 - \rho_0.$ (2) Assume that $|\frac{\partial V_n(\pi,1)}{\partial \pi}| \leq \kappa (\rho_1 - \rho_0).$ (3) Compute the partial derivatives of $\mathcal{L}^1V_{n+1}(\pi,1),$ $\mathcal{L}^0V_{n+1}(\pi,1)$ and $V_{n+1}(\pi,1).$ (4) Compute upper bound on the derivatives using the inequalities $\rho(\pi)\geq \rho_0,$ $1-\rho(\pi)\geq 1-\rho_1,$ and $p_{00} > \Gamma_{0}(\pi) > \Gamma_{1}^1(\pi) > p_{10}.$ (5) Use the principle of induction to claim the result.[$\square$]{}
The quantity $D(\pi) = \mathcal{L}^1 V(\pi,1)-\mathcal{L}^0V(\pi,1)$ gives the advantage of choosing to play the arm in belief state $\pi$. The following lemma states that this advantage decreases as the belief (probability of arm being in bad state) increases.
(Advantage of playing the arm when available is monotonic in belief) $D(\pi):=\mathcal{L}^1 V(\pi,1)-\mathcal{L}^0V(\pi,1)$ is a decreasing in $\pi \in [0,1],$ if $\rho_1>\rho_0 $ and either of the following is true,
1. ${0<p_{00}-p_{10}<1/5},$
2. or $0<p_{10}-p_{00}<1/3.$
\[lemma:advantage\]
*Proof Sketch :* The key steps are as follows. 1) From Lemma \[lemma:derivativewrt\_pi\_bound\], the derivatives of the action value functions with respect to belief $\pi$ are bounded. 2) Compute the derivative of $D(\pi).$ It can be shown to be negative under the stated conditions.
$D(\pi)$ has at most one root in $\pi \in (0,1).$ \[rem:dpiroot\]
If $\rho_1>\rho_0 $ and either ${0<p_{00}-p_{10}<1/5}$ or $0<p_{10}-p_{00}<1/3,$ is true, then the optimal policy is of threshold type.
Suppose $\mathcal{L}^1V(\pi,1)>\mathcal{L}^0V(\pi,1),$ at $\pi = 0 .$ That is, playing the arm is advantageous than not playing it. From lemma \[lemma:advantage\], $D(\pi)$ can have at most one root in $[0,1].$\
Case 1) $D(\pi)$ has a root in $[0,1]:$ From Lemma \[lemma:advantage\], we know that this advantage decreases as $\pi$ increases. So, there exists a $\pi_T \in(0,1) : D(\pi) = \mathcal{L}^1V(\pi,1) < \mathcal{L}^0V(\pi,1)<0, \forall \pi > \pi_T .$ Hence the policy is of threshold type by definition \[def:threhold\].\
Case 2) $D(\pi)$ has no root in $(0,1):$ This means $D(\pi)>0,\pi \in (0,1).$ Hence the optimal policy always choose to play the arm and is threshold type by definition. Similar arguments can be made when $\mathcal{L}^1V(\pi,1)<\mathcal{L}^0V(\pi,1),$ to claim the result.
Indexability
------------
For a given subsidy $w,$ let $\mathcal{G}(w)$ be a set formed by members $(\pi,y)$ of perceived state space $S = [0,1]\times \{0,1\}$ for which not playing the arm when available is optimal. That is,
\(w) = [(,y){ \[0,1\]\_[y=0]{} } ]{}\
[{\[0,1\]\_[y=1]{} : \^1 V(,1) \^0V(,1)}]{} .
(Indexability) The arm is indexable if the set $\mathcal{G}(w)$ is increasing in $w
\in \mathbb{R}.$
Indexability ensures the existence of a unique index. Hence, proving indexability is necessary for the application of Whittle’s index policy as a solution to the CRMAB problem.
For any $\pi\in[0,1],w\in \mathbb{R},$ $\left|\frac{\partial \mathcal{L}^0 V(\pi,1)}{\partial w} \right|,$ $\left|\frac{\partial \mathcal{L}^1 V(\pi,1)}{\partial w}\right|$ and $\left|\frac{\partial V(\pi,1)}{\partial w}\right|$ are bounded by $\frac{1}{1-\beta}.$ \[lemma:derivativewrtwbound\]
Again, we will prove this using the principle of mathematical induction. Let $\mathcal{L}^0V_1(\pi,1) = w,$ $\mathcal{L}^1V_1(\pi,1) = \rho(\pi),$ $V_1(\pi,1) = \max\{\mathcal{L}^0V_1(\pi,1),\mathcal{L}^1V_1(\pi,1)\}.$ Clearly, $\left|\frac{\partial \mathcal{L}^0 V_1(\pi,1)}{\partial w} \right|,$ $\left|\frac{\partial \mathcal{L}^1 V_1(\pi,1)}{\partial w}\right|$, $\left|\frac{\partial V_1(\pi,1)}{\partial w}\right|$ are $<$ $\frac{1}{1-\beta}.$ Assume $\left|\frac{\partial \mathcal{L}^0 V_n(\pi,1)}{\partial w} \right|,$ $\left|\frac{\partial \mathcal{L}^1 V_n(\pi,1)}{\partial w}\right|$, $\left|\frac{\partial V_n(\pi,1)}{\partial w}\right|$ are $<$ $\frac{1}{1-\beta}.$ Similarly,
Hence, $\frac{\partial V_{n+1}(\pi,1)}{\partial w}$ is also bounded by $\frac{1}{1-\beta}.$
Further, as the value functions are non-negative and their are convergent with $lim_{n\rightarrow \infty}V_n(\pi,y) = V(\pi,y),$ our claim holds.
The action value functions $\mathcal{L}_{w}^{1}V(\pi,1)$ and $\mathcal{L}_{w}^{0}V(\pi,1),$ $w\in \mathbb{R}$ are respectively non-decreasing and strictly increasing in subsidy $w.$
The following lemma from [@Meshram18] is needed to prove indexability.
Let $\pi_{T} := \inf \{0\leq \pi \leq 1 : \mathcal{L}^1 V(\pi,1) \le \mathcal{L}^0 V(\pi,1)\} \in [0,1].$ If $\frac{\partial \mathcal{L}^1 V(\pi,1)}{\partial w}\left| _{\pi=\pi_{T}(w)} < \frac{\partial \mathcal{L}^0 V(\pi,1)}{\partial w}\right|_{\pi=\pi_{T}(w)},$ then $\pi_{T}(w)$ is a monotonically decreasing function of $w.$ \[lemma:thresholddecreasesinpi\]
*Proof sketch:* This proof is by contradiction. Assume that thresholds $\pi_T(w) < \pi_T(w')$ for $w<w',$ under given ‘if’ condition. By definition of threshold, $\mathcal{L}_{\pi_T(w)}^1 V(\pi,1) = \mathcal{L}_{\pi_T(w)}^0 V(\pi,1).$ For some $w' = w+\epsilon,$ $\epsilon\in (0,c),c<1,$ we have $\mathcal{L}_{\pi_T(w')}^1 V(\pi,1) \geq \mathcal{L}_{\pi_T(w')}^0 V(\pi,1).$ This means that $\frac{\partial \mathcal{L}^1 V(\pi,1)}{\partial w}\left| _{\pi=\pi_{T}(w)} > \frac{\partial \mathcal{L}^0 V(\pi,1)}{\partial w}\right|_{\pi=\pi_{T}(w)},$ contradicts our assumption. [$\square$]{}
Let $D(\pi,w) := \mathcal{L}_{w}^{1}V(\pi,1)-\mathcal{L}_{w}^{0}V(\pi,1).$
The constrained arm of a partially observable restless single armed bandit with bounded subsidy $w\in [w_l,w_h]$ is indexable if $0<p_{00}-p_{10}<1/5$ or $0<p_{10}-p_{00}<1/3$ and $\beta \in (0,1).$
The proof proceeds in the following steps. (1) From Lemma \[lemma:valfuncprops\] and Lemma \[lemma:derivativewrtwbound\], it can be seen that the functions $\mathcal{L}_{w}^{a}V(\pi,y)$ $a\in\mathcal{A}_y$ are convex and Lipschitz in $w.$ This implies that they are absolutely continuous. (2) It means $D(\pi,w)$ is absolutely continuous, which implies that it is differentiable w.r.t $w$ almost everywhere in the interval $[w_l,w_h],$ for all $\pi \in [0,1].$ (3) This implies that the threshold $\pi_T(w):= \{\pi\in[0,1]\mid D(\pi,w) = 0\}$ is absolutely continuous on $[w_l,w_h];$ hence, $\pi_T(w)$ is differentiable w.r.t $w$ almost everywhere. (4) From Remark 2, $D(\pi,w)$ is decreasing in $w;$ hence, $\frac{\partial D}{\partial w}\leq 0$ almost everywhere in $[w_l,w_h].$ This implies $\frac{\partial \pi_T(w)}{\partial w}\leq 0.$ Now, using Lemma \[lemma:thresholddecreasesinpi\] we can say that $\pi_T(w)$ decreases with $w.$ This means as subsidy $w$ increases, the set $\mathcal{G}(w)$ also increases. Hence, the arm is indexible.
Computing Whittle’s index
-------------------------
(Whittle’s index) For a given belief $\pi \in [0,1],$ Whittle’s index $W(\pi)$ is the minimum subsidy for which, not playing the arm will be the optimal action. $$W(\pi) = \inf\{w\in\mathbb{R}:\mathcal{L}_w^0V(\pi,1)\geq\mathcal{L}_w^1V(\pi,1) \}$$
We will now look at an algorithm for computing Whittle’s index. Algorithm \[algo:WI\] is based on two timescale stochastic approximation.
for $\pi\in\mathcal{G}([0,1])$\
$w_t \gets w_0$;\
h
Notice that the algorithm runs on two timescales. Along the faster timescale the values of $\mathcal{L}_w^0V(\pi,1)$, $\mathcal{L}_w^1V(\pi,1)$ are updated, while the value of $w_t$ is updated along the slower one. It is a well known result in stochastic approximation that, such two-timescale algorithms converge if the sequence $ \alpha_t $ is decreasing, $ \sum_t \alpha_t = \infty $ and $\sum_t \alpha_t^2 <\infty.$ This convergence is almost sure as shown in Theorem $2$, [@Borkar08 Chapter 6]. If $ \alpha_t $ is replaced with a tiny constant value $ \alpha, $ there is convergence with high probability; see [@Borkar08 Section 9.3].
Bounds on optimal value functions {#sec:bounds}
=================================
In this section we shall derive upper bounds on the optimal value function of a CRMAB. First, we shall compare its value function to that of a RMAB (unconstrained). In the following discussion we use the terms ‘unconstrained restless bandits’ and ‘restless bandits’ interchangeably.
Relation between value functions of RMAB and CRMAB
--------------------------------------------------
Let $U(\pi)$ be the value function of an restless single armed bandit which is always available. $U(\pi)$ is the solution of the following dynamic program.
U\_S() = () +\
[U\_[NS]{}() = w + U(\^0\_1()) ]{},\
[U() = {U\_S(),U\_[NS]{}()}.]{}
The following Lemma states that for the same Markov chain parameters, the optimal value of the restless single armed bandit is greater than that of constrained restless single armed bandit.
\[lemma:unconst\] For any given set of parameters $p_{00},$ $p_{1,0},$ $\rho_0,$ $\rho_1,$ $\eta_0,$ $\eta_1,$ $w,$ each of the following statements is true.
1. For belief update rules $\gamma^0_1(\pi) = \pi p_{00} + (1-\pi)p_{10}$ and $\gamma^0_0(\pi) = q,$ the inequality $U(\pi)\geq V(\pi,y),$ holds $\forall \pi\in \Pi_\Gamma ,y\in\{0,1\},$ where
2. If belief update rule $\gamma^0_1(\pi) = \gamma^0_0(\pi) = \pi p_{00} + (1-\pi)p_{10},$ the inequality $U(\pi)\geq V(\pi,y),$ holds $\forall \pi\in[0,1],y\in\{0,1\}.$
This is proved by induction.\
1) Consider $\pi\in\Pi_\Gamma,$ let $U_{S,1}(\pi) = \rho(\pi),$ $U_{NS,1}(\pi) = w,$ and $U_1(\pi) = \max\{U_{S,1}(\pi),U_{NS,1}(\pi)\} = \max\{\rho(\pi),w\}.$ Also, $\mathcal{L}^1V_1(\pi,1) = \rho(\pi)
,$ $\mathcal{L}^0V_1(\pi,1) = w,$ $\mathcal{L}^0V_1(\pi,0) = w,$ and $V_1(\pi,1) = \max\{\mathcal{L}^0V_1(\pi,1),\mathcal{L}^1V_1(\pi,1)\} = \max\{\rho(\pi),w\}. $ $V_1(\pi,0) = w.$ Then, assuming $U_{S,n}\geq \mathcal{L}^1V_n(\pi,1),$ $U_{NS,n}\geq \mathcal{L}^0V_n(\pi,1)$ and $U_{NS,n}\geq \mathcal{L}^0V_n(\pi,0),$ it can be shown that $U_{S,n+1}\geq \mathcal{L}^1V_{n+1}(\pi,1),$ $U_{NS,n+1}\geq \mathcal{L}^0V_{n+1}(\pi,1)$ and $U_{NS,n+1}\geq \mathcal{L}^0V_{n+1}(\pi,0).$ Then, by induction the result follows. The second part can also be proved similarly.
Bounds on value functions
-------------------------
We shall now derive an upper bound on the value function of the constrained bandit. The Lagrangian relaxation provides an upper bound on the value function of the original problem. This has been studied for weakly couple Markov decision processes by [@Hawkins03] and [@Adelman08]. We extend this idea for CRMABs with partially observable states to derive an upper bound.
Let us now look at the constrained multi-armed bandit problem as a set of $N$ single armed bandits. We will be slightly abusing the notation in order to keep the mathematical expressions simpler; any change in notation is mentioned.
The CRMAB problem can be described as follows. Given belief vector ${\boldsymbol{\pi}}\in [0,1]^N$ and availability vector ${\boldsymbol{y}}\in\{0,1\}^N,$ find $J({\boldsymbol{\pi}},{\boldsymbol{y}})$ satisfying
J(,) = \_[\_]{} R(,,) +\
\_[,S\_]{}[( ,|,, )]{} J((),)\
[s.t.` ` \_1 = M, ` `\_ := \_[y\_1]{}\_[y\_2]{}...\_[y\_n]{}.]{} \[eqn:crmab-wcpomdp\]
Here, ${\boldsymbol{\Gamma^o}}$ is the belief (vector) update rule for observation vector ${\boldsymbol{o}}.$ So, ${\boldsymbol{\Gamma}}^{o_n}$ is the belief update rule based on observation $o_n$ for arm $n.$ And ${\boldsymbol{\Gamma^o}}:S_{{\boldsymbol{o}}}\mapsto [0,1]^N.$ Here, $S_{{\boldsymbol{o}}}$ is the observation set with the set of all possible observation vectors. $S_{o_n}$ is the set of possible observations for arm $n.$ An observation vector ${\boldsymbol{o}}$ also contains some ‘no observation’ elements corresponding to the unplayed arms.
The Lagrangian relaxation of the above optimization problem is written as
\[eqn:crmab-Lag\_relaxed\] J\^(,) = \_[\_]{} R(,,) +\
+ \_[,S\_]{}[( ,|,, )]{} J\^((),)\
[>0.]{}
The following Lemma states that the Lagrange relaxed value function of CRMAB can be written as a linear combination of value functions of $N$ constrained single armed bandits.
\[lemma:Lagrangian\_decouple\]
J\^(,) = + \_[n=1]{}\^[N]{} J\^(\_n,y\_n), \[eqn:lemma-Lagrangian\_decouple\]
where,
We need to show that the right hand side (RHS) of can be obtained by substituting the RHS of in the RHS of . It suffices to show that the following expression equals $0.$
The proof of Lemma \[lemma:Lagrangian\_decouple\] is also valid for restless multi-armed bandits with constrained availability (studied in [@Varun18]) which allow the play of unavailable arms.
The Lagrangian relaxed value function for the restless bandit is given as follows (in [@Kaza19]).
U\^() = + \_[n=1]{}\^[N]{} U\^(\_n),
The inequality $J^{\lambda}({\boldsymbol{\pi}},{\boldsymbol{y}}) \leq U^{\lambda}({\boldsymbol{\pi}})$ holds for each of the following cases.
1. $\gamma_0^0(\pi) = q,$ $\gamma^0_1(\pi) = \pi p_{00} + (1-\pi)p_{10},$ $\forall \pi\in\Pi_\Gamma,y\in\{0,1\},$
2. $\gamma_0^0(\pi) = \gamma^0_1(\pi) = \pi p_{00} + (1-\pi)p_{10},$ $\forall \pi\in[0,1],y\in\{0,1\}$
From Lemma \[lemma:unconst\] we know that the value functions of constrained restless single armed bandits are upper bounded by those of restless single armed bandits. It follows that their summation as given in Lemma \[lemma:Lagrangian\_decouple\] is also similarly bounded.
Numerical Experiments {#sec:simulation}
=====================
The main objective in this section is to apply the proposed model to different scenarios and evaluate its performance. We evaluate the performance of the Whittle’s index policy (WI), modified Whittle index policy (MWI) and myopic policy in terms of their value (discounted cumulative reward).
Modified Whittle index (MWI) is a less complex alternative to Whittle’s index considered in [@Brown17; @Kaza19]. However, its performance is found to be highly sensitive to problem parameters, in case of RMABs [@Kaza19]. It is defined for MDPs with finite horizon. The value of MWI at time $t$ is given as $m_t(\pi) = \mathcal{L}_{m_{t+1}}^1V(\pi,1)-\mathcal{L}_{m_{t+1}}^0V(\pi,1).$
Let us first mention a few things about the simulation setup and numerical examples. The policies are evaluated for different bandit instances (a parameter set is called an instance). A bandit instance is specified by giving the values of 1) number of arms $N,$ 2) state transition probabilities of arms $p^n_{ij}(y,a),$ 3) availability probabilities of arms $\theta^a_n(y),$ 4) reward structure $\eta_{n,i},$ 5) success probabilities $\rho_n(i).$ For each bandit instance, the value function of each policy is computed, and averaged over numerous sample sequences of states and arm availability. We will now see the results of five experiments which will provide insight into the performance of various policies. Experiments $1$ $\&$ $2$ consider a $15$-armed bandit instance with same transition matrices and rewards, for stochastic and semi-deterministic availability models, respectively. Experiments $3,$ $4$ $\&$ $5$ consider several $100$-armed bandit instances with randomly generated transition matrices and rewards for stochastic and semi-deterministic availability models.
### Experiment $1$ - Moderately sized system with stochastic availability
We consider a $15$-armed bandit instance with stochastic availability model. The parameter set is given in Table \[Exp1\_parameters\]. The first five arms are always available while the remaining are available according to action dependent probabilities.
Arm $[\theta_1^1, \theta^0_1, \theta^0_0]$ $\rho_0$ $\rho_1$ $\eta_0$ $\eta_1$ $p_{00}$ $p_{10}$
----- ---------------------------------------- ---------- ---------- ---------- ---------- ---------- ----------
1 $[1, 1, 1]$ $0$ $1$ $0$ $0.65$ $0.2$ $0.5$
2 $[1, 1, 1]$ $0$ $1$ $0$ $0.7$ $0.3$ $0.5$
3 $[1, 1, 1]$ $0$ $1$ $0$ $0.75$ $0.4$ $0.3$
4 $[1, 1, 1]$ $0$ $1$ $0$ $0.8$ $0.5$ $0.4$
5 $[1, 1, 1]$ $0$ $1$ $0$ $0.85$ $0.3$ $0.3$
6 $[0.25, 0.8, 0.9]$ $0.1$ $0.9$ $0.1$ $0.9$ $0.2$ $0.8$
7 $[0.3, 0.9, 0.8]$ $0.1$ $0.7$ $0.1$ $0.7$ $0.3$ $0.7$
8 $[0.4, 0.75, 0.7]$ $0.1$ $0.8$ $0.1$ $0.8$ $0.4$ $0.6$
9 $[0.5, 0.7, 0.4]$ $0.2$ $0.7$ $0.2$ $0.7$ $0.5$ $0.5$
10 $[0.6, 0.8, 0.8]$ $0.1$ $0.7$ $0.1$ $0.7$ $0.3$ $0.5$
11 $[0.7, 0.8, 0.7]$ $0.2$ $0.6$ $0.2$ $0.6$ $0.3$ $0.3$
12 $[0.5, 0.5, 0.5]$ $0.2$ $0.8$ $0.2$ $0.8$ $0.6$ $0.4$
13 $[0.8, 0.3, 0.4]$ $0.3$ $0.9$ $0.3$ $0.9$ $0.7$ $0.3$
14 $[0.8, 0.4, 0.2]$ $0.2$ $0.9$ $0.2$ $0.9$ $0.8$ $0.2$
15 $[0.7, 0.6, 0.6]$ $0.3$ $0.95$ $0.3$ $0.95$ $0.9$ $0.2$
: Experiment 1: Parameter set[]{data-label="Exp1_parameters"}
Fig. \[fig:reward-armchoice-Ex1\]a) and Table \[Exp1\_results\] show the discounted cumulative rewards achieved by various policies. Notice that WI and myopic are equivalent in terms of value generated, but not necessarily so, in terms of their arm choices. This is unlike the case of restless bandits in [@Meshram18; @Kaza19], where numerical experiments demonstrated WI to be better than myopic policy for moderately sized systems. Further, both WI and myopic almost reach up to the Lagrangian bound. Hence, it can be expected that their performance is close to optimal.
$L_b$ WI MWI Myopic Random
--------- -------- -------- -------- -------- --------
$Value$ $65.7$ $64.7$ $57.4$ $64.3$ $48.0$
: Experiment $1$: Discounted Cumulative Rewards from various polices, with random initial belief.[]{data-label="Exp1_results"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
![ Experiment $1$ - Stochastic availability: a) discounted cumulative rewards as function of sessions for different policies and b) arm choice fraction for each arm with different policies. []{data-label="fig:reward-armchoice-Ex1"}](CRMAB_N_15_WI_MWI_ran_new1.eps "fig:") ![ Experiment $1$ - Stochastic availability: a) discounted cumulative rewards as function of sessions for different policies and b) arm choice fraction for each arm with different policies. []{data-label="fig:reward-armchoice-Ex1"}](CRMAB_N_15_WI_MWI_armchoice_ran_new1.eps "fig:")
a) b)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
### Experiment $2$ - Moderately sized system with semi-deterministic availability
We again consider a $15$-armed bandit instance with semi-deterministic availability model. The parameters used for this experiment are same as in Experiment $1$ (Table \[Exp1\_parameters\]), except for the availability parameters. Recall that semi-deterministic availability is characterized by parameters $[\theta_1^1, \theta^0_1, T_0].$ Here, $\theta_1^1, \theta^0_1$ are same as in Table \[Exp1\_parameters\], and $T_0$ is chosen to be $3$ slots. The discounted cumulative rewards achieved by various policies are shown in Table \[Exp2\_results\]. Again, the ordering on policy performance is same as in Experiment 1.
$L_b$ WI MWI Myopic Random
------- -------- -------- -------- -------- --------
Value $65.7$ $64.3$ $61.4$ $63.5$ $48.1$
: Experiment $2$ - Semi-deterministic availability: Discounted Cumulative Rewards from various polices, with random initial belief.[]{data-label="Exp2_results"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- --
![ Experiment $2$ - semi-deterministic availability: a) discounted cumulative rewards as function of sessions for different policies and b) arm choice fraction for each arm with different policies. []{data-label="fig:reward-armchoice-Ex2"}](CRMAB_sd_N_15_WI_MWI_ran_new1.eps "fig:") ![ Experiment $2$ - semi-deterministic availability: a) discounted cumulative rewards as function of sessions for different policies and b) arm choice fraction for each arm with different policies. []{data-label="fig:reward-armchoice-Ex2"}](CRMAB_sd_N_15_WI_MWI_armchoice_ran_new1.eps "fig:")
a) b)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- --
### Experiment $3$ - Large systems with unconstrained availability
We consider two sets of $100$-armed bandit instances with unconstrained availability, *i.e.*, arms are always available. The first set of instances has a contiguous reward structure, *i.e.,* $\eta_0$ of the arms is generated randomly from $[0,1],$ and then $ \eta_1$ is picked from $(\eta_0,1].$ The second set has a partitioned reward structure, *i.e.,* $\eta_0 \in [0,0.3]$ and $\eta_1\in [0.5,1].$ All transition probabilities are randomly generated such that the first $50$ arms are positively correlated $(p_{0,0}>p_{1,0})$ and the other half are negatively correlated $(p_{0,0}<p_{1,0})$. Fig. \[fig:100arms\_Ex-3-allavail\]\[top\] shows the comparison for the case $\rho_0=0$ and $\rho_1=1.$ For this case, it can be seen that the WI and myopic policies are equivalent in terms of value generated. In this case the Whittle’s indices of arms can be computed with closed form expressions. Fig. \[fig:100arms\_Ex-3-allavail\]\[bottom\] shows the comparison for arbitrary values of $\rho.$ In this case, as computation of Whittle’s index is cumbersome, only myopic and MWI are evaluated. In [@Meshram18; @Kaza19], numerical experiments demonstrated the better performance of WI compared to myopic policy, for moderately sized systems. However, in systems in large systems, as the differences between rewards of arms gets smaller, the advantage of WI over myopic tends to reduce.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Experiment 3 - Unconstrained Restless Bandits : Values achieved by various policies when all arms are always available, for different sets of $100$-armed bandit instances, with contiguous reward structure (left) and with partitioned reward structure (right). The case for $\rho_0=0$ and $\rho_1=1$ is shown at the top, and the case for arbitrary $0\leq \rho_0<\rho_1\leq 1$ is shown at the bottom. []{data-label="fig:100arms_Ex-3-allavail"}](CRMAB_N_100_K_always_100_rhos01_Trail.eps "fig:") ![Experiment 3 - Unconstrained Restless Bandits : Values achieved by various policies when all arms are always available, for different sets of $100$-armed bandit instances, with contiguous reward structure (left) and with partitioned reward structure (right). The case for $\rho_0=0$ and $\rho_1=1$ is shown at the top, and the case for arbitrary $0\leq \rho_0<\rho_1\leq 1$ is shown at the bottom. []{data-label="fig:100arms_Ex-3-allavail"}](CRMAB_N_100_K_always_100_rhos01_Tr.eps "fig:")
![Experiment 3 - Unconstrained Restless Bandits : Values achieved by various policies when all arms are always available, for different sets of $100$-armed bandit instances, with contiguous reward structure (left) and with partitioned reward structure (right). The case for $\rho_0=0$ and $\rho_1=1$ is shown at the top, and the case for arbitrary $0\leq \rho_0<\rho_1\leq 1$ is shown at the bottom. []{data-label="fig:100arms_Ex-3-allavail"}](100arms_always_Trail1.eps "fig:") ![Experiment 3 - Unconstrained Restless Bandits : Values achieved by various policies when all arms are always available, for different sets of $100$-armed bandit instances, with contiguous reward structure (left) and with partitioned reward structure (right). The case for $\rho_0=0$ and $\rho_1=1$ is shown at the top, and the case for arbitrary $0\leq \rho_0<\rho_1\leq 1$ is shown at the bottom. []{data-label="fig:100arms_Ex-3-allavail"}](100arms_always_Tr1.eps "fig:")
a) b)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
### Experiment $4$ - Large systems with stochastic availability
We consider two sets $100$-armed bandit instances with stochastic availability. Same instances are used as in Experiment 3, except for the availability parameters. The availability probabilities $[\theta_1^1, \theta^0_1, \theta^0_0]$ are randomly generated from $[0,1].$ In this experiment we only evaluate the performances of modified Whittle’s index and myopic policies, as computing indices for large systems is cumbersome. Furthermore, as observed in Experiments 1, 2, we can expect the performance of WI to be similar to that of myopic. Hence, we compare only myopic and MWI policies.
Fig. \[fig:100arms\_Ex-4\] shows the values of discounted cumulative rewards achieved by MWI and myopic policies for different $100$-armed bandit instances, for contiguous and partitioned reward structures.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Experiment $4$ - Stochastic availability model : Values achieved by myopic and MWI policies for two sets of $100$-armed bandit instances, one set with contiguous reward structure and the other with partitioned reward structure.[]{data-label="fig:100arms_Ex-4"}](CRMAB_N_100_ran_Trail_new1.eps "fig:") ![ Experiment $4$ - Stochastic availability model : Values achieved by myopic and MWI policies for two sets of $100$-armed bandit instances, one set with contiguous reward structure and the other with partitioned reward structure.[]{data-label="fig:100arms_Ex-4"}](CRMAB_N_100_ran_Tr_new1.eps "fig:")
a) b)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
### Experiment $5$ - Large system with semi-deterministic availability
We again consider two sets $100$-armed bandit instances, but with semi-deterministic availability. As in Experiment $4,$ the first set of instances has contiguous reward structure with $0<\eta_0<\eta_1<1,$ and the second set has a partitioned reward structure with $\eta_0 \in [0,0.3]$ and $\eta_1\in [0.5,1].$ The rewards, transition probabilities are same as in Experiment $4,$ so are $\theta_1^1, \theta^0_1.$ $T_0$ is chosen to be $3$ slots for all the arms. Fig. \[fig:100arms\_sd\_Ex-5\] shows the discounted cumulative rewards for MWI and myopic policies for different instances, for contiguous and partitioned reward structures. Again, the ordering on policy performances is same as in case of the stochastic availability model.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
![ Experiment $5$ - Semi-deterministic availability model : Values achieved by myopic and MWI policies for two sets of $100$-armed bandit instances, one set with contiguous reward structure and the other with partitioned reward structure.[]{data-label="fig:100arms_sd_Ex-5"}](CRMAB_sd_N_100_ran_Trail_new1.eps "fig:") ![ Experiment $5$ - Semi-deterministic availability model : Values achieved by myopic and MWI policies for two sets of $100$-armed bandit instances, one set with contiguous reward structure and the other with partitioned reward structure.[]{data-label="fig:100arms_sd_Ex-5"}](CRMAB_sd_N_100_ran_Tr_new1.eps "fig:")
a) b)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
Conclusion {#sec:conc}
==========
In this paper, the problem of constrained restless multi-armed bandits is studied. These constraints are in the form of time varying availability of arms which can either be stochastic or semi-deterministic. Some numerical experiments show that for moderately sized systems both Whittle’s index policy and myopic policy are close to the upper bound on the value function. Experiments also suggest that myopic policy performs almost as well as Whittle’s index policy.
A useful research direction would be to study variations on myopic policy such as finite step look ahead policies, as robust low complexity alternatives to Whittle’s index policy. Another future direction would be towards developing learning algorithms for scenarios where the systems parameters are unknown.
|
---
abstract: 'We consider level crossing in the background of the sphaleron barrier for nondegenerate fermions. The mass splitting within the fermion doublets allows only for an axially symmetric ansatz for the fermion fields. In the background of the sphaleron we solve the partial differential equations for the fermion functions. We find little angular dependence for our choice of ansatz. We therefore propose a good approximate ansatz with radial functions only. We generalize this approximate ansatz with radial functions only to fermions in the background of the sphaleron barrier and argue, that it is a good approximation there, too.'
author:
- |
[**Guido Nolte**]{}\
Fachbereich Physik, Universität Oldenburg, Postfach 2503\
D-26111 Oldenburg, Germany
- |
[**Jutta Kunz**]{}\
Fachbereich Physik, Universität Oldenburg, Postfach 2503\
D-26111 Oldenburg, Germany\
and\
Instituut voor Theoretische Fysica, Rijksuniversiteit te Utrecht\
NL-3508 TA Utrecht, The Netherlands
- |
[**Burkhard Kleihaus**]{}\
Fachbereich Physik, Universität Oldenburg, Postfach 2503\
D-26111 Oldenburg, Germany
title: NONDEGENERATE FERMIONS IN THE BACKGROUND OF THE SPHALERON BARRIER
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Introduction
============
The explanation of the observed baryon asymmetry of the universe represents a challenging problem. Although far from solving this highly complex problem, we know at least what features a theory must have to allow for an explanation. It is therefore remarkable that the standard model fulfills all three Sakharov-conditions to generate the observed baryon asymmetry: C and CP violation, a first order phase transition and non-conservation of baryon number [@sak].
Here we are concerned with the violation of baryon number (or more generally fermion number) in the standard model. It was discovered by ’t Hooft [@hooft] as a consequence of the Adler-Bell-Jackiw anomaly present in chiral gauge theories. In particular ’t Hooft studied the fermion number violation induced by vacuum to vacuum tunneling processes described by instantons, resulting in extremely small tunneling rates.
In Weinberg-Salam theory topologically distinct vacua are separated by finite energy barriers. The height of the barriers is given by the energy of the sphaleron, an unstable solution of the static field equations [@man; @km]. Thus the sphaleron determines the minimal energy needed for a classically allowed vacuum to vacuum transition. The probability for a transition is expected to be enhanced significantly, if enough energy is put into the system under consideration, either in suitable (future) accelerators or at high temperatures in the early universe \[5-10\].
While the barrier is traversed baryon number violation may be seen explicitly by analyzing the corresponding Dirac equation in the bosonic background fields. The lowest positive energy continuum state becomes continuously deformed along the barrier until it reaches the negative energy continuum, passing zero energy precisely at the top of the energy barrier, at the sphaleron \[11-15\]. Investigating the whole spectrum of the Dirac equation shows, that along the barrier in fact all levels become continuously deformed into the next lower levels, resulting finally in an identical spectrum, where only the number of occupied levels above the Dirac sea has decreased by one [@dia].
These calculations \[11-16\] are based on the approximation, that the fermion doublets are degenerate in mass (and that the Weinberg angle may be set to zero [@kkb1; @kkb2]), allowing for a spherically symmetric ansatz for the fermion wave function. For the physical situation of highly nondegenerate fermion masses (at least for the heavy flavours) an analogous calculation is far more involved, since the spherically symmetric ansatz fails and the equations of motion cannot be reduced to ordinary differential equations. (This is in contrast to the case of instantons [@dm1].)
Here we consider an axially symmetric ansatz for the fermion fields in the background of the sphaleron barrier. The ansatz is chosen in such a way, that it is “almost spherically symmetric”, in the sense that the functions involved have little angular dependence. Due to the symmetry of the sphaleron the ansatz simplifies considerably in the background field of the sphaleron. In this case we solve numerically the full set of partial differential equations for the fermion functions. We then consider a set of approximate ordinary differential equations for the fermion functions, finding almost identical solutions. Because of the numerical complexity involved in solving the full set of partial differential equations in the background of the sphaleron barrier, we consider in this general case only an approximate set of ordinary differential equations for radial fermion functions. We argue that these equations represent a good approximation as well.
In section 2 we briefly review the Weinberg-Salam Lagrangian (for vanishing mixing angle) for nondegenerate fermion doublets. In section 3 we present our axially symmetric ansatz for the fermions, constructed as a generalization of the usual spherically symmetric ansatz. In section 4 we consider fermions in the background of the sphaleron. We derive the equations of motion, present the solutions of the full set of partial differential equations, and compare with the solutions of the set of approximate ordinary differential equations. In section 5 we consider fermions in the background of the sphaleron barrier. We present our conclusions in section 6.
**Weinberg-Salam Lagrangian**
=============================
We start with the bosonic sector of the Weinberg-Salam theory in the limit of vanishing Weinberg angle, where the electromagnetic field decouples and can be set to zero, $${\cal L}_{\rm b} = -\frac{1}{4} F_{\mu\nu}^a F^{\mu\nu,a}
+ (D_\mu \Phi)^{\dagger} (D^\mu \Phi)
- \lambda (\Phi^{\dagger} \Phi - \frac{1}{2}v^2 )^2
\$$ with the field strength tensor $$F_{\mu\nu}^a=\partial_\mu V_\nu^a-\partial_\nu V_\mu^a
+ g \epsilon^{abc} V_\mu^b V_\nu^c
\ ,$$ and the covariant derivative $$D_{\mu} = \partial_{\mu}
-\frac{1}{2}ig \tau^a V_{\mu}^a
\ .$$ The ${\rm SU(2)_L}$ gauge symmetry is spontaneously broken due to the non-vanishing vacuum expectation value $v$ of the Higgs field $$\langle \Phi \rangle = \frac{v}{\sqrt2}
\left( \begin{array}{c} 0\\1 \end{array} \right)
\ ,$$ leading to the boson masses $$M_W = M_Z =\frac{1}{2} g v \ , \ \ \ \ \ \
M_H = v \sqrt{2 \lambda}
\ .$$ We employ the values $M_W=80 \ {\rm GeV}$, $g=0.65$.
For vanishing mixing angle, considering only one fermion doublet, the fermion Lagrangian reads $$\begin{aligned}
{\cal L}_{\rm f} & = &
\bar q_L i \gamma^\mu D_\mu q_L
+ \bar q_R i \gamma^\mu \partial_\mu q_R
\nonumber \\
& - & f^{(u)} (\bar q_L
\tilde \Phi u_R +\bar u_R \tilde \Phi^\dagger q_L)
- f^{(d)} (\bar d_R \Phi^\dagger q_L
+\bar q_L \Phi d_R)
\ , \end{aligned}$$ where $q_L$ denotes the lefthanded doublet $(u_L,d_L)$, while $q_R$ abbreviates the righthanded singlets $(u_R,d_R)$, with $\tilde \Phi = i \tau_2 \Phi^*$. The fermion masses are given by $$M_{u,d}=\frac{1}{\sqrt{2}}f^{(u,d)} v
\ .$$
The fermion equations read in dimensionless coordinates (chosen in units of $M_W$) $$\big(i\frac{\partial}{\partial t}+i\sigma^i
\frac{\partial}{\partial x^i}+\frac{1}{2}\tau^a V_i^a\sigma^i\big)q_L
-(m M+\Delta m M \tau_z)q_R=0
\label{f1}$$ and $$\big(i\frac{\partial}{\partial t}-i\sigma^i
\frac{\partial}{\partial x^i})q_R
-(m M^\dagger+\Delta m \tau_z M^\dagger)q_L=0 \ ,
\label{f2}$$ where $M$ is the Higgsfield matrix defined by $$\Phi=\frac{v}{\sqrt{2}}M
\left(\begin{tabular}{c}0\\ 1\end{tabular}\right) \ ,$$ and $m$ and $\Delta m$ are the average fermion mass and half the mass difference (in units of $M_W$) $$\ m = (M_u+M_d)/(2M_W) \nonumber \ ,$$ $$\Delta m = (M_u-M_d)/(2M_W) \nonumber \ .$$
Ansatz
======
For the gauge and Higgs fields along the sphaleron barrier we take the usual spherically symmetric ansatz in the temporal gauge $$\begin{aligned}
V_i^a & = & \frac{1-f_A(r)}{gr} \varepsilon_{aij}\hat r_j
+ \frac{f_B(r)}{gr} (\delta_{ia}-\hat r_i \hat r_a)
+ \frac{f_C(r)}{gr} \hat r_i \hat r_a \ ,
\\
V_0^a & = & 0\ , \\
\Phi & = & \frac{v}{\sqrt {2}}
\Bigl(H(r) + i \vec \tau \cdot \hat r K(r)\Bigr)
\left( \begin{array}{c} 0\\1 \end{array} \right)
\ . \end{aligned}$$ Due to a residual gauge degree of freedom we are free to choose the gauge $f_C=0$.
To construct an appropriate ansatz for nondegenerate fermions we begin by recalling the spherically symmetric ansatz for degenerate fermions with $\Delta m=0$ \[11-16,20\], containing four radial functions, $$q_L(\vec r\,,t) = e^{-i\omega t} M_W^{\frac{3}{2}}
\bigl[ G_L(r)
+ i \vec \sigma \cdot \hat r F_L(r) \bigr] \chi_{\rm h}
\ , \label{ans1}$$ $$q_R(\vec r\,,t) = e^{-i\omega t} M_W^{\frac{3}{2}}
\bigl[ G_R(r)
+ i \vec \sigma \cdot \hat r F_R(r) \bigr] \chi_{\rm h}
\ , \label{ans2}$$ where the normalized hedgehog spinor $\chi_{\rm h}$ satisfies the spin-isospin relation $$\vec \sigma \chi_{\rm h} + \vec \tau \chi_{\rm h} = 0
\ .$$ The generalized axially symmetric ansatz contains the spherically symmetric ansatz, where the four functions $G_L$, $F_L$, $G_R$ and $F_R$ now depend on the variables $r$ and $\theta$. Because of the presence of the $\tau_z$-terms in the field equations (\[f1\])-(\[f2\]) for $\Delta m \ne 0$, we need to ‘double’ the ansatz by adding terms of the same structure, but with $\chi_{\rm h}$ replaced by $\tau_z \chi_{\rm h}$, involving the four new ($r$ and $\theta$-dependent) functions $\Delta G_L$, $\Delta F_L$, $\Delta G_R$ and $\Delta F_R$. The ansatz now contains eight functions, which are in general complex and $\theta$-dependent, caused by various occurrences of the nonvanishing anticommutator $[\vec{\tau}\cdot\hat{r},\tau_z]_+=2\cos{\theta}$ in the equations of motion. Considering the $\theta$-dependence of the functions, the real part is even in $\cos\theta$ while the imaginary part is odd. This then suggests the following parametrization of the general axially symmetric ansatz, involving 16 real functions of the variables $r$ and $p=\cos^2\theta$, $$\begin{aligned}
q_L(\vec r\,,t)& =& e^{-i\omega t} M_W^{\frac{3}{2}}
\Bigl(
\bigl[ {G_L^1}(r,p)+i\cos(\theta){G_L^2}(r,p)
+ i \vec \sigma \cdot \hat r
({F_L^1}(r,p)+i\cos(\theta){F_L^2}(r,p))
\bigr]\nonumber\\&&
+\tau_z\bigl[ {\Delta G_L^1}(r,p)+i\cos(\theta){\Delta G_L^2}(r,p)
\nonumber\\&&
+ i \vec \sigma \cdot \hat r
({\Delta F_L^1}(r,p)+i\cos(\theta){\Delta F_L^2}(r,p))
\bigr]
\Bigr)
\chi_{\rm h}
\ , \label{ans3} \end{aligned}$$ $$\begin{aligned}
q_R(\vec r\,,t)& =& e^{-i\omega t} M_W^{\frac{3}{2}}
\Bigl(
\bigl[ {G_R^1}(r,p)+i\cos(\theta){G_R^2}(r,p)
+ i \vec \sigma \cdot \hat r
({F_R^1}(r,p)+i\cos(\theta){F_R^2}(r,p))
\bigr]\nonumber\\&&
+\tau_z\bigl[ {\Delta G_R^1}(r,p)+i\cos(\theta){\Delta G_R^2}(r,p)
\nonumber\\&&
+ i \vec \sigma \cdot \hat r
({\Delta F_R^1}(r,p)+i\cos(\theta){\Delta F_R^2}(r,p))
\bigr]
\Bigr)
\chi_{\rm h} \ .
\ \label{ans4} \end{aligned}$$
The choice of ansatz (\[ans3\])-(\[ans4\]) is not unique. We have also considered alternative parametrizations of the axially symmetric fermion ansatz. These involve different fermion functions, uniquely related to the above fermion functions. The crucial advantage of the ansatz (\[ans3\])-(\[ans4\]) lies in the observation, that its fermion functions have only a very weak angular dependence in the background field of the sphaleron, as shown below. This is in contrast to the alternative parametrizations considered.
Sphaleron
=========
We first consider fermions in the background of the sphaleron. Since the background field barrier is symmetric about the sphaleron, the fermion eigenvalue is precisely zero at the sphaleron \[11-16\], also for nondegenerate fermion masses. As for degenerate fermion masses, the fermion ansatz (\[ans3\])-(\[ans4\]) then simplifies significantly in the background field of the sphaleron. This is due to the parity reflection symmetry of the sphaleron, for which the functions $f_B$ and $H$ vanish, resulting in the decoupling of eight of the 16 functions. These functions, $F_L^1$, $G_L^2$, $\Delta F_L^1$, $\Delta G_L^2$ and $F_R^1$, $G_R^2$, $\Delta F_R^1$, $\Delta G_R^2$, can therefore consistently be set to zero. After dropping the number index on the remaining eight functions the set of partial differential equations in the variables $r$ and $p$ reads $$\begin{aligned}
0&=&-{G_R}'+\frac{2}{r}p\frac{\partial}{\partial p} {G_R}
+\frac{1}{r}(1+2p\frac{\partial}{\partial p}){\Delta F_R}
-m K{G_L}\nonumber\\&&
+\Delta m K{\Delta G_L}-2pKm{\Delta F_L} \ ,\label{eq1} \\
0&=&-{\Delta G_R}'+\frac{1}{r}(1+2p\frac{\partial}{\partial p}){F_R}
+\frac{2}{r}p\frac{\partial}{\partial p} {\Delta G_R}
+m K{\Delta G_L}\nonumber\\&&
-\Delta m K{G_L}-2pK\Delta m{\Delta F_L} \ ,\label{eq2} \\
0&=&{F_R}'+\frac{1}{r}(3+2p\frac{\partial}{\partial p}){F_R}
+\frac{2}{r}\frac{\partial}{\partial p}{\Delta G_R}
-\Delta m K{\Delta F_L}+m K ({F_L} + 2{\Delta G_L}) \ ,\label{eq3} \\
0&=&{\Delta F_R}'+\frac{1}{r}(3+2p\frac{\partial}{\partial p}){\Delta F_R}
+\frac{2}{r}\frac{\partial}{\partial p}{G_R}
-m K{\Delta F_L}+\Delta m K ({F_L} + 2{\Delta G_L}) \ ,\label{eq4} \\
0&=&{G_L}'-\frac{2}{r}p\frac{\partial}{\partial p} {G_L}
-\frac{1}{r}(1+2p\frac{\partial}{\partial p}){\Delta F_L}
+m K{G_R} +\Delta m K{\Delta G_R} \nonumber\\&&+2pK(m{\Delta F_R}
+\Delta m{F_R})+\frac{1-f_A}{r}({G_L}+p{\Delta F_L}) \ ,\label{eq5} \\
0&=&{\Delta G_L}'
-\frac{2}{r}p\frac{\partial}{\partial p} {\Delta G_L}
-\frac{1}{r}(1+2p\frac{\partial}{\partial p}){F_L}
-K(m {\Delta G_R}+\Delta m {G_R}) \ ,\label{eq6} \\
0&=&-{F_L}'-\frac{1}{r}(3+2p\frac{\partial}{\partial p}){F_L}
-\frac{2}{r}\frac{\partial}{\partial p}{\Delta G_L}
-m K{F_R}-\Delta m K{\Delta F_R} \nonumber\\&&-2K(m{\Delta G_R}
+\Delta m{G_R} )+\frac{1-f_A}{r}({F_L}+{\Delta G_L}) \ ,\label{eq7} \\
0&=&-{\Delta F_L}'-\frac{1}{r}(3+2p\frac{\partial}{\partial p}){\Delta F_L}
-\frac{2}{r}\frac{\partial}{\partial p}{G_L} +\Delta m K{F_R}+m K{\Delta F_R}
\ . \label{eq8} \end{aligned}$$
Inspection of the equations shows, that only three equations, eqs. (\[eq1\]),(\[eq2\]) and (\[eq5\]), contain $p$-dependent terms, when the terms involving the partial derivative with respect to $p$, present in all eight equations, are not considered. In fact only three functions occur with a prefactor $p$. These are ${F_R}, {\Delta F_R}$ and ${\Delta F_L}$. If these three functions are small, then the ansatz is approximately spherically symmetric in the sense, that all functions have little angular dependence. In the following we show, that this is indeed the case.
Let us denote the three functions ${F_R}, {\Delta F_R}$ and ${\Delta F_L}$ as $b$, as ‘bad’ functions, and the other five functions as $g$, as ‘good’ functions. First we note, that we could set all three bad functions $b$ consistently equal to zero, if the source term $$s=-K({F_L}+2{\Delta G_L})
\label{s}$$ for the bad functions $F_R$ and $\Delta F_R$ in eqs. (\[eq3\]) and (\[eq4\]) did vanish. Then the five good functions $g$ were pure radial functions. Let us therefore inspect this source term more closely and split it into two terms, $s=a_1-a_2$, with $$\ \ a_1=-K{F_L} \ ,$$ and $$a_2=2K{\Delta G_L} \ .$$ If $a_1=a_2$, the source term vanishes. We now argue that $a_1$ and $a_2$ are approximately equal. Setting the bad functions ${F_R}, {\Delta F_R}$ and ${\Delta F_L}$ equal to zero, and neglecting terms with prefactors $\frac{1}{r}$, for large $r$ eqs. (\[eq6\]) and (\[eq7\]) reduce to $${\Delta G_L}'=K(m{\Delta G_R}+\Delta m{G_R}) \ ,$$ and $${F_L}'=-2K(m{\Delta G_R}+\Delta m{G_R}) \ .$$ With the proper boundary conditions at infinity we thus find for large $r$ for the solutions the desired behaviour, ${F_L}=-2{\Delta G_L}$, i.e. the source term vanishes there. On the other hand, for small $r$ the source term vanishes, since the function $K$ vanishes. In the intermediate region the size of the source term needs numerical analysis.
We have solved the set of partial differential equations in the background of the sphaleron numerically for various values of the average mass $m$ and the mass difference $\Delta m$. Let us consider a typical numerical result. In Fig. 1 we show the ‘good’ lefthanded functions, $G_L$, $\Delta G_L$ and $F_L$, with normalization $G_L(0)=1$, for three values of the angle $\theta$ ($\theta=0$, $\pi/4$ and $\pi/2$) for the mass parameters $m=0.5$ and $\Delta m=0.25$. The $\theta$-dependence of the functions is too small to be seen in the figure, being on the order of $10^{-4}$. The corresponding bad lefthanded function $\Delta F_L$ is very small, indeed. For the case considered it is less then $5\cdot 10^{-4}$, i.e. two orders of magnitude smaller than the good functions, with almost no $\theta$-dependence at all.
These results suggest to approximate all functions by radial functions. We have therefore obtained a new set of ordinary differential equations by integrating out the $\theta$-dependence in the energy density, before variation with respect to the fermion functions. The resulting equations then differ only in prefactors for the three bad functions, apart from the absence of the partial derivatives with respect to $p$. In block form the approximate set of differential equations reads $$\left(
\begin{tabular}{c}
$g'$ \\
$b'$ \\
\end{tabular}
\right)
=
\left(
\begin{tabular}{cc}
$A$&$B$ \\
$C$&$D$ \\
\end{tabular}
\right)
\left(
\begin{tabular}{c}
$g$ \\
$b$ \\
\end{tabular}
\right)
\ ,$$ where $A$ is a 5 by 5 matrix, $B$ is a 5 by 3 matrix etc. The vector $Cg$ represents the source terms of the good functions $g$ for the bad functions $b$. (It is identical in both sets of equations.) These source terms are $ms$, $\Delta m s$ and zero for $F_R$, $\Delta F_R$ and $\Delta F_L$, respectively, with the source $s$ defined in eq. (\[s\]).
Solving the approximate set of ordinary differential equations leads to results almost identical to those of the full partial differential equations. This is demonstrated in Fig. 1, where also the approximate good lefthanded functions $G_L$, $\Delta G_L$ and $F_L$, with normalization $G_L(0)=1$, are shown. The difference of the approximate functions and the exact functions is too small to be seen in the figure, being on the order of $10^{-3}$. The bad lefthanded function $\Delta F_L$ is less then $10^{-4}$.
Thus the exact calculation and the radial approximation result in almost identical results, and the bad functions are very small, indeed. We are therefore free to present in the following only results obtained with the approximate calculation. In Figs. 2-4 we show the same good lefthanded functions, $G_L$, $\Delta G_L$ and $F_L$, as in Fig. 1 for the same value of the average mass $m=0.5$, but for three different values of the mass difference, $\Delta m=0.25$, 0.5, and 0.75. Fig. 5 is the corresponding figure for the good righthanded function $G_R$. The functions ${G_L}$ and ${G_R}$ are the only functions which do not vanish in the limit $\Delta m=0$. All other functions, which vanish for $\Delta m=0$, are approximately proportional to $\Delta m$ as seen in Figs. 3 and 4. Finally in Fig. 6 we demonstrate the approximate cancellation of the source terms $a_1$ and $a_2$, responsible for the fact that the bad functions are very small.
Sphaleron Barrier
=================
Let us now consider nondegenerate fermions in the background of the sphaleron barrier. Along the barrier we expect a smooth transition of one fermion level from the positive continuum to the negative continuum. In the case of degenerate fermion masses, all fermion levels change along the barrier to the respective next lower level [@dia], thus only one level crosses zero, and the spectrum exhibits no crossing of any two levels. Expecting the same qualitative behaviour of the spectrum in the case of nondegenerate masses, the lowest free fermion level, corresponding to the lower mass fermion of the doublet, then should cross zero.
In the general background of the sphaleron barrier the full ansatz, eqs. (\[ans3\])-(\[ans4\]), is needed. The background fields along the barrier may be taken from the extremal path calculations [@kb3] or, as done here, from the gradient approach [@nk1]. The set of partial equations for the 16 real fermionic functions of the variables $r$ and $p$ reads $$\begin{aligned}
0&=&\omega {F_R^1}-{G_R^1}'+\frac{2}{r}p\frac{\partial}{\partial p} {G_R^1}
+\frac{1}{r}(1+2p\frac{\partial}{\partial p}){\Delta F_R^2}
-m(K{G_L^1}+H{F_L^1})\nonumber\\ &&
-\Delta m(-K{\Delta G_L^1}+H{\Delta F_L^1})-2pKm{\Delta F_L^2} \ ,
\label{g1} \\
0&=&\omega {F_R^2}-{G_R^2}'+\frac{1}{r}(1+2p\frac{\partial}{\partial p}){G_R^2}
-\frac{2}{r}\frac{\partial}{\partial p}{\Delta F_R^1}
-\Delta m(-K{\Delta G_L^2}+H{\Delta F_L^2}) \nonumber\\ &&
-m(K{G_L^2}+H{F_L^2}-2K{\Delta F_L^1}) \ , \label{g2} \\
0&=&\omega {\Delta F_R^1}-{\Delta G_R^1}'+\frac{1}{r}(1+2p\frac{\partial}
{\partial p}){F_R^2}+\frac{2}{r}p\frac{\partial}{\partial p}
{\Delta G_R^1}- m(-K{\Delta G_L^1}+H{\Delta F_L^1})\nonumber\\ &&
-\Delta m(K{G_L^1}+H{F_L^1})-2pK\Delta m{\Delta F_L^2} \ , \label{g3} \\
0&=&\omega {\Delta F_R^2}-{\Delta G_R^2}'+\frac{1}{r}(1+2p\frac{\partial}
{\partial p}){\Delta G_R^2}-\frac{2}{r}\frac{\partial}{\partial p}
{F_R^1}- m(-K{\Delta G_L^2}+H{\Delta F_L^2})\nonumber\\ &&
-\Delta m(K{G_L^2}+H{F_L^2}-2K{\Delta F_L^1}) \ , \label{g4} \\
0&=&\omega {G_R^1} +{F_R^1}'
+\frac{1}{r}(2+2p\frac{\partial}{\partial p}){F_R^1}
-\frac{1}{r}(1+2p\frac{\partial}{\partial p}){\Delta G_R^2}
-m(H{G_L^1}-K{F_L^1})\nonumber\\ &&
-\Delta m(H{\Delta G_L^1}+K{\Delta F_L^1})-2pKm{\Delta G_L^2} \ , \label{g5} \\
0&=&\omega {G_R^2}+{F_R^2}'+\frac{1}{r}(3+2p\frac{\partial}{\partial p}){F_R^2}
+\frac{2}{r}\frac{\partial}{\partial p}{\Delta G_R^1}
-\Delta m(H{\Delta G_L^2}+K{\Delta F_L^2}) \nonumber\\ &&
-m(H{G_L^2}-K{F_L^2} -2K{\Delta G_L^1})\ , \label{g6} \\
0&=&\omega {\Delta G_R^1}+{\Delta F_R^1}'
+\frac{1}{r}(2+2p\frac{\partial}{\partial p}){\Delta F_R^1}
-\frac{1}{r}(1+2p\frac{\partial}{\partial p}){G_R^2}
-\Delta m(H{G_L^1}-K{F_L^1})\nonumber\\ &&
-m(H{\Delta G_L^1}+K{\Delta F_L^1})-2pK\Delta m{\Delta G_L^2} \ , \label{g7} \\
0&=&\omega {\Delta G_R^2}+{\Delta F_R^2}'
+\frac{1}{r}(3+2p\frac{\partial}{\partial p}){\Delta F_R^2}
+\frac{2}{r}\frac{\partial}{\partial p}{G_R^1}
-m(H{\Delta G_L^2}+K{\Delta F_L^2}) \nonumber\\ &&
-\Delta m(H{G_L^2}-K{F_L^2}-2K{\Delta G_L^1}) \ , \label{g8} \\
0&=&\omega {F_L^1}+{G_L^1} '-\frac{2}{r}p\frac{\partial}{\partial p} {G_L^1}
-\frac{1}{r}(1+2p\frac{\partial}{\partial p}){\Delta F_L^2}
+m(K{G_R^1} -H{F_R^1})
+\Delta m(K{\Delta G_R^1}-H{\Delta F_R^1})\nonumber\\ &&
+2pK(m{\Delta F_R^2}
+\Delta m{F_R^2})+\frac{1-f_A}{r}({G_L^1}+p{\Delta F_L^2})+\frac{f_B}{r}(
{F_L^1}-p{\Delta G_L^2}) \ , \label{g9} \\
0&=&\omega {F_L^2}+{G_L^2}'
-\frac{1}{r}(1+2p\frac{\partial}{\partial p}){G_L^2}+
\frac{2}{r}\frac{\partial}{\partial p} {\Delta F_L^1}
+ m(K{G_R^2}-H{F_R^2})
+\Delta m(K{\Delta G_R^2}-H{\Delta F_R^2})\nonumber\\ &&
-2K(m{\Delta F_R^1}
+\Delta m{F_R^1})+\frac{1-f_A}{r}({G_L^2}-{\Delta F_L^1})+\frac{f_B}{r}(
{F_L^2}+{\Delta G_L^1}) \ , \label{g10} \\
0&=&\omega {\Delta F_L^1}+{\Delta G_L^1}'
-\frac{1}{r}(1+2p\frac{\partial}{\partial p})
{F_L^2}-\frac{2}{r}p\frac{\partial}{\partial p} {\Delta G_L^1}-
m(K{\Delta G_R^1}+H{\Delta F_R^1})\nonumber\\ &&
-\Delta m(K{G_R^1} +H{F_R^1}) \ , \label{g11} \\
0&=&\omega {\Delta F_L^2}+{\Delta G_L^2}'
-\frac{1}{r}(1+2p\frac{\partial}{\partial p})
{\Delta G_L^2}+\frac{2}{r}\frac{\partial}{\partial p} {F_L^1} \nonumber\\&&
-m(K{\Delta G_R^2} +H{\Delta F_R^2})-\Delta m(K{G_R^2}+H{F_R^2}) \ ,
\label{g12} \\
0&=&\omega {G_L^1}-{F_L^1}'-\frac{1}{r}(2+2p\frac{\partial}{\partial p}){F_L^1}
+\frac{1}{r}(1+2p\frac{\partial}{\partial p}){\Delta G_L^2}
-m(H{G_R^1} +K{F_R^1})
-\Delta m(H{\Delta G_R^1}\nonumber\\ &&
+K{\Delta F_R^1})
+2pK(m{\Delta G_R^2}
+\Delta m{G_R^2})+\frac{1-f_A}{r}({F_L^1}-p{\Delta G_L^2})
-\frac{f_B}{r}({G_L^1}+p{\Delta F_L^2}) \ , \label{g13} \\
0&=&\omega {G_L^2}-{F_L^2}'-\frac{1}{r}(3+2p\frac{\partial}{\partial p}){F_L^2}
-\frac{2}{r}\frac{\partial}{\partial p}{\Delta G_L^1}
-m(H{G_R^2}+K{F_R^2})
-\Delta m(H{\Delta G_R^2}+K{\Delta F_R^2})\nonumber\\ &&
-2K(m{\Delta G_R^1} +\Delta m{G_R^1} )
+\frac{1-f_A}{r}({F_L^2}+{\Delta G_L^1})
+\frac{f_B}{r}( -{G_L^2}+{\Delta F_L^1}) \ , \label{g14} \\
0&=&\omega {\Delta G_L^1}-{\Delta F_L^1}'
-\frac{1}{r}(2+2p\frac{\partial}{\partial p}){\Delta F_L^1}
+\frac{1}{r}(1+2p\frac{\partial}{\partial p}){G_L^2}
-\Delta m(H{G_R^1} -K{F_R^1})\nonumber\\ &&
-m(H{\Delta G_R^1}-K{\Delta F_R^1}) \ , \label{g15} \\
0&=&\omega {\Delta G_L^2}-{\Delta F_L^2}'
-\frac{1}{r}(3+2p\frac{\partial}{\partial p})
{\Delta F_L^2} -\frac{2}{r}\frac{\partial}{\partial p}{G_L^1}
-\Delta m(H{G_R^2}-K{F_R^2})\nonumber\\ &&
-m(H{\Delta G_R^2}-K{\Delta F_R^2}) \ . \label{g16}\end{aligned}$$
These equations are analogous in structure to the equations in the sphaleron background, with all relevant features ‘doubled’. Now six equations contain $p$-dependent terms (apart from the terms containing partial derivatives with respect to $p$). These are eqs. (\[g1\]), (\[g3\]), (\[g5\]), (\[g7\]), (\[g9\]) and (\[g13\]). And six functions occur with a prefactor $p$, these are ${F_R^2}, {\Delta F_R^2}$, ${\Delta F_L^2}$, ${G_R^2}, {\Delta G_R^2}$ and ${\Delta G_L^2}$, the six ‘bad’ functions, $b$. The other ten functions are the ‘good’ functions, $g$. Again, if the bad functions are small, all functions have little angular dependence, and an approximation with radial functions only will be good.
Let us therefore inspect the two source terms for the bad functions, $$s_1= H{G_L^2}-K{F_L^2}-2K{\Delta G_L^1} \ ,$$ $$s_2= K{G_L^2}+H{F_L^2}-2K{\Delta F_L^1} \ ,$$ occurring in eqs. (\[g6\]), (\[g8\]), and in (\[g2\]), (\[g4\]), respectively, and split these two source terms according to $s_1=a_1-a_2$, with $$\ \ \ a_1=H{G_L^2}-K{F_L^2} \ ,$$ $$a_2=2K{\Delta G_L^1} \ ,$$ and $s_2=b_1-b_2$, with $$\ \ b_1=K{G_L^2}+H{F_L^2} \ ,$$ $$b_2=2K{\Delta F_L^1} \ .$$ If both source terms are small, then the bad functions are small, and consequently the angular dependence of all 16 fermion functions is small.
Due to its great complexity, we have not yet attempted to solve the full set of 16 coupled partial differential equations numerically. Instead we have from the beginning resorted to the study of the approximate set of 16 ordinary differential equations, obtained by integrating out the angular dependence in the energy density. But even this approximate set of 16 ordinary differential equations has resisted a numerical solution along the full sphaleron barrier. Only by setting two of the 16 radial functions explicitly to zero, namely the supposedly small bad functions ${\Delta G_L^2}$ and ${\Delta G_R^2}$, we have succeeded in constructing the fermion solution along the sphaleron barrier. (Note, that $\Delta G_L^2$ has no source term.)
Without the solution of the partial diffential equations to compare with, the quality of the approximate solution is not known along the full barrier, away from the sphaleron. At the sphaleron the approximation is excellent, and it should remain good close to the sphaleron. Away from the sphaleron, however, we can at least make a consistency check for the radial approximation used, by inspecting the source terms $s_1$ and $s_2$ in this approximation. Numerical analysis shows, that the source terms are indeed small. In Fig. 7 we show as a typical example along the barrier the source terms $b_1$ and $b_2$ for the Chern-Simons number $N_{CS}=0.4$ and the mass parameters $m=0.5$ and $\Delta m=0.25$. While the cancellation of the source terms $a_1$ and $a_2$ remains as good along the barrier as it is at the sphaleron (shown in Fig. 6), the cancellation of the additional source terms $b_1$ and $b_2$ is even much better. This indicates, that the bad functions are indeed small compared to the good functions. The radial approximation therefore should be good along the full sphaleron barrier.
Let us then discuss the level crossing along the sphaleron barrier, as obtained with the approximate radial set of equations. In Fig. 8 we present the fermion eigenvalue along the barrier for an average mass of $m=2$ and for several values of the mass difference, $\Delta m =0.5$, 1.0 and 1.5. The eigenvalue starts from the positive continuum at the lower mass (1.5, 1.0 and 0.5, respectively), and reaches the negative continuum at the corresponding negative value. The bigger the mass splitting, i.e. the smaller the lower mass, the later the fermion level leaves the continuum to become bound, analogous to the case of degenerate fermion masses \[14,15,20\].
For degenerate fermion masses the fermion wavefunction is determined by the hedgehog spinor $\chi_{\rm h}$, giving both isospin components of the fermion doublet an equal amplitude along the sphaleron barrier. For nondegenerate fermion masses this is no longer the case. Let us define the up-part of the fermion wavefunction along the barrier as $$\frac{<P\Psi,P\Psi>}{<\Psi,\Psi>} \ ,$$ where $P$ projects out the upper isospin component. (Note, that this definition of the up-part is not gauge invariant.) For degenerate fermions the up-part is everywhere one half. For nondegenerate fermions the up-part along the barrier depends on the size of the mass splitting, as shown in Fig. 9 (for the mass parameters employed also in Fig. 8). The up-part dominates slightly in the vicinity of the sphaleron and clearly disappears when the vacua are reached. Remarkably, the point where the down-part equals the up-part only depends on $m$ and not on $\Delta m$.
We finally address the question, how to best approximate a fermion solution in the physical situation of nondegenerate fermion masses by a far simpler solution, obtained in the approximation of degenerate fermion masses, in the vicinity of the sphaleron. In the physical case the mass parameters $m$ and $\Delta m$ determine the nondegenerate masses, $m+\Delta m$ and $m-\Delta m$. Close to the continuum clearly the lower mass, $m-\Delta m$, is the relevant fermion mass. In the vicinity of the sphaleron, however, it is the average mass $m$, which matters. In fact, the average mass $m$ of the nondegenerate case mostly leads to an excellent approximation for the fermion eigenvalue in the vicinity of the sphaleron, when employed in the far simpler calculations with degenerate fermion mass. This is demonstrated in Fig. 10, where we compare the nondegenerate case $m=2$, $\Delta m=1$ with the degenerate cases $m=1$, $\Delta m=0$ and $m=2$, $\Delta m=0$. Having the same average mass, the fermion eigenvalues in the nondegenerate case, and in the second degenerate case, agree very well in the vicinity of the sphaleron. In Fig. 11 we present the slope of the fermion eigenvalue at the sphaleron as a function of the mass difference, for three values of the average mass, $m=0.5$, 1 and 2. We observe, that the slope is fairly independent of the mass difference $\Delta m$ for not too large values of the average mass $m$.
Conclusion
==========
We have considered level crossing in the background field of the sphaleron barrier for fermion doublets with nondegenerate masses. The mass splitting necessitates a generalized ansatz for the fermions, possessing only axial symmetry. We have proposed a particular parametrization of the axially symmetric ansatz, containing 16 real functions of the two variables $r$ and $p=\cos^2\theta$. The structure of the ansatz chosen is based on the structure of the spherically symmetric ansatz, which represents its simple limit for vanishing mass splitting. This particular parametrization has the great advantage, that it leads to fermion functions with little angular dependence in the background of the sphaleron, and (supposedly) also along the full sphaleron barrier.
In the background field of the sphaleron the proposed ansatz simplifies considerably. It leads to a set of eight partial differential equations. We have solved these equations numerically, finding that the resulting fermion functions have very little angular dependence. The reason lies in the structure of the equations for this particular choice of ansatz. Only three functions occur with an angular dependent prefactor $p$ (apart from partial derivative terms), and there is a single source term for these three functions. Since this source term is small, these three functions, which introduce explicit angular dependence into the equations, are small, and consequently all eight functions have only little angular dependence.
We have then proposed an approximate ansatz with radial functions only. Integrating out the angular dependence in the energy density, leads to a new approximate set of ordinary differential equations. Solving these numerically, we find that the solutions are in excellent agreement with those of the full calculation. Thus we have an excellent radial approximation for nondegenerate fermion masses at the sphaleron.
In the general case of fermions in the background of the sphaleron barrier, we have found the same structure of the equations as in the sphaleron case, but with all relevant features ‘doubled’, since the ansatz no longer simplifies. As yet we have only solved the approximate set of ordinary differential equations, obtained by integrating out the angular dependence in the energy density (and then setting two of the small functions explicitly to zero). Without the solution of the set of partial differential equations to compare with, we do not know the quality of the approximation away from the sphaleron. However, we have made a consistency check by evaluating the two source terms for the six functions, which introduce explicit angular dependence into the equations. Since the source terms are small, these functions are small, and consequently all functions have only little angular dependence. We therefore argue, that the radial approximation employed should be good along the full barrier.
Considering level crossing along the barrier, we have observed that the fermion mode which crosses zero energy at the sphaleron reaches the continua at the lower fermion mass, as expected. Finally we have shown, that in the vicinity of the sphaleron the eigenvalue for nondegenerate fermions with average mass $m$ and mass difference $\Delta m$, may be well approximated by the eigenvalue obtained with the far simpler calculation, involving only degenerate fermions with the average mass $m$. With respect to the large splitting of the top and bottom quark masses, this suggests to rather use half the top quark mass in approximate calculations with degenerate fermions.
[000]{}
A. D. Sakharow, JETP Letters 5 (1967) 5
G. ’t Hooft, Symmetry breaking through Bell-Jackiw Anomalies, Phys. Rev. Lett. 37 (1976) 8.
N. S. Manton, Topology in the Weinberg-Salam theory, Phys. Rev. D28 (1983) 2019.
F. R. Klinkhamer, and N. S. Manton, A saddle-point solution in the Weinberg-Salam theory, Phys. Rev. D30 (1984) 2212.
A. Ringwald, Rate of anomalous electroweak baryon and lepton number violation at finite temperature, Phys. Lett. B201 (1988) 510.
M. Mattis, and E. Mottola, eds., “Baryon Number Violation at the SSC?”, World Scientific, Singapore (1990).
P. Arnold, and L. McLerran, Sphalerons, small fluctuations, and baryon-number violation in electroweak theory, Phys. Rev. D36 (1987) 581.
P. Arnold, and L. McLerran, The sphaleron strikes back: A response to objections to the sphaleron approximation, Phys. Rev. D37 (1988) 1020.
L. Carson, X. Li, L. McLerran, and R.-T. Wang, Exact computation of the small-fluctuation determinant around a sphaleron, Phys. Rev. D42 (1990) 2127.
E. W. Kolb, and M. S. Turner, “The Early Universe”, Addison-Wesley Publishing Company, Redwood City (1990).
C. R. Nohl, Bound-state solutions of the Dirac equation in extended hadron models, Phys. Rev. D12 (1975) 1840.
J. Boguta, and J. Kunz, Hadroids and sphalerons, Phys. Lett. B154 (1985) 407.
A. Ringwald, Sphaleron and level crossing, Phys. Lett. B213 (1988) 61.
J. Kunz, and Y. Brihaye, Fermions in the background of the sphaleron barrier, Phys. Lett. B304 (1993) 141.
J. Kunz, and Y. Brihaye, Level crossing along sphaleron barriers, Phys. Rev. D50 (1994) 1051.
D. Diakonov, M. Polyakov, P. Sieber, J. Schaldach, and K. Goeke, Fermion sea along the sphaleron barrier, Phys. Rev. D49 (1994) 6864.
B. Kleihaus, J. Kunz, and Y. Brihaye, The electroweak sphaleron at physical mixing angle, Phys. Lett. B273 (1991) 100.
J. Kunz, B. Kleihaus, and Y. Brihaye, Sphalerons at finite mixing angle, Phys. Rev. D46 (1992) 3587.
B. Kastening, Fermionic instanton zero modes in models with spontaneous symmetry breaking and broken custodial SU(2) symmetry, Phys. Lett. B291 (1992) 92.
G. Nolte, and J. Kunz, Gradient approach to the sphaleron barrier, Phys. Rev. D51 (1995) 3061.
|
---
abstract: 'We present an adaptation of the rotation-corrected, $m$-averaged spectrum technique designed to observe low signal-to-noise-ratio, low-frequency solar p modes. The frequency shift of each of the $2l+1$ $m$ spectra of a given ($n,l$) multiplet is chosen that [maximizes the likelihood]{} of the $m$-averaged spectrum. A high signal-to-noise ratio can result from combining individual low signal-to-noise-ratio, individual-$m$ spectra, none of which would yield a strong enough peak to measure. We apply the technique to GONG and MDI data and show that it allows us to measure modes with lower frequencies than those obtained with classic peak-fitting analysis of the individual-$m$ spectra. We measure their central frequencies, splittings, asymmetries, lifetimes, and amplitudes. The low-frequency, low- and intermediate-angular degrees rendered accessible by this new method correspond to modes that are sensitive to the deep solar interior down to the core ($l \leq 3$) and to the radiative interior ($4 \leq l \leq 35$). Moreover, the low-frequency modes have deeper upper turning points, and are thus less sensitive to the turbulence and magnetic fields of the outer layers, as well as uncertainties in the nature of the external boundary condition. As a result of their longer lifetimes (narrower linewidths) at the same signal-to-noise ratio the determination of the frequencies of lower-frequency modes is more accurate, and the resulting inversions should be more precise.'
author:
- 'D. Salabert, J. Leibacher, T. Appourchaux, and F. Hill'
title: |
Measurement of low signal-to-noise-ratio solar p modes\
in spatially-resolved helioseismic data
---
Introduction
============
Our knowledge of the structure and dynamics of the solar interior has been considerably improved by the use of measurements of the properties of the normal modes of oscillation of the Sun. However, the Sun’s interior is far from being fully understood, and better measurements of the mode parameters will also help to better understand the mode excitation and damping mechanisms as well as the physical properties of the outer layers by better constraining the turbulence models. A large number of predicted acoustic oscillation modes, defined by their radial orders ($n$) and their angular degrees ($l$), are not yet observed in the low-frequency range (i.e., approximately below 1800 $\mu$Hz), because the amplitude of the acoustic modes decreases as the mode inertia increases as the frequency decreases, while the solar noise from incoherent, convective motions increases: thus the signal-to-noise ratio (SNR) of those modes is progressively reduced. Moreover, these low-frequency p modes have very long lifetimes, as much as several years, which results in very narrow linewidths, hence precise frequency measurements. Thanks to the long-duration helioseismic observations collected by the spaced-based instruments Michelson Doppler Imager (MDI) [@scherrer95] and Global Oscillations at Low Frequencies (GOLF) [@gabriel95] onboard the Solar and Heliospheric Observatory (SOHO) spacecraft, and by the ground-based, multi-site Global Oscillation Network Group (GONG) [@harvey96] and Birmingham Solar Oscillations Network (BiSON) [@chaplin96], the frequency resolution is continuously improving and the observation of lower radial-order solar p modes is becoming possible. Their precise mode parameter determination is of great interest for improving our resolution throughout the solar interior because they cover a broad range of horizontal phase velocity, and thus a broad range of depths of penetration. Moreover, these low-frequency modes have lower reflection points in the outer part of the Sun, which make them less sensitive to the turbulence and the magnetic fields in the outer layers, where the physics is poorly understood.
The usual mode-fitting analysis consists of fitting the $2l+1$ individual-$m$ spectra of a given multiplet ($n,l$), either individually or simultaneously. Such fitting methods fail to obtain reliable estimates of the mode parameters when the SNR [of the individual-$m$ spectra]{} is low. Instead, various pattern-recognition techniques have been developed in an effort to reveal the presence of modes in the low-frequency range [see e.g., @schou98; @app00; @chaplin02; @broomhall07 and references therein]. In the case of spatially-resolved helioseismic data (such as GONG and MDI observations), $m$-averaged spectra appeared to be a powerful tool, since for a given multiplet ($n,l$), there exist $2l+1$ individual-$m$ spectra, which can result in an average spectrum with a SNR $\gg 1$ once the individual-$m$ spectra are corrected for the rotation- and structure-induced [frequency]{} shifts. The $m$-averaged spectra were employed early in the development of helioseismology by @brown85, but were replaced by fitting the $m$ spectra individually as the quality and the SNR of the data improved. However, years later, in order to fully take advantage of the long-duration helioseismic GONG and MDI instruments and reach lower frequencies in the solar oscillation spectrum, @schou98 [@schou02; @schou04] and @app00 used the $m$-averaged spectra corrected by the modeled solar rotation to detect new low radial-order p modes and to set upper limits on the detectability of the g modes. These authors demonstrated the potential advantage of such rotation-corrected, $m$-averaged spectra.
We present here an adaptation of the $m$-averaged spectrum technique in which the $m$-dependent shift parameters are determined by maximizing the quality of the resulting average spectrum. The analysis is performed on long-duration time series of the spatially-resolved helioseismic GONG and MDI observations of the low- and medium-angular degrees ($1 \leq l \leq 35$). This range of oscillation multiplets samples the radiative interior down to the solar core. In Sec. \[sec:obs\], we introduce the different datasets used in this analysis. In Sec. \[sec:method\], we describe this new technique in order to observe low signal-to-noise-ratio, low-frequency p modes, explaining the different steps of the analysis from the mode detection to peak-fitting. In Sec. \[sec:compa\], we demonstrate that this method allows us to successfully measure lower-frequency modes than those obtained from classic peak-fitting analysis of the individual-$m$ spectra by comparing with other measurements obtained from coeval datasets. In Sec. \[sec:data\_results\], we present the mode parameters of these long-lived, low-frequency acoustic modes down to $\approx$ 850 $\mu$Hz extracted from the analysis of 3960 days of GONG observations using the $m$-averaged spectrum technique. Finally, Sec. \[sec:conc\] summarizes our conclusions.
Observations {#sec:obs}
============
Details of the spatially-resolved helioseismic observations collected by both GONG and MDI used for this work (the starting and ending dates, and their corresponding duty cycles) are given in Table \[table:series\]. Coeval 2088-day observations of GONG and MDI were analyzed for oscillation multiplets with angular degrees from $l=1$ to $l=35$, and are then directly compared to those of @korz05 for $l \leq 25$ measurements of the same datasets. We also applied the analysis to 3960 days of GONG data ($1 \leq l \leq 35$), which constitutes so far the longest time series ($\approx$ 11 years, spanning most of solar cycle 23) of spatially-resolved observations analyzed.
--------------- ------------ ------------- ------
3960-day GONG 1995 May 7 2006 Mar 9 84.6
2088-day GONG 1996 May 1 2002 Jan 17 83.7
2088-day MDI 1996 May 1 2002 Jan 17 88.9
--------------- ------------ ------------- ------
: Details of the long GONG and MDI analyzed time series
\[table:series\]
Method {#sec:method}
======
An $m$-averaged spectrum corresponds to the average of the $2l+1$ individual-$m$ components of an oscillation multiplet ($n,l$), thus reducing the non-coherent noise. Before averaging, each $m$ spectrum of a given mode ($n, l$) is shifted by a frequency that compensates for the effect of differential rotation and structural effects on the frequencies. The $m$-averaged spectrum concentrates, for a given multiplet ($n, l$), all of the $2l+1$ $m$ components, as it would be if the Sun were a purely-spherical, non-rotating object. Thus, the average of the $2l+1$ individual-$m$ spectra considerably improves the SNR of the resulting $m$-averaged spectrum.
Determination of the shifts {#ssec:det_acoef}
---------------------------
The $m$-averaged spectrum is obtained by finding the estimates of the splitting coefficients, commonly called $a$-coefficients, which [maximizes the likelihood]{} of the $m$-averaged spectrum. The $a$-coefficients are individually estimated through an iterative process, with the initial values taken from a model. Thus, for a given mode ($n,l$), the frequency shift $\delta\nu_{n l m}$ is parameterized by a set of coefficients, as:
$$\delta\nu_{n l m} = \sum_{i=1}^{i_{max}}{a_i(n,l) P^{i}_{l,m}},
\label{eq:poly}$$
where $a_i(n,l)$ are the splitting coefficients, and $P^i_{l,m}$ corresponds to the Clebsch-Gordan polynomial expansion as defined by @ritz91. In this definition, the odd orders of the $a$-coefficients describe the effects of solar rotation, while the even orders correspond to departures from spherical symmetry in the solar structure as well as to quadratic effects of rotation. Each $a_i$ is chosen to [maximize the likelihood]{} of the $m$-averaged spectrum. This is performed through an iterative procedure. For a particular order $i$ of the coefficients $a_i$, a range of values is scanned around its initial value, while the other $a_{i'\neq i}$s are kept fixed to their previously estimated values.
For each scanned value of $a_i$, the individual-$m$ spectra are shifted by the corresponding Clebsch-Gordan polynomials, and the mean of these $2l+1$ shifted spectra is taken. The mean power spectrum is then fitted using a Maximum-Likelihood Estimator (MLE) minimization as described in Sec \[ssec:extraction\] and its likelihood determined.
For a Monte-Carlo simulation, the left panel of Fig. \[fig:fom\] shows the variation of the likelihood from the MLE minimization as a function of the first splitting coefficient $a_1$, showing a well defined minimum which represents the best value of $a_1$. The artificial power spectra were simulated following the methodology described in @fierry98. We also examined the sensitivity of the mode linewidth and the entropy as criteria for determining the best shifts. [In our case, the entropy [@shannon48] can be seen as a measure of randomness in the $m$-averaged spectrum, $\mathcal{S}$, and is defined as $-\sum [\mathcal{S} \times \ln \mathcal{S}]$.]{}
Both linewidth and entropy show well defined minima around the input value of $a_1$ (middle and right panels of Fig. \[fig:fom\] respectively). Indeed, the $m$-averaged spectrum gets narrower as $a_1$ converges to its input values of 0.4 $\mu$Hz and $a_1 = 400$ nHz. Similar variations are obtained for all the $a_i$s. As detailed in Appendix \[sec:foms\], the use of these different criteria to determine the best estimates of the $a$-coeficients returns consistent results.
The iteration is performed until the difference between two iterations in each of the computed $a_i$ coefficients falls below a given threshold (such as 0.25$\sigma$ in the case of $a_1$). Also, [in order to remove any outliers]{}, some quality checks are performed after each measure of an $a_i$ which needs to fall within a constrained range of values. [For example, a $\pm$15% window around its theoretical expectation is used for $a_1$.]{} Here, we fitted only the six first $a_i$ in the Clebsch-Gordan expansion, even though the quality of the data supports the determination of higher-order coefficients.
[Finally, low SNR peaks in the $m$-averaged spectrum (after adjustment) are tested against the H0 hypothesis. In the framework of that hypothesis, the resulting spectra are tested against a statistics pertaining to pure noise ($\chi^2$ with 2(2$l$+1) d.o.f). This test has been widely applied to helioseismic observations in the search for long-lived, low radial-order p modes and g modes [see e.g., @app00].]{} In the present analysis, we rejected peaks that have a greater than 10% chance of being [due to noise]{} in the [238]{} analyzed windows, [each containing 288 frequency bins. Here the fixed number of bins was chosen because we know that the range of theoretical frequency lie within 1.5 $\mu$Hz or so]{}. Figures \[fig:spec\_l3\] and \[fig:spec\_l16\] illustrate the advantage of using the $m$-averaged spectrum technique in the case of two oscillation multiplets for 2088 days of GONG data, where the $m$-averaged spectra before and after the correction for the splitting coefficients are shown. These examples show the $m$-averaged spectra of the modes $l=3$, $n=5$ at $\approx$ 1015.0 $\mu$Hz (Fig. \[fig:spec\_l3\]), and $l=16$, $n=4$ at $\approx$ 1293.8 $\mu$Hz (Fig. \[fig:spec\_l16\]), as well as the corresponding $m-\nu$ diagrams. These two examples were chosen to demonstrate the performance at different SNR levels. The corresponding 10% probability levels are given. The $m-\nu$ diagrams in the case of the mode $l=3$, $n=5$ (right panels on Fig. \[fig:spec\_l3\]) do not show any high SNR structure before or after correction. However, the $m$-averaged spectrum after correction clearly shows the target mode (lower left-panel on Fig. \[fig:spec\_l3\]), with an unambiguous detection level. The mode $l=16$, $n=4$ presents a higher SNR (Fig. \[fig:spec\_l16\]) and its $m-\nu$ diagram shows that the individual-$m$ spectra line up after correction (lower-right panel on Fig. \[fig:spec\_l16\]). The estimated splitting coefficients of the low-frequency modes with $1\leq l \leq35$ measured in the 3960-day GONG dataset are shown in Fig. \[fig:acoefs3960d\] as a function of frequency and $\nu/L$ (with $L=\sqrt{l(l+1)}$), which is approximately proportional to the sound speed at the mode’s inner turning point. Modes with selected ranges of radial orders are represented with different colors and symbols.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
Extraction of the mode parameters {#ssec:extraction}
---------------------------------
For a given mode $(n,l)$, the best estimates of the splitting coefficients determined as discussed in Sec \[ssec:det\_acoef\] are used to calculate its $m$-averaged spectrum. When $N$ independent power spectra are averaged together, the statistics of the mean power spectrum corresponds to a $\chi^2$ with $2\times N$ degree-of-freedom (d.o.f.) statistics. @app03 demonstrated that the mean of $2l+1$ independent power densities, which has a $\chi^2$ with more than 2 d.o.f. statistics, can be correctly fitted with a Maximum-Likelihood Estimator (MLE) minimization code developed for spectra following a $\chi^2$ with 2 d.o.f. statistics. The asymmetric Lorentzian model of @nigam98 was used to describe the $m$-averaged spectrum, as:
$$P_{n,l}(\nu) = H_{n,l} \frac{(1+\alpha_{n,l} x_{n,l})^2+\alpha_{n,l}^2}{1+x_{n,l}^2} + B_{n,l},
\label{eq:mlemodel}$$
where
$$x_{n,l}=\frac{2(\nu-\nu_{n,l})}{\Gamma_{n,l}}.$$
Then, for a given mode ($n,l$), the central frequency, the Full-Width-at-Half-Maximum (<span style="font-variant:small-caps;">fwhm</span>), and the power height of the spectral density are respectively $\nu_{n,l}$, $\Gamma_{n,l}$, and $H_{n,l}$. The peak asymmetry is described by the parameter $\alpha_{n,l}$, while $B_{n,l}$ represents an additive, constant background level in the fitted window. The first spatial leaks ($\delta l=0$, $\delta m=\pm2$), commonly called $m$-leaks, are also included in the fitting model and added to Eq. \[eq:mlemodel\]. The frequencies of the $m$-leaks are set from the central frequency of the target mode using the previously measured splitting coefficients (Sec. \[ssec:det\_acoef\]). Their peak asymmetries are assumed to be the same as that of the target mode, while their <span style="font-variant:small-caps;">fwhm</span>s are a free parameter of the fit and different from the target mode. The amplitude of the $m$-leaks is specified to be a fixed fraction of the central peak, which was estimated from the leakage matrix developed especially for the GONG [@hill98] and MDI (J. Schou, private communication) data. The first spatial leaks in the $m$-averaged spectrum were determined by averaging for a given multiplet ($n,l$) the $\delta m \pm 2$ leaks over the entire $2l+1$ spectra.
The size of the fitting window, [$\Omega_\nu$]{}, is proportional to the first estimates of the mode [width, $\Gamma_{n,l}$,]{} and centered around the frequency of the target mode. [It is defined as: $$\Omega_\nu = 20\sqrt{\Gamma_{n,l}^2 + \Delta\nu_r^2} + \Delta_{\delta m},$$ where $\Delta\nu_r$ is the frequency resolution of the power spectrum. The first spatial leaks are always included in the fitting range by adding the offset $\Delta_{\delta m} = 800$ nHz. The multiplicative factor $20$ ensures a good sampling of the mode profile in the low-frequency range. A comparable definition of the fitting window was adopted by @korz05.]{} Bad fits were removed based on a set of quality criteria based on the fitted mode parameters and associated uncertainties, [such as, (1) the error of the mode frequency must be less than its mode width; (2) the SNR must be larger than 1; and (3) the mode width must be larger than the frequency resolution.]{} A discussion on the impact of the fitting model (asymmetry, spatial leaks) on the extracted mode parameters used to describe the $m$-averaged spectrum can also be found in the Appendix \[sec:impmodel\].
Figure \[fig:mspec\] shows examples of the $m$-averaged power spectra for four different radial orders $n$ of the multiplet $l=17$, and the corresponding best MLE fits, which included the mode asymmetry and the $\delta m \pm 2$ spatial leaks. The blending of the first $m$ leaks is particularly clear as the linewidths increase with increasing frequency.
Mode parameter and $a$-coefficient uncertainties {#ssec:error}
------------------------------------------------
The mode parameter uncertainties are established in the usual manner by the inverse of the covariance matrix. However, because the $m$-averaged spectrum is fitted using a MLE minimization and, as explained in @app03, the formal uncertainties must be normalized by the square root of the number of averaged spectra, i.e., in our case, by $\sqrt{2l+1}$. But this a-posteriori error normalization is correct only if the $2l+1$ spectra of a given ($n,l$) mode have the same variance (or SNR). Since the condition of equal SNR among the $m$ spectra within a multiplet is not satisfied in our case, the uncertainties of the mode parameters have to be taken as a first approximation only. However, Monte-Carlo simulations show that this error normalization holds even in the case of $m$-dependent SNR (see Sec. \[ssec:montecarlo\]).
-- -- -- --
-- -- -- --
It can also be derived that the errors on the $a$-coefficients can be estimated as follows: $$\sigma^{-2}_{a_i}=\frac{l^2}{2l+1}\Big(\sum_m[P_{l,m}^i(m/l)]^2\Big)\sigma^{-2}_{\nu_0},
\label{eq:a_error}$$
where $i$ is the $a$-coefficient order and $P_{l,m}^i$ the associated Clebsch-Gordan polynomials. The derivation of Eq. \[eq:a\_error\] is detailed in Appendix \[sec:a\_errors\].
### $m/l$ dependence of the signal-to-noise ratio
Figure \[fig:mdep\] shows the dependence in $m/l$ of the SNR in the GONG data. This was obtained with modes observed in the 3960-day GONG dataset below 2000 $\mu$Hz and of angular degree up to $l=35$. Both mode amplitude and background noise depends on the azimuthal order $m$ and can be described with polynomials with only even terms, the polynomials being different for both parameters. Note that any frequency dependence of the $m/l$ dependence is averaged out in Fig. \[fig:mdep\].
The $m/l$ dependence of the SNR implies that the $a$-coefficients are not exactly orthogonal and that their errors are correlated (see Appendix \[sec:a\_errors\]). However, as a first order approximation, the errors on the $a$-coefficients can be estimated by using Eq. \[eq:a\_error\] (see Sec. \[ssec:montecarlo\]).
![Signal-to-noise ratio for modes up to $l=35$ as a function of $m/l$, obtained with the 3960-day GONG dataset.[]{data-label="fig:mdep"}](fig6.eps)
### Validation of the error estimates: Monte-Carlo simulations {#ssec:montecarlo}
The formal uncertainties of the mode parameters and of the $a$-coefficients were verified through Monte-Carlo simulations. The artificial power spectra were simulated following the methodology described in @fierry98. In a first series of simulated spectra, the $m$ dependence in amplitude within a given multiplet ($n,l$) was introduced, the SNR being symmetric in $|m|$ around the $m=0$ spectrum. In a second series, no $m$ dependence was introduced, i.e., a constant SNR over $m$. The mean values of the formal errors returned by the MLE minimization were compared to the RMS value of the corresponding fitted parameter. The Monte-Carlo simulations showed that in both cases the formal uncertainties of the $m$-averaged spectra determined as in Sec \[ssec:error\] using a MLE minimization are a very good approximation of the errors.
Comparison with other measurements {#sec:compa}
==================================
Comparison with spatially-resolved observations ($l\leq25$)
-----------------------------------------------------------
GONG and MDI use two independent peak-finding approaches to extract the mode parameters. Developed in the early 1990s, and mostly unchanged since, they provide mode parameters on a routine basis. Time series of 108 days are used by the GONG project [@anderson90], while the MDI project uses 72-day time series [@schou99]. Recently, @korz05 developed a new and independent peak-finding method of the individual-$m$ spectra, optimized to take advantage of the long, spatially-resolved, helioseismic time series available today from both projects.
{width="\textwidth"}
### Mode detection: $l-\nu$ diagram
@korz05 applied his peak-fitting to extract the low- and medium-degree ($l \leq 25$) mode parameters from both GONG and MDI observations using one 2088-day long time series, as well as using five overlapping segments of 728 days. In order to compare our results obtained with the $m$-averaged spectrum technique, we applied the procedure described in Sec. \[sec:method\] to the same 2088 days of GONG and MDI observations (Table \[table:series\]). Figure \[fig:lnu2088d\] shows the $l-\nu$ diagrams of the low-frequency modes measured with the two different analyses in the case of the 2088-day GONG ([*left panel*]{}) and MDI ([*right panel*]{}) datasets. The modes measured by @korz05 with a classic peak-fitting method of the individual-$m$ spectra are represented by the open circles. We considered that a given mode ($n,l$) from @korz05 was detected when at least two of the $2l+1$ $m$ spectra were successfully fitted, which is enough to obtain estimates of the corresponding central frequency and first splitting coefficient $a_1$. The red dots represent modes measured with the $m$-averaged spectrum technique which were not observed by @korz05. A significantly larger number of low-frequency modes in the 2088-day GONG and MDI datasets (respectively 45 and 14 new modes) down to $\approx$ 900 $\mu$Hz can be measured using the $m$-averaged spectrum technique.
### Mode parameter and uncertainty comparisons
In order to check the accuracy of the technique and to identify any potential bias in our analysis, we compare the central frequencies and splitting coefficients obtained by the two methods. The individual-$m$ frequencies of @korz05 were fitted using a Clebsch-Gordan polynomial expansion [@ritz91] in order to estimate the corresponding central frequencies and $a$-coefficients of each $(n,l$) multiplet. The formal uncertainties of the individual-$m$ frequencies were used as fitting weights. The left panel on Fig. \[fig:hist2088d\] shows the distribution of the differences in central frequencies below $\approx$ 1800 $\mu$Hz of the common modes between the 2088-day GONG estimates measured using the $m$-averaged spectrum technique and from @korz05 (as represented on Fig. \[fig:lnu2088d\]), demonstrating that there is no frequency dependence over the analyzed low-frequency range. The distribution was fitted by a Gaussian function, and its associated parameters (mean, standard deviation) are indicated on Fig. \[fig:hist2088d\]. While, on average, the GONG central frequencies obtained using the $m$-averaged spectrum technique are less than 1 nHz smaller than @korz05’s estimates, this offset is not significant — the corresponding standard deviation being about 5 times larger. The MDI frequencies estimated with the $m$-averaged spectrum technique give comparable, insignificant mean differences with @korz05. Similar results are obtained with the splitting coefficients.
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We also compared the low frequencies ($\nu \lesssim 1800\mu$Hz, see Fig. \[fig:lnu2088d\]) estimated in both the 2088-day GONG and MDI datasets using the $m$-averaged spectrum technique. The right panel of Fig. \[fig:hist2088d\] represents the distribution of the frequency differences of the common modes, in the sense GONG minus MDI. The mean difference value is of -0.17 $\pm$ 1.99 nHz, i.e. the GONG and MDI low-frequency modes are essentially the same. The mean difference in @korz05’s central frequencies between the 2088-day GONG and MDI datasets for modes below 1800 $\mu$Hz is of 0.35 $\pm$ 5.40 nHz. The splitting coefficient estimates are also consistent between the two datasets with in the case of the $a_1$ coefficient a mean difference of -0.04 $\pm$ 0.31 nHz.
{width="50.00000%"} {width="50.00000%"}
The left panel of Fig. \[fig:nu\_2088d\] shows the formal $1\sigma$ uncertainties of the central frequencies of the measured common modes between the 2088-day GONG dataset analyzed in the present analysis and the coeval 2088-day GONG dataset from @korz05 up to $\approx$ 1800 $\mu$Hz. Our estimates of the frequency uncertainties are much smaller than those quoted by @korz05. However, Fig. 10 in @korz05 suggests that the errors are overestimated, and that his results “might be too conservative". @korz08 reported that in the case of a 2088-day long time series, as a first estimate, a multiplicative factor of 0.75 needs to be applied to the frequency uncertainties reported in @korz05. However, despite these uncertainty scaling issues, while the @korz05’s uncertainties show an increase with decreasing frequency from $\approx$ 1500 $\mu$Hz, the uncertainties returned from the $m$-averaged spectrum technique do not show this increase, thanks to the higher SNR of the $m$-averaged spectrum than for the individual-$m$ spectra.
The uncertainties on the $a$-coefficients returned by the $m$-averaged spectrum technique are also smaller than the ones obtained by fitting the individual-$m$ spectra, as shown on the right panel of Fig. \[fig:nu\_2088d\] in the case of the $a_1$ coefficients. As for the frequencies, the $a$-coefficients of the modes below $\approx$ 1500 $\mu$Hz are better constrained using the $m$-averaged spectrum technique.
Comparison with Sun-as-a-star observations ($l\leq3$)
-----------------------------------------------------
The spatially-resolved GONG and MDI instruments are not optimized to observe low-degree solar p modes below $l\leq3$, unlike the Sun-as-a-star, integrated-light instruments such as the space-based instrument GOLF onboard SOHO and the ground-based, multi-site BiSON network. The low-degree modes are of particular interest as they reach the very deep interior of the Sun. However, the spatially-resolved observations are still able to observe such low-degree oscillations.
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![$l-\nu$ diagram of the low-frequency, low-degree ($1\leq l \leq 3$) modes measured with the integrated-light instruments GOLF and BiSON (open circles) by R. A García, P. Boumier, and W. J. Chaplin (private communications) and with the spatially-resolved instruments GONG and MDI using the $m$-averaged spectrum technique (red dots), over comparable periods of time.[]{data-label="fig:lnu_lowdeg"}](fig11.eps){width="50.00000%"}
Low-degree ($l \leq 3$) modes down to $\approx$ 1000 $\mu$Hz are observed in both GONG and MDI data with the $m$-averaged spectrum technique as illustrated in Figs. \[fig:lnu2088d\] and \[fig:odect\]. In order to test the capability and the precision of the $m$-averaged spectrum technique to observe low-degree, low-frequency modes in spatially-resolved data, measurements obtained for $\approx$ 11 years of the Sun-as-a-star GOLF and BiSON instruments were compared with the 3960-day GONG dataset and the 2088-day GONG and MDI datasets. The GOLF data were independently analyzed by two mode-fitting algorithms (R. A. García, private communication[^1], and P. Boumier, private communication[^2]). The BiSON observations come from a combination of frequencies obtained with different long time series analyzed in order to measure low-frequency modes [W. J. Chaplin, private communication. See also @chaplin02; @broomhall07]. However, while imaging instruments give us access to all of the $2l+1$ individual-$m$ components, only $l+1$ components can be clearly observed with integrated-sunlight technique. These various components have different spatial structure over the solar surface, which can lead to differences in the extracted central frequencies of the multiplet. @chaplin04 and @app07 derived expressions to predict the differences between the low-degree frequencies extracted from spatially-resolved (as MDI and GONG) and Sun-as-a-star (as GOLF and BiSON) observations. However, in the following, as a first approximation, we compared directly the extracted mode parameters.
The comparisons of the estimated mode frequencies and $a_1$ rotational splittings between the common low-degree ($1\leq l \leq3$), low-frequency modes in the two types of observation are shown on the left and right panels respectively of Fig. \[fig:diff\_fulldisk\]. The three different datasets and analysis methods give consistent results, for both the frequency and the splitting coefficient $a_1$. Of course, this is only assuming that the different subsets of observed multiplets from both types of observational technique “see" the same central frequencies.
Thanks to decade-long available datasets, the low-degree, low-frequency modes are today measured lower than 1200 $\mu$Hz with high precision, demonstrated by the consistency in the extracted parameters from different instruments using distinct and independent analysis. Figure \[fig:diff\_fulldisk\] also demonstrates that spatially-resolved observations can provide as accurate measurements of the low-degree modes as the Sun-as-a-star instruments do. Moreover, the $m$-averaged spectrum technique allows the observation of lower radial-order $l=3$ modes than the integrated-light GOLF and BiSON observations, for commensurate observation lengths, thanks to the observations of the $2l+1$ components (Fig. \[fig:lnu\_lowdeg\]).
{width="\textwidth"}
Mode parameters of the low-frequency oscillations {#sec:data_results}
=================================================
The $m$-averaged spectrum technique has been applied to 3960 days of GONG observations (see Sec. \[sec:obs\]), spanning most of the 11 years of solar cycle 23. The analysis covered low-frequency modes with angular degrees from $l=1$ to $l=35$. Oscillation multiplets well below 1000 $\mu$Hz were detected with good precision, such as the modes $l=4$, $n=4$ at $\approx$ 913.5 $\mu$Hz; $l=9$, $n=3$ at $\approx$ 930.5 $\mu$Hz; $l=16$, $n=2$ at $\approx$ 912.1 $\mu$Hz; or $l=31$, $n=1$ at $\approx$ 907.5 $\mu$Hz. Some examples are illustrated in Figs. \[fig:spec\_l3\] and \[fig:spec\_l16\]. These low horizontal-phase-velocity modes do not penetrate deeply into the Sun, but their very high inertias afford higher precision frequencies for the inversions. It is clear from Sec. \[sec:compa\] that this method allows us to observe modes that are otherwise lost in the background of each individual-$m$ spectrum of a given multiplet ($n,l$), and thus unobservable with a classic peak-fitting analysis. The $l-\nu$ diagram of the observed low-frequency modes ($1 \leq l \leq 35$) down to $n=1$ and $\approx$ 850 $\mu$Hz in the 3960-day GONG dataset and 2088-day GONG and MDI datasets with the $m$-averaged spectrum technique is shown on Fig. \[fig:odect\].
Mode linewidths, heights, and background levels
-----------------------------------------------
Figure \[fig:fwhm\] shows the fitted mode <span style="font-variant:small-caps;">fwhm</span>s $\Gamma_{n,l}$ ([*upper-left panel*]{}) and mode heights $H_{n,l}$ ([*upper-right panel*]{}) of the measured low-frequency oscillations. The fitted background level is also represented on the right panel. The <span style="font-variant:small-caps;">fwhm</span>s and heights are extremely valuable tests of models of the physical processes responsible for the mode damping and excitation by the turbulent convective motions in the outer layers of the Sun: the mode damping is inversely related to the <span style="font-variant:small-caps;">fwhm</span> of the mode, and the mode excitation is proportional to the mode height times the mode <span style="font-variant:small-caps;">fwhm</span> squared [for a detailed description, see, e.g., @salabert06]. [The leveling off of the mode widths observed below $\approx$ 1100 $\mu$Hz, despite their dispersion becoming larger, could be a resolution effect, the peaks being then so narrow that the limiting resolution of the spectrum becomes an issue. Moreover, @schou04 did not observe such behaviour at low frequency in MDI data with a 2952-day time series.]{}
As indicated by different colors and symbols on Fig. \[fig:fwhm\], the fitted mode widths follow ridges for equal radial orders $n$. This dependence on angular degree ($l$) is directly related to the mode inertia ($I$) in terms of a power law, as illustrated on the lower-left panel of Fig. \[fig:fwhm\]. The $l$ dependence in the mode <span style="font-variant:small-caps;">fwhm</span>s is removed when represented as a function of the mode inertia $I$.
{width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"}
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Mode asymmetry
--------------
The mode parameters extracted through the routine GONG and MDI peak-fitting pipelines are obtained by use of symmetric Lorenztian profiles (@anderson90 and @schou92 respectively). However, it was demonstrated that ignoring the peak asymmetry in the description of the acoustic modes leads to biais in the estimated mode parameters [see Appendix \[sec:impmodel\] and @thiery00]. Today, most of the estimates of the mode asymmetries have been restricted to low degrees ($l \leq3$) only, from Sun-as-a-star, integrated-sunlight observations. However, @korz05 used asymmetric profiles and presented estimates of the peak asymmetry for modes with angular degrees $1\leq l \leq25$, obtained with GONG and MDI observations. Recently, @larson08 are planning to reprocess all the MDI medium-$l$ data including a set of corrections and improvements (such as the mode asymmetry) in the MDI pipeline algorithm itself.
The asymmetry parameter ($\alpha_{n,l}$) in the low-frequency range, obtained by fitting the 3960-day GONG $m$-averaged spectrum ($1 \leq l \leq 35$), is shown on the lower-right panel of Fig. \[fig:fwhm\]. The extracted peak asymmetry is well constrained down to $\approx$ 1400 $\mu$Hz, with a mean value of about $-0.044\pm0.002$, and no discernable $l$ dependence. The average asymmetry observed in the $m$-averaged spectrum is consistent with other measurements. For instance, the mean value observed by @korz05 was about $-0.04$ for modes below 2000 $\mu$Hz and $l \leq 25$, once his estimates are transformed back into the @nigam98’s definition of the peak asymmetry. A comparable mean value is also observed at the lowest frequencies for which asymmetries were reported in Sun-as-a-star, integrated-sunlight observations [e.g., @thiery00].
Mode frequencies
----------------
Figure \[fig:freq\_th\] shows the frequency differences (in $\mu$Hz) between the fitted low-frequency modes observed in the 3960-day GONG dataset using the $m$-averaged spectrum technique and the corresponding theoretical values calculated from Christensen-Dalsgaard’s model S [@jcd96]. The corresponding frequency uncertainties were multiplied by 20 to render them visible. These comparisons are represented as a function of the angular degree ([*left panel*]{}), of the frequency ([*middle panel*]{}), and of the inner turning point ([*right panel*]{}). Modes of equal radial orders are connected. As these differences between observed and theoretical frequencies show, there is still room to improve the model of solar internal structure. Note that the right panel on Fig. \[fig:freq\_th\] illustrates also the wide range of depths of penetration that these low-frequency modes cover.
![$l - \nu$ diagram of the new low-frequency p modes observed in spatially-resolved data in the range of angular degrees $1\leq l \leq35$ (black dots: observed in the 3960-day GONG dataset / green dots: observed in the 2088-day GONG dataset / red dots: modes observed in the 2088-day MDI dataset). The corresponding frequency uncertainties were multiplied by $2\times10^4$. The already known modes are represented by the open circles, and the predicted modes by the crosses. The ridges of same radial order are also indicated from $n=1$ to $n=8$.[]{data-label="fig:newmodes"}](fig15.eps){width="50.00000%"}
Conclusion and discussion {#sec:conc}
=========================
We presented here an adaptation of the rotation-corrected, $m$-averaged spectrum technique to observe low signal-to-noise-ratio, low-frequency solar p modes in spatially-resolved helioseismic data. For a given multiplet ($n,l$), the shift coefficients describing the differential rotation- and structural-induced effects are chosen to [maximize the likelihood]{} of the $m$-averaged spectra. The average of the $2l+1$ individual-$m$ spectra can result in a high signal-to-noise ratio when the individual-$m$ spectra have a too low signal-to-noise ratio to be successfully fitted. This technique was applied to long time series of the spatially-revolved GONG and MDI observations for low-frequency modes (i.e., approximately below 1800 $\mu$Hz) with low- and intermediate-angular degrees ($1\leq l \leq35$). We demonstrated that it allows us to measure lower frequency modes than with classic peak-fitting analysis of the individual-$m$ spectra. Figure \[fig:newmodes\] shows the new low-frequency solar p modes observed in spatially-resolved data using the $m$-averaged spectrum technique in long time series of both GONG and MDI observations. Their central frequencies and splitting $a_1$ coefficients, as well as their associated uncertainties are indicated in Table \[table:newmodes\]. These normal modes of oscillation were predicted but were not measured previously. The potential of the $m$-averaged spectrum technique for increasing our knowledge of the solar interior is clearly illustrated on Fig. \[fig:newmodes\], p modes well below 1000 $\mu$Hz being measured with a high accuracy [thanks to their longer lifetimes]{}. We also demonstrated that the $m$-averaged spectrum technique returns unbiased results with no systematic differences with other long-duration measurements, which also include the asymmetry in the mode profile description.
The oscillation parameters of these low signal-to-noise-ratio, low-frequency modes, such as their central frequencies, splittings, asymmetries, lifetimes, and heights were measured. These low-frequency p modes contribute to improve our resolution throughout the solar interior since they sample a large range of penetration depths. Moreover, because these modes have lower upper turning points in the outer part of the Sun, they are less sensitive to the turbulence and magnetic fields in the outer layers, which should make them extremely valuable for the study of the physical processes responsible for the oscillation excitation and damping by the turbulent convection.
[We would like to recall that @schou92’s peak-finding approach consists in fitting the individual-$m$ spectra simultaneously by using a model in which the shift coefficients are introduced, while in the present technique, the best shifts are determined first, based on the calculation of figure-of-merits (Sec. \[ssec:det\_acoef\] and Appendix \[sec:foms\]), and then the rotation-corrected, $m$-averaged spectrum is fitted (Sec. \[ssec:extraction\]).]{}
The development of the $m$-averaged spectrum technique towards both higher frequencies and larger angular degrees is one of the next step to be addressed, as also the analysis of shorter datasets, such as the canonical 108- and 72-day time series.
---- --- ---------------------- ---------------------
1 7 1185.599 $\pm$ 0.005 431.491 $\pm$ 6.161
2 7 1250.555 $\pm$ 0.003 428.263 $\pm$ 2.286
3 5 1015.046 $\pm$ 0.005 430.154 $\pm$ 2.471
4 4 913.477 $\pm$ 0.004 420.055 $\pm$ 1.594
4 5 1062.140 $\pm$ 0.002 429.275 $\pm$ 0.614
5 4 954.560 $\pm$ 0.002 430.712 $\pm$ 0.596
6 4 992.412 $\pm$ 0.002 431.906 $\pm$ 0.502
6 5 1145.074 $\pm$ 0.002 432.019 $\pm$ 0.464
7 4 1028.156 $\pm$ 0.003 430.292 $\pm$ 0.740
8 4 1062.338 $\pm$ 0.002 434.087 $\pm$ 0.459
9 3 930.540 $\pm$ 0.002 430.045 $\pm$ 0.363
11 3 987.206 $\pm$ 0.002 436.639 $\pm$ 0.344
12 3 1013.572 $\pm$ 0.001 435.158 $\pm$ 0.172
13 3 1038.795 $\pm$ 0.001 435.900 $\pm$ 0.156
16 2 912.080 $\pm$ 0.002 436.331 $\pm$ 0.213
17 2 931.609 $\pm$ 0.002 435.855 $\pm$ 0.180
18 2 950.625 $\pm$ 0.002 436.627 $\pm$ 0.146
19 2 969.222 $\pm$ 0.002 438.012 $\pm$ 0.149
27 1 856.964 $\pm$ 0.002 437.741 $\pm$ 0.123
35 1 954.940 $\pm$ 0.002 440.118 $\pm$ 0.091
---- --- ---------------------- ---------------------
: Set of the new low-frequency solar p modes observed in the GONG and MDI datasets with the $m$-averaged spectrum technique in the range $1\leq l \leq35$.
\[table:newmodes\]
This work utilizes data obtained by the Global Oscillation Network Group (GONG) program, managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísica de Canarias, and Cerro Tololo Interamerican Observatory. The GOLF and MDI instruments onboard SOHO are cooperative efforts to whom we are indebted. SOHO is a project of international collaboration between ESA and NASA. BiSON is funded by the Science Technology and Facilities Council (STFC). We thank the members of the BiSON team, and colleagues at the host institutes at each of the BiSON sites. The authors are particularly grateful to S. G. Korzennik for providing us with the 2088-day MDI dataset, and to J. Schou for the MDI leakage matrix. The authors thank R. A García, P. Boumier, and W. J. Chaplin for providing estimates of GOLF and BiSON mode frequencies and $a_1$ rotational splittings observed in decade-long time series. The authors also thank S. J. Jiménez-Reyes and J. Schou for their useful comments on the manuscript, [and S. G. Korzennik for helpful discussions during the various stages of this work.]{} D. S. acknowledges the support of the NASA SEC GIP grant NAG5-11703. This work has been partially funded by the grant PNAyA2007-62650 of the Spanish National Research Plan.
Figures-of-merit and determination of the $a$-coefficients {#sec:foms}
==========================================================
The best estimates of the splitting $a$-coefficients are obtained by [maximizing the likelihood]{} of the $m$-averaged spectrum (see Sec. \[ssec:det\_acoef\]) through the calculation of a figure-of-merit (FOM). However, as shown on Fig. \[fig:fom\], other criteria to define a FOM can be used, such as the narrowest peak (i.e., the minimum mode linewidth), or the minimum entropy of the resulting $m$-averaged spectrum. In order to compare the actual mode parameters and associated uncertainties obtained with two different definitions of the FOM, we applied the $m$-averaged spectrum technique to the 3960-day GONG dataset by using both the [maximum likelihood]{} and the narrowest peak in the $m$-averaged spectrum as FOM. Figure \[fig:compfom\] shows the corresponding central frequencies, and the odd $a_1$, $a_3$, and $a_5$ splitting coefficients of the common, measured low-frequency p modes. The associated formal uncertainties are also represented. The two FOMs return consistent mode parameters within the error bars, the difference between the two being within the $3\sigma$ limit for all of the mode parameters.
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Impact of the fitting model used {#sec:impmodel}
================================
As a test of the dependence of the measured frequencies on the fitting model used to describe the $m$-averaged spectra, we fitted the $m$-averaged spectra using three different models: an asymmetric Lorenztian profile (Eq. \[eq:mlemodel\]) including the closest $\delta m \pm 2$ spatial leaks (hereafter called [*A2*]{} and used as the reference model); a symmetric Lorenztian profile including the $\delta m \pm 2$ spatial leaks (hereafter [*S2*]{}); and an asymmetric Lorenztian profile (Eq. \[eq:mlemodel\]) but omitting the neighbouring $\delta m \pm 2$ spatial leaks (hereafter [*A*]{}). Figure \[fig:nunoasym\_2088d\] shows the differences as a function of frequency in the 2088-day GONG low frequencies estimated using the $m$-averaged technique between [*S2*]{} and [*A2*]{} (red dots), and between [*A*]{} and [*A2*]{} (black plus signs), in both cases [*A2*]{} being the reference model. Ignoring the peak asymmetry in the fitting model leads to a systematic underestimation of the mode frequency as the frequency increases, the effect becoming particularly large above $\approx$ 1400 $\mu$Hz (red dots). The differences become much larger than $3\sigma$, for example, at $\approx$ 1800 $\mu$Hz, the fitted frequencies between [*S2*]{} and [*A2*]{} are about $20\sigma$ apart. These results obtained for modes below 2000 $\mu$Hz confirm previous observations, e.g. @thiery00 who analyzed low-degree modes above 2000 $\mu$Hz in 805 days of GOLF data.
On the other hand, omitting the spatial leaks has no effect below $\approx$ 1600 $\mu$Hz, as they become well separated from the main peak because the corresponding mode linewidths are much smaller than their frequency separation. As the frequency increases, the mode linewidths increase, and ignoring the spatial leaks in the fitting model of the $m$-averaged spectrum between about 1600 and 2000 $\mu$Hz leads to an underestimation of the target mode frequency, the maximum difference occuring around 1800 $\mu$Hz. The frequency separation between the target mode and the $m$-leaks then becomes comparable to their linewidths and the lines blend together in the $m$-averaged spectrum. Above 2000 $\mu$Hz, this underestimation seems to vanish. Indeed, at that frequency range, the mode linewidths are much larger than the frequency separation, and the first spatial leaks (at least) are totally blended into the target mode in the $m$-averaged spectrum, having a much lower impact on the frequency determination. However, the effect of ignoring the peak asymmetry is much larger than that from ignoring the $m$ leaks even in the frequency range where the $m$ leaks have the strongest impact. For instance, at 1800 $\mu$Hz, the effect on the frequency underestimation by ignoring the mode asymmetry is about seven times larger than by ignoring the $m$ leaks.
As an example of the other mode parameters, the right panel of Fig. \[fig:nunoasym\_2088d\] shows the differences in the extracted mode linewidths between the different fitting models. The color code is the same as for the differences in frequency represented on the left panel of Fig. \[fig:nunoasym\_2088d\]. Ignoring the presence of the $m$ leaks in the fitting model leads to a 35% overestimation at most of the extracted linewidths in the low-frequency range showing a maximum mismatch around 1900 $\mu$Hz. Interestingly, if the $m$ leaks are omitted, the linewidths are underestimated below $\approx$ 1600 $\mu$Hz, showing a maximum 10% underestimation around 1500 $\mu$Hz. On the other hand, ignoring the peak asymmetry has a very small influence on the fitted linewidths in the low-frequency range. However, above $\approx$ 1800 $\mu$Hz, the linewidths extracted using an asymmetric profile are systematically larger than the ones returned using a symmetric profile.
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Derivation of errors from an $m$-averaged spectrum {#sec:a_errors}
==================================================
The derivation of the errors of the mode central frequencies and of the $a$-coefficients measured from the $m$-averaged spectrum technique is detailed here.
Approximation of the statistics of the $m$-averaged spectrum
------------------------------------------------------------
The $m$-averaged spectrum is obtained from the summation of $2l+1$ spectra assumed to be with $\chi^2$ with 2 d.o.f statistics each having a [*different*]{} mean or signal-to-noise ratio. All of the individual $m$-spectra are independent from each other. The solar background noise is assumed to depend on $m$ with a polynomial with only even terms (0, 2, etc...). The amplitude of the modes is assumed to depend on $m$ with a different polynomial also with even terms (0, 2, etc...). In a first step, the $a_i$-coefficients are calculated to [maximize the likelihood]{} of the resulting $m$-averaged spectrum.
Using @app03, we can derive an approximation of the statistics of the summation of the $2l+1$ spectra. The statistics of the $m$-averaged spectrum ${\cal S}$ can be approximated by a Gamma law given by: $$p({\cal S})=\frac{\lambda^{\nu_1}}{\Gamma(\nu_1)} {\cal S}^{\nu_1-1}e^{-\lambda {\cal S}}
\label{approx1}$$ The mean and $\sigma$ are given by: $$E[{\cal S}]=\frac{\nu_1}{\lambda}\,\,{\rm and}\,\,\sigma^2=\frac{\nu_1}{\lambda^2}$$ $\lambda$ and $\nu_1$ are then derived from the mean and $\sigma$ as: $$\lambda=\frac{E[{\cal S}]}{\sigma^2}\,\,{\rm and}\,\,\nu_1=\frac{E[{\cal S}]^2}{\sigma^2}
\label{approx2}$$ In our case, the mean $E[{\cal S}]$ and $\sigma$ are given by: $$E[{\cal S}]=\sum_{m=-l}^{m= l} f_{m}\,\,{\rm and}\,\,\sigma=\sqrt{\sum_{m=-l}^{m=l} f_{m}^2}
\label{approx3}$$ where $f_m$ is the power spectrum for azimuthal order $m$ which can expressed as: $$f_m(\nu,\nu_0,a_i)=B_m(\nu)+A_m(\nu,\nu_0,a_i)$$ where $\nu$ is the frequency, $\nu_0$ the central frequency , $a_i$ are the usual Ritzwoller-Lavely coefficients, $B_m$ is the background noise, $A_m$ is the profile of the mode (the linewidth and amplitude have been omitted for simplifying the notation). We can write the noise as: $$B_m(\nu)={\cal B}(\nu)(1+g_B(m))$$ where the $m$-dependence is assumed to be independent of frequency. $g_B(m)$ is such that: $$E[B_m(\nu)]=(2l+1){\cal B}(\nu)
\label{approx3}$$ If the correction of the $a_i$ has been done properly, to the first order the $m$-averaged spectrum is independent of the $a_i$. We can write the mode amplitude as: $$A_m(\nu)={\cal A}(\nu)(1+h_A(m))$$ where the $m$-dependence is assumed to be independent of frequency. $h_A(m)$ is such that $$E[A_m(\nu)]=(2l+1){\cal A}(\nu)
\label{approx3}$$ Then we find: $$E[{\cal S}]=(2l+1)({\cal A}(\nu)+{\cal B}(\nu))$$ and $$\sigma^2=(2l+1)\left[{\cal A}^2(\nu)\left(1+\alpha\right)+{\cal B}^2(\nu)\left(1+\beta\right)+2{\cal A}{\cal B}(\nu)\left(1+\rho\right)\right]$$ with $$\alpha=\frac{1}{2l+1}\sum_{m=-l}^{m= l} h^2_A(m)$$ and $$\beta=\frac{1}{2l+1}\sum_{m=-l}^{m= l} g^2_B(m)$$ $$\rho=\frac{1}{2l+1}\sum_{m=-l}^{m= l} h_A(m)g_B(m)$$ we finally get for $\lambda$ and $\nu_1$ the following: $$\lambda=\frac{{\cal A}(\nu)+{\cal B}(\nu)}{{\cal A}^2(\nu)\left(1+\alpha\right)+{\cal B}^2(\nu)\left(1+\beta\right)+2{\cal A}{\cal B}(\nu)\left(1+\rho\right)}$$ and $\nu_1$ as: $$\nu_1=\frac{(2l+1)({\cal A}(\nu)+{\cal B}(\nu))^2}{{\cal A}^2(\nu)\left(1+\alpha\right)+{\cal B}^2(\nu)\left(1+\beta\right)+2{\cal A}(\nu){\cal B}(\nu)\left(1+\rho\right)}$$ After simplification we get: $$\lambda=\frac{{\cal A}(\nu)+{\cal B}(\nu)}{({\cal A}(\nu)+{\cal B}(\nu))^2+\alpha {\cal A}^2(\nu)+\beta {\cal B}^2(\nu)+ 2 \rho {\cal A}(\nu){\cal B}(\nu)}$$ and $$\nu_1=\frac{(2l+1)({\cal A}(\nu)+{\cal B}(\nu))^2}{({\cal A}(\nu)+{\cal B}(\nu))^2+\alpha {\cal A}^2(\nu)+\beta {\cal B}^2(\nu)+ 2 \rho {\cal A}(\nu){\cal B}(\nu)}$$ Using the dependence observed in the GONG data, we have $\alpha \approx 0.17$, $\beta \approx 0.035$ and $\rho \approx 0.075$. They are sufficiently small such that we have: $$\lambda \approx \frac{1}{{\cal A}(\nu)+{\cal B}(\nu)}$$ and $$\nu_1 \approx (2l+1)$$ then we find the following statistics for the $m$-averaged spectrum: $$p({\cal S})=\frac{1}{{\Gamma(2l+1)}}\frac{{\cal S}^{2l}}{({\cal A}(\nu)+{\cal B}(\nu))^{2l+1}} e^{-\frac{{\cal S}}{ {\cal A}(\nu)+{\cal B}(\nu)}}
\label{approx1}$$ After a change of variable $u={\cal S}/(2l+1)$, we have $$p(u) \propto \frac{1}{({\cal A}(\nu)+{\cal B}(\nu))^{2l+1}} e^{-\frac{(2l+1)u}{ {\cal A}(\nu)+{\cal B}(\nu)}}
\label{approx1}$$ When we use MLE, we minimize the following $${\cal L}(\nu,\nu_0,a_i)=-\ln p(u) = -(2l+1) \left[\ln ({\cal A}(\nu)+{\cal B}(\nu))+\frac{u}{({\cal A}(\nu)+{\cal B}(\nu))}\right] +....
\label{eq:eq_mle}$$ which shows that using the MLE applied to a $\chi^2$ with 2 d.o.f as prescribed by @app03 is in the case of the $m$-averaged spectrum a good approximation. It is not an approximation when averaging several power spectra of identical mean (or variance), i.e. when $\alpha=\beta=\rho=0$. Note that what we minimize is the sum over a range of frequency that can be approximated as: $$L(\nu_0,a_i)=\int {\cal L}(\nu,\nu_0,a_i) {\rm d}\nu$$
Error bars on the central frequencies
-------------------------------------
Error bars for frequency are derived from the inverse of the Hessian (second derivative of $L$) as: $$\sigma^{-2}_{\nu_0}=\frac{\partial^2 {L}}{\partial \nu^2_0}.$$ @toutain94 showed that we could express the error bars as a function of the mode profile ${\cal P}$ (=${\cal A}+{\cal B}$) as: $$\sigma^{-2}_{\nu_0}=T(2l+1)\int \frac{1}{{\cal P}^2(\nu)}\left(\frac{\partial{\cal P}}{\partial \nu_0}\right)^2 {\rm d}{\nu},
\label{eq:eq26}$$ where $T$ is the observation time. The $2l+1$ factor is due to the fact that the likelihood is $2l+1$ times larger than the likelihood of @toutain94 (cf Eq. 23). Eq. (\[eq:eq26\]) shows that the error bars on the frequencies in the $m$-averaged spectrum will be $\sqrt{2l+1}$ smaller than for the [*mean*]{} of the individual modes. In deriving Eq. (\[eq:eq26\]), we assumed that $\langle u \rangle={\cal P}$. This is an approximation good enough for getting the error bars on the frequency but not on the $a_i$.
Error bars on the $a$-coefficients
----------------------------------
The error bars on the $a$-coefficients are derived from the inverse of the Hessian (second derivative of $L$) as: $$h_{ij}=\frac{\partial^2 {L}}{{\partial a_i}{\partial a_j}}.$$
As shown by @toutain94, these coefficients can be related to the mode profile as using Eq. (\[eq:eq\_mle\]): $$h_{ij}=T \sum_m \int \frac{1}{{\cal P}_m^2(\nu)}\frac{\partial {\cal P}_m}{{\partial a_i}}\frac{\partial {\cal P}_m}{{\partial a_j}} {\rm d}\nu,$$ using the following property: $$\frac{\partial {\cal P}_m}{{\partial a_i}} =\frac{\partial {\cal P}_m}{{\partial \nu_0}} l P_{l,m}^i(m/l),$$ where $P_{l,m}^i(m/l)$ are the Ritzwoller-Lavely polynomials. And we finally get: $$h_{ij}=T l^2 \sum_m P_{l,m}^i(m/l)P_{l,m}^j(m/l) \int \frac{1}{{\cal P}_m^2(\nu)}\left(\frac{\partial {\cal P}_m}{{\partial \nu_0}} \right)^2 {\rm d}\nu.
\label{eq:eq_32}$$ We recognize the error bars of the frequency for the $m$ spectrum depending on the inverse of the signal-to-noise ratio $\beta_m$ [as in @lib92]: $$\sigma^{-2}_m=T \int \frac{1}{{\cal P}_m^2(\nu)}\left(\frac{\partial {\cal P}_m}{{\partial \nu_0}} \right)^2 {\rm d}\nu.$$ Finally Eq. (\[eq:eq\_32\]) can be written as: $$h_{ij}=l^2 \left(\sum_m P_{l,m}^i(m/l)P_{l,m}^j(m/l)\right) \sigma_m^{-2}.$$ The errors $\sigma_{a_1}$ scale like $l^{-\frac{3}{2}}$ [as in @veitzer93]. If the SNR is the same for all $m$, then we have $\sigma_{\nu_0}^{-2}=(2l+1)\sigma_{m}^{-2}$. Thus, by simply using the orthogonality property of the $P_{l,m}^i$ polynomials, and as given in Sec. \[ssec:error\] (Eq. \[eq:a\_error\]), we obtain the following expression to calculate the error bars of the $a$-coefficients in the $m$-averaged spectrum: $$\sigma_{a_i}^{-2}=\frac{l^2}{2l+1} \left(\sum_m \left[P_{l,m}^i(m/l)\right]^2\right) \sigma^{-2}_{\nu_0}.$$ All terms off of the diagonal are zero. Of course, when the SNR varies with $m$, the off-diagonal terms are non-zero and correlations appear.
Anderson, E. R., Duvall, T. L., Jr., & Jefferies, S. M. 1990, , 364, 699
Appourchaux, T. 2003, , 412, 903
Appourchaux, T., & Chaplin, W. J. 2007, , 469, 1151
Appourchaux, T., et al. 2000, , 538, 401
Broomhall, A. M., Chaplin, W. J., Elsworth, Y., & Appourchaux, T. 2007, , 379, 2
Brown, T. M. 1985, , 317, 591
Chaplin, W. J., Appourchaux, T., Elsworth, Y., Isaak, G. R., Miller, B. A., & New, R. 2004, , 424, 713
Chaplin, W. J., Elsworth, Y., Isaak, G. R., Marchenkov, K. I., Miller, B. A., New, R., Pinter, B., & Appourchaux, T. 2002, , 336, 979
Chaplin, W. J., et al. 1996, , 168, 1
Christensen-Dalsgaard, J., et al. 1996, Science, 272, 1286
Fierry Fraillon, D., Gelly, B., Schmider, F. X., Hill, F., Fossat, E., & Pantel, A. 1998, , 333, 362
Gabriel, A. H., et al. 1995, , 162, 61
Harvey, J. W., et al. 1996, Science, 272, 1284
Hill, F., & Howe, R. 1998, Structure and Dynamics of the Interior of the Sun and Sun-like Stars, 418, 225
Larson, T. P., & Schou, J. 2008, in L. Gizon (ed.), Helioseismology, Asteroseismology, and MHD Connections, Journal of Physics (in press)
Libbrecht, K. G. 1992, , 387, 712
Nigam, R., & Kosovichev, A. G. 1998, , 505, L51
Korzennik, S. G. 2008, Astron. Nachr., 329, 453
Korzennik, S. G. 2005, , 626, 585
Ritzwoller, M. H., & Lavely, E. M. 1991, , 369, 557
Salabert, D., & Jim[é]{}nez-Reyes, S. J. 2006, , 650, 451
Scherrer, P. H., et al. 1995, , 162, 129
Schou, J. 2004, in SOHO 14/GONG 2004 Workshop, Helio- and Asteroseismology: Towards a Golden Future, ed. D. Danesy (ESA SP-559), 134
Schou, J. 2002, in SOHO 11 Symposium, From Solar Min to Max: Half a Solar Cycle with SOHO, ed. A. Wilson (ESA SP-508), 99
Schou, J. 1999, , 523, L181
Schou, J. 1998, in SOHO 6/GONG 98 Workshop, Structure and Dynamics of the Interior of the Sun and Sun-like Stars (ESA SP-418), 341
Schou, J. 1992, Ph.D. Thesis, Aarhus Univ.
Shannon, C. E. 1948, Bell System Technical Journal, vol. 27, pp. 379-423, 623-656
Thiery, S., et al. 2000, , 355, 743
Toutain, T., & Appourchaux, T. 1994, , 289, 649
Veitzer, S. A., Tomczyk, S., & Schou, J. 1993, GONG 1992. Seismic Investigation of the Sun and Stars, 42, 465
[^1]: From 1996 April 11 to 2006 April 18.
[^2]: From 1996 April 11 to 2006 May 23.
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---
abstract: 'Probabilistic models are conceptually powerful tools for finding structure in data, but their practical effectiveness is often limited by our ability to perform inference in them. Exact inference is frequently intractable, so approximate inference is often performed using Markov chain Monte Carlo (MCMC). To achieve the best possible results from MCMC, we want to efficiently simulate many steps of a rapidly mixing Markov chain which leaves the target distribution invariant. Of particular interest in this regard is how to take advantage of multi-core computing to speed up MCMC-based inference, both to improve mixing and to distribute the computational load. In this paper, we present a parallelizable Markov chain Monte Carlo algorithm for efficiently sampling from continuous probability distributions that can take advantage of hundreds of cores. This method shares information between parallel Markov chains to build a scale-location mixture of Gaussians approximation to the density function of the target distribution. We combine this approximation with a recently developed method known as elliptical slice sampling to create a Markov chain with no step-size parameters that can mix rapidly without requiring gradient or curvature computations.'
author:
- |
Robert Nishihara rkn@eecs.berkeley.edu\
Department of Electrical Engineering and Computer Science\
University of California\
Berkeley, CA 94720, USA Iain Murray i.murray@ed.ac.uk\
School of Informatics\
University of Edinburgh\
Edinburgh EH8 9AB, UK Ryan P. Adams rpa@seas.harvard.edu\
School of Engineering and Applied Sciences\
Harvard University\
Cambridge, MA 02138, USA
bibliography:
- 'refs.bib'
title: Parallel MCMC with Generalized Elliptical Slice Sampling
---
Markov chain Monte Carlo, parallelism, slice sampling, elliptical slice sampling, approximate inference
Introduction
============
Probabilistic models are fundamental tools for machine learning, providing a coherent framework for finding structure in data. In the Bayesian formulation, learning is performed by computing a representation of the posterior distribution implied by the data. Unobserved quantities of interest can then be estimated as expectations of various functions under this posterior distribution.
These expectations typically correspond to high-dimensional integrals and sums, which are usually intractable for rich models. There is therefore significant interest in efficient methods for approximate inference that can rapidly estimate these expectations. In this paper, we examine Markov chain Monte Carlo (MCMC) methods for approximate inference, which estimate these quantities by simulating a Markov chain with the posterior as its equilibrium distribution. MCMC is often seen as a principled “gold standard” for inference, because (under mild conditions) its answers will be correct in the limit of the simulation. However, in practice, MCMC often converges slowly and requires expert tuning. In this paper, we propose a new method to address these issues for continuous parameter spaces. We generalize the method of *elliptical slice sampling* [@Murray2010] to build a new efficient method that: 1) mixes well in the presence of strong dependence, 2) does not require hand tuning, and 3) can take advantage of multiple computational cores operating in parallel. We discuss each of these in more detail below.
Many posterior distributions arising from real-world data have strong dependencies between variables. These dependencies can arise from correlations induced by the likelihood function, redundancy in the parameterization, or directly from the prior. One of the primary challenges for efficient Markov chain Monte Carlo is making large moves in directions that reflect the dependence structure. For example, if we imagine a long, thin region of high density, it is necessary to explore the length in order to reach equilibrium; however, random-walk methods such as Metropolis–Hastings (MH) with spherical proposals can only diffuse as fast as the narrowest direction allows [@Neal1995]. More efficient methods such as Hamiltonian Monte Carlo [@Duane1987; @Neal2011; @Girolami2011] avoid random walk behavior by introducing auxiliary “momentum” variables. Hamiltonian methods require differentiable density functions and gradient computations.
In this work, we are able to make efficient long-range moves—even in the presence of dependence—by building an approximation to the target density that can be exploited by elliptical slice sampling. This approximation enables the algorithm to consider the general shape of the distribution without requiring gradient or curvature information. In other words, it encodes and allows us to make use of global information about the distribution as opposed to the local information used by Hamiltonian Monte Carlo. We construct the algorithm such that it is valid regardless of the quality of the approximation, preserving the guarantees of approximate inference by MCMC.
One of the limitations of MCMC in practice is that it is often difficult for non-experts to apply. This difficulty stems from the fact that it can be challenging to tune Markov transition operators so that they mix well. For example, in Metropolis–Hastings, one must come up with appropriate proposal distributions. In Hamiltonian Monte Carlo, one must choose the number of steps and the step size in the simulation of the dynamics. For probabilistic machine learning methods to be widely applicable, it is necessary to develop black-box methods for approximate inference that do not require extensive hand tuning. Some recent attempts have been made in the area of adaptive MCMC [@Roberts2006; @Haario2005], but these are only theoretically understood for a relatively narrow class of transition operators (for example, not Hamiltonian Monte Carlo). Here we propose a method based on slice sampling [@Neal2003], which uses a local search to find an acceptable point, and avoid potential issues with convergence under adaptation.
In all aspects of machine learning, a significant challenge is exploiting a computational landscape that is evolving toward parallelism over single-core speed. When considering parallel approaches to MCMC, we can readily identify two ends of a spectrum of possible solutions. At one extreme, we could run a large number of independent Markov chains in parallel [@Rosenthal2000; @Bradford1996]. This will have the benefit of providing more samples and increasing the accuracy of the end result, however it will do nothing to speed up the convergence or the mixing of each individual chain. The parallel algorithm will run up against the same limitations faced by the non-parallel version. At another extreme, we could develop a single-chain MCMC algorithm which parallelizes the individual Markov transitions in a problem-specific way. For instance, if the likelihood is expensive and consists of many factors, the factors can potentially be computed in parallel. See @Suchard2010 [@Tarlow2012a] for examples. Alternatively, some Markov chain transition operators can make use of multiple parallel proposals to increase their efficiency, such as multiple-try Metropolis–Hastings [@Liu2000].
We propose an intermediate algorithm to make effective use of parallelism. By sharing information between the chains, our method is able to mix faster than the naïve approach of running independent chains. However, we do not require fine-grained control over parallel execution, as would be necessary for the single-chain method. Nevertheless, if such local parallelism is possible, our sampler can take advantage of it. Our general objective is a black-box approach that mixes well with multiple cores but does not require the user to build in parallelism at a low level.
The structure of the paper is as follows. In Section \[sec:background\], we review slice sampling [@Neal2003] and elliptical slice sampling [@Murray2010]. In Section \[sec:gess\], we show how an elliptical approximation to the target distribution enables us to generalize elliptical slice sampling to continuous distributions. In Section \[sec:parallelism\], we describe a natural way to use parallelism to dynamically construct the desired approximation. In Section \[sec:related\_work\], we discuss related work. In Section \[sec:experiments\], we evaluate our new approach against other comparable methods on several typical modeling problems.
Background {#sec:background}
==========
Throughout this paper, we will use $\mathcal N({\bf
x};\boldsymbol\mu,\boldsymbol\Sigma)$ to denote the density function of a Gaussian with mean $\boldsymbol\mu$ and covariance $\boldsymbol\Sigma$ evaluated at a point ${\bf x} \in
\mathbb R^D$. We will use $\mathcal
N(\boldsymbol\mu,\boldsymbol\Sigma)$ to refer to the distribution itself. Analogous notation will be used for other distributions. Throughout, we shall assume that we wish to draw samples from a probability distribution over $\mathbb R^D$ whose density function is $\pi$. We sometimes refer to the distribution itself as $\pi$.
The objective of Markov chain Monte Carlo is to formulate transition operators that can be easily simulated, that leave $\pi$ invariant, and that are ergodic. Classical examples of MCMC algorithms are Metropolis–Hastings [@Metropolis1953; @Hastings1970] and Gibbs Sampling [@Geman1984]. For general overviews of MCMC, see @tierney-1994a [@andrieu-etal-2003a; @HandbookMCMC]. Simulating such a transition operator will, in the limit, produce samples from $\pi$, and these can be used to compute expectations under $\pi$. Typically, we only have access to an unnormalized version of $\pi$. However, none of the algorithms that we describe require access to the normalization constant, and so we will abuse notation somewhat and refer to the unnormalized density as $\pi$.
Slice Sampling
--------------
Slice sampling [@Neal2003] is a Markov chain Monte Carlo algorithm with an adaptive step size. It is an auxiliary-variable method, which relies on the observation that sampling $\pi$ is equivalent to sampling the uniform distribution over the set $S =
\{({\bf x},y):~0\le~y~\le~\pi({\bf x})\}$ and marginalizing out the $y$ coordinate (which in this case is accomplished simply by disregarding the $y$ coordinate). Slice sampling accomplishes this by alternately updating ${\bf x}$ and $y$ so as to leave invariant the distributions $p({\bf x} \given y)$ and $p(y \given {\bf x})$ respectively. The key insight of slice sampling is that sampling from these conditionals (which correspond to uniform “slices” under the density function) is potentially much easier than sampling directly from $\pi$.
Updating $y$ according to $p(y \given {\bf x})$ is trivial. The new value of $y$ is drawn uniformly from the interval $(0, \pi({\bf
x}))$. There are different ways of updating ${\bf x}$. The objective is to draw uniformly from among the “slice” $\{{\bf x} : \pi({\bf
x}) \ge y\}$. Typically, this is done by defining a transition operator that leaves the uniform distribution on the slice invariant. @Neal2003 describes such a transition operator: first, choose a direction in which to search, then place an interval around the current state, expand it as necessary, and shrink it until an acceptable point is found. Several procedures have been proposed for the expansion and contraction phases.
Less clear is how to choose an efficient direction in which to search. There are two approaches that are widely used. First, one could choose a direction uniformly at random from all possible directions [@MacKay2003]. Second, one could choose a direction uniformly at random from the $D$ coordinate directions. We consider both of these implementations later, and we refer to them as *random-direction slice sampling* (RDSS) and *coordinate-wise slice sampling* (CWSS), respectively. In principle, any distribution over directions can be used as long as detailed balance is satisfied, but it is unclear what form this distribution should take. The choice of direction has a significant impact on how rapidly mixing occurs. In the remainder of the paper, we describe how slice sampling can be modified so that candidate points are chosen to reflect the structure of the target distribution.
Elliptical Slice Sampling
-------------------------
Elliptical slice sampling [@Murray2010] is an MCMC algorithm designed to sample from posteriors over latent variables of the form $$\label{ess}
\pi({\bf x}) \propto L({\bf x}) \, \mathcal N({\bf x};\boldsymbol\mu, \boldsymbol\Sigma),$$ where $L$ is a likelihood function, and $\mathcal N(\boldsymbol\mu,
\boldsymbol\Sigma)$ is a multivariate Gaussian prior. Such models, often called *latent Gaussian models*, arise frequently from Gaussian processes and Gaussian Markov random fields. Elliptical slice sampling takes advantage of the structure induced by the Gaussian prior to mix rapidly even when the covariance induces strong dependence. The method is easier to apply than most MCMC algorithms because it has no free tuning parameters.
Elliptical slice sampling takes advantage of a convenient invariance property of the multivariate Gaussian. Namely, if ${\bf x}$ and ${\boldsymbol\nu}$ are independent draws from $\mathcal
N(\boldsymbol\mu, \boldsymbol\Sigma)$, then the linear combination $$\label{ess2}
{\bf x}' = ({\bf x} - \boldsymbol\mu)\cos \theta + ({\boldsymbol\nu} - \boldsymbol\mu)\sin \theta + \boldsymbol\mu$$ is also (marginally) distributed according to $\mathcal
N(\boldsymbol\mu, \boldsymbol\Sigma)$ for any $\theta \in
[0,2\pi]$. Note that ${\bf x'}$ is nevertheless correlated with ${\bf
x}$ and ${\boldsymbol\nu}$. This correlation has been previously used to make perturbative Metropolis–Hastings proposals in latent Gaussian models [@Neal1998; @Adams2009], but elliptical slice sampling uses it as the basis for a rejection-free method.
The elliptical slice sampling transition operator considers the locus of points defined by varying $\theta$ in Equation . This locus is an ellipse which passes through the current state ${\bf x}$ as well as through the auxiliary variable $\boldsymbol\nu$. Given a random ellipse induced by $\boldsymbol\nu$, we can slice sample $\theta \in [0,2\pi]$ to choose the next point based purely on the likelihood term. The advantage of this procedure is that the ellipses will necessarily reflect the dependence induced by strong Gaussian priors and that the user does not have to choose a step size.
More specifically, elliptical slice sampling updates the current state ${\bf x}$ as follows. First, the auxiliary variable $\boldsymbol\nu \sim \mathcal
N(\boldsymbol\mu,\boldsymbol\Sigma)$ is sampled to define an ellipse via Equation , and the value $u \sim
\textnormal{Uniform}[0,1]$ is sampled to define a likelihood threshold. Then, a sequence of angles $\{\theta_k\}$ are chosen according to a slice-sampling procedure described in Algorithm \[alg:ess-update\]. These angles specify a corresponding sequence of proposal points $\{ {\bf x}_k'\}$. We update the current state ${\bf x}$ by setting it equal to the first proposal point ${\bf
x}_k'$ satisfying the slice-sampling condition $L({\bf
x}_k')>uL({\bf x})$. The proof of the validity of this algorithm is given in @Murray2010. Intuitively, the pair $({\bf
x},\boldsymbol\nu)$ is updated to a pair $({\bf
x}',\boldsymbol\nu')$ with the same joint prior probability, and so slice sampling only needs to compare likelihood ratios. The new point ${\bf x}'$ is given by Equation , while $\boldsymbol\nu'=(\boldsymbol\nu-\boldsymbol\mu)\cos\theta-({\bf
x}-\boldsymbol\mu)\sin\theta + \boldsymbol\mu$ is never used and need not be computed.
Figure \[fig:ess\_ellipses\] depicts random ellipses produced by elliptical slice sampling superimposed on background points from some target distribution. This diagram illustrates the idea that the ellipses produced by elliptical slice sampling reflect the structure of the distribution. The full algorithm is shown in Algorithm \[alg:ess-update\].
[0.5]{} ![Background points are drawn independently from a probability distribution, and five ellipses are created by elliptical slice sampling. The distribution in question is a two-dimensional multivariate Gaussian. In this example, the same distribution is used as the prior for elliptical slice sampling. [**(a)**]{} Shows the ellipses created by elliptical slice sampling. [**(b)**]{} Shows the values of ${\bf x}$ (depicted as $\mathlarger{\mathlarger{\mathlarger{\circ}}}$) and $\boldsymbol\nu$ (depicted as $\textnormal{+}$) corresponding to each elliptical slice sampling update. The values of ${\bf x}$ and ${\boldsymbol\nu}$ with a given color correspond to the ellipse of the same color in [**(a)**]{}.[]{data-label="fig:ess_ellipses"}](figures/ess_ellipses-eps-converted-to "fig:")
[0.5]{} ![Background points are drawn independently from a probability distribution, and five ellipses are created by elliptical slice sampling. The distribution in question is a two-dimensional multivariate Gaussian. In this example, the same distribution is used as the prior for elliptical slice sampling. [**(a)**]{} Shows the ellipses created by elliptical slice sampling. [**(b)**]{} Shows the values of ${\bf x}$ (depicted as $\mathlarger{\mathlarger{\mathlarger{\circ}}}$) and $\boldsymbol\nu$ (depicted as $\textnormal{+}$) corresponding to each elliptical slice sampling update. The values of ${\bf x}$ and ${\boldsymbol\nu}$ with a given color correspond to the ellipse of the same color in [**(a)**]{}.[]{data-label="fig:ess_ellipses"}](figures/ess_ellipses_points-eps-converted-to "fig:")
Current state ${\bf x}$, Gaussian parameters $\boldsymbol\mu$ and $\boldsymbol\Sigma$, log-likelihood function $\log L$ New state ${\bf x'}$, with stationary distribution proportional to $\mathcal N({\bf x};\boldsymbol\mu,\boldsymbol\Sigma)L({\bf x})$ $\boldsymbol\nu \sim \mathcal N(\boldsymbol\mu, \boldsymbol\Sigma)$ $u \sim \textnormal{Uniform}[0,1]$ $\log y \leftarrow \log L({\bf x}) + \log u$ $\theta \sim \textnormal{Uniform}[0,2\pi]$ $[\theta_{\min},\theta_{\max}] \leftarrow [\theta-2\pi, \theta]$ ${\bf x'} \leftarrow ({\bf x} - \boldsymbol\mu)\cos\theta + (\boldsymbol\nu - \boldsymbol\mu) \sin\theta + \boldsymbol\mu$ $\theta_{\min} \leftarrow \theta$ $\theta_{\max} \leftarrow \theta$ $\theta \sim \textnormal{Uniform}[\theta_{\min},\theta_{\max}]$ 6
Generalized Elliptical Slice Sampling {#sec:gess}
=====================================
In this section, we generalize elliptical slice sampling to handle arbitrary continuous distributions. We refer to this algorithm as *generalized elliptical slice sampling* (GESS). In this section, our target distribution will be a continuous distribution over $\mathbb R^D$ with density $\pi$. In practice, $\pi$ need not be normalized.
Our objective is to reframe our target distribution so that it can be efficiently sampled with elliptical slice sampling. One possible approach is to put $\pi$ in the form of Equation by choosing some approximation $\mathcal N(\boldsymbol\mu,\boldsymbol\Sigma)$ to $\pi$ and writing $$\pi({\bf x}) = R({\bf x}) \, \mathcal N({\bf x};\boldsymbol\mu, \boldsymbol\Sigma),$$ where $$R({\bf x}) = \frac{\pi({\bf x})}{\mathcal N({\bf x};\boldsymbol\mu,\boldsymbol\Sigma)}$$ is the residual error of our approximation to the target density. Note that $\mathcal N({\bf x};\boldsymbol\mu,\boldsymbol\Sigma)$ is an approximation rather than a prior and that $R$ is not a likelihood function, but since the equation has the correct form, this representation enables us to use elliptical slice sampling.
[1.0]{} ![Samples are drawn from a Gaussian with zero mean and unit variance using elliptical slice sampling with various Gaussian approximations. These trace plots show how sampling behavior depends on how heavy the tails of the approximation are relative to how heavy the tails of the target distribution are. We plot one of every ten samples.[]{data-label="fig:trace_plots"}](figures/trace_plot-2-eps-converted-to "fig:")
[1.0]{} ![Samples are drawn from a Gaussian with zero mean and unit variance using elliptical slice sampling with various Gaussian approximations. These trace plots show how sampling behavior depends on how heavy the tails of the approximation are relative to how heavy the tails of the target distribution are. We plot one of every ten samples.[]{data-label="fig:trace_plots"}](figures/trace_plot-1-eps-converted-to "fig:")
[1.0]{} ![Samples are drawn from a Gaussian with zero mean and unit variance using elliptical slice sampling with various Gaussian approximations. These trace plots show how sampling behavior depends on how heavy the tails of the approximation are relative to how heavy the tails of the target distribution are. We plot one of every ten samples.[]{data-label="fig:trace_plots"}](figures/trace_plot0-eps-converted-to "fig:")
[1.0]{} ![Samples are drawn from a Gaussian with zero mean and unit variance using elliptical slice sampling with various Gaussian approximations. These trace plots show how sampling behavior depends on how heavy the tails of the approximation are relative to how heavy the tails of the target distribution are. We plot one of every ten samples.[]{data-label="fig:trace_plots"}](figures/trace_plot1-eps-converted-to "fig:")
[1.0]{} ![Samples are drawn from a Gaussian with zero mean and unit variance using elliptical slice sampling with various Gaussian approximations. These trace plots show how sampling behavior depends on how heavy the tails of the approximation are relative to how heavy the tails of the target distribution are. We plot one of every ten samples.[]{data-label="fig:trace_plots"}](figures/trace_plot2-eps-converted-to "fig:")
Applying elliptical slice sampling in this manner will produce a correct algorithm, but it may mix slowly in practice. Difficulties arise when the target distribution has much heavier tails than does the approximation. In such a situation, $R({\bf x})$ will increase rapidly as ${\bf x}$ moves away from the mean of the approximation. To illustrate this phenomenon, we use this approach with different approximations to draw samples from a Gaussian in one dimension with zero mean and unit variance. Trace plots are shown in Figure \[fig:trace\_plots\]. The subplot corresponding to variance $0.01$ illustrates the problem. Since $R$ explodes as $|{\bf
x}|$ gets large, the Markov chain is unlikely to move back toward the origin. On the other hand, the size of the ellipse is limited by a draw from the Gaussian approximation, which has low variance in this case, so the Markov chain is also unlikely to move away from the origin. The result is that the Markov chain sometimes gets stuck. In the subplot corresponding to variance $0.01$, this occurs between iterations $400$ and $630$.
In order to resolve this pathology and extend elliptical slice sampling to continuous distributions, we broaden the class of allowed approximations. To begin with, we express the density of the target distribution in the more general form $$\label{general-ess}
\pi({\bf x}) \propto R({\bf x}) \int \mathcal N({\bf x} ; \boldsymbol\mu(s), \boldsymbol\Sigma(s)) \, \phi(\mathrm{d}s),$$ where the integral represents a scale-location mixture of Gaussians (which serves as an approximation to $\pi$), and where $\phi$ is a measure over the auxiliary parameter $s$. As before, $R$ is the residual error of the approximation. Here, $\phi$ can be chosen in a problem-specific way, and any residual error between $\pi$ and the approximation will be compensated for by $R$. Equation is quite flexible. Below, we will choose the measure $\phi$ so as to make the approximation a multivariate $t$ distribution, but there are many other possibilities. For instance, taking $\phi$ to be a combination of point masses will make the approximation a discrete mixture of Gaussians.
Through Equation , we can view $\pi({\bf x})$ as the marginal density of an augmented joint distribution over ${\bf x}$ and $s$. Using $\lambda$ to denote the density of $\phi$ with respect to the base measure over $s$ (this is fully general because we have control over the choice of base measure), we can write $$p({\bf x}, s) = R({\bf x}) \, \mathcal N({\bf x};\boldsymbol\mu(s),\boldsymbol\Sigma(s)) \, \lambda(s) .$$ Therefore, to sample $\pi$, it suffices to sample ${\bf x}$ and $s$ jointly and then to marginalize out the $s$ coordinate (by simply dropping the $s$ coordinate). We update these components alternately so as to leave invariant the distributions $$\label{x-given-s}
p({\bf x} \given s) \propto R({\bf x}) \, \mathcal N({\bf x};\boldsymbol\mu(s),\boldsymbol\Sigma(s))$$ and $$\label{s-given-x}
p(s \given {\bf x}) \propto \mathcal N({\bf x};\boldsymbol\mu(s),\boldsymbol\Sigma(s)) \, \lambda(s) .$$ Equation has the correct form for elliptical slice sampling and can be updated according to Algorithm \[alg:ess-update\]. Equation can be updated using any valid Markov transition operator.
We now focus on a particular case in which the update corresponding to Equation is easy to simulate and in which we can make the tails as heavy as we desire, so as to control the behavior of $R$. A simple and convenient choice is for the scale-location mixture to yield a multivariate $t$ distribution with degrees-of-freedom parameter $\nu$: $$\mathcal T_{\nu}({\bf x};\boldsymbol\mu,\boldsymbol\Sigma) = \int_0^{\infty} \textnormal{IG}(s;\tfrac{\nu}{2},\tfrac{\nu}{2}) \, \mathcal N({\bf x};\boldsymbol\mu,s\boldsymbol\Sigma) \, \mathrm{d}s ,$$ where $\lambda$ becomes the density function of an inverse-gamma distribution: $$\text{IG}(s;\alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}s^{-\alpha-1}e^{-\beta/s} .$$ Here $s$ is a positive real-valued scale parameter. Now, in the update $p(s \given {\bf x})$, we observe that the inverse-gamma distribution acts as a conjugate prior (whose “prior” parameters are $\alpha=\frac{\nu}{2}$ and $\beta=\frac{\nu}{2}$), so $$p(s\given {\bf x}) = \textnormal{IG}(s;\alpha',\beta')$$ with parameters $$\begin{aligned}
\alpha' & = & \frac{D+\nu}{2} \,\,\,\,\,\,\,\, \text{and} \label{IG-params-alpha} \\
\beta' & = & \frac12(\nu+({\bf x}-\boldsymbol\mu)^{\sf T}\boldsymbol\Sigma^{-1}({\bf x}-\boldsymbol\mu)) . \label{IG-params-beta}\end{aligned}$$ We can draw independent samples from this distribution [@Devroye1986].
Combining these update steps, we define the transition operator $S({\bf x} \to {\bf x'}; \nu,
\boldsymbol\mu,\boldsymbol\Sigma)$ to be the one which draws $s \sim
\textnormal{IG}(s;\alpha',\beta')$, with $\alpha'$ and $\beta'$ as described in Equations and , and then uses elliptical slice sampling to update ${\bf x}$ so as to leave invariant the distribution defined in Equation , where $\boldsymbol\mu(s)=\boldsymbol\mu$ and $\boldsymbol\Sigma(s)=s\boldsymbol\Sigma$. From the above discussion, it follows that the stationary distribution of $S({\bf x}
\to {\bf x'} ;\nu,\boldsymbol\mu,\boldsymbol\Sigma)$ is $\pi$. Figure \[fig:vis\_update\] illustrates this transition operator.
Current state ${\bf x}$, multivariate $t$ parameters $\nu, \boldsymbol\mu, \boldsymbol\Sigma$, dimension $D$, a routine $\textnormal{ESS}$ that performs an elliptical slice sampling update New state ${\bf x'}$ $\alpha' \leftarrow \frac{D+\nu}{2}$ $\beta' \leftarrow \frac12(\nu + ({\bf x}-\boldsymbol\mu)^{\sf T}\boldsymbol\Sigma^{-1}({\bf x}-\boldsymbol\mu))$ $s \sim \textnormal{IG}(\alpha',\beta')$ $\log L \leftarrow \log \pi - \log \mathcal T$ ${\bf x'} \leftarrow \textnormal{ESS}({\bf x}, \boldsymbol\mu, s\boldsymbol\Sigma, \log L)$
[0.5]{} ![The gray points were drawn independently from a two-dimensional Gaussian to show the mode and shape of the corresponding density function. [**(a)**]{} Shows the contours of a multivariate $t$ approximation to this distribution. [**(b)**]{} Shows a sample update step using the transition operator $S({\bf x}\to~{\bf x'};\nu,\boldsymbol\mu,\boldsymbol\Sigma)$. The blue point represents the current state. The yellow point defines an ellipse and is drawn from the Gaussian distribution corresponding to the scale $s$ drawn from the appropriate inverse-gamma distribution. The red point corresponds to the new state and is sampled from the given ellipse.[]{data-label="fig:vis_update"}](figures/t_contours-eps-converted-to "fig:")
[0.5]{} ![The gray points were drawn independently from a two-dimensional Gaussian to show the mode and shape of the corresponding density function. [**(a)**]{} Shows the contours of a multivariate $t$ approximation to this distribution. [**(b)**]{} Shows a sample update step using the transition operator $S({\bf x}\to~{\bf x'};\nu,\boldsymbol\mu,\boldsymbol\Sigma)$. The blue point represents the current state. The yellow point defines an ellipse and is drawn from the Gaussian distribution corresponding to the scale $s$ drawn from the appropriate inverse-gamma distribution. The red point corresponds to the new state and is sampled from the given ellipse.[]{data-label="fig:vis_update"}](figures/t_update-eps-converted-to "fig:")
Building the Approximation with Parallelism {#sec:parallelism}
===========================================
Up to this point, we have not described how to choose the multivariate $t$ parameters $\nu$, $\boldsymbol\mu$, and $\boldsymbol\Sigma$. These choices can be made in many ways. For instance, we may choose the maximum likelihood parameters given samples collected during a burn in period, we may build a Laplace approximation to the mode of the distribution, or we may use variational approaches. Note that this algorithm is valid regardless of the particular choice we make here. In this section, we discuss a convenient way to use parallelism to dynamically choose these parameters without requiring tuning runs or exploratory analysis of the distribution. This method creates a large number of parallel chains, each producing samples from $\pi$, and it divides them into two groups. The need for two groups of Markov chains is not immediately obvious, so we motivate our approach by first discussing two simpler algorithms that fail in different ways.
Naïve Approaches
----------------
We begin with a collection of $K$ parallel chains. Let ${\mathcal
X~=~\{{\bf x}_1, \ldots, {\bf x}_K\} }$ denote the current states of the chains. We observe that $\mathcal X$ may contain a lot of information about the shape of the target distribution. We would like to define a transition operator $Q(\mathcal X \to \mathcal X')$ that uses this information to intelligently choose the multivariate $t$ parameters $\nu$, $\boldsymbol\mu$, and $\boldsymbol\Sigma$ and then uses these parameters to update each ${\bf x}_k$ via generalized elliptical slice sampling. Additionally, we would like $Q$ to have two properties. First, each ${\bf x}_k$ should have the marginal stationary distribution $\pi$. Second, we should be able to parallelize the update of $\mathcal X$ over $K$ cores.
Here we describe two simple approaches for parallelizing generalized elliptical slice sampling, each of which lacks one of the desired properties. The first approach begins with $K$ parallel Markov chains, and it requires a procedure for choosing the multivariate $t$ parameters given $\mathcal X$ (for example, maximum likelihood estimation). In this setup, $Q$ uses this procedure to determine the multivariate $t$ parameters $\nu_{\mathcal
X}$, $\boldsymbol\mu_{\mathcal X}$, $\boldsymbol\Sigma_{\mathcal X}$ from $\mathcal X$ and then applies $S({\bf x} \to {\bf x'};
\nu_{\mathcal X},\boldsymbol\mu_{\mathcal
X},\boldsymbol\Sigma_{\mathcal X})$ to each ${\bf x}_k$ individually. These updates can be performed in parallel, but the variables ${\bf x}_k$ no longer have the correct marginal distributions because of the coupling between the chains introduced by the approximation (this update violates detailed balance).
A second approach creates a valid MCMC method by including the multivariate $t$ parameters in a joint distribution $$\label{eq:naive_parallel}
p( \mathcal X, \nu, \boldsymbol\mu, \boldsymbol\Sigma) = p(\nu, \boldsymbol\mu, \boldsymbol\Sigma \given \mathcal X) \left[ \prod_{k=1}^K \pi({\bf x}_k) \right] .$$ Note that in Equation , each ${\bf x}_k$ has marginal distribution $\pi$. We can sample this joint distribution by alternately updating the variables and the multivariate $t$ parameters so as to leave invariant the conditional distributions $p(\mathcal X
\given \nu, \boldsymbol\mu, \boldsymbol\Sigma)$ and $p(\nu,
\boldsymbol\mu, \boldsymbol\Sigma \given \mathcal X)$. Ideally, we would like to update the collection $\mathcal X$ by updating each ${\bf x}_k$ in parallel. However, we cannot easily parallelize the update in this formulation because of the factor of $p(\nu,
\boldsymbol\mu, \boldsymbol\Sigma \given \mathcal X)$, which nontrivially couples the chains.
The Two-Group Approach
----------------------
Our proposed method creates a transition operator $Q$ that satisfies both of the desired properties. That is, each ${\bf x}_k$ has marginal distribution $\pi$, and the update can be efficiently parallelized. This method circumvents the problems of the previous approaches by maintaining two groups of Markov chains and using each group to choose multivariate $t$ parameters to update the other group. Let $\mathcal X = \{{\bf x}_1, \ldots, {\bf x}_{K_1}\}$ and $\mathcal Y = \{{\bf y}_1, \ldots, {\bf y}_{K_2}\}$ denote the states of the Markov chains in these two groups (in practice, we set $K_1=K_2=K$, where $K$ is the number of available cores). The stationary distribution of the collection is $$\Pi(\mathcal X, \mathcal Y) = \Pi_1(\mathcal X)\Pi_2(\mathcal Y) = \left[\prod_{k=1}^{K_1} \pi({\bf x}_k)\right] \left[\prod_{k=1}^{K_2} \pi({\bf y}_k)\right] .$$ By simulating a Markov chain which leaves this product distribution invariant, this method generates samples from the target distribution. Our Markov chain is based on a transition operator, $Q$, defined in two parts. The first part of the transition operator, $Q_1$, uses $\mathcal Y$ to determine parameters $\nu_{\mathcal Y}$, $\boldsymbol\mu_{\mathcal Y}$, and $\boldsymbol\Sigma_{\mathcal Y}$. It then uses these parameters to update $\mathcal X$. The second part of the transition operator, $Q_2$, uses $\mathcal X$ to determine parameters $\nu_{\mathcal X}$, $\boldsymbol\mu_{\mathcal X}$, and $\boldsymbol\Sigma_{\mathcal X}$. It then uses these parameters to update $\mathcal Y$. The transition operator $Q$ is the composition of $Q_1$ and $Q_2$. The idea of maintaining a group of Markov chains and updating the states of some Markov chains based on the states of other Markov chains has been discussed in the literature before. For example, see @Zhang2011 [@gilks1994].
In order to make these descriptions more precise, we define $Q_1$ as follows. First, we specify a procedure for choosing the multivariate $t$ parameters given the population $\mathcal Y$. We use an extension of the expectation-maximization algorithm [@Liu1995] to choose the maximum-likelihood multivariate $t$ parameters given the data $\mathcal Y$. The details of this algorithm are described in Algorithm \[alg:t-params\] in the Appendix. More concretely, we choose $$\nu_{\mathcal Y}, \boldsymbol\mu_{\mathcal Y},\boldsymbol\Sigma_{\mathcal Y} = \argmax_{\nu, \boldsymbol\mu,\boldsymbol\Sigma} \prod_{k=1}^{K_2}\mathcal T_{\nu}({\bf y}_{k}\,;\,\boldsymbol\mu,\boldsymbol\Sigma) .$$ After choosing $\nu_{\mathcal Y}$, $\boldsymbol\mu_{\mathcal Y}$, and $\boldsymbol\Sigma_{\mathcal Y}$ in this manner, we update $\mathcal X$ by applying the transition operator $S({\bf x} \to~{\bf x'} ;\nu_{\mathcal Y}, \boldsymbol\mu_{\mathcal Y},\boldsymbol\Sigma_{\mathcal Y})$ to each ${\bf x}_k \in \mathcal X$ in parallel. The operator $Q_2$ is defined analogously.
Two groups of states $\mathcal X = \{{\bf x}_1, \ldots, {\bf x}_{K_1}\}$ and $\mathcal Y = \{{\bf y}_1, \ldots, {\bf y}_{K_2}\}$, a subroutine $\textnormal{FIT-MVT}$ which takes data and returns the maximum-likelihood $t$ parameters, a subroutine $\textnormal{GESS}$ which performs a generalized elliptical slice sampling update Updated groups $\mathcal X'$ and $\mathcal Y'$ $\nu, \boldsymbol\mu, \boldsymbol\Sigma \leftarrow \textnormal{FIT-MVT}(\mathcal Y)$ ${\bf x}_k' = \textnormal{GESS}({\bf x}_k, \nu, \boldsymbol\mu, \boldsymbol\Sigma)$ $\mathcal X' \leftarrow \{{\bf x}_1', \ldots, {\bf x}_{K_1}'\}$ $\nu, \boldsymbol\mu, \boldsymbol\Sigma \leftarrow \textnormal{FIT-MVT}(\mathcal X')$ ${\bf y}_k' = \textnormal{GESS}({\bf y}_k, \nu, \boldsymbol\mu, \boldsymbol\Sigma)$ $\mathcal Y' \leftarrow \{{\bf y}_1', \ldots, {\bf y}_{K_2}'\}$
In the case where the number of chains in the collection $\mathcal Y$ is less than or close to the dimension of the distribution, the particular algorithm that we use to choose the parameters [@Liu1995] may not converge quickly (or at all). Suppose we are in the setting where $K<2D$. In this situation, we can perform a regularized estimate of the parameters. We describe this procedure below. The choice $K<2D$ probably overestimates the regime in which the algorithm for fitting the parameters performs poorly. The particular algorithm that we use appears to work well as long as $K\ge D$.
Let $\bar{\bf y}$ be the mean of $\mathcal Y$, and let $\{{\bf v}_1, \ldots, {\bf v}_{J}\}$ be the first ${J}$ principal components of the set $\{{\bf y}_1-\bar{\bf y},\ldots,{\bf y}_K-\bar{\bf y}\}$, where ${J}=\lfloor \tfrac{K}{2}\rfloor$, and let $V=\textnormal{span}({\bf v}_1,\ldots,{\bf v}_{J})$. Let ${\bf A}$ be the $D\times {J}$ matrix defined by ${\bf A}{\bf e}_j={\bf v}_j$, where ${\bf e}_j$ is the $j$th standard basis vector in $\mathbb R^{J}$. This map identifies $\mathbb R^{J}$ with $V$.
Let the set $\hat{\mathcal Y}$ consist of the projections of the elements of $\mathcal Y$ onto $\mathbb R^{J}$ by $\hat{\bf y}_k={\bf A}^{\mathsf T}{\bf y}_k$. Using the algorithm from @Liu1995, fit the multivariate $t$ parameters $\nu_{\hat{\mathcal Y}}$, $\boldsymbol\mu_{\hat{\mathcal Y}}$ and, $\boldsymbol\Sigma_{\hat{\mathcal Y}}$ to $\hat{\mathcal Y}$. At this point, we have constructed a ${J}$-dimensional multivariate $t$ distribution, but we would like a $D$-dimensional one. We construct the desired distribution by rotating back to the original space. Concretely, we can set $$\begin{aligned}
\nu_{\mathcal Y} & = & \nu_{\hat{\mathcal Y}}\\
\boldsymbol\mu_{\mathcal Y} & = & {\bf A} \, \boldsymbol\mu_{\hat{\mathcal Y}} + \bar{{\bf y}}\\
\boldsymbol\Sigma_{\mathcal Y} & = & {\bf A} \, \boldsymbol\Sigma_{\hat{\mathcal Y}} \, {\bf A}^{\mathsf T} + \epsilon \, {\bf I}_D,\end{aligned}$$ where ${\bf I}_D$ is the $D\times D$ identity matrix and $\epsilon$ is the median entry on the diagonal of $\boldsymbol\Sigma_{\hat{\mathcal Y}}$. We add a scaled identity matrix to the covariance parameter to avoid producing a degenerate distribution. The choice of $\epsilon$ is based on intuition about typical values of the variance of $\pi$ in the directions orthogonal to $V$.
We emphasize that the nature of the procedure for fitting a multivariate $t$ distribution to some points is not important to our algorithm. One could devise more sophisticated approaches drawing on ideas from the literature on high-dimensional covariance estimation, see @Ravikumar2011 for instance, but we merely choose a simple idea that seems to work in practice. Since our default choice (if there are at least $2D$ chains, then choose the maximum-likelihood parameters, otherwise project to a lower dimension, choose the maximum-likelihood parameters, and then pad the diagonal of the covariance parameter) works well, the fact that one could design a more sophisticated procedure does not compromise the tuning-free nature of our algorithm.
Correctness
-----------
To establish the correctness of our algorithm, we treat the collection of chains as a single aggregate Markov chain, and we show that this aggregate Markov chain with transition operator $Q$ correctly samples from the product distribution $\Pi$.
We wish to show that $Q_1$ and $Q_2$ preserve the invariant distributions $\Pi_1$ and $\Pi_2$ respectively. As the two cases are identical, we consider only the first. We have $$\begin{aligned}
\int \Pi_1(\mathcal X) \, Q_1(\mathcal X \to \mathcal X') \, \mathrm{d}\mathcal X & = & \int \Pi_1(\mathcal X) \, Q_1(\mathcal X \to \mathcal X' \given \nu_{\mathcal Y}, \boldsymbol\mu_{\mathcal Y}, \boldsymbol\Sigma_{\mathcal Y}) \, \mathrm{d}\mathcal X \\
& = & \prod_{k=1}^{K_1} \left[ \int \pi({\bf x}_k) \, S({\bf x}_k \to {\bf x}_k'; \nu_{\mathcal Y},\boldsymbol\mu_{\mathcal Y},\boldsymbol\Sigma_{\mathcal Y}) \, \mathrm{d}{\bf x}_k \right] \\
& = & \Pi_1(\mathcal X') .\end{aligned}$$ The last equality uses the fact that $\pi$ is the stationary distribution of $S({\bf x} \to {\bf x'} ;\nu_{\mathcal
Y},\boldsymbol\mu_{\mathcal Y},\boldsymbol\Sigma_{\mathcal Y})$, so we see that $Q$ leaves the desired product distribution invariant.
Within a single chain, elliptical slice sampling has a nonzero probability of transitioning to any region that has nonzero probability under the posterior, as described by @Murray2010. The transition operator $Q$ updates the chains in a given group independently of one another. Therefore $Q$ has a nonzero probability of transitioning to any region that has nonzero probability under the product distribution. It follows that the transition operator is both irreducible and aperiodic. These conditions together ensure that this Markov transition operator has a unique invariant distribution, namely $\Pi$, and that the distribution over the state of the Markov chain created from this transition operator will converge to this invariant distribution [@Roberts2004]. It follows that, in the limit, samples derived from the repeated application of $Q$ will be drawn from the desired distribution.
Cost and Complexity
-------------------
There is a cost to the construction of the multivariate $t$ approximation. Although the user has some flexibility in the choice of $t$ parameters, we fit them with the iterative algorithm described by @Liu1995 and in Algorithm \[alg:t-params\] of the Appendix. Let $D$ be the dimension of the distribution and let $K$ be the number of parallel chains. Then the complexity of each iteration is $O(D^3K)$, which comes from the fact that we invert a $D \times D$ matrix for each of the $K$ chains. Empirically, Algorithm \[alg:t-params\] appears to converge in a small number of iterations when the number of parallel Markov chains in each group exceeds the dimension of the distribution. As described in the next section, this cost can be amortized by reusing the same approximation for multiple updates. On the challenging distributions that most interest us, the cost of constructing the approximation (when amortized in this manner), will be negligible compared to the cost of evaluating the density function.
An additional concern is the overhead from sharing information between chains. The chains must communicate in order to build a multivariate $t$ approximation, and so the updates must be synchronized. Since elliptical slice sampling requires a variable amount of time, updating the different chains will take different amounts of time, and the faster chains may end up waiting for the slower ones. We can mitigate this cost by performing multiple updates between such periods of information sharing. In this manner, we can perform as much computation as we want between synchronizations without compromising the validity of the algorithm. As we increase the number of updates performed between synchronizations, the fraction of time spent waiting will diminish.
The time measured in our experiments is wall-clock time, which includes the overhead from constructing the approximation and from synchronizing the chains.
Reusing the Approximation {#reusing_approx}
-------------------------
Here we explain that reusing the same approximation is valid. To illustrate this point, let the transition operators $Q_1$ and $Q_2$ be defined as before. In our description of the algorithm, we defined the transition operator $Q$ as the composition ${Q = Q_2Q_1}$. However, both $Q_1$ and $Q_2$ preserve the desired product distribution, so we may use any transition operator of the form ${Q =
Q_2^{r_2}Q_1^{r_1}}$, where this notation indicates that we first apply $Q_1$ for $r_1$ rounds and then we apply $Q_2$ for $r_2$ rounds. As long as $r_2, r_1 \ge 1$, the composite transition operator is ergodic. When we apply $Q_1$ multiple times in a row, the states $\mathcal Y$ do not change, so if $Q_1$ computes $\nu_{\mathcal
Y}$, $\boldsymbol\mu_{\mathcal Y}$, and $\boldsymbol\Sigma_{\mathcal
Y}$ deterministically from $\mathcal Y$, then we need only compute these values once. Reusing the approximation works even if $Q_1$ samples $\nu_{\mathcal Y}$, $\boldsymbol\mu_{\mathcal Y}$, and $\boldsymbol\Sigma_{\mathcal Y}$ from some distribution. In this case, we can model the randomness by introducing a separate variable $r_{\mathcal Y}$ in the Markov chain, and letting $Q_1$ compute $\nu_{\mathcal Y}$, $\boldsymbol\mu_{\mathcal Y}$, and $\boldsymbol\Sigma_{\mathcal Y}$ deterministically from $\mathcal
Y$ and $r_{\mathcal Y}$.
Our algorithm maintains two collections of Markov chains, one of which will always be idle. Therefore, each collection can take advantage of all available cores. Given $K$ cores, it makes sense to use two collections of $K$ Markov chains. In general, it seems to be a good idea to sample equally from both collections so that the chains in both collections burn in.
To motivate reusing the approximation, we demonstrate the effect of reusing the approximation for different numbers of iterations on a Gaussian distribution in $100$ dimensions (the same one that we use in Section \[sec:scaling\_experiments\]). For each value of $i$ from $1$ to $4$, we sample this distribution for $10^4$ iterations and we reuse each approximation for $10^i$ iterations. We show plots of the running time of GESS and the convergence of the approximation for different values of $i$. Figure \[fig:reuse\_approx\] shows how the amount of time required by GESS changes as we vary $i$, and how the covariance matrix parameter of the fitted multivariate $t$ approximation changes over time for the different values of $i$. We summarize the covariance matrix parameter by its trace $\text{tr}(\boldsymbol\Sigma)$. The figure shows that increasing the number of iterations for which we reuse the approximation can dramatically reduce the amount of time required by GESS. It also shows that if we rebuild the approximation frequently, the approximation will settle on a reasonable approximation in fewer iterations. However, there is little difference between rebuilding the approximation every $10$ iterations versus every $100$ iterations (in terms of the number of iterations required), while there is a dramatic difference in the time required.
[0.5]{} ![We used GESS to sample a multivariate Gaussian distribution in $100$ dimensions for $10^4$ iterations. We repeated this procedure four times, reusing the approximation for $10^1$, $10^2$, $10^3$, and $10^4$ iterations. [**(a)**]{} Shows the durations (in seconds) of the sampling procedures as we varied the number of iterations for which we reused the approximation. [**(b)**]{} Shows how $\text{tr}(\boldsymbol\Sigma)$ changes over time in the four different settings.[]{data-label="fig:reuse_approx"}](figures/repeats_times-eps-converted-to.pdf "fig:")
[0.5]{} ![We used GESS to sample a multivariate Gaussian distribution in $100$ dimensions for $10^4$ iterations. We repeated this procedure four times, reusing the approximation for $10^1$, $10^2$, $10^3$, and $10^4$ iterations. [**(a)**]{} Shows the durations (in seconds) of the sampling procedures as we varied the number of iterations for which we reused the approximation. [**(b)**]{} Shows how $\text{tr}(\boldsymbol\Sigma)$ changes over time in the four different settings.[]{data-label="fig:reuse_approx"}](figures/repeats_traces-eps-converted-to.pdf "fig:")
Related Work {#sec:related_work}
============
Our work uses updates on a product distribution in the style of Adaptive Direction Sampling [@gilks1994], which has inspired a large literature of related methods. The closest research to our work makes use of slice-sampling based updates of product distributions along straight-line directions chosen by sampling pairs of points [@MacKay2003; @braak2006]. The work on elliptical slice sampling suggests that in high dimensions larger steps can be taken along curved trajectories, given an appropriate Gaussian fit. Using closed ellipses also removes the need to set an initial step size or to build a bracket.
The recent affine invariant ensemble sampler [@goodman2010] also uses Gaussian fits to a population, in that case to make Metropolis proposals. Our work differs by using a scale-mixture of Gaussians and elliptical slice sampling to perform updates on a variety of scales with self-adjusting step-sizes. Rather than updating each member of the population in sequence, our approach splits the population into two groups and allows the members of each group to be updated in parallel.
Population MCMC with parallel tempering [@Friel2008] is another parallel sampling approach that involves sampling from a product distribution. It uses separate chains to sample a sequence of distributions interpolating between the target distribution and a simpler distribution. The different chains regularly swap states to encourage mixing. In this setting, samples are generated only from a single chain, and all of the others are auxiliary. However, some tasks such as computing model evidence can make use of samples from all of the chains [@Friel2008].
Recent work on Hamiltonian Monte Carlo has attempted to reduce the tuning burden [@Hoffman2014]. A user friendly tool that combines this work with a software stack supporting automatic differentiation is under development [@Stan2012]. We feel that this alternative line of work demonstrates the interest in more practical MCMC algorithms applicable to a variety of continuous-valued parameter spaces and is very promising. Our complementary approach introduces simpler algorithms with fewer technical software requirements. In addition, our two-population approach to parallelization could be applied with whichever methods become dominant in the future.
Experiments {#sec:experiments}
===========
In this section, we compare Algorithm \[alg:parallel-update\] with other parallel MCMC algorithms by measuring how quickly the Markov chains mix on a number of different distributions. Second, we compare how the performance of Algorithm \[alg:parallel-update\] scales with the dimension of the target distribution, the number of cores used, and the number of chains used per core.
These experiments were run on an EC2 cluster with $5$ nodes, each with two eight-core Intel Xeon E5-2670 CPUs. We implement all algorithms in Python, using the IPython environment [@Perez2007] for parallelism.
Comparing Mixing {#sec:mixing}
----------------
We empirically compare the mixing of the parallel MCMC samplers on seven distributions. We quantify their mixing by comparing the effective number of samples produced by each method. This quantity can be approximated as the product of the number of chains with the effective number of samples from the product distribution. We estimate the effective number of samples from the product distribution by computing the effective number of samples from its sequence of log likelihoods. We compute effective sample size using R-CODA [@coda], and we compare the results using two metrics: effective samples per second and effective samples per density function evaluation (in the case of Hamiltonian Monte Carlo, we count gradient evaluations as density function evaluations).
In each experiment, we run each algorithm with $100$ parallel chains. Unless otherwise noted, we burn in for $10^4$ iterations and sample for $10^5$ iterations. We run five trials for each experiment to estimate variability.
Figure \[fig:results\] shows the average effective number of samples, with error bars, according to the two different metrics. Bars are omitted where the sequence of aggregate log likelihoods did not converge according to the Geweke convergence diagnostic [@Geweke1992]. We diagnose this using the tool from R-CODA [@coda].
### Samplers Considered
We compare generalized elliptical slice sampling (GESS) with parallel versions of several different sampling algorithms.
First, we consider random-direction slice sampling (RDSS) [@MacKay2003] and coordinate-wise slice sampling (CWSS) [@Neal2003]. These are variants of slice sampling which differ in their choice of direction (a random direction versus a random axis-aligned direction) in which to sample. RDSS is rotation invariant like GESS, but CWSS is not.
In addition, we compare to a simple Metropolis–Hastings (MH) [@Metropolis1953] algorithm whose proposal distribution is a spherical Gaussian centered on the current state. A tuning period is used to adjust the MH step size so that the acceptance ratio is as close as possible to the value $0.234$, which is optimal in some settings [@Roberts1998]. This tuning is done independently for each chain. We also compare to an adaptive MCMC (AMH) algorithm following the approach in @Roberts2006 in which the covariance of a Metropolis–Hastings proposal is adapted to the history of the “Markov” chain.
We also compare to the No-U-Turn sampler [@Hoffman2014], which is an implementation of Hamiltonian Monte Carlo (HMC) combined with procedures to automatically tune the step size parameter and the number of steps parameter. Due to the large number of function evaluations per sample required by HMC, we run HMC for a factor of $10$ or $100$ fewer iterations in order to make the algorithms roughly comparable in terms of wall-clock time. Though we include the comparisons, we do not view HMC as a perfectly comparable algorithm due to its requirement that the density function of the target distribution be differentiable. Though the target distribution is often differentiable in principle, there are many practical situations in which the gradient is difficult to access, either by manual computation or by automatic differentiation, possibly because evaluating the density function requires running a complicated black-box subroutine. For instance, in computer vision problems, evaluating the likelihood function may require rendering an image or running graph cuts. See @Tarlow2012b or @Lang2012 for examples.
We compare to parallel tempering (PT) [@Friel2008], using each Markov chain to sample the distribution at a different temperature (if the target distribution has density $\pi({\bf x})$, then the distribution “at temperature $t$” has density proportional to $\pi({\bf x})^{1/t}$) and swapping states between the Markov chains at regular intervals. Samples from the target distribution are produced by only one of the chains. Using PT requires the practitioner to pick a temperature schedule, and doing so often requires a significant amount of experimentation [@Neal2001]. We follow the practice of @Friel2008 and use a geometric temperature schedule. As with HMC, we do not view PT as entirely comparable in the absence of an automatic and principled way to choose the temperatures of the different Markov chains. One of the main goals of GESS is to provide a black-box MCMC algorithm that imposes as few restrictions on the target distribution as possible and that requires no expertise or experimentation on the part of the user.
### Distributions
In this section, we describe the different distributions that we use to compare the mixing of our samplers.
[*Funnel:*]{} A ten-dimensional funnel-shaped distribution described in @Neal2003. The first coordinate is distributed normally with mean zero and variance nine. Conditioned on the first coordinate $v$, the remaining coordinates are independent identically-distributed normal random variables with mean zero and variance $e^v$. In this experiment, we initialize each Markov chain from a spherical multivariate Gaussian centered on the origin.
[*Gaussian Mixture:*]{} An eight-component mixture of Gaussians in eight dimensions. Each component is a spherical Gaussian with unit variance. The components are distributed uniformly at random within a hypercube of edge length four. In this experiment, we initialize each Markov chain from a spherical multivariate Gaussian centered on the origin.
[*Breast Cancer:*]{} The posterior density of a linear logistic regression model for a binary classification problem with thirty explanatory variables (thirty-one dimensions) using the Breast Cancer Wisconsin data set [@Street1993]. The data set consists of $569$ data points. We scale the data so that each coordinate has unit variance, and we place zero-mean Gaussian priors with variance $100$ on each of the regression coefficients. In this experiment, we initialize each Markov chain from a spherical multivariate Gaussian centered on the origin.
[*German Credit:*]{} The posterior density of a linear logistic regression model for a binary classification problem with twenty-four explanatory variables (twenty-five dimensions) from the UCI repository [@Frank2010]. The data set consists of $1000$ data points. We scale the data so that each coordinate has unit variance, and we place zero-mean Gaussian priors with variance $100$ on each of the regression coefficients. In this experiment, we initialize each Markov chain from a spherical multivariate Gaussian centered on the origin.
[*Stochastic Volatility:*]{} The posterior density of a simple stochastic volatility model fit to synthetic data in fifty-one dimensions. This distribution is a smaller version of a distribution described in @Hoffman2014. In this experiment, we burn-in for $10^5$ iterations and sample for $10^5$ iterations. We initialize each Markov chain from a spherical multivariate Gaussian centered on the origin and we take the absolute value of the first parameter, which is constrained to be positive.
[*Ionosphere:*]{} The posterior density on covariance hyperparameters for Gaussian process regression applied to the Ionosphere data set [@Sigillito1989]. We use a squared exponential kernel with thirty-four length-scale hyperparameters and $100$ data points. We place exponential priors with rate $0.1$ on the length-scale hyperparameters. In this experiment, we burn-in for $10^4$ iterations and sample for $10^4$ iterations. We initialize each Markov chain from a spherical multivariate Gaussian centered on the vector $(1, \ldots,
1)^{\mathsf T}$.
[*SNP Covariates:*]{} The posterior density of the parameters of a generative model for gene expression levels simulated in thirty-eight dimensions using actual genomic sequences from $480$ individuals for covariate data [@Engelhardt2014]. In this experiment, we burn-in for $2000$ iterations and sample for $10^4$ iterations. We initialize each Markov chain from a spherical multivariate Gaussian centered on the origin and we take the absolute value of the first three parameters, which are constrained to be positive.
### Mixing Results
[0.5]{} ![The results of experimental comparisons of seven parallel MCMC methods on seven distributions. Each figure shows seven groups of bars, (one for each distribution) and the vertical axis shows the effective number of samples per unit cost. Error bars are included. Bars are omitted where the method failed to converge according to the Geweke diagnostic [@Geweke1992]. The costs are *per second* (left) and *per density function evaluation* (right). Mean and standard error for five runs are shown. Each group of bars has been rescaled for readability: the number beneath each group gives the effective samples corresponding to CWSS, which always has height one. []{data-label="fig:results"}](figures/samples_per_sec-eps-converted-to "fig:")
[0.5]{} ![The results of experimental comparisons of seven parallel MCMC methods on seven distributions. Each figure shows seven groups of bars, (one for each distribution) and the vertical axis shows the effective number of samples per unit cost. Error bars are included. Bars are omitted where the method failed to converge according to the Geweke diagnostic [@Geweke1992]. The costs are *per second* (left) and *per density function evaluation* (right). Mean and standard error for five runs are shown. Each group of bars has been rescaled for readability: the number beneath each group gives the effective samples corresponding to CWSS, which always has height one. []{data-label="fig:results"}](figures/samples_per_eval-eps-converted-to "fig:")
The results of the mixing experiments are shown in Figure \[fig:results\]. For the most part, GESS sampled more effectively than the other algorithms according to both metrics. The poor performance of PT can be attributed to the fact that PT only produces samples from one of its chains, unlike the other algorithms, which produce samples from $100$ chains. HMC also performed well, although it failed to converge on the SNP Covariates distribution. The density function of this particular distribution is only piecewise continuous, with the discontinuities arising from thresholding in the model. In this case, the gradient and curvature largely reflect the prior, whereas the likelihood mostly manifests itself in the discontinuities of the distribution.
One reason for the rapid mixing of GESS is that GESS performs well even on highly-skewed distributions. RDSS, CWSS, MH, and PT propose steps in uninformed directions, the vast majority of which lead away from the region of high density. As a result, these algorithms take very small steps, causing successive states to be highly correlated. In the case of GESS, the multivariate $t$ approximation builds information about the global shape of the distribution (including skew) into the transition operator. As a consequence, the Markov chain can take long steps along the length of the distribution, allowing the Markov chain to mix much more rapidly. Skewed distributions can arise as a result of the user not knowing the relative length scales of the parameters or as a result of redundancy in the parameterization. Therefore, the ability to perform well on such distributions is frequently relevant.
These results show that a multivariate $t$ approximation to the target distribution provides enough information to greatly speed up the mixing of the sampler and that this information can be used to improve the convergence of the sampler. These improvements occur on top of the performance gain from using parallelism.
Scaling the Number of Cores {#sec:scaling_experiments}
---------------------------
[1.0]{}
[ X|XXXXX ]{} $D=50$ & $K=C$ & $K=2C$ & $K=3C$ & $K=4C$ & $K=5C$\
$C=20$ & ${\color{blue}\scriptstyle{10^{-0.5} \pm 10^{-0.4}}}$ & ${\color{blue}\scriptstyle{10^{-1.2} \pm 10^{-1.5}}}$ & ${\color{blue}\scriptstyle{10^{-1.5} \pm 10^{-1.8}}}$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-1.8}}}$ & ${\color{blue}\scriptstyle{10^{-1.6} \pm 10^{-1.6}}}$\
$C=40$ & ${\color{blue}\scriptstyle{10^{-0.8} \pm 10^{-0.9}}}$ & ${\color{blue}\scriptstyle{10^{-2.6} \pm 10^{-2.6}}}$ & ${\color{blue}\scriptstyle{10^{-1.9} \pm 10^{-1.9}}}$ & ${\color{blue}\scriptstyle{10^{-1.8} \pm 10^{-1.8}}}$ & ${\color{blue}\scriptstyle{10^{-2.4} \pm 10^{-2.6}}}$\
$C=60$ & ${\color{blue}\scriptstyle{10^{-1.6} \pm 10^{-1.5}}}$ & ${\color{blue}\scriptstyle{10^{-1.6} \pm 10^{-1.7}}}$ & ${\color{blue}\scriptstyle{10^{-2.1} \pm 10^{-2.2}}}$ & ${\color{blue}\scriptstyle{10^{-2.1} \pm 10^{-2.2}}}$ & ${\color{blue}\scriptstyle{10^{-2.2} \pm 10^{-2.4}}}$\
$C=80$ & ${\color{blue}\scriptstyle{10^{-1.3} \pm 10^{-1.1}}}$ & ${\color{blue}\scriptstyle{10^{-2.4} \pm 10^{-2.8}}}$ & ${\color{blue}\scriptstyle{10^{-2.4} \pm 10^{-2.4}}}$ & ${\color{blue}\scriptstyle{10^{-2.1} \pm 10^{-2.4}}}$ & ${\color{blue}\scriptstyle{10^{-2.3} \pm 10^{-2.5}}}$\
$C=100$ & ${\color{blue}\scriptstyle{10^{-1.6} \pm 10^{-1.7}}}$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-1.7}}}$ & ${\color{blue}\scriptstyle{10^{-2.0} \pm 10^{-2.0}}}$ & ${\color{blue}\scriptstyle{10^{-2.2} \pm 10^{-2.4}}}$ & ${\color{blue}\scriptstyle{10^{-2.5} \pm 10^{-2.3}}}$
[1.0]{}
[ X|XXXXX ]{} $D=100$ & $K=C$ & $K=2C$ & $K=3C$ & $K=4C$ & $K=5C$\
$C=20$ & $\scriptstyle{10^{+0.3} \pm 10^{+0.2}}$ & ${\color{blue}\scriptstyle{10^{-1.3} \pm 10^{-2.2}}}$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-2.1}}}$ & ${\color{blue}\scriptstyle{10^{-1.9} \pm 10^{-2.2}}}$ & ${\color{blue}\scriptstyle{10^{-2.4} \pm 10^{-3.5}}}$\
$C=40$ & ${\color{blue}\scriptstyle{10^{-1.1} \pm 10^{-1.1}}}$ & ${\color{blue}\scriptstyle{10^{-1.9} \pm 10^{-2.1}}}$ & ${\color{blue}\scriptstyle{10^{-2.5} \pm 10^{-3.2}}}$ & ${\color{blue}\scriptstyle{10^{-2.5} \pm 10^{-2.6}}}$ & ${\color{blue}\scriptstyle{10^{-2.7} \pm 10^{-3.0}}}$\
$C=60$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-2.0}}}$ & ${\color{blue}\scriptstyle{10^{-2.5} \pm 10^{-2.8}}}$ & ${\color{blue}\scriptstyle{10^{-2.8} \pm 10^{-3.0}}}$ & ${\color{blue}\scriptstyle{10^{-2.9} \pm 10^{-3.4}}}$ & ${\color{blue}\scriptstyle{10^{-2.9} \pm 10^{-3.1}}}$\
$C=80$ & ${\color{blue}\scriptstyle{10^{-2.1} \pm 10^{-2.7}}}$ & ${\color{blue}\scriptstyle{10^{-2.7} \pm 10^{-2.8}}}$ & ${\color{blue}\scriptstyle{10^{-2.7} \pm 10^{-3.0}}}$ & ${\color{blue}\scriptstyle{10^{-2.9} \pm 10^{-3.2}}}$ & ${\color{blue}\scriptstyle{10^{-3.1} \pm 10^{-4.0}}}$\
$C=100$ & ${\color{blue}\scriptstyle{10^{-2.4} \pm 10^{-2.6}}}$ & ${\color{blue}\scriptstyle{10^{-2.8} \pm 10^{-3.3}}}$ & ${\color{blue}\scriptstyle{10^{-3.0} \pm 10^{-3.5}}}$ & ${\color{blue}\scriptstyle{10^{-3.0} \pm 10^{-3.6}}}$ & ${\color{blue}\scriptstyle{10^{-2.9} \pm 10^{-3.0}}}$
[1.0]{}
[ X|XXXXX ]{} $D=150$ & $K=C$ & $K=2C$ & $K=3C$ & $K=4C$ & $K=5C$\
$C=20$ & $\scriptstyle{10^{+2.3} \pm 10^{+1.4}}$ & $\scriptstyle{10^{+2.3} \pm 10^{+1.7}}$ & $\scriptstyle{10^{+1.4} \pm 10^{+1.0}}$ & $\scriptstyle{10^{+0.5} \pm 10^{+0.2}}$ & ${\color{blue}\scriptstyle{10^{-0.7} \pm 10^{-1.0}}}$\
$C=40$ & $\scriptstyle{10^{+2.1} \pm 10^{+1.6}}$ & ${\color{blue}\scriptstyle{10^{-0.1} \pm 10^{-0.0}}}$ & ${\color{blue}\scriptstyle{10^{-1.1} \pm 10^{-1.2}}}$ & ${\color{blue}\scriptstyle{10^{-1.4} \pm 10^{-1.4}}}$ & ${\color{blue}\scriptstyle{10^{-1.8} \pm 10^{-1.7}}}$\
$C=60$ & $\scriptstyle{10^{+1.3} \pm 10^{+0.7}}$ & ${\color{blue}\scriptstyle{10^{-1.2} \pm 10^{-1.2}}}$ & ${\color{blue}\scriptstyle{10^{-1.6} \pm 10^{-1.5}}}$ & ${\color{blue}\scriptstyle{10^{-1.9} \pm 10^{-2.0}}}$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-1.6}}}$\
$C=80$ & ${\color{blue}\scriptstyle{10^{-0.0} \pm 10^{-0.0}}}$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-1.8}}}$ & ${\color{blue}\scriptstyle{10^{-2.2} \pm 10^{-2.3}}}$ & ${\color{blue}\scriptstyle{10^{-1.9} \pm 10^{-2.0}}}$ & ${\color{blue}\scriptstyle{10^{-2.1} \pm 10^{-2.6}}}$\
$C=100$ & ${\color{blue}\scriptstyle{10^{-0.7} \pm 10^{-1.0}}}$ & ${\color{blue}\scriptstyle{10^{-1.8} \pm 10^{-2.1}}}$ & ${\color{blue}\scriptstyle{10^{-1.9} \pm 10^{-2.1}}}$ & ${\color{blue}\scriptstyle{10^{-2.0} \pm 10^{-2.1}}}$ & ${\color{blue}\scriptstyle{10^{-2.3} \pm 10^{-2.3}}}$
[1.0]{}
[ X|XXXXX ]{} $D=200$ & $K=C$ & $K=2C$ & $K=3C$ & $K=4C$ & $K=5C$\
$C=20$ & $\scriptstyle{10^{+2.8} \pm 10^{+2.5}}$ & $\scriptstyle{10^{+3.0} \pm 10^{+2.4}}$ & $\scriptstyle{10^{+3.1} \pm 10^{+2.1}}$ & $\scriptstyle{10^{+3.1} \pm 10^{+1.9}}$ & $\scriptstyle{10^{+3.0} \pm 10^{+1.5}}$\
$C=40$ & $\scriptstyle{10^{+3.1} \pm 10^{+1.6}}$ & $\scriptstyle{10^{+3.1} \pm 10^{+1.7}}$ & $\scriptstyle{10^{+2.7} \pm 10^{+1.6}}$ & $\scriptstyle{10^{+1.1} \pm 10^{+0.6}}$ & ${\color{blue}\scriptstyle{10^{-1.4} \pm 10^{-1.6}}}$\
$C=60$ & $\scriptstyle{10^{+3.1} \pm 10^{+1.6}}$ & $\scriptstyle{10^{+2.6} \pm 10^{+1.8}}$ & ${\color{blue}\scriptstyle{10^{-0.6} \pm 10^{-0.8}}}$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-2.0}}}$ & ${\color{blue}\scriptstyle{10^{-2.0} \pm 10^{-2.8}}}$\
$C=80$ & $\scriptstyle{10^{+3.1} \pm 10^{+1.7}}$ & $\scriptstyle{10^{+0.7} \pm 10^{+0.1}}$ & ${\color{blue}\scriptstyle{10^{-1.7} \pm 10^{-2.3}}}$ & ${\color{blue}\scriptstyle{10^{-1.9} \pm 10^{-1.9}}}$ & ${\color{blue}\scriptstyle{10^{-2.1} \pm 10^{-2.5}}}$\
$C=100$ & $\scriptstyle{10^{+3.0} \pm 10^{+2.1}}$ & ${\color{blue}\scriptstyle{10^{-1.4} \pm 10^{-1.6}}}$ & ${\color{blue}\scriptstyle{10^{-2.3} \pm 10^{-2.8}}}$ & ${\color{blue}\scriptstyle{10^{-2.0} \pm 10^{-2.6}}}$ & ${\color{blue}\scriptstyle{10^{-2.3} \pm 10^{-2.9}}}$
[1.0]{}
[ X|XXXXX ]{} $D=250$ & $K=C$ & $K=2C$ & $K=3C$ & $K=4C$ & $K=5C$\
$C=20$ & $\scriptstyle{10^{+3.5} \pm 10^{+2.0}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.5}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.7}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.4}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.6}}$\
$C=40$ & $\scriptstyle{10^{+3.5} \pm 10^{+2.3}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.3}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.6}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+2.1}}$ & $\scriptstyle{10^{+3.6} \pm 10^{+1.8}}$\
$C=60$ & $\scriptstyle{10^{+3.5} \pm 10^{+2.0}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.6}}$ & $\scriptstyle{10^{+3.6} \pm 10^{+2.1}}$ & $\scriptstyle{10^{+3.6} \pm 10^{+2.4}}$ & $\scriptstyle{10^{+2.3} \pm 10^{+1.9}}$\
$C=80$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.6}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.9}}$ & $\scriptstyle{10^{+3.5} \pm 10^{+2.2}}$ & $\scriptstyle{10^{+1.1} \pm 10^{+0.8}}$ & ${\color{blue}\scriptstyle{10^{-0.8} \pm 10^{-0.9}}}$\
$C=100$ & $\scriptstyle{10^{+3.5} \pm 10^{+1.8}}$ & $\scriptstyle{10^{+3.6} \pm 10^{+2.0}}$ & $\scriptstyle{10^{+2.2} \pm 10^{+1.7}}$ & $\scriptstyle{10^{+0.3} \pm 10^{+0.2}}$ & ${\color{blue}\scriptstyle{10^{-0.1} \pm 10^{-0.2}}}$
We wish to explore the performance of GESS as a function of the dimension $D$ of the target distribution, the number $C$ of cores available, and the number $K$ of parallel chains. In this experiment, we consider all $125$ triples $(D,C,K)$ such that $$\begin{aligned}
D & \in & \{50,100,150,200,250\} \\
C & \in & \{20,40,60,80,100\} \\
K & \in & \{C,2C,3C,4C,5C\} .\end{aligned}$$ It makes sense to let $K$ be an integer multiple of $C$ so that each core will be tasked with updating the same number of chains (the experiments in Section \[sec:mixing\] set $K$ equal to $C$).
For each triple $(D,C,K)$, we sample a $D$-dimensional multivariate Gaussian distribution centered on the origin whose precision matrix was generated from a Wishart distribution with identity scale matrix and $D$ degrees of freedom. The distributions used in this experiment were modeled off of one of the distributions considered in @Hoffman2014. We initialize GESS from a broad spherical Gaussian distribution centered on the origin, and we run GESS for $500$ seconds. The first half of the resulting samples are discarded, and the second half of the resulting samples are used to estimate the vector $\boldsymbol\sigma=(\sigma_1,\ldots,\sigma_D)$, where $\sigma_d$ is the marginal standard deviation of the $d$th coordinate. For each triple $(D,C,K)$, we run five trials. Figure \[fig:scaling\_table\] shows the resulting average squared error in the empirical estimate of $\boldsymbol\sigma$ after $500$ seconds. Error bars are included as well.
When aggregating samples from $K$ independent Markov chains, we would expect the squared error of our estimator to decrease at the rate $1/K$. However, in the setting of GESS, additional chains not only provide additional samples, but may enable the construction of a more accurate approximation to the target distribution thereby enabling the other chains to sample more effectively. In some situations, the presence of additional chains can even enable the sampler to converge in situations where it otherwise would not.
We can see this effect in Figure \[fig:scaling\_table\]. Singling out the column corresponding to $D=200$ and $K=3C$, we notice that using either $20$ or $40$ cores, GESS fails to estimate $\boldsymbol\sigma$, indeed the Markov chain fails to burn in during the allotted time (the average squared errors are $10^{3.1}$ and $10^{2.7}$ respectively). However, once we increase the number of cores to $60$, $80$, and $100$, GESS provides an accurate estimate of $\boldsymbol\sigma$ (the average squared errors are $10^{-0.6}$, $10^{-1.7}$, and $10^{-2.3}$ respectively). In this case, increasing the number of cores enabled our estimator to converge. This property contrasts sharply with many other approaches to parallel sampling. If a single Markov chain running MH will not converge, then one-hundred chains running MH will not converge either.
Discussion
==========
In this paper, we generalized elliptical slice sampling to handle arbitrary continuous distributions using a scale mixture of Gaussians to approximate the target distribution. We then showed that parallelism can be used to dynamically choose the parameters of the scale mixture of Gaussians in a way that encodes information about the shape of the target distribution in the transition operator. The result is Markov chain Monte Carlo algorithm with a number of desirable properties. In particular, it mixes well in the presence of strong dependence, it does not require hand tuning, and it can be parallelized over hundreds of cores.
We compared our algorithm to several other parallel MCMC algorithms in a variety of settings. We found that generalized elliptical slice sampling (GESS) mixed more rapidly than the other algorithms on a variety of distributions, and we found evidence that the performance of GESS can scale superlinearly in the number of available cores.
One possible area of future work is reducing the overhead from the information sharing. In Section \[reusing\_approx\] we remarked that the synchronization requirement leads to faster chains waiting for slower chains. There are a number of factors which contribute to the difference in speed from chain to chain. Most obviously, some chains may be running on faster machines. More subtly, the slice sampling procedure performs a variable number of function evaluations per update, and the average number of required updates may be a function of location. For instance, Markov chains whose current states lie in narrow portions of the distribution may require more function evaluations per update. In each situation, the chains with the rapid updates end up waiting for the chains with the slower updates, leaving some processors idle. We imagine that a cleverly-engineered system would be able to account for the potentially different update speeds, perhaps by sending the chains in the narrower parts of the distribution to the faster machines or by allowing the slower chains to spawn multiple threads. Properly done, the performance gain in wall-clock time due to using GESS should approach the gain as measured by function evaluations.
In addition to using parallelism to distribute the computational load of MCMC, we saw that our algorithm was able to use information from the parallel chains to speed up mixing. One area of future work is extending the algorithm to take advantage of a greater number of cores. The magnitude of this performance gain depends on the accuracy of our multivariate $t$ approximation, which will increase, to a point, as the number of available cores grows. However, there is a limit to how well a multivariate $t$ distribution can approximate an arbitrary distribution. We chose to use the multivariate $t$ distribution because it has the flexibility to capture the general allocation of probability mass of a given distribution, but it is too coarse to capture more complex features such as the locations of multiple modes. After some point, the approximation will not significantly improve. A more general approach would be to use a scale-location mixture of Gaussians, which could accurately approximate a much larger class of distributions. The idea of approximating the target distribution with a mixture of Gaussians has been explored by @Schmidler2010 in the context of adaptive Metropolis–Hastings. We leave it to future work to explore this more general setting.
Appendix A {#sec:appendix .unnumbered}
==========
In Algorithm \[alg:t-params\], we detail the algorithm for estimating the maximum likelihood multivariate $t$ parameters $\nu$, $\boldsymbol\mu$, $\boldsymbol\Sigma$ from @Liu1995.
$I$ points ${\bf x}_i$ (each $D$ dimensional) Maximum likelihood multivariate $t$ parameters $\nu$, $\boldsymbol\mu$, $\boldsymbol\Sigma$ $t \leftarrow 0$ Initialize $\nu^{(0)}$, $\boldsymbol\mu^{(0)}$, and $\boldsymbol\Sigma^{(0)}$ Compute the distances from each point ${\bf x}_i$ to $\boldsymbol\mu^{(t)}$ with respect to $\boldsymbol\Sigma^{(t)}$ $$\delta_i^{(t)} = \left({\bf x}_i - \boldsymbol\mu^{(t)}\right)^{\mathsf T}\left(\boldsymbol\Sigma^{(t)}\right)^{-1}\left({\bf x}_i - \boldsymbol\mu^{(t)}\right)$$
Set $$w_i^{(t+1)} = \frac{\nu^{(t)} + D}{\nu^{(t)} + \delta_i^{(t)}}$$
Update the mean and covariance parameters via $$\begin{aligned}
\boldsymbol\mu^{(t+1)} & = &\frac{\sum_{i=1}^I w_i^{(t+1)} {\bf x}_i}{ \sum_{i=1}^I w_i^{(t+1)}} \\
\boldsymbol\Sigma^{(t+1)} & = & \frac{1}{I}\sum_{i=1}^I w_i^{(t+1)}\left( {\bf x}_i - \boldsymbol\mu^{(t)}\right) \left({\bf x}_i - \boldsymbol\mu^{(t)}\right)^{\mathsf T}\end{aligned}$$
Using the updated mean and covariance parameters, recompute the distance $$\delta_i^{(t+1)} = \left({\bf x}_i - \boldsymbol\mu^{(t+1)}\right)^{\mathsf T}\left(\boldsymbol\Sigma^{(t+1)}\right)^{-1}\left({\bf x}_i - \boldsymbol\mu^{(t+1)}\right)$$
Let $\psi$ be the digamma function, and let $$w_i = \frac{\nu + D}{\nu + \delta_i^{(t+1)}}$$
Set $\nu^{(t+1)}$ by solving for $\nu$ in the equation $$-\psi\left(\frac{\nu}{2}\right) + \log\left(\frac{\nu}{2}\right) + \frac{1}{I} \sum_{i=1}^I \left(\log\left(w_i\right) - w_i\right) + \psi\left( \frac{\nu + D}{2} \right) - \log\left( \frac{\nu + D}{2} \right) = -1$$
$t \leftarrow t + 1$
$\nu^{(t)}$, $\boldsymbol\mu^{(t)}$, and $\boldsymbol\Sigma^{(t)}$
0.2in
|
---
abstract: 'We introduce an enriched analogue of Lam and Pylyavskyy’s theory of set-valued $P$-partitions. An an application, we construct a $K$-theoretic version of Stembridge’s Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this algebra is generated by Ikeda and Naruse’s shifted stable Grothendieck polynomials. We give the first proof that the natural skew analogues of these power series are also symmetric. A central tool in our constructions is a “$K$-theoretic” Hopf algebra of labeled posets, which may be of independent interest. Our results also lead to some new explicit formulas for the involution $\omega$ on the ring of symmetric functions.'
author:
- |
Joel Brewster Lewis\
Department of Mathematics\
George Washington University\
[jblewis@gwu.edu]{}
- |
Eric Marberg\
Department of Mathematics\
HKUST\
[eric.marberg@gmail.com]{}
bibliography:
- 'shifted\_stable\_grothendieck.bib'
title: 'Enriched set-valued $P$-partitions and shifted stable Grothendieck polynomials'
---
Introduction
============
Stanley developed the now-classical theory of *(ordinary) $P$-partitions* in [@St1]. These are certain maps from (finite) labeled posets to the positive integers $\PP :=\{1,2,3,\dots\}$. They can be seen as generalizations of semistandard Young tableaux, and provide a streamlined description of the tableau generating functions for the skew Schur functions $s_{\lambda/\mu}$.
Following Stanley’s work, the theory of $P$-partitions has been generalized and extended in a number of ways. Stembridge [@Stembridge1997a] introduced *enriched $P$-partitions*, which are certain maps from labeled posets to the *marked integers* $\MM := \{ 1' < 1 < 2'<2< \dots\}$. They can be seen as generalizations of semistandard shifted (marked) tableaux, whose generating functions give the *Schur $P$- and $Q$-functions* $P_\lambda$ and $Q_\lambda$. In [@LamPyl], Lam and Pylyavskyy defined *set-valued $P$-partitions*, which are maps that now take finite nonempty subsets of $\PP$ as values. These maps are generalizations of semistandard set-valued tableaux, which play a role in the generating functions for the *stable Grothendieck polynomials* $G^{(\beta)}_\lambda$ studied in [@Buch2002; @BKSTY; @FominKirillov94].
Our goal in this article is to provide the enriched counterpart to Lam and Pylyavskyy’s definition. Stembridge [@Stembridge1997a] remarks that “almost every aspect of the theory of ordinary $P$-partitions has an enriched counterpart.” In a very satisfying sense, it turns out that most features of enriched $P$-partitions similarly have a set-valued extension. In particular, we introduce a theory of *enriched set-valued $P$-partitions*. These are certain maps that take finite nonempty subsets of $\MM$ as values. They can be viewed as generalizations of semistandard shifted set-valued tableaux, which appear in the combinatorial generating functions for Ikeda and Naruse’s *$K$-theoretic Schur $P$- and $Q$-functions* $\bGP_\lambda$ and $\bGQ_\lambda$ studied in [@HKPWZZ; @IkedaNaruse; @NN2017; @NN2018; @Naruse2018].
One motivation for this project was to compute $\omega(\bGP_\lambda)$ and $\omega(\bGQ_\lambda)$, where $\omega$ denotes the usual involution of the ring of symmetric functions that maps $s_\lambda \mapsto s_{\lambda^T}$ for all partitions $\lambda$. It turns out that we can obtain explicit formulas for $\omega$ evaluated at $\bGP_\lambda$ and $\bGQ_\lambda$ by decomposing the latter power series into quasisymmetric functions attached to enriched set-valued $P$-partitions on which $\omega$ acts in a more transparent manner. Our results along these lines appear at end of this paper in Section \[antipode-sect\].
There are a few other reasons to be interested in enriched set-valued analogues of $P$-partitions. They suggest a straightforward definition of $K$-theoretic Schur $P$- and $Q$-functions indexed by skew shapes. These skew generalizations do not seem to have been considered previously; we establish some of their fundamental properties, like symmetry. Our definition also indicates a good notion of *$K$-theoretic Schur $S$-functions*. The theory of enriched set-valued $P$-partitions leads, moreover, to the construction of a Hopf algebra $\mcoPeak$, whose elements we call *multipeak quasisymmetric functions*. This is a $K$-theoretic analogue of Stembridge’s algebra of peak quasisymmetric functions, and is perhaps of independent interest.
In [@LamPyl], Lam and Pylyavskyy study a diagram of six Hopf algebras related to the $K$-theory of the Grassmannian. There are two shifted variants of this diagram, one for the orthogonal Grassmannian and one for the Lagrangian Grassmannian. The planned sequel to this paper will study the “shifted $K$-theoretic” Hopf algebras in these diagrams. The algebra $\mcoPeak$ and its dual (a $K$-theoretic analogue of the peak algebra) will figure prominently in the shifted diagrams.
We now summarize our main results and outline the rest of this paper. Section \[prelim-sect\] gives some background on Hopf algebras in the category of *linearly compact modules* and on *combinatorial Hopf algebras*. In Section \[mlpset-sect\], we introduce a “$K$-theoretic” Hopf algebra of labeled posets and use this object to recover several constructions of Lam and Pylyavskyy, including the stable Grothendieck polynomials. Section \[main-sect\] contains our main definitions and results related to enriched $P$-partitions and associated quasisymmetric functions. We define the skew analogues of Ikeda and Naruse’s $K$-theoretic Schur $P$- and $Q$-functions in Section \[shifted-stable-sect\]. In Section \[sym-sect\], we prove the symmetry of these power series and characterize the subalgebra that they generate. Finally, Section \[antipode-sect\] leverages our results to derive explicit formulas for some notable involutions on quasisymmetric functions.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The first author was partially supported by an ORAU Powe award. The second author was partially supported by Hong Kong RGC Grant ECS 26305218. We are grateful to Zach Hamaker, Hiroshi Naruse, Brendan Pawlowski, and Alex Yong for helpful comments.
Preliminaries {#prelim-sect}
=============
Completions {#completions-sect}
-----------
In this article, we are often concerned with rings of formal power series of unbounded degree that are “too large” to belong to the category of free modules. To define monoidal structures on these objects, we need to work in the following slightly more exotic setting.
Fix an integral domain $R$ and write $\otimes = \otimes_R$ for the usual tensor product. An *$R$-algebra* is an $R$-module $A$ with $R$-linear product $\nabla : A\otimes A \to A$ and unit $\iota : R\to A$ maps. Dually, an *$R$-coalgebra* is an $R$-module $A$ with $R$-linear coproduct $\Delta : A \to A\otimes A$ and counit $\epsilon : A \to R$ maps. The (co)product and (co)unit maps must satisfy several natural associativity axioms; see [@GrinbergReiner §1] for the complete definitions. (Co)algebras form a category in which morphisms are $R$-linear maps commuting with the (co)unit and (co)product maps.
An $R$-module $A$ that is simultaneously an $R$-algebra and an $R$-coalgebra is an *$R$-bialgebra* if the coproduct and counit maps are algebra morphisms (equivalently, the product and unit are coalgebra morphisms). Suppose $A$ is an $R$-bialgebra with structure maps $\nabla$, $\iota$, $\Delta$, and $\epsilon$. Let $\End(A)$ denote the set of $R$-linear maps $A \to A$. This set is an $R$-algebra with product given by the linear map with $
f\otimes g \mapsto \nabla \circ (f\otimes g) \circ \Delta
$ for $f,g \in \End(A)$ and unit given by the linear map with $1_R \mapsto \iota\circ \epsilon$. The bialgebra $A$ is a *Hopf algebra* if $\id : A \to A$ has a (necessarily unique) two-sided multiplicative inverse $\antipode : A \to A$ in the algebra $\End(A)$, in which case we call $\antipode$ the *antipode* of $A$.
Consider a free $R$-module $A$, and fix a basis $\{a_i\}_{i \in I}$ for $A$. Suppose that $B$ is an $R$-module and $\langle\cdot,\cdot\rangle : A \times B \to R$ is a *nondegenerate* $R$-bilinear form, in the sense that $b\mapsto \langle \cdot,b\rangle$ is a bijection $B\to \Hom_R(A,R)$. Then for each $j \in I$, there exists a unique element $b_j \in B$ with $\langle a_i, b_j \rangle = \delta_{ij}$ for all $i \in I$, and we can identify $B$ with the product $\prod_{j \in I} Rb_j$, which we view as the set of arbitrary $R$-linear combinations of the elements $\{b_j\}_{j\in I}$. We refer to $\{b_i\}_{i \in I}$ as a *pseudobasis* for $B$; some authors call this a *continuous basis*.
\[lc-ex0\] Let $A = R[x]$ and $B=R[[x]]$. Define $\langle\cdot,\cdot\rangle : A \times B \to R$ to be the nondegenerate $R$-bilinear form with $\left\langle \sum_{n\geq 0} r_n x^n, \sum_{n \geq 0} s_n x^n\right\rangle = \sum_{n\geq 0} r_n s_n$. The set $\{x^n\}_{n\geq 0}$ is a basis for $A$ and a pseudobasis for $B$.
Endow $R$ with the discrete topology. The *linearly compact topology* on $B$ [@Dieudonne §I.2] is the coarsest topology in which the maps $\langle a_i, \cdot \rangle : B \to R$ are all continuous. If we identify $B \cong \prod_{j \in I} R b_j$ and give each $Rb_j$ the discrete topology, then this is the usual product topology. The linearly compact topology depends on $\langle\cdot,\cdot\rangle$ but not on the choice of basis for $A$. It is discrete if $A$ has finite rank. We refer to $B$, equipped with this topology, as a *linearly compact $R$-module*, and say that $B$ is the *dual* of $A$ with respect to the form $\langle\cdot,\cdot\rangle$.
We will often abbreviate by writing “LC-” in place of “linearly compact.” LC-modules modules form a category in which morphisms are continuous $R$-linear maps.
\[lc-ex1\] With $A = R[x]$ and $B = R[[x]]$ as in Example \[lc-ex0\], a basis of open subsets in the LC-topology is given by sets of power series in $R[[x]]$ whose coefficients are fixed in a finite set of degrees and unconstrained elsewhere. We view formal power series rings in multiple variables as LC-modules similarly.
Suppose $A$ is a free $R$-module with basis $S$. Let $B$ be the $R$-module of arbitrary $R$-linear combinations of elements of $S$, equipped with the nondegenerate bilinear form $ A \times B \to R$ making $S$ orthonormal. We say that $B$ is the *completion* of $A$ with respect to $S$. This is a linearly compact $R$-module with $S$ as a pseudobasis.
Let $B$ and $B'$ be linearly compact $R$-modules dual to free $R$-modules $A$ and $A'$, and write $\langle\cdot,\cdot\rangle$ for both of the associated bilinear forms. Every $R$-linear map $\phi : A' \to A$ has a unique adjoint $\psi : B\to B'$ such that $\langle \phi(a), b\rangle = \langle a,\psi(b)\rangle$ for all $a \in A'$ and $b \in B$. A linear map $B \to B'$ is continuous if and only if it arises as the adjoint of some linear map $A' \to A$.
The *completed tensor product* of $B$ and $B'$ is the $R$-module $$B \htimes B' := \Hom_R(A\otimes A',R),$$ given the LC-topology from the tautological pairing $(A\otimes A') \times \Hom_R(A\otimes A',R) \to R$. If $\{b_i\}_{i \in I}$ and $\{b'_j \}_{j \in J}$ are pseudobases for $B$ and $B'$, then we can realize $B\htimes B'$ concretely as the linearly compact $R$-module with the set of tensors $\{ b_i \otimes b_j'\}_{(i,j) \in I \times J}$ as a pseudobasis. There is an inclusion $B\otimes B' \hookrightarrow B\htimes B'$, which is an isomorphism if and only if $A$ or $A'$ has finite rank.
If we view $R[[x]]$ and $R[[y]]$ as linearly compact $R$-modules as in Example \[lc-ex1\] then $R[[x]] \otimes R[[y]] \neq R[[x]]\htimes R[[y]] \cong R[[x,y]]$, where we consider $R[[x,y]]$ as the linearly compact $R$-module dual to $R[x,y]$.
If $\iota : B \to R$ and $\nabla : B \htimes B \to B$ are continuous linear maps, then these maps are the adjoints of unique linear maps $\epsilon : R \to A$ and $\Delta : A \to A \otimes A$, and we say that $(B,\nabla,\iota)$ is an *LC-algebra* if $(A,\Delta,\epsilon)$ is an $R$-coalgebra. Similarly, we say that continuous linear maps $\Delta : B \to B\htimes B$ and $\epsilon : B \to R$ make $B$ into an *LC-coalgebra* if $\Delta$ and $\epsilon$ are the adjoints of the product and unit maps of an $R$-algebra structure on $A$. *LC-bialgebras* and *LC-Hopf algebras* are defined analogously. In each case we say that the (co, bi, Hopf) algebra structures on $A$ and $B$ are duals of each other. If $B$ is an LC-Hopf algebra then its antipode is defined to be the adjoint of the antipode of the Hopf algebra $A$.
One can reformulate these definitions in terms of commutative diagrams; see [@Marberg2018 §2 and §3]. Linearly compact (co, bi, Hopf) algebras form a category in which morphisms are continuous linear maps commuting with (co)products and (co)units. The completed tensor product of two linearly compact (co, bi, Hopf) algebras is naturally a linearly compact (co, bi, Hopf) algebra.
\[lc-ex4\] Again let $A = R[x]$ and $B = R[[x]]$ but now suppose $R = \ZZ[\beta]$. Define $\iota : R \to B$ to be the obvious inclusion and let $\epsilon : B \to R$ be the map setting $x=0$. Let $\nabla : B \htimes B \to B$ be the usual multiplication map but define $\Delta_\beta : B \to B\htimes B$ to be the continuous algebra homomorphism with $\Delta_\beta(x) = x\otimes 1 +1\otimes x + \beta x \otimes x .$ The operations $\iota$ and $\epsilon$ restrict to maps $R \to A$ and $A \to R$. Define $\Delta : A \to A\otimes A$ as the linear map with $\Delta(x^n) = \sum_{i+j = n} x^i\otimes x^j$ and let $\nabla_\beta : A \otimes A \to A$ be the commutative, associative, linear map whose $(n-1)$-fold iteration maps \[fold-eq\] \_\^[(n - 1)]{} : x (x-) (x-2) (x-(n-1)) n! x\^n and which is such that $\nabla_\beta \circ (\id \otimes \iota) : A\otimes R \to A$ and $\nabla_\beta \circ (\iota \otimes \id) : R\otimes A \to A$ are the canonical isomorphisms. (The existence and uniqueness of $\nabla_\beta$ is not obvious, but follows as an interesting, fairly straightforward exercise.)
The triple $(A,\nabla_\beta,\iota)$ is automatically an algebra, and one can show that $\epsilon$ and $\nabla_\beta$ are algebra homomorphisms, so $(A,\nabla_\beta,\iota,\Delta,\epsilon)$ is a bialgebra. One can check, moreover, that the dual of this bialgebra structure via the form in Example \[lc-ex1\] is precisely $(B,\nabla,\iota,\Delta_\beta,\epsilon)$, which is thus an LC-bialgebra.
There are several ways of seeing that $A$ is a Hopf algebra and computing its antipode $\antipode$. Since $\iota \circ \epsilon = \nabla_\beta \circ (\id \otimes \antipode) \circ \Delta$, we must have $\antipode(x) = -x$. Using this and the fact that $\antipode$ is an algebra anti-automorphism, one can show that $$\antipode(x^n) = (-1)^n x(x+\beta)^{n-1}$$ for $n>0$. It follows by duality that $B$ is an LC-Hopf algebra whose antipode $\hat \antipode : B\to B$ is the continuous linear map with $\hat \antipode(1) = 1$ and $$\hat \antipode(x^m) = \sum_{n\geq m} (-1)^n \tbinom{n-1}{m-1}\beta^{n-m} x^n = (\tfrac{-x}{1+\beta x})^m$$ for $m>0$. It is interesting to note that $\nabla$ and $\Delta_\beta$ restrict to well-defined maps $A \otimes A \to A$ and $A \to A\otimes A$, which give $A$ a second bialgebra structure. This bialgebra is not a Hopf algebra, however, since $\hat \antipode$ is not a map $A \to A$.
In a few places we will encounter the following construction, where the added complications of linear compactness seem a little superfluous. Suppose $H$ is a $\NN$-graded connected Hopf algebra, with finite graded rank and with a homogeneous basis $S$. Let $\hat H$ be the completion of $H$ with respect to $S$. There is an inclusion $H\hookrightarrow \hat H$ and all Hopf structure maps automatically extend to continuous linear maps, making $\hat H$ into an LC-Hopf algebra (namely, the one dual to the graded dual of $H$). Significantly, LC-Hopf algebras arising as completions in this way often have interesting subalgebras that are not themselves completions.
If we set $\beta=0$ in Example \[lc-ex4\], then $A=\ZZ[x]$ becomes a $\NN$-graded connected Hopf algebra with finite graded rank. The completion of this Hopf algebra with respect to $S = \{x^n\}_{n\geq 0}$ can be identified with the (proper) LC-Hopf subalgebra of $B=\ZZ[[x]]$ with pseudobasis $\{ n!\cdot x^n\}_{n\geq 0}$.
Quasisymmetric functions
------------------------
Continue to let $R$ be an integral domain, and suppose $x_1$, $x_2$, …are commuting indeterminates. A power series $f \in R[[x_1,x_2,\dots]]$ is *quasisymmetric* if for any choice of exponents $a_1,a_2,\dots,a_k \in \PP$, the coefficients of $x_1^{a_1}x_2^{a_2}\cdots x_{k}^{a_k}$ and $x_{i_1}^{a_1}x_{i_2}^{a_2}\cdots x_{i_k}^{a_k}$ in $f$ are equal for all $i_1<i_2<\dots<i_k$.
Let $\mQSym_R$ denote the $R$-module of all quasisymmetric power series in $R[[x_1,x_2,\dots]]$. Let $\QSym_R$ denote the submodule of power series in $\mQSym_R$ of bounded degree.
A *composition* is a finite sequence of positive integers $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_l)$. If $n = \alpha_1 + \alpha_2 + \dots+ \alpha_l$ then we write $\alpha \vDash n$ and and set $|\alpha| := n$. The *monomial quasisymmetric function* of a nonempty composition $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_l)$ is $$M_\alpha := \sum_{i_1<i_2<\dots<i_l} x_{i_1}^{\alpha_1} x_{i_2}^{\alpha_2}\cdots x_{i_l}^{\alpha_l} \in \QSym_R.$$ In particular, when $\alpha =\emptyset$ is empty, we have $M_\emptyset = 1$. Then $\QSym_R$ is a graded ring that is free as an $R$-module with the set of power series $M_\alpha$ as a homogeneous basis. We identify $\mQSym_R$ with the corresponding completion. This makes $\mQSym_R$ into a linearly compact $R$-module, whose topology coincides with the subspace topology induced by the LC-module $R[[x_1,x_2,\dots]]$.
Let $\alpha'\alpha''$ denote the concatenation of two compositions $\alpha'$ and $\alpha''$. There is a unique $R$-linear map $\Delta : \QSym_R \to \QSym_R \otimes \QSym_R$ such that $
\Delta(M_\alpha) = \sum_{\alpha =\alpha'\alpha''} M_{\alpha'} \otimes M_{\alpha''}
$ for each composition $\alpha$. Let $\epsilon : \QSym_R \to R$ be the linear map with $M_{\emptyset} \mapsto 1$ and $M_\alpha\mapsto 0$ for all nonempty compositions $\alpha$. With this coproduct and counit, $\QSym_R$ becomes a graded, connected Hopf algebra [@GrinbergReiner §5.1]. (For a discussion of its antipode, see Section \[antipode-subsection\].) Since $\QSym_R$ has finite graded rank, its product and coproduct extend to continuous linear maps $\mQSym_R \htimes \mQSym_R \to\mQSym_R$ and $\mQSym_R \to \mQSym_R \htimes \mQSym_R$ making $\mQSym_R$ into an LC-Hopf algebra. This algebra has an important universal property, which we presently describe.
Suppose $H $ is a linearly compact $R$-bialgebra with product $\nabla$, coproduct $\Delta$, unit $\iota$, and counit $\epsilon$. Let $\XX(H)$ denote the set of continuous algebra morphisms $\zeta : H \to R[[t]]$ with $\zeta(\cdot )|_{t=0}=\epsilon$. This set is a monoid under the *convolution product* $ \zeta * \zeta' := \nabla_{R[[t]]} \circ (\zeta \htimes \zeta')\circ \Delta$ with unit element $\iota\circ \epsilon$. If $H$ has an antipode $\antipode$ then then $\zeta\circ\antipode$ is the inverse of $\zeta \in \XX(H)$ under $*$, so in this case $\XX(H)$ is a group. This leads to a natural extension of Aguiar, Bergeron, and Sottile’s notion of a *combinatorial Hopf algebra* [@ABS] to linearly compact modules.
If $H$ is an LC-bialgebra (respectively, LC-Hopf algebra) and $\zeta \in \XX(H)$, then we refer to $(H,\zeta)$ as a *combinatorial LC-bialgebra* (respectively, *combinatorial LC-Hopf algebra*). Such pairs form a category in which morphisms $(H,\zeta) \to (H',\zeta')$ are LC-bialgebra morphisms $\phi : H \to H'$ with $\zeta = \zeta'\circ \phi$.
We view $\mQSym_R$ as a combinatorial LC-Hopf algebra with respect to the *universal zeta function* $\zetaq : \mQSym_R \to R[[t]]$ given by $\zetaq(f) = f(t,0,0,\dots)$. One has $
\zetaq(M_\alpha) = t^{|\alpha|}
$ for $\alpha\in\{\emptyset,(1),(2),(3),\dots\}$ and $\zetaq(M_\alpha)=0$ for all other compositions $\alpha$.
The next theorem, which is a mild of generalization of [@ABS Thm. 4.1], shows that $(\mQSym_R, \zetaq)$ is a terminal object in the category of combinatorial LC-bialgebras. Given an LC-bialgebra $H$, a character $\zeta \in \XX(H)$, and a nontrivial composition $\alpha
=(\alpha_1,\alpha_2,\dots,\alpha_k)$, let $\zeta_\alpha : H \to R$ denote the map sending $h \in H$ to the coefficient of $t^{\alpha_1}\otimes t^{\alpha_2}\otimes \cdots \otimes t^{\alpha_k}$ in $\zeta^{\otimes k} \circ \Delta^{(k-1)}(h) \in R[[t]]^{\htimes k}$, where $\Delta^{(0)} := \id$. When $\alpha=\emptyset$ is empty, let $\zeta_\emptyset = \epsilon$.
\[abs-thm\] If $(H,\zeta)$ is a combinatorial LC-bialgebra then there exists a unique morphism $\Phi : (H,\zeta) \to (\mQSym_R,\zetaq)$, given explicitly by the map with $\Phi(h) = \sum_\alpha \zeta_\alpha(h) M_\alpha$ for $h \in H$, where the sum is over all compositions $\alpha$.
By extending the ring of scalars, any such morphism $\Phi$ extends to a morphism over the field of fractions of $R$, and this extension satisfies $\zetaq \circ \Phi = \zeta$. By [@Marberg2018 Thm. 7.8], there is a unique such morphism $\Phi$ defined over a field, and this morphism has the given formula $\Phi(h) = \sum_\alpha \zeta_\alpha(h) M_\alpha$. Since this formula is in fact defined over $R$, the result follows.
Suppose $R=\ZZ[\beta]$ and $H = \ZZ[\beta][[x]]$ with the LC-Hopf algebra structure in Example \[lc-ex4\]. Each map $\zeta : H \to \ZZ[\beta][[t]]$ in $\XX(H)$ is uniquely determined by its value at $x$, and there exists $\zeta \in \XX(H)$ with $\zeta(x) = f \in \ZZ[[t]]$ if and only if $f \in t\ZZ[\beta][[t]]$ is a power series with no constant term. If $\zeta \in \XX(H)$ is the trivial isomorphism mapping $x \mapsto t$, then the morphism $\Phi$ in Theorem \[abs-thm\] sends $x \mapsto M_{(1)} + \beta M_{(1,1)} + \beta^2 M_{(1,1,1)} + \dots$.
For the rest of this article, we fix the ring of scalars to be $R=\ZZ[\beta]$, where $\beta$ is an indeterminate, and define \[qsym-beta-convention-eq\] := \_ := \_. All combinatorial Hopf algebras are assumed to be defined over $\ZZ[\beta]$.
Set-valued $P$-partitions {#mlpset-sect}
=========================
In this section, we introduce a combinatorial LC-Hopf algebra on labeled posets. Applying the canonical morphism from this object to $\mQSym$ produces a family of interesting quasisymmetric functions. recovering Lam and Pylyavskyy’s generating functions for set-valued $P$-partitions [@LamPyl].
Labeled posets
--------------
Let $P$ be a finite poset with an injective labeling map $\gamma : P \to \ZZ$. We refer to the pair $(P,\gamma)$ as a *labeled poset*.
We say that $t \in P$ *covers* $s \in P$ and write $s\lessdot t$ if $\{x \in P : s \leq x < t\} = \{s\}$. Two labelings $\gamma$ and $\delta$ of $P$ are *equivalent* if, whenever $s\lessdot t$ in $P$, we have $\gamma(s) > \gamma(y)$ if and only if $\delta(s) > \delta(t)$. Labeled posets $(P,\gamma)$ and $(Q,\delta)$ are *isomorphic* if there is a poset isomorphism $\phi : P \xrightarrow{\sim} Q$ such that $\gamma$ and $ \delta\circ \phi$ are equivalent labelings of $P$. Denote the isomorphism class of $(P,\gamma)$ by $[(P,\gamma)]$.
The isomorphism class $[(P, \gamma)]$ may be represented uniquely by adding an orientation to the Hasse diagram of $P$, as follows: for each covering relation $x\lessdot y$, there is an edge $x\to y$ if $\gamma(x) > \gamma(y)$ and an edge $x \leftarrow y$ if $\gamma(x) < \gamma(y)$.
\[ori-ex\] Let $P$ be the labeled poset with four elements $s_1,s_2,s_3,s_4$ and four covering relations $s_1\lessdot s_2$ and $s_1\lessdot s_3$ and $s_2\lessdot s_4$ and $s_3\lessdot s_4$, and let $\gamma : P \to \ZZ$ be defined by $\gamma(s_1) = 5$ and $\gamma(s_i) = i$ for $i\in\{2,3,4\}$. Then $[(P, \gamma)]$ is represented by the oriented Hasse diagram $$\begin{tikzpicture}[baseline=(c2.base), xscale=0.4, yscale=0.3]
\tikzset{edge/.style = {->}}
\node (c4) at (0,3) {$s_4$};
\node (c3) at (2,0) {$s_3$};
\node (c2) at (-2,0) {$s_2$};
\node (c1) at (0,-3) {$s_1$};
\draw[edge] (c1) -- (c2);
\draw[edge] (c1) -- (c3);
\draw[edge] (c4) -- (c2);
\draw[edge] (c4) -- (c3);
\end{tikzpicture}$$
Define $\mLPSet$ to be the linearly compact $\ZZ[\beta]$-module with a pseudobasis given by the isomorphism classes of all (finite) labeled posets.
There is a natural Hopf structure on $\mLPSet$, which we now describe. Define the disjoint union of labeled posets $(P,\gamma)$ and $(Q,\delta)$ to be the labeled poset $(P,\gamma)\sqcup (Q,\delta) := (P\sqcup Q, \gamma\sqcup \delta)$, where $P \sqcup Q$ is the usual disjoint poset union and $\gamma \sqcup \delta : P \sqcup Q \to \ZZ$ is the labeling map \[sqcup-eq\] ()(s) :=
\(s) - \_[xP]{} (x)&\
(s) - \_[x Q]{} (x)+ 1 &
, so that the oriented Hasse diagram of the isomorphism class $[(P\sqcup Q,\gamma\sqcup \delta)]$ is the disjoint union of the oriented Hasse diagrams of $[(P,\gamma)]$ and $[(Q,\delta)]$.
If $S \subseteq P$ is any subset of a labeled poset $(P, \gamma)$ then $(S,\gamma|_S)$ is itself a labeled poset, where $S$ inherits the partial order of $P$. To unclutter our notation, we will henceforth write $(S,\gamma)$ in place of $(S,\gamma|_S)$. A subposet $S \subseteq P$ is a *lower set* (respectively, *upper set*) if for all $x,y \in P$ with $x<y$, $y \in S$ $\Rightarrow$ $x\in S$ (respectively, $x \in S$ $\Rightarrow$ $y \in S$). A subset $S\subseteq P$ is an *antichain* if no elements $x,y \in S$ satisfy $x<y$ in $P$. Given labeled posets $(P,\gamma)$ and $(Q,\delta)$, define $$\nabla([(P,\gamma)] \otimes [(Q,\delta)]) := [(P,\gamma)\sqcup (Q,\delta)]$$ and $$\Delta([(P,\gamma)]) := \sum_{S\cup T = P} \beta^{|S\cap T|} \cdot[(S,\gamma)] \otimes [(T,\gamma)]$$ where the sum is over all ordered pairs $(S,T)$ of subsets of $P$ such that $S$ is a lower set, $T$ is an upper set, $P = S\cup T$, and $S\cap T$ is an antichain. Since there are only finitely many isomorphism classes of labeled posets of a given size, these operations extend to continuous linear maps $\nabla : \mLPSet \htimes \mLPSet \to \mLPSet$ and $\Delta : \mLPSet \to \mLPSet \htimes \mLPSet$.
\[mlp-delta-ex\] The value of $\Delta([(P,\gamma)])$ for $P = \{ a < b\}$ with $\gamma(a) < \gamma(b)$ is $$\ba
\Delta\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (b) at (0,0) {$b$};
\node (a) at (0,-2) {$a$};
\draw[edge] (a) -- (b);
\end{tikzpicture}\)
&=
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (a) at (0,-2) {\ };
\node (b) at (0,0) {\ };
\end{tikzpicture}\)
\otimes
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (b) at (0,0) {$b$};
\node (a) at (0,-2) {$a$};
\draw[edge] (a) -- (b);
\end{tikzpicture}\)
+
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (a) at (0,-2) {$a$};
\node (b) at (0,0) {\ };
\end{tikzpicture}\)
\otimes
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (a) at (0,-2) {\ };
\node (b) at (0,0) {$b$};
\end{tikzpicture}\)
+
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (b) at (0,0) {$b$};
\node (a) at (0,-2) {$a$};
\draw[edge] (a) -- (b);
\end{tikzpicture}\)
\otimes
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (a) at (0,-2) {\ };
\node (b) at (0,0) {\ };
\end{tikzpicture}\)
\\&\quad+
\beta \(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (a) at (0,-2) {$a$};
\node (b) at (0,0) {\ };
\end{tikzpicture}\)
\otimes
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (b) at (0,0) {$b$};
\node (a) at (0,-2) {$a$};
\draw[edge] (a) -- (b);
\end{tikzpicture}\)
+
\beta
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (b) at (0,0) {$b$};
\node (a) at (0,-2) {$a$};
\draw[edge] (a) -- (b);
\end{tikzpicture}\)
\otimes
\(\begin{tikzpicture}[baseline=(c.base), xscale=0.6, yscale=0.4]
\tikzset{edge/.style = {<-}}
\node (c) at (0, -1) {};
\node (a) at (0,-2) {\ };
\node (b) at (0,0) {$b$};
\end{tikzpicture}\)
.
\ea$$
Write $\iota : \ZZ[\beta]\to \mLPSet$ for the linear map that sends $1$ to the isomorphism class of the empty labeled poset. Write $\epsilon : \mLPSet \to\ZZ[\beta]$ for the continuous linear map whose value at $[(P,\gamma)] $ is $1$ if $|P|=0$ and $0$ otherwise.
\[mlpset-thm\] With respect to the operations $\nabla$, $\Delta$, $\iota$, $\epsilon$ just given, the $\ZZ[\beta]$-module $\mLPSet$ is a commutative LC-Hopf algebra.
Every covering relation in $P \sqcup Q$ is a covering relation in either $P$ or $Q$, and the relative order of labels is not altered by the disjoint union. Hence $[(P, \gamma) \sqcup (Q, \delta)] = [(Q, \delta) \sqcup (P, \gamma)]$ and so the product $\Delta$ is commutative.
Let $\LPSet$ denote the free $\ZZ[\beta]$-module with a basis given by all isomorphism classes of labeled posets $[(P,\gamma)]$. For $n \in \NN$, let $\LPSet_n$ denote the $\ZZ[\beta]$-submodule of $\LPSet$ spanned by isomorphism classes $[(P,\gamma)]$ with $|P|=n$.
Consider the nondegenerate bilinear form $ \LPSet \times \mLPSet \to \ZZ[\beta]$ making obvious the (pseudo)bases of isomorphism classes dual to each other. Let $\nabla^\vee : \LPSet \to \LPSet \otimes \LPSet$, $\Delta^\vee : \LPSet \otimes \LPSet \to \LPSet$, $\iota^\vee : \LPSet \to \ZZ[\beta]$, and $\epsilon^\vee : \ZZ[\beta] \to \LPSet$ denote the respective adjoints of $\nabla$, $\Delta$, $\iota$, and $\epsilon$ relative to this form. One has $\iota^\vee = \epsilon|_{\LPSet}$ and $\epsilon^\vee = \iota|_{\LPSet}$, while $$\nabla^\vee([(P,\gamma)]) = \sum_{\substack{S\sqcup T = P \\ \text{(as posets)}}} [(S,\gamma)]\otimes [(T,\gamma)].$$ It is slightly more complicated, but still straightforward, to write down a similar formula for $\Delta^\vee$; the important observation is that $\nabla^\vee$ maps $\LPSet_n \to \bigoplus_{i+j =n} \LPSet_i \otimes \LPSet_j$.
To show that $\mLPSet$ is an LC-Hopf algebra, it suffices to show that the maps $\Delta^\vee$, $\nabla^\vee$, $\epsilon^\vee$, and $\iota^\vee$ make $\LPSet$ into an ordinary Hopf algebra. It is a routine calculation to show that $\LPSet$ is at least a bialgebra. Moreover, $\LPSet$ is evidently graded and connected as a coalgebra and filtered as an algebra. It follows that $f: = \id - \epsilon^\vee \circ \iota^\vee$ is *locally $\star$-nilpotent* in the sense of [@GrinbergReiner Rem. 1.4.23], i.e., that for each $x \in \LPSet$, $ (\Delta^\vee)^{(k-1)} \circ f^{\otimes k} \circ (\nabla^\vee)^{(k-1)}(x) = 0$ for some sufficiently large $k=k(x)$. The bialgebra $\LPSet$ therefore has an antipode given by Takeuchi’s formula [@GrinbergReiner Prop. 1.4.22], and we conclude that $\LPSet$ is a Hopf algebra, as needed.
One can obtain an antipode formula for $\mLPSet$ by taking the adjoint of Takeuchi’s antipode formula for $\LPSet$. Both formulas involve substantial cancellation of terms. This raises the following problem, which seems to be open:
Find a cancellation-free formula for the antipode of $\mLPSet$.
A solution to this problem would generalize the antipode formulas in [@AguiarArdila §15.3], which roughly correspond to the case when $\beta=0$.
Up to equivalence, there is a unique labeled poset $(P,\gamma)$ in which $P$ is an $n$-element antichain. The continuous linear map sending $x^n$ to this poset is an injective morphism of LC-Hopf algebras $\ZZ[\beta][[x]] \to \mLPSet$, where $\ZZ[\beta][[x]]$ has the LC-Hopf structure described in Example \[lc-ex4\].
Define $\zetaLP : \mLPSet \to \ZZ[\beta][[t]]$ to be the continuous linear map with \[incr-zeta-eq\] (\[(P,)\]) =
t\^[|P|]{} &\
0 &.
This is a particularly natural algebra morphism, which makes $(\mLPSet,\zetaLP)$ into a combinatorial LC-Hopf algebra. Theorem \[abs-thm\] asserts that there is a unique morphism $
(\mLPSet,\zetaLP) \to (\mQSym,\zetaq)
$, and it is an interesting and *a priori* nontrivial problem to evaluate this map at the isomorphism class of a given labeled poset. We turn to this problem in the next section.
Set-valued $P$-partitions {#sv-sect}
-------------------------
*Set-valued $P$-partitions* are certain maps assigning sets of integers to the vertices of a labeled poset $(P,\gamma)$. It will turn out that these maps exactly parametrize the monomials appearing in the quasisymmetric generating function associated to $[(P,\gamma)]$ by the unique morphism $
(\mLPSet,\zetaLP) \to (\mQSym,\zetaq)
$.
Let $\PSet$ denote the set of finite, nonempty subsets of $\PP$. Given $S,T \in \PSet$, write $S\prec T$ if $\max(S) < \min(T)$ and $S\preceq T$ if $\max(S) \leq \min(T)$. (In particular, $S \preceq S$ if and only if $|S| = 1$.) For $S \in \PSet$, define $x^S = \prod_{i \in S} x_i$.
\[svp-def\] Let $(P,\gamma)$ be a labeled poset. A *set-valued $(P,\gamma)$-partition* is a map $\sigma : P \to \PSet$ such that for each covering relation $s\lessdot t$ in $P$ one has $\sigma(s) \preceq \sigma(t)$, with $\sigma(s) \prec \sigma(t)$ if $\gamma(s) > \gamma(t)$.
Suppose $P = \{1<2<3\}$ is a 3-element chain and $\gamma(1) < \gamma(2) > \gamma(3)$. If $\sigma$ is a set-valued $(P,\gamma)$-partition, then $(\sigma(1), \sigma(2), \sigma(3))$ could be $(\{ 2\},\{2,3\}, \{4,5\})$ or $(\{1\},\{2,4\}, \{6\})$, but not $(\{1,2\},\{3\}, \{3,4\})$.
Lam and Pylyavskyy [@LamPyl] introduced this definition while studying a “$K$-theoretic” analogue of the Malvenuto–Reutenaurer Hopf algebra of permutations. The idea generalizes the classical notion of a *$P$-partition* from [@St1], which is just a set-valued partition whose values are all singleton sets.
Let $\tA(P,\gamma)$ denote the set of all set-valued $(P,\gamma)$-partitions. The *length* of $\sigma \in \tA(P,\gamma)$ is the nonnegative integer $|\sigma| := \sum_{s \in P} |\sigma(s)|$, while the *weight* of $\sigma$ is the monomial $x^\sigma := \prod_{s \in P} x^{\sigma(s)}$. We define the *set-valued weight enumerator* of $(P,\gamma)$ to be the quasisymmetric formal power series \[tgam-eq\] ([P,]{}) := \_[(P,)]{} \^[|| - |P|]{} x\^. The power series $\Gamma^{(1)}(P,\gamma)$ obtained from by setting $\beta=1$ is denoted $\tilde K_{P,\gamma}$ in [@LamPyl §5.3]. These generating functions (and by extension, the sets $\tA(P,\gamma)$) are natural objects to consider on account of the following theorem.
\[<-thm\] The continuous linear map with $[(P,\gamma)] \mapsto \tGamma(P,\gamma)$ for each labeled poset $(P,\gamma)$ is the unique morphism of combinatorial LC-Hopf algebras $(\mLPSet,\zetaLP) \to (\mQSym,\zetaq).$
If $(P,\gamma) \cong (Q,\delta)$ then clearly $\tGamma(P,\gamma) = \tGamma(Q,\delta)$, so the continuous linear map $\mLPSet \to \mQSym$ described in the theorem is at least well-defined.
Fix a labeled poset $(P,\gamma)$. For each $k \in \NN$, let $\sP_k$ denote the set of $k$-tuples $(P_1,\dots, P_k)$ of nonempty sets with $P_1 \cup \dots \cup P_k = P$ such that
if $s \in P_i$ and $t \in P_j$ where $i<j$ then $t \not < s$ in $P$, and
if $s,t \in P_i$ and $s \lessdot t$ in $P$ then $\gamma(s) <\gamma(t)$.
Also let $\sI_k$ be the set of $k$-tuples of positive integers $(i_1,\dots,i_k)$ with $i_1<\dots<i_k$. According to Theorem \[abs-thm\], the unique morphism of combinatorial LC-Hopf algebras $ (\mLPSet,\zetaLP) \to (\mQSym,\zetaq)$ is the continuous linear map with $$[(P,\gamma)] \mapsto \sum_{k \in \NN} \sum_{(P_1, \dots, P_k) \in \sP_k} \sum_{(i_1,\dots, i_k) \in \sI_k}
\beta^{\sum_i |P_i| - |P|} x_{i_1}^{|P_1|} \cdots x_{i_k}^{|P_k|}.$$ We claim that the right hand expression is equal to $ \tGamma(P,\gamma)$. Given tuples $\pi = (P_1,\dots,P_k) \in \sP_k$ and $I = (i_1<\dots<i_k) \in \sI_k$, define $\sigma : P \to \PSet$ to be the map with $\sigma(s) = \{ i_j : 1 \leq j \leq k \text{ and }s \in P_j\}$. If $s\lessdot t$ in $P$ and $ i,j $ are indices such that $s \in P_i$ and $t \in P_j$, then property (a) implies that $i \leq j$, so $\sigma(s) \preceq \sigma(t)$; moreover, if $\gamma(s) > \gamma(t)$, then property (b) implies that $s$ and $t$ do not belong to the same $P_i$ and so $\sigma(s) \prec \sigma(t)$. Thus $\sigma \in \tA(P,\gamma)$, and we have $|\sigma| = \sum_i |P_i|$ and $x^\sigma = x_{i_1}^{|P_1|} \cdots x_{i_k}^{|P_k|}$.
It suffices to show that $(\pi,I) \mapsto \sigma$ is a bijection $\bigsqcup_{k \in \NN} \sP_k \times \sI_k \to \tA(P,\gamma)$. This is straightforward; the inverse map is $\sigma \mapsto (\pi,I)$ where $I$ is the sequence of elements in $\bigcup_{s \in P} \sigma(s)$ arranged in order, and $\pi = (P_1,P_2,\dots,P_{|I|})$ is the tuple in which $P_i$ is the set of $s \in P$ such that $\sigma(s)$ contains the $i$th element of $I$. It is easy to deduce from Definition \[svp-def\] that $\sigma \in \tA(P,\gamma)$ implies $\pi \in \sP_k$.
As one application of Theorem \[<-thm\], we obtain some new (co)product formulas for the quasisymmetric functions $\tGamma(P,\gamma)$.
\[sv-products-cor\] Suppose $(P,\gamma)$ and $(Q,\delta)$ are labeled posets. Then $$\tGamma(P,\gamma)\cdot \tGamma(Q,\delta) = \tGamma((P,\gamma)\sqcup (Q,\delta))$$ and $$\Delta(\tGamma(P,\gamma)) = \sum_{S\cup T = P} \beta^{|S\cap T|} \cdot \tGamma(S,\gamma)\otimes
\tGamma( T,\gamma)$$ where the sum is over all ordered pairs $(S,T)$ of subsets of $P$ such that $S$ is a lower set, $T$ is an upper set, $P = S\cup T$, and $S\cap T$ is an antichain.
Multifundamental quasisymmetric functions {#multi-sect}
-----------------------------------------
In this section we investigate the properties of the generating functions $\tGamma(P,\gamma)$ in the special case when $P$ is linearly ordered. One can show that these quasisymmetric functions form a pseudobasis of $\mQSym$; following [@LamPyl], we refer to them as *multifundamental quasisymmetric functions*.
Fix an arbitrary labeled poset $(P,\gamma)$. Lam and Pylyavskyy show in [@LamPyl §5.3] that both $\tA(P,\gamma)$ and $\tGamma(P,\gamma)$ are controlled by the following objects:
A finite sequence $w=(w_1,w_2,\dots,w_N)$ is a *linear multiextension* of $P$ if it holds that $P = \{w_1,w_2,\dots,w_N\}$, $w_i \neq w_{i+1}$ for each $1 \leq i < N$, and $\{ i : w_i = a\} \prec \{ i : w_i = b\}$ whenever $a\lessdot b$ in $P$.
(This differs superficially from Lam and Pylyavskyy’s definition in [@LamPyl §5.3]; what they call a linear multiextension is the map sending $s \in P$ to $\{ i : w_i = s\}$ rather than the sequence $(w_1,w_2,\dots,w_N)$.)
Let $\tL(P)$ denote the set of linear multiextensions of $P$. This set has a unique element if $P$ is a chain and is infinite otherwise. For each integer $N \in \NN$, let $[N]:= \{1<2<\dots<N\}$ be the usual $N$-element chain. Given a finite sequence $w=(w_1,w_2,\dots,w_N)$ with an injective map $\gamma : \{w_1,w_2,\dots,w_N\}\to\ZZ$, let $\delta : [N] \to [N]$ denote the unique bijection with $\delta(i) > \delta(j)$ for $ i < j $ if and only if $\gamma(w_i) > \gamma(w_{j})$, and define $$\tA(w,\gamma) := \tA([N], \delta)
\qquand
\tGamma(w,\gamma):=\tGamma([N],\delta).$$ Let $\ell(w) :=N$ denote the length of the finite sequence $w$. The following is a sort of “Fundamental Lemma of Set-Valued $(P,\gamma)$-Partitions.”
\[p-thm1\] For each labeled poset $(P,\gamma)$, there is a length- and weight-preserving bijection $\tA(P,\gamma) \xrightarrow{\sim} \bigsqcup_{w \in \tL(P)} \tA(w,\gamma)$, and so $$\tGamma({P,\gamma}) = \sum_{w \in \tL(P)} \beta^{\ell(w)-|P|} \tGamma({w,\gamma}).$$
To be precise, [@LamPyl Thm. 5.6] is the case of the preceding result with $\beta=1$; however, both statements have the same proof.
Fix a sequence $w$ with $N=\ell(w)$ and an injective map $\gamma : \{w_1,w_2,\dots,w_N\} \to \ZZ$. Define $\Des(w,\gamma) := \{ i\in[N-1] : \gamma(w_i) > \gamma(w_{i+1})\}$. We then have (w,) = { : \[N\] :(1)&(2)…(N)\
(i) &(i+1)i (w,) }.This suggests the following definition. For a tuple of sets $S=(S_1,S_2,\dots, S_N)$, let $x^S := \prod_i x^{S_i}$ and $|S| := \sum_i |S_i|$. For a composition $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_k)$, let $$I(\alpha) := \{\alpha_1, \alpha_1+\alpha_2,\dots, \alpha_1+\alpha_2 + \dots + \alpha_{k-1}\}.$$ Then $\alpha \mapsto I(\alpha)$ is a bijection from compositions of $N$ to subsets of $[N-1]$. Define the *multifundamental quasisymmetric function* of $\alpha \vDash N$ to be \[lbeta-eq\] L\^[()]{}\_:= \_[ ]{} \^[|S| - N]{} x\^S where in the sum each $S_i$ belongs to $\PSet$. The following holds by definition.
\[by-def-prop\] Fix a choice of $(w,\gamma)$ as above, and let $\alpha \vDash \ell(w)$ be the unique composition such that $I(\alpha)= \Des(w,\gamma)$. Then $L^{(\beta)}_\alpha = \tGamma(w,\gamma)$.
Write $\leq$ for the usual *refinement order* on compositions, so that $\alpha \leq \alpha'$ if and only if $|\alpha| = |\alpha'|$ and $I(\alpha) \subseteq I(\alpha')$. Setting $\beta=0$ in gives the *fundamental quasisymmetric function* $
L_\alpha := L_\alpha^{(0)} %= \sum_{\substack{ i_1 \leq i_2 \leq \dots \leq i_n \\ i_j<i_{j+1}\text{ if }j \in I(\alpha)}} x_{i_1}x_{i_2}\cdots x_{i_n}
= \sum_{\alpha \leq \alpha'} M_{\alpha'} \in \QSym.$ If $\beta$ has degree zero and $x_i$ has degree one, then $L_\alpha$ is the nonzero homogeneous component of $L^{(\beta)}_\alpha$ of lowest degree. Since the power series $L_\alpha$ form a basis of $\QSym$, it follows that the functions $\{L^{(\beta)}_\alpha\}$ are a pseudobasis of $\mQSym$.
\[lbeta-ex\] If $\alpha =(2,1)$ then is the sum over all $S_1 \preceq S_2 \prec S_3$ with $S_i \in \PSet$. This translates to the somewhat more explicit formula $$L^{(\beta)}_{(2,1)} = \sum_{n} \sum_{i_1 < i_2<\dots <i_n} \beta^{n-3} \(\tbinom{n-1}{2}+ \beta \sum_{1 \leq j <k\leq n} x_{i_j} \)x_{i_1}x_{i_2} \cdots x_{i_n}.$$ Setting $\beta=0$ gives $L_{(2,1)} =
%\sum_{i<j<k} x_i x_jx_k + \sum_{i<j} x_i x_j^2 =
M_{(1,1,1)} + M_{(2,1)}$ as expected.
Lam and Pylyavskyy [@LamPyl §5.3] refer to the specializations $\tilde L_\alpha := L_\alpha^{(1)}$ as *multifundamental quasisymmetric functions*. One recovers $L_\alpha^{(\beta)}$ from $\tilde L_\alpha$ by substituting $x_i \mapsto \beta x_i$ and then dividing by $\beta^{|\alpha|}$; that is, we have \[tilde-l-eq\] \^[||]{} L\_\^[()]{} = L\_(x\_1, x\_2, …).This lets one rewrite any identities in [@LamPyl] involving $\tilde L_\alpha$ in terms of $L^{(\beta)}_\alpha$.
\[product-rmk\] One can use Corollary \[sv-products-cor\] to obtain (co)product formulas for the multifundamental quasisymmetric functions $L^{(\beta)}_\alpha$. Lam and Pylyavskyy have already derived such formulas in [@LamPyl §5.4]; their results are stated in terms of the functions $\tilde L_\alpha := L^{(1)}_\alpha$, but easily translate to $L^{(\beta)}_\alpha$ via . The coproduct $\Delta\(L^{(\beta)}_\alpha\)$ is always a finite $\NN[\beta]$-linear combination of tensors of the form $L^{(\beta)}_{\alpha'}\otimes L^{(\beta)}_{\alpha''}$. By contrast, the product $\nabla\(L^{(\beta)}_{\alpha'}\otimes L^{(\beta)}_{\alpha''}\)$ is an infinite linear combination of $L^{(\beta)}_{\alpha}$’s if $\alpha'$ and $\alpha''$ are both nonempty.
Stable Grothendieck polynomials {#stable-sect}
-------------------------------
A primary motivation for the definition of set-valued $P$-partitions comes from the labeled posets associated with Young diagrams of partitions. As Lam and Pylyavskyy note in [@LamPyl], the weight enumerators of these labeled posets are essentially the *stable Grothendieck polynomials* introduced by Buch in [@Buch2002]. We review these here.
Let $\lambda =(\lambda_1\geq \lambda_2 \geq \dots \geq 0)$ and $\mu = (\mu_1\geq \mu_2\geq \dots \geq 0)$ be (integer) partitions with $\mu \subseteq \lambda$, i.e., with $\mu_i \leq \lambda_i$ for all $i$. The *skew diagram* of $\lambda/\mu$ is $$\D_{\lambda/\mu} := \{ (i,j) \in \PP\times \PP : \mu_i < j \leq \lambda_i\}.$$ Let $\D_\lambda = \D_{\lambda/\emptyset}$. We consider $\D_{\lambda/\mu}$ to be partially ordered with $(i,j) \leq (i',j')$ if $i \leq i'$ and $j \leq j'$. Let $n = |\lambda|-|\mu|$ and fix a bijection $\theta : \D_{\lambda/\mu} \to [n]$ with \[canonical-eq\] (i,j) < (i,j+1) (i,j) > (i+1,j) for all relevant positions in $\D_{\lambda/\mu}$. (The choice of $\theta$ is unimportant, since all such labelings are equivalent.) For example, if $\lambda =(5,4,2)$ and $\mu=(2,1)$ then the oriented Hasse diagram representing $(\D_{\lambda/\mu},\theta)$ is \[dlam-eq\]
(center) at (0, -1) ; (a) at (0,0) [$(1,5)$]{}; (b) at (1,-1) [$(1,4)$]{}; (c) at (2,0) [$(2,4)$]{}; (d) at (2,-2) [$(1,3)$]{}; (e) at (3,-1) [$(2,3)$]{}; (g) at (4,-2) [$(2,2)$]{}; (h) at (5,-1) [$(3,2)$]{}; (i) at (6,-2) [$(3,1)$]{}; (a) – (b); (b) – (d); (c) – (e); (e) – (g); (h) – (i); (b) – (c); (d) – (e); (g) – (h);
The elements of $\tA(\D_{\lambda/\mu},\theta)$ may be identified with *semistandard set-valued tableaux* of shape $\lambda/\mu$ as defined in [@Buch2002 §3], i.e., fillings of $\D_{\lambda/\mu}$ by nonempty finite subsets of positive integers that are weakly increasing (in the sense of $\preceq$) along rows and strictly increasing (in the sense of $\prec$) along columns. Let $$\SetSSYT(\lambda/\mu) := \tA(\D_{\lambda/\mu},\theta).$$ The *stable Grothendieck polynomial* of $\lambda/\mu$ is then \[tk-eq\] G\^[()]{}\_[/]{} := ([\_[/]{},]{}) = \_[T (/)]{} \^[|T|-|/|]{} x\^T. Both definitions are independent of the choice of $\theta$.
The power series $G^{(\beta)}_{\lambda/\mu}$ is symmetric in the $x_i$ variables, though of unbounded degree (see [@Buch2002 §2 and Thm. 3.1]). The formula sometimes appears in the literature with $\beta$ set to $\pm 1$ [@Buch2002; @BuchSamuel; @LamPyl]. There is no loss of generality in making such a specialization, but it is more convenient to work with a generic parameter. Setting $\beta=0$ in gives the skew Schur function $s_{\lambda/\mu}$.
Say that a set-valued tableau $T \in \SetSSYT(\lambda/\mu)$ is *standard* if its entries are disjoint sets, *not containing any consecutive integers*, with union $\{1,2,\dots,N\}$ for some $N\geq n$. We identify the set $\SetSYT(\lambda/\mu)$ of standard set-valued tableaux of shape $\lambda/\mu$ with the set $\tL(\D_{\lambda/\mu})$ of linear multiextensions of $\D_{\lambda/\mu}$: each $(w_1,w_2,\dots,w_N) \in \tL(\D_{\lambda/\mu})$ corresponds to the standard set-valued tableau with $i$ in box $w_i$. Then it follows from Theorem \[p-thm1\] that \[tilde-G-eq\] G\^[()]{}\_[/]{} = \_f\^\_[/]{} \^[|| - |/|]{} L\^[()]{}\_, where the sum is over compositions $\alpha$ and $ f^\alpha_{\lambda/\mu}$ is the number of standard set-valued tableaux $T \in \SetSYT(\lambda/\mu)$ with $\Des(T,\theta)=I(\alpha)$ and $|T|=|\alpha|$.
Enriched $P$-partitions {#main-sect}
=======================
In [@Stembridge1997a], Stembridge introduced an “enriched” analogue of ordinary (i.e., not set-valued) $P$-partitions as a means of constructing skew Schur $Q$-functions. Stembridge’s theory admits a $K$-theoretic analogue, which we describe in this section. This leads to interesting quasisymmetric generalizations of Ikeda and Naruse’s *shifted stable Grothendieck polynomials* [@IkedaNaruse], discussed in Section \[shifted-stable-sect\].
Labeled posets revisited
------------------------
We begin by embedding the LC-Hopf algebra $\mLPSet$ from Section \[mlpset-sect\] in a larger algebra. Given a labeled poset $(P,\gamma)$, a *valley* $v \in P$ is an element with the property that $\gamma(x) > \gamma(v)$ for all covers $x\lessdot v$ in $P$ and $\gamma(v) < \gamma(y)$ for all covers $v \lessdot y$ in $P$. Define $\ValSet(P,\gamma)$ to be the set of valleys of $(P, \gamma)$.
Let $\mLPSet^+$ be the linearly compact $\ZZ[\beta]$-module with a pseudobasis consisting of the isomorphism classes $[(P,\gamma,V)]$ for labeled posets $(P,\gamma)$ and subsets $V\subseteq \ValSet(P,\gamma)$, where we define $(P,\gamma,V) \cong (P',\gamma',V')$ if there exists an isomorphism of labeled posets $ (P,\gamma) \xrightarrow{\sim} (P',\gamma')$ taking $V$ to $V'$.
The Hopf structure on $\mLPSet$ extends to $\mLPSet^+$ as follows. Let $\nabla : \mLPSet^+ \htimes \mLPSet^+ \to \mLPSet^+$ denote the continuous linear map with $$\nabla([(P,\gamma,V)]\otimes [(Q,\delta,W)]) := [(P,\gamma,V)\sqcup (Q,\delta,W)]$$ where $(P,\gamma,V)\sqcup (Q,\delta,W) := (P\sqcup Q,\gamma\sqcup \delta,V\sqcup W)$, with $\gamma\sqcup \delta$ as in . Let $\Delta : \mLPSet^+ \to \mLPSet^+ \htimes \mLPSet^+$ be the continuous linear map with $$\Delta([(P,\gamma,V)]) := \sum_{S\cup T = P}\beta^{|S\cap T|}\cdot [(S,\gamma,V\cap S)] \otimes [(T,\gamma,V\cap T)]$$ where the sum is over all ordered pairs $(S,T)$ of subsets of $P$ such that $S$ is a lower set, $T$ is an upper set, $P = S\cup T$, and $S\cap T$ is an antichain. Finally, write $\iota : \ZZ[\beta]\to \mLPSet^+$ for the linear map with $1\mapsto [(\varnothing, \varnothing,\varnothing)]$ and $\epsilon : \mLPSet^+ \to\ZZ[\beta]$ for the continuous linear map whose value at $[(P,\gamma,V)] $ is $1$ if $P$ is empty and $0$ otherwise.
With respect to the operations $\nabla$, $\Delta$, $\iota$, $\epsilon$ just given, the $\ZZ[\beta]$-module $\mLPSet^+$ is a commutative LC-Hopf algebra.
The proof is the same as for Theorem \[mlpset-thm\], *mutatis mutandis*.
Identifying $[(P,\gamma)] = [(P,\gamma,\varnothing)]$ lets us view $\mLPSet \subset \mLPSet^+$ as LC-Hopf algebras.
We now define a map $\uzetaLP$ that turns $\mLPSet^+$ into a combinatorial LC-Hopf algebra. It is slightly different from the structure on $\mLPSet$ considered in the previous section.
The map $\zetaLP$ defined in extends to a continuous algebra morphism $\mLPSet^+ \to \ZZ[\beta][[t]]$ with $\zetaLP([(P,\gamma,V)]) := \zetaLP([(P,\gamma)])$. Define $\dzetaLP$ to be the continuous algebra morphism $\mLPSet^+ \to \ZZ[\beta][[t]]$ with $$\dzetaLP([(P,\gamma,V)]) := \begin{cases}
t^{|P|} &\text{if $V=\varnothing$ and $\gamma(x) > \gamma(y)$ for all $x\lessdot y$ in $P$}, \\
0&\text{otherwise}.
\end{cases}$$ Finally, write $\uzetaLP : \mLPSet^+ \to \ZZ[\beta][[t]]$ for the convolution product \[tilde-unimodal-zeta-eq\] := = \_ (). Next, we derive a more explicit formula for this algebra morphism.
For a labeled poset $(P,\gamma)$, let $\PeakSet(P,\gamma)$ denote the set of elements $y \in P$ for which there exist elements $x,z \in P$ with $x\lessdot y \lessdot z$ and $\gamma(x) < \gamma(y) > \gamma(z)$.
\[><-prop\] Let $(P,\gamma)$ be a labeled poset and $V \subseteq \ValSet(P,\gamma)$. Then $$\uzetaLP([(P,\gamma,V)]) =
\begin{cases}
t^{|P|} (2+\beta t)^{|\ValSet(P,\gamma)| - |V|}&\text{if }\PeakSet(P,\gamma)=\varnothing, \\
0&\text{otherwise}.
\end{cases}$$
Each pair $(S, T)$ of subsets of $P$ such that $S$ is a lower set, $T$ is an upper set, $P = S\cup T$, and $S\cap T$ is an antichain contributes \[term-eq\] \^[|ST|]{} (\[(S,,VS)\]) (\[(T,)\]) to the value of $\uzetaLP([(P,\gamma,V)])$.
If there is some element $y \in \PeakSet(P,\gamma)$, then for every such pair $(S, T)$, either $y \in S$ and there is $x\lessdot y$ in $S$ with $\gamma(x) <\gamma(y)$ or $y \in T$ and there is $z \gtrdot y$ in $T$ with $\gamma(y) > \gamma(z)$. Thus one of $\dzetaLP([(S,\gamma,V\cap S)])$ or $ \zetaLP([(T,\gamma)])$ is always zero, and consequently $\uzetaLP([(P,\gamma,V)])=0$, as claimed.
Otherwise, we may assume $\PeakSet(P,\gamma) = \varnothing$. We construct a collection of pairs $(S, T)$ of subsets of $P$, as follows:
(i) if there is $z \gtrdot y $ in $P$ and $\gamma(y) > \gamma(z)$ then $y \in S$;
(ii) if there is $x \lessdot y$ in $P$ and $\gamma(x) < \gamma(y)$ then $y \in T$;
(iii) if $y \in V$ then $y \in T$; and
(iv) if $y \in \ValSet(P,\gamma)\setminus V$ then we may assign $y$ to $S$, to $T$, or to both.
Since $\PeakSet(P,\gamma) = \varnothing$, these rules are disjoint. Suppose $(S, T)$ is one of the pairs so constructed. If $y \in S$ and $x\lessdot y$, then rules (ii) and (iv) imply that $\gamma(x) > \gamma(y)$, and rule (i) implies that $x \in S$. Thus $S$ is a lower set with $\dzetaLP([(S,\gamma,\varnothing)]) = t^{|S|}$. It follows similarly that $T$ is an upper set with $ \zetaLP([(T,\gamma)])=t^{|T|}$. Finally, it is clear that $S\cap T$ is an antichain contained in $\ValSet(P,\gamma)\setminus V$. Thus, the total contribution of the pair $(S, T)$ is $t^{|P|} (\beta t)^{|S\cap T|}$. Moreover, any pair $(S, T)$ with nonzero contribution must be constructed in this way. Since constructing such a pair $(S,T)$ is equivalent to choosing, independently for each $v \in \ValSet(P,\gamma)\setminus V$, whether $v \in S \setminus T$ or $v \in T \setminus S$ or $v \in S\cap T$, the proposition follows.
Enriched $P$-partitions {#enriched-p-partitions}
-----------------------
In Section \[sv-sect\], we encountered set-valued $P$-partitions by considering Theorem \[abs-thm\] applied to the combinatorial LC-Hopf algebra $(\mLPSet,\zetaLP)$. To obtain “enriched” analogues of those definitions, we proceed in a similar way but now consider $(\mLPSet^+,\uzetaLP)$ in place of $(\mLPSet,\zetaLP)$.
Let $\MM :=\{1'<1<2'<2<\dots\}$ denote the totally ordered *marked alphabet*. Let $\MSet$ denote the set of finite, nonempty subsets of $\MM$. For $i \in\PP$, let $|i'|: = |i| = i$, and define $x^S := \prod_{i \in S} x_{|i|}$ for $S \in \MSet$. For $S,T \in \MSet$, write $S\prec T$ if $\max(S) < \min(T)$ and $S \preceq T$ if $\max(S) \leq \min(T)$ (just as before).
\[enriched-def\] Let $(P,\gamma)$ be a labeled poset. An *enriched set-valued $(P,\gamma)$-partition* is a map $\sigma : P \to \MSet$ such that for each covering relation $s\lessdot t$ in $P$, one has $\sigma(s) \preceq \sigma(t)$ and the following properties hold:
if $\gamma(s) < \gamma(t)$ then $\sigma(s)\cap \sigma(t) \subset \{1,2,3,\dots\}$, and
if $\gamma(s) > \gamma(t)$ then $\sigma(s)\cap \sigma(t) \subset \{1',2',3',\dots\}$.
This is a generalization of Stembridge’s definition of an *enriched $P$-partition* [@Stembridge1997a §2], which corresponds to the case when $|\sigma(s)| =1$ for all $s \in P$. (For other generalizations of this notion, see [@Petersen].) A set-valued partition in the sense of Definition \[svp-def\] is an enriched set-valued $(P,\gamma)$-partition with values in $\PSet$.
Suppose $P = \{1<2<3\}$ and $\gamma(1) < \gamma(2) > \gamma(3)$. If $\sigma $ is an enriched set-valued $(P,\gamma)$-partition, then possible values for the sequence $(\sigma(1), \sigma(2), \sigma(3))$ include $(\{ 2',2\},\{2,3'\}, \{3',3\})$, $(\{2'\},\{2,3'\}, \{3,4',4\})$, and $(\{1',2'\}, \{3'\},\{3'\})$, but not $(\{1,2'\},\{2',3'\}, \{3,4'\})$ or $(\{1,2'\},\{2,3\}, \{3,4'\})$.
For a given labeled poset $(P,\gamma)$, let $\tE(P,\gamma)$ denote the set of all enriched set-valued $(P,\gamma)$-partitions. For each subset $V \subseteq \ValSet(P,\gamma)$, define (P,, V) := { (P,) : (v) v V}, so that $\tE(P,\gamma) = \tE(P,\gamma,\varnothing)$. Define the *length* and *weight* of $\sigma \in \tE(P,\gamma)$ to be $|\sigma| := \sum_{s \in P} |\sigma(s)|$ and $x^\sigma := \prod_{s \in P} x^{\sigma(s)}$. The *enriched set-valued weight enumerator* of the triple $(P,\gamma,V)$ is the quasisymmetric formal power series (P,,V) := \_[(P,,V)]{} \^[||-|P|]{} x\^.Two special cases of this construction deserve their own notation, namely, let (P,) := (P,,) (P,) := (P,,(P,)). These definitions are natural in view of the following analogue of Theorem \[<-thm\].
\[><-thm\] The continuous linear map with $[(P,\gamma,V)] \mapsto \tOmega(P,\gamma,V)$ for each labeled poset $(P,\gamma)$ and subset $V\subseteq\ValSet(P,\gamma)$ is the unique morphism of combinatorial LC-Hopf algebras $(\mLPSet^+,\uzetaLP) \to (\mQSym,\zetaq).$
The result follows by a calculation similar to the proof of Theorem \[<-thm\]. Fix a labeled poset $(P,\gamma)$ and a subset $V \subseteq\ValSet(P,\gamma)$. For each $k \in \NN$, let $\sQ_k$ denote the set of $2k$-tuples $\pi=(Q_1',Q_1,\dots, Q_k',Q_k)$ of sets with $Q_i' \cup Q_i\neq \varnothing$ for all $i\in[k]$ and $Q_1'\cup Q_1 \cup \dots \cup Q_k'\cup Q_k = P$ such that
if $s \in Q_i'\cup Q_i$ and $t \in Q_j'\cup Q_j$ where $i<j$ then $t \not < s$ in $P$,
if $s \in Q_i'$ and $t \in Q_i$, then $t \not < s$ in $P$,
if $s,t \in Q_i'$ and $s \lessdot t$ in $P$ then $\gamma(s) >\gamma(t)$,
if $s,t \in Q_i$ and $s \lessdot t$ in $P$ then $\gamma(s) <\gamma(t)$, and
$V \subseteq P \setminus (Q_1' \cup Q_2' \cup \dots \cup Q_k')$. Define $\sI_k$ to be the set of $k$-tuples of positive integers $I=(i_1,i_2,\dots,i_k)$ with $i_1<i_2<\dots<i_k$. Given $\pi \in \sQ_k$ and $I \in \sI_k$, let $|\pi| := |Q_1'| + |Q_1| + \dots + | Q_k'| + |Q_k|$ and $x^{(\pi,I)} := x_{i_1}^{|Q_1'| + |Q_1|} \cdots x_{i_k}^{|Q_k'| + |Q_k|}$. According to Theorem \[abs-thm\], the unique morphism $\Phi : (\mLPSet^+,\zetaLP) \to (\mQSym,\zetaq)$ is the continuous linear map with $$[(P,\gamma,V)] \mapsto \sum_{k \in \NN} \sum_{(\pi,I) \in \sQ_k \times \sI_k}
\beta^{|\pi| - |P|} x^{(\pi,I)} .$$ We claim that the right side is $ \tOmega(P,\gamma,V)$. For $\pi = (Q_1',Q_1,\dots,Q_k',Q_k) \in \sQ_k$ and $I = (i_1<\dots<i_k) \in \sI_k$, define $\sigma : P \to \MSet$ to be the map with $$\sigma(s) = \{i_j' : j \in [k]\text{ and } s \in Q_j'\} \cup \{ i_j : j \in [k]\text{ and }s \in Q_j\}
\quad\text{for $s \in P$.}$$ Properties (a) and (b) imply that if $s,t\in P$ are such that $s\lessdot t$ then $\sigma(s) \preceq \sigma(t)$. Given this fact, it is easy to see that properties (c)-(e) imply that $\sigma \in \tE(P,\gamma,V)$. Clearly $|\sigma| = |\pi|$ and $x^\sigma = x^{(\pi,I)}$.
It suffices to show that $(\pi,I) \mapsto \sigma$ is a bijection $\bigsqcup_{k \in \NN} \sQ_k \times \sI_k \to \tE(P,\gamma,V)$. This is straightforward: the inverse map is $\sigma \mapsto (\pi,I)$ where $I$ is the sequence of elements in $\{i_1<i_2<\dots<i_k\} := \bigcup_{s \in P} \{ |i| : i \in \sigma(s)\}$ arranged in order, and $\pi = (Q_1',Q_1,\dots,Q_k',Q_k)$ is the tuple in which $Q_j'$ and $Q_j$ are the sets consisting of the elements $s \in P$ with $i_j' \in \sigma(s)$ and $i_j \in \sigma(s)$, respectively.
The theorem implies an analogue of Corollary \[sv-products-cor\].
\[esv-products-cor\] If $(P,\gamma,V)$ and $(Q,\delta,W)$ are labeled posets then $$\tOmega(P,\gamma,V)\cdot \tOmega(Q,\delta,W) = \tOmega((P,\gamma,V)\sqcup (Q,\delta,W))$$ and $$\Delta(\tOmega(P,\gamma,V)) = \sum_{S\cup T = P} \beta^{|S\cap T|}\cdot\tOmega(S,\gamma,V\cap S)\otimes
\tOmega(T,\gamma,V\cap T)$$ where the sum is over all ordered pairs $(S,T)$ of subsets of $P$ such that $S$ is a lower set, $T$ is an upper set, $P = S\cup T$, and $S\cap T$ is an antichain.
We take note of a few special cases of Corollary \[esv-products-cor\]. Taking $V=\varnothing$ gives ((P,)) = \_[ST = P]{} \^[|ST|]{}(S,)( T,)with the summation as above. The same formula does not hold if we replace $\tOmega$ by $\oOmega$, since one may have $\ValSet(P,\gamma) \cap S \subsetneq \ValSet(S,\gamma)$. However, it does hold that (P,)(Q,) = ((P,)(Q,));this follows by setting $V =\ValSet(P, \gamma)$ and $W = \ValSet(Q, \delta)$.
Multipeak quasisymmetric functions {#multipeak-sect}
----------------------------------
For an arbitrary labeled poset $(P, \gamma)$, the structures of $\tE(P,\gamma)$ and $\tOmega(P,\gamma)$ are again determined by the set $\tL(P)$ of linear multiextensions of $P$.
Given an arbitrary finite sequence $w=(w_1,w_2,\dots,w_N)$ with an injective map $\gamma : \{w_1,w_2,\dots,w_N\} \to \ZZ$, let $\delta : [N] \to [N]$ denote the unique bijection such that if $i<j$ then \[delta-eq\] (i)>(j) (w\_i) > (w\_[j]{}), and define $$\tE(w,\gamma) := \tE([N],\delta)
\qquand
\tOmega(w,\gamma) := \tOmega([N],\delta).$$ The set $\tE(w,\gamma)$ consists of all maps $\sigma : [N] \to \MSet$ with $\sigma(1) \preceq \dots \preceq \sigma(N)$ such that, for each $i \in [N-1]$, the following properties hold:
if $i\notin \Des(w,\gamma)$ then $\sigma(w_i)\cap \sigma(w_{i+1})\subset \{1,2,3,\dots\}$, and
if $i\in \Des(w,\gamma)$ then $\sigma(w_i) \cap \sigma(w_{i+1}) \subset \{1',2',3',\dots\}$.
\[ep-thm1\] For any labeled poset $(P,\gamma)$, there is a length- and weight-preserving bijection $\tE(P,\gamma) \xrightarrow{\sim} \bigsqcup_{w \in \tL(P)} \tE(w,\gamma)$ and consequently $$\tOmega({P,\gamma}) = \sum_{w \in \tL(P)} \beta^{\ell(w) -|P|} \tOmega({w,\gamma}).$$
Setting $\beta=0$ here recovers [@Stembridge1997a Lem. 2.1].
To make sense of the constructions that follow, it may be helpful to consult Example \[enriched-ex\]. Fix $\sigma \in \tE(P,\gamma)$. For each $i \in \PP$, let $$a^{(i)} = (a^{(i)}_1,a^{(i)}_2,\dots,a^{(i)}_{M_i})
\quand
b^{(i)} = (b^{(i)}_1,b^{(i)}_2,\dots,b^{(i)}_{N_i})$$ be the sequences of elements $s \in P$ with $i' \in \sigma(s)$ and $i \in \sigma(s)$, respectively, arranged so that $
\gamma(a^{(i)}_1)> \dots > \gamma(a^{(i)}_{M_i})
$ and $
\gamma(b^{(i)}_1)< \dots < \gamma(b^{(i)}_{N_i}).
$ The concatenation $a^{(1)} b^{(1)} a^{(2)} b^{(2)}\cdots$ may have adjacent repeated entries, but omitting these repetitions produces a linear multiextension $w=(w_1,\dots,w_N) \in \tL(P)$, and there is a unique non-decreasing surjective map $ \fk t : [\sum_i M_i + \sum_i N_i ] \to [N]$ such that $a^{(1)} b^{(1)} a^{(2)} b^{(2)}\cdots = (w_{\fk t(1)}, w_{\fk t(2)}, w_{\fk t(3)}, \cdots)$. For each $j \in [N]$, define $\tau(j) \in \MSet$ to be the set containing $i'$ for $i \in \PP$ if and only if $$j \in \fk t\( M_1 + N_1 + \dots + M_{i-1} + N_{i-1} + [M_i]\)$$ and containing $i \in \PP$ if and only if $$j \in \fk t\( M_1 + N_1 + \dots + M_{i-1} + N_{i-1} + M_i + [N_i]\).$$ In other words, $\tau(j)$ contains $i'$ (respectively, $i$) precisely when $a^{(i)}$ (respectively, $b^{(i)}$) contributes the $j$th entry of $w$. This defines a map $\tau : [N] \to \MSet$.
By construction, $\tau(j) \preceq \tau(j+1)$ for all $j \in [N-1]$. Write $\delta : [N] \to [N]$ for the unique bijection satisfying for $i < j$. If $\max(\tau(j)) = \min(\tau(j+1)) = i \in \PP$, then $b^{(i)}$ contains $(w_j,w_{j+1})$ as a consecutive subsequence, so $\gamma(w_j) < \gamma(w_{j+1})$ and $\delta(j) < \delta(j+1)$. If $\max(\tau(j)) = \min(\tau(j+1)) = i'$ for some $i \in \PP$, then $u^{(i)}$ contains $(w_j,w_{j+1})$ as a consecutive subsequence, so $\gamma(w_j) > \gamma(w_{j+1})$ and $\delta(j) > \delta(j+1)$. We conclude that $\tau \in \tE(w,\gamma)$.
Given $w \in \tL(P)$ and $\tau \in \tE(w,\gamma)$, we reconstruct $\sigma$ as the map $s \mapsto \bigcup_{j: w_j = s} \tau(j)$. The correspondence $\sigma \mapsto (w,\tau)$ is then a bijection from $\cA(P,\gamma)$ to the set of pairs $(w,\tau)$ with $w \in \tL(P)$ and $\tau \in \tE(w,\gamma)$. Since it holds by construction that $|\sigma| = |\tau|$ and $x^\sigma = x^\tau$, the theorem follows.
\[enriched-ex\] Let $(P, \gamma)$ be the labeled poset of Example \[ori-ex\], so that $P=\{s_1,s_2,s_3,s_4\}$ with covering relations $s_1\lessdot s_2$ and $s_1\lessdot s_3$ and $s_2\lessdot s_4$ and $s_3\lessdot s_4$, and $\gamma(s_1) = 5$ and $\gamma(s_i) = i$ for $i\in\{2,3,4\}$. The map $\sigma : P \to \MSet$ with $$\sigma(s_1) = \{ 1',1,2'\},
\quad
\sigma(s_2) = \{ 2', 2, 3'\},
\quad
\sigma(s_3) = \{2', 3\},
\quad
\sigma(s_4) = \{3\}$$ is an element of $\tE(P,\gamma)$. In the notation of the proof of Theorem \[ep-thm1\], we have $a^{(1)} =b^{(1)}= (s_1)$, $a^{(2)} = (s_1,s_3,s_2)$, $b^{(2)} = a^{(3)} = (s_2)$, $b^{(3)} = (s_3,s_4)$, and $a^{(i)} = b^{(i)} = \emptyset$ for $i>3$. Thus $$\ba
a^{(1)} b^{(1)} a^{(2)} b^{(2)}\cdots
&=
(s_1;s_1;s_1,s_3,s_2; s_2;s_2;s_3,s_4),
\\
w &= (s_1,s_3,s_2, s_3,s_4),
\\
(\delta(1),\delta(2),\delta(3),\delta(4),\delta(5)) &= (5, 2,1,3,4).
\ea$$ The non-decreasing surjective map $\fk t :\{1,2,\dots,9\} \to \{1,2,3,4,5\}$ has $$(\fk t(1), \fk t(2),\dots, \fk t(9)) = (1, 1, 1, 2, 3, 3, 3, 4, 5)$$ and $\tau : \{1,2,3,4,5\} \to \MSet$ is the map with $$\tau(1) = \{1',1,2'\},
\quad
\tau(2) = \{2'\},
\quad
\tau(3) = \{2',2,3'\},
\quad
\tau(4) = \tau(5)=\{3\}.$$ As expected, we have $\tau \in \tE(w,\gamma)$ and $|\sigma|=|\tau| = 9$ with $x^\sigma = x^\tau = x_1^2 x_2^4 x_3^3$.
Fix a sequence $w$ of length $N$ and an injective map $\gamma : \{w_1,w_2,\dots,w_N\} \to \ZZ$. It turns out that $\tOmega(w,\gamma)$ depends only on the *peak set* of $w$, given by $$\ba
\PeakSet(w,\gamma) &:= \{ i : 1 < i < N,\ \gamma(w_{i-1}) \leq \gamma(w_i) > \gamma(w_{i+1})\}
\\&
= \{ i \in\Des(w,\gamma): i > 1, i-1\notin \Des(w,\gamma)\}.
\ea$$ (Since we allow $w$ to have repeated entries, the inequality “$\leq$” on the first line is meaningful and may sometimes be an equality.) To express this precisely, we introduce an “enriched” analogue of $L^{(\beta)}_\alpha$.
The set $\PeakSet(w,\gamma)$ is a finite subset of $\PP$ that does not contain $1$ or any two consecutive integers; we refer to such sets as *peak sets*. A *peak composition* is a composition $\alpha$ for which $I(\alpha)$ is a peak set. Equivalently, $\alpha$ is a peak composition if and only if $\alpha_i \geq 2$ for $1\leq i < \ell(\alpha)$. We define the *multipeak quasisymmetric function* of a peak composition $\alpha \vDash N$ to be \[kbeta-eq\] K\^[()]{}\_:= \_[ S ]{} \^[|S|-N]{} x\^S where the sum is over $N$-tuples $S=(S_1 , S_2 , \dots, S_N)$ of sets $S_i \in \MSet$ with
- $S_1 \preceq S_2 \preceq \dots\preceq S_N$,
- $S_i \cap S_{i+1} \subset \{1',2',3',\dots\}$ if $i \in I(\alpha)$, and
- $S_i \cap S_{i+1} \subset \{1,2,3,\dots\}$ if $i \notin I(\alpha)$.
\[kbeta-ex\] If $\alpha=(2,1)$ then is the sum over all triples $S_1 \preceq S_2 \preceq S_3$ in $\MSet$ with $S_1 \cap S_2 \subset \{1,2,\dots\}$ and $S_2 \cap S_3\subset \{1',2',\dots\}$. One can show that in this case is equivalent to the formula $$K^{(\beta)}_{(2,1)}
=\sum_{n} \sum_{i_1 <i_2<\dots <i_n} \beta^{n-3} \(\tbinom{n-1}{2} +\beta \sum_{1\leq j < k \leq n} x_{i_j} \oplus x_{i_k} \) \prod_{j=1}^n x_{i_j}\oplus x_{i_j}$$ where $x\oplus y := x + y + \beta xy$. There is an amusing formal similarity between this expression and the one for $L^{(\beta)}_{(2,1)}$ in Example \[lbeta-ex\].
Here we have an analogue of Proposition \[by-def-prop\]:
\[k1-prop\] Fix a choice of $(w,\gamma)$ as above, and let $\alpha \vDash \ell(w)$ be the unique composition such that $I(\alpha)= \PeakSet(w,\gamma)$. Then $K^{(\beta)}_\alpha = \tOmega(w,\gamma)$.
Let $w$ be a sequence of length $N$, let $\gamma : \{w_1,w_2,\dots,w_N\} \to \ZZ$ be an injective map, and let $\alpha\vdash N$ be the composition with $I(\alpha) = \PeakSet(w,\gamma)$. If $\PeakSet(w,\gamma) = \Des(w,\gamma)$ then $\tOmega(w,\gamma) = K^{(\beta)}_\alpha$ holds by definition.
Suppose $1<i<N$ is such that $i-1,i \in \Des(w,\gamma)$ and $i+1 \notin\Des(w,\gamma)$. Let $w'$ be a word of length $N$ with an injective map $\gamma' : \{w_1',w_2',\dots,w_N'\}\to\ZZ $ such that $\Des(w',\gamma') = \Des(w,\gamma) \setminus \{i\}$. Then $\PeakSet(w',\gamma') = \PeakSet(w,\gamma) \neq \Des(w,\gamma)$ and $\sigma(i) \cap \sigma(i+1) \subseteq\{1',2',\dots\}$ for all $\sigma \in \tE(w,\gamma)$. It suffices to show that $\tOmega(w,\gamma) = \tOmega(w',\gamma')$, since iterating this identity lets us assume without loss of generality that $\PeakSet(w,\gamma) = \Des(w,\gamma)$.
To this end, it is enough to construct a length- and weight-preserving bijection $\tE(w,\gamma) \xrightarrow{\sim} \tE(w',\gamma')$. For $\sigma \in \tE(w,\gamma)$, define $\sigma' : [N] \to \MSet$ as follows. Set $\sigma'(j) = \sigma(j)$ for $j \notin \{i,i+1\}$, and if $\max(\sigma(i)) < \min(\sigma(i+1)) $ then define $$\sigma'(i) = \sigma(i)
\quand \sigma'(i+1) = \sigma(i+1).$$ Suppose $\max(\sigma(i)) = \min(\sigma(i+1)) = a'$ for $a \in \PP$. If $a \in \sigma(i+1)$ then set $$\sigma'(i) = \sigma(i) \sqcup \{a\}\quand \sigma'(i+1) = \sigma(i+1) \setminus\{a'\}$$ and if $a \notin \sigma(i+1)$ then set $$\sigma'(i) = \sigma(i)\setminus\{a'\} \sqcup \{a\} \quand \sigma'(i+1) = \sigma(i+1)\setminus\{a'\} \sqcup \{a\}.$$ The resulting map $\sigma'$ is an element of $\tE(w',\gamma')$ with $|\sigma| = |\sigma'|$ and $x^\sigma = x^{\sigma'}$ and it is easy to see that $\sigma \mapsto \sigma'$ is a bijection $\tE(w,\gamma) \xrightarrow{\sim} \tE(w',\gamma')$, as needed.
Setting $\beta=0$ in recovers Stembridge’s definition [@Stembridge1997a §2.2] of the *peak quasisymmetric function* $
K_\alpha := K^{(0)}_\alpha =\sum_{\alpha'} 2^{\ell(\alpha')} M_{\alpha'} \in \QSym$, where the sum is over all compositions $\alpha' \vDash |\alpha|$ with $I(\alpha) \subseteq I(\alpha') \cup (I(\alpha')+1)$. If $\beta$ has degree zero and $x_i$ has degree one, then $K_\alpha$ is the nonzero homogeneous component of $K^{(\beta)}_\alpha$ of lowest degree.
The peak quasisymmetric functions are a basis for a subalgebra of $\QSym$ [@Stembridge1997a Thm. 3.1]. The power series $\{ K^{(\beta)}_\alpha\}$ are therefore a pseudobasis for a linearly compact $\ZZ[\beta]$-submodule of $\mQSym$.
There is a slight variant of the preceding constructions which is also of interest. Let $w=(w_1,w_2,\dots,w_N)$ be an arbitrary finite sequence along with an injective map $\gamma : \{w_1,w_2,\dots,w_N\} \to \ZZ$. Extending our earlier notation, let $$\oOmega(w,\gamma) := \oOmega([N],\delta)$$ where $\delta : [N] \to [N]$ is again the map satisfying for all $1\leq i < j \leq N$. For a peak composition $\alpha \vDash N$, define \[okbeta-eq\] \^[()]{}\_:= \_S \^[|S|-N]{} x\^S , where the sum is over $N$-tuples $S=(S_1 , S_2 , \dots, S_N)$ of sets $S_i \in \MSet$ with
- $S_1 \preceq S_2 \preceq \dots\preceq S_N$,
- $S_i \prec S_{i+1}$ if $i \in I(\alpha)$, and $S_{i+1} \subseteq \PP$ if $i \in \{0\} \cup I(\alpha)$, and
- $S_i \cap S_{i+1} \subseteq \PP$ if $i \notin I(\alpha)$.
These are the same as the tuples indexing the sum in except we require the set $S_{i+1}$ to contain only unprimed numbers if $i \in \{0\} \cup I(\alpha)$.
If $\alpha=(2,1)$ then is the sum over all triples $S_1 \preceq S_2 \prec S_3$ with $S_1, S_3\in \PSet$ and $S_2 \in \MSet$. In this case is equivalent to $$\oK^{(\beta)}_{(2,1)}
=\sum_{n} \sum_{i_1 <i_2<\dots <i_n} \sum_{1\leq j < k \leq n} \beta^{n-3} \cdot \Pi(i;j,k)\cdot x_{i_1}x_{i_2}\cdots x_{i_n}$$ where we define $$\Pi(i;j,k) := \begin{cases}
(1+\beta x_{i_{j}}) \cdot \prod_{j<t<k} (2 + \beta x_{i_{t}} ) \cdot (1+\beta x_{i_k}) &\text{if $j+1<k$}\\
\beta\cdot (x_{i_j} + x_{i_{j+1}} + \beta x_{i_j} x_{i_{j+1}}) & \text{if }j+1=k.
\end{cases}$$
We have a second analogue of Proposition \[by-def-prop\]:
\[k2-prop\] Fix a choice of $(w,\gamma)$ as above, and let $\alpha \vDash \ell(w)$ be the unique composition such that $I(\alpha)= \PeakSet(w,\gamma)$. Then $\oK^{(\beta)}_\alpha = \oOmega(w,\gamma)$.
Our argument is similar to the proof of Proposition \[k1-prop\] but relies on a different bijection. Let $w=(w_1,w_2,\dots,w_N)$ be a finite sequence of length $N$, let $\gamma : \{w_1,w_2,\dots,w_N\} \to \ZZ$ be an injective map, and let $\alpha \vDash N$ be the composition with $I(\alpha) = \PeakSet(w,\gamma)$. We have $\oOmega(w,\gamma) = \oK^{(\beta)}_\alpha$ by definition if $\PeakSet(w,\gamma) = \Des(w,\gamma)$. Let $\oE(w, \gamma)
= \cE([N],\delta,\ValSet([N],\delta))$ where $\delta : [N]\to[N]$ satisfies , so that $\oOmega(w, \gamma) = \sum_{\sigma \in \oE(w,\gamma)} \beta^{|\sigma|-N} x^\sigma$.
Suppose $1<i<N$ is such that $i-1,i \in \Des(w,\gamma)$ and $i+1 \notin\Des(w,\gamma)$. Define $w'$ and $\gamma'$ as in the proof of Proposition \[k1-prop\] so that $\Des(w',\gamma') = \Des(w,\gamma) \setminus \{i\}$ and $\PeakSet(w',\gamma') = \PeakSet(w,\gamma) \neq \Des(w,\gamma)$. It suffices to show that $\oOmega(w,\gamma) = \oOmega(w',\gamma')$, and we do so by constructing a length- and weight-preserving bijection $\oE(w,\gamma) \xrightarrow{\sim} \oE(w',\gamma')$.
We have $i+1 \in \ValSet(w,\gamma) :=
\{ j \in [N] : \gamma(w_{j-1}) > \gamma(w_j) \leq \gamma(w_{j+1})\}$, where we define $\gamma(w_0) = \gamma(w_{N+1}) := \infty$. More specifically, it holds that $$\ValSet(w',\gamma') = \ValSet(w,\gamma)\setminus\{i+1\} \sqcup \{i\}.$$ Let $\sigma \in \tE(w,\gamma)$ and observe that necessarily $$\sigma(i) \cap \sigma(i+1) \subset \{1',2',\dots\}
\quand
\sigma(i+1) \subset \PP.$$ We must therefore have $\sigma(i) \prec \sigma(i+1)$. We define a map $\sigma' : [N] \to \MSet$ as follows. Set $\sigma'(j) = \sigma(j)$ for $j \notin \{i,i+1\}$. If $\sigma(i) \subseteq \PP$ then define $$\sigma'(i) = \sigma(i)
\quand \sigma'(i+1) = \sigma(i+1).$$ Otherwise, there exists a smallest $a\in\PP$ with $a' \in \sigma(i) \cap \{1',2',\dots\}$. In this case, if $b := \min(\sigma(i+1)) \in \PP$ and $b' \in \sigma(i)$ then we set $$\ba
\sigma'(i) &= \{ x \in \sigma(i) : x < a'\} \sqcup \{a\},\\
\sigma'(i+1) &=\{ x \in \sigma(i) : a' < x\} \sqcup \sigma(i+1),
\ea$$ while if $b' \notin \sigma(i)$ then we set $$\ba
\sigma'(i) &= \{ x \in \sigma(i) : x < a'\} \sqcup \{a\},\\
\sigma'(i+1) &=\{ x \in \sigma(i) : a' < x\} \sqcup (\sigma(i+1)\setminus\{b\}) \sqcup \{b'\}.
\ea$$ The resulting map $\sigma'$ is an element of $\oE(w',\gamma')$ with $|\sigma| = |\sigma'|$ and $x^\sigma = x^{\sigma'}$ and one can check that $\sigma \mapsto \sigma'$ is a bijection $\oE(w,\gamma) \xrightarrow{\sim} \oE(w',\gamma')$, as needed.
Setting $\beta=0$ in recovers the power series denoted $\oK_\alpha$ in [@Stembridge1997a], which is the nonzero homogeneous component of $\oK^{(\beta)}_\alpha$ of lowest degree. Since $\oK_\alpha = 2^{-\ell(\alpha)} K_\alpha$, the power series $\{ \oK^{(\beta)}_\alpha\}$ are a pseudobasis for a (different) linearly compact $\ZZ[\beta]$-submodule of $\mQSym$.
Unlike $\oK_\alpha = \oK^{(0)}_\alpha$ and $K_\alpha=K^{(0)}_\alpha$, the power series $\oK^{(\beta)}_\alpha$ and $K^{(\beta)}_\alpha$ are not scalar multiples of each other. We investigate their precise relationship next.
Poset operators
---------------
Fix a labeled poset $(P,\gamma)$ and let $V\subseteq \ValSet(P,\gamma)$. It is not hard to show that we always have $\Omega^{(0)}(P,\gamma,V) = 2^{-|V|} \Omega^{(0)}(P,\gamma)$. The relationship between $\tOmega(P,\gamma,V)$ and $\tOmega(P,\gamma)$ is more complicated, involving the *vertex doubling operators* $\fkD_v$ that we now define.
Given $v \in P$, define $\fkD_v(P,\gamma) := (Q,\delta)$ to be the labeled poset with the following properties.
- As a set, $Q = P\sqcup\{ v'\}$ is the disjoint of union of $P$ and a new element $v'$.
- The order on $Q$ is the one extending the order on $P$ such that $v\lessdot v'$ and for every $x \in P \setminus \{v\}$, $v'$ is related to $x$ in the same way that $v$ is.
- The labeling map $\delta$ satisfies $\delta(v) =\gamma(v)$, $\delta(v') = \gamma(v)+1$, $\delta(x) = \gamma(x)+1$ for $x \in P$ with $\gamma(v) < \gamma(x)$, and $\delta(x) = \gamma(x)$ for all other $x \in P$.
\[poset-ex1\] If we represent labeled posets as oriented Hasse diagrams as in Example \[ori-ex\], then the doubling operator $\fkD_v$ acts as follows: $$\fkD_{v_2}\(\begin{tikzpicture}[baseline=(b.base), xscale=0.6, yscale=0.5]
\tikzset{edge/.style = {<-}}
\node (c1) at (-2,3) {$v_3$};
\node (c2) at (2,3) {$v_4$};
\node (b) at (0,0) {$v_2$};
\node (a) at (0,-3) {$v_1$};
\node (d) at (-2, 0) {$v_5$};
\draw[edge] (b) -- (a);
\draw[edge] (b) -- (c1);
\draw[edge] (c2) to (b);
\draw[edge] (a) to (d);
\draw[edge] (d) to (c1);
\end{tikzpicture}\)\ =\
\begin{tikzpicture}[baseline=(b.base), xscale=0.6, yscale=0.5]
\tikzset{edge/.style = {<-}}
\node (c1) at (-2,3) {$v_3$};
\node (c2) at (2,3) {$v_4$};
\node (b) at (0,0) {};
\node (b2) at (0,1) {$v_2'$};
\node (b1) at (0,-1) {$v_2$};
\node (a) at (0,-3) {$v_1$};
\node (d) at (-2, 0) {$v_5$};
\draw[edge] (a) to (d);
\draw[edge] (d) to (c1);
\draw[edge] (b1) -- (a);
\draw[edge] (b2) -- (c1);
\draw[edge] (b1) -- (b2);
\draw[edge] (c2) -- (b2);
\end{tikzpicture}$$
If $P$ is a chain (i.e., linearly ordered) then $\fkD_v(P,\gamma)$ is also a chain.
\[fkd-lem\] Let $(P,\gamma)$ be a labeled poset. Suppose $v \in V \subseteq \ValSet(P,\gamma)$ and $(Q,\delta) = \fkD_v(P,\gamma)$. There is then a length- and weight-preserving bijection $$\tE(P,\gamma,V\setminus\{v\}) \xrightarrow{\sim} \tE(P,\gamma,V) \sqcup \tE(P,\gamma,V) \sqcup \tE(Q,\delta,V).$$ Consequently, $\tOmega(P,\gamma,V\setminus\{v\}) = 2 \cdot \tOmega(P,\gamma,V) + \beta \cdot \tOmega(Q,\delta,V)$.
Let $\cE'(P,\gamma,V)$ be the set of enriched set-valued $(P,\gamma)$-partitions $\tau \in \cE(P,\gamma,V\setminus\{v\})$ such that $\max(\tau(v))$ is the unique primed entry of $\tau(v)$. Removing the prime from $\max(\tau(v))$ defines a length- and weight-preserving bijection $\cE'(P,\gamma,V) \to \cE(P,\gamma,V)$. We describe a bijection $\tE(P,\gamma,V\setminus\{v\}) \to \tE(P,\gamma,V) \sqcup \tE'(P,\gamma,V)
\sqcup \tE(Q,\delta,V)$.
Fix $\sigma \in \tE(P,\gamma,V\setminus\{v\})$ and define $\sigma' $ as follows. If $\sigma \in \tE(P,\gamma,V) \sqcup \tE'(P,\gamma,V)$, then we set $\sigma' = \sigma$. Otherwise, $\sigma(v)$ contains a least primed integer $a' \neq \max(\sigma(v))$, and we define $\sigma' \in \tE(Q,\delta,V)$ to be the map with $$\sigma'(v) = \{ x \in \sigma(v) : x < a' \} \sqcup \{a\}
\quand
\sigma'(v') = \{ x \in \sigma(v) : a' < x\}$$ and $\sigma'(s) = \sigma(s)$ for all $s \in P \setminus \{v\} = Q\setminus\{v,v'\}$. It is easy to see that $\sigma \mapsto \sigma'$ is a length- and weight-preserving bijection of the desired type.
Fix a labeled poset $(P, \gamma)$, a subset $V \subseteq \ValSet(P, \gamma)$, and a vertex $v \in P$. Let $(Q,\delta) := \fkD_v (P,\gamma)$. We use the notational shorthand $\fkD_v \cdot \tOmega(P,\gamma, V)$ to mean $$\fkD_v \cdot \tOmega(P,\gamma, V) := \tOmega(Q,\delta, V),$$ so that $
\fkD_v \cdot \oOmega(P, \gamma) = \oOmega(\fkD_v(P, \gamma))
$ and $
\fkD_v \cdot \tOmega(P, \gamma) = \tOmega(\fkD_v(P, \gamma)).
$ In this shorthand, one has by simple telescoping that $$(2 + \beta \fkD_v) \cdot \frac{1}{2} \sum_{n=0}^\infty (-\beta/2)^n \fkD_v^n \cdot \tOmega(P,\gamma, V) = \tOmega(P,\gamma, V)$$ in $\mQSym_{\QQ[\beta]}$, and we define $(2 + \beta \fkD_v)^{-1} :=\frac{1}{2} \sum_{n=0}^\infty (-\beta/2)^n \fkD_v^n$.
\[op-thm\] Let $(P,\gamma)$ be a labeled poset, $V$ a subset of $\ValSet(P, \gamma)$, and $U = \ValSet(P, \gamma) \setminus V$. Then $$\ba
\tOmega(P,\gamma,V) &= \prod_{u \in U} (2 + \beta \fkD_u)\cdot \oOmega(P,\gamma)
\\&= \prod_{v \in V} (2 + \beta \fkD_v)^{-1}\cdot \tOmega(P,\gamma).
\ea$$ In particular, we have $\tOmega(P,\gamma) = \prod_{v \in \ValSet(P,\gamma)} (2 + \beta \fkD_v)\cdot \oOmega(P,\gamma).$
Lemma \[fkd-lem\] implies that $(2 + \beta \fkD_u) \cdot \tOmega(P,\gamma, V\sqcup\{u\}) = \tOmega(P,\gamma,V)$ if $u \in U$. The result follows by repeating this observation.
As a corollary, we deduce that each $K^{(\beta)}_\alpha$ is a finite $\ZZ[\beta]$-linear combination of $\oK^{(\beta)}_{\alpha}$’s while each $\oK^{(\beta)}_{\alpha}$ is an infinite $\QQ[\beta]$-linear combination of $K^{(\beta)}_{\alpha}$’s.
\[equiexp-cor\] If $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_n)$ is a peak composition then $$K^{(\beta)}_\alpha = \sum_{\delta \in \{0,1\}^n} 2^{n-|\delta|} \beta^{|\delta|} \oK^{(\beta)}_{\alpha+\delta}
\quand \oK^{(\beta)}_\alpha = \tfrac{1}{2^n}\sum_{\delta \in (\NN)^n} (-\beta/2)^{|\delta|} K^{(\beta)}_{\alpha+\delta}.$$
Given Propositions \[k1-prop\] and \[k2-prop\], it is straightforward to deduce these identities from Theorem \[op-thm\].
Quasisymmetric subalgebras
--------------------------
As noted in Section \[multipeak-sect\], both $\{K^{(\beta)}_\alpha\}$ and $\{ \oK^{(\beta)}_\alpha\}$ (with $\alpha$ ranging over all peak compositions) are linearly independent families in $\mQSym$. Moreover, all $x$-monomials appearing in $K^{(\beta)}_\alpha$ and $\oK^{(\beta)}_\alpha$ have degree at least $|\alpha|$. We may therefore make the following definitions.
Let $\mcoPeak $ and $ \omcoPeak$ denote the linearly compact $\ZZ[\beta]$-modules with the multipeak quasisymmetric functions $\{ K^{(\beta)}_\alpha\}$ and $\{ \oK^{(\beta)}_\alpha\}$ ($\alpha$ ranging over all peak compositions) as respective pseudobases. Define $\mcoPeak_{\QQ[\beta]}$ to be the linearly compact $\QQ[\beta]$-module with $\{ K^{(\beta)}_\alpha\}$ as a pseudobasis.
\[inter-thm\] Both $\mcoPeak$ and $\omcoPeak$ are LC-Hopf subalgebras of $\mQSym$, and $\omcoPeak = \mQSym \cap \mcoPeak_{\QQ[\beta]} \supseteq \mcoPeak$.
On setting $\beta=0$, this reduces to [@Stembridge1997a Cor. 3.4], whose proof is similar.
The modules $\mcoPeak$ and $\omcoPeak$ are LC-Hopf subalgebras of $\mQSym$ since (in view of Theorem \[ep-thm1\] and the results in the previous section) they are the images of $\mLPSet $ and $\mLPSet^+$ under the LC-Hopf algebra morphism described in Theorem \[><-thm\].
Corollary \[equiexp-cor\] implies that $\omcoPeak \subseteq \mQSym \cap \mcoPeak_{\QQ[\beta]}$. Write $\prec$ for the linear order on compositions with $\alpha \prec \alpha'$ if $|\alpha| < |\alpha'|$ or if $|\alpha| =|\alpha'|$ and $\alpha$ exceeds $\alpha'$ in lexicographic order. By , if $\alpha$ is a peak composition then $\oK^{(\beta)}_\alpha \in M_\alpha + \sum_{\alpha \prec \alpha'} \ZZ[\beta] M_{\alpha'}$, so any $\QQ[\beta]$-linear combination of $\oK^{(\beta)}_\alpha$’s in $\mQSym$ must have coefficients in $\ZZ[\beta]$. Thus, in view of Corollary \[equiexp-cor\], the reverse inclusion $\omcoPeak \supseteq \mQSym \cap \mcoPeak_{\QQ[\beta]}$ also holds.
\[omco-cor\] If $V \subseteq \ValSet(P,\gamma)$ then $\tOmega(P,\gamma,V) \in \omcoPeak$.
This follows from Theorem \[inter-thm\] since $\tOmega(P,\gamma,V)$ belongs to $\mQSym$ by definition and to $\mcoPeak_{\QQ[\beta]}$ by Theorem \[op-thm\].
As in Example \[kbeta-ex\], let $x \oplus y := x + y + \beta xy$. Also define $\ominus x := \frac{-x}{1+\beta x}$, so that $x\oplus (\ominus x) = 0$. Elements of $\mcoPeak_{\QQ[\beta]}$ satisfy the following cancellation law; setting $\beta=0$ in this statement recovers [@Stembridge1997a Lem. 3.7].
\[q-cancel-lem\] If $f \in \mcoPeak_{\QQ[\beta]}$ then $f(t,\ominus t,x_3,x_4,\dots) = f(x_3,x_4,\dots)$ where $t$ is an indeterminate commuting with each $x_i$.
If $f \in \ZZ[\beta][[x_1,x_2,\dots]]$ then let $f(x_1,x_2,\dots,x_n)\in \ZZ[\beta][x_1,x_2,\dots,x_n]$ denote the polynomial obtained by setting $x_i=0$ for all $i>n$. Suppose $(P,\gamma)$ is a labeled poset. It follows from the definition of $\tOmega(P,\gamma)$ that $$\tOmega(P,\gamma) = \sum_{S\cup T = P}
\beta^{|S\cap T|}\cdot
\tOmega(S,\gamma)(x_1,x_2)\cdot \tOmega(T,\gamma)(x_3,x_4,\dots)$$ where the sum is over all ordered pairs $(S,T)$ of subsets of $P$ such that $S$ is a lower set, $T$ is an upper set, $P = S\cup T$, and $S\cap T$ is an antichain. To prove the lemma, it is therefore enough to check that $\tOmega(P,\gamma)(t,\ominus t)=0$ whenever $P$ is nonempty. Let $\alpha$ be a nonempty peak composition. By Theorem \[ep-thm1\] and Proposition \[k1-prop\], it suffices to show that $K^{(\beta)}_\alpha(t,\ominus t) =0$.
It is clear from that $K^{(\beta)}_\alpha(x_1,x_2) = 0$ if $\ell(\alpha)>2$. If $\alpha$ has a single part, then in the notation of [@IkedaNaruse] one has $K^{(\beta)}_{\alpha} = \bGQ_\alpha$ (see [@IkedaNaruse Thm. 9.1] and Section \[shifted-stable-sect\] below) and the desired identity is [@IkedaNaruse Prop. 3.1]. Assume $\alpha = (j,N-j)\vDash N$ has two parts with $j\geq 2$. From , it is easy to see that $K^{(\beta)}_{\alpha}(x_1,x_2)= \sum_\sigma \beta^{|\sigma|-N} x^\sigma$ where the sum is over all maps $\sigma:[N]\to \{\text{nonempty subsets of }\{1',1,2',2\}\}$ such that
- $\sigma(1)$ is $\{1'\}$ or $\{1\}$ or $\{1', 1\}$, and $\sigma(i) =\{1\}$ for $1<i<j$,
- $\sigma(j)$ is $\{1\}$ or $\{2'\}$ or $\{1,2'\}$, and
- $\sigma(j+1)$ is $\{2'\}$ or $\{2\}$ or $\{2',2\}$, and $\sigma(i) = \{2\}$ for $j+1<i \leq N$.
Thus $K^{(\beta)}_{\alpha}(x_1,x_2) = (2x_1 + \beta x_1^2) x_1^{j-2} (x_1 + x_2 + \beta x_1 x_2) (2x_2 + \beta x_2^2) x_2^{N-j-1}$, which we can rewrite as $K^{(\beta)}_{\alpha}(x_1,x_2) = (x_1\oplus x_1)(x_1\oplus x_2)(x_2\oplus x_2) x_1^{j-2} x_2^{N-j-1}$. Since $t \oplus (\ominus t) = 0$, we have $K^{(\beta)}_\alpha(t,\ominus t) = 0$ as needed.
The LC-Hopf subalgebra $\mcoPeak$ is also a quotient of $\mQSym$. Define $$\ttheta : \mQSym \to \mcoPeak$$ to be the continuous $\ZZ[\beta]$-linear map with $L^{(\beta)}_\alpha \mapsto K^{(\beta)}_{\Lambda(\alpha)}$ for each composition $\alpha$, where $\Lambda(\alpha)$ is the peak composition of $|\alpha|$ characterized by \[Lambda-eq\] I(()) = { i I() : i > 1, i-1 I()}.If $\Des(w,\gamma) = I(\alpha)$ then $\PeakSet(w,\gamma) = I(\Lambda(\alpha))$, and if $\alpha$ is already a peak composition then $\Lambda(\alpha) = \alpha$.
\[ttheta-cor\] The map $\ttheta$ sends $\tGamma(P,\gamma)\mapsto \tOmega(P,\gamma)$ for all labeled posets $(P,\gamma)$ and is a surjective morphism of LC-Hopf algebras $ \mQSym \to \mcoPeak$.
Theorems \[p-thm1\] and \[ep-thm1\] show that $\ttheta(\tGamma(P,\gamma)) = \tOmega(P,\gamma)$ for all labeled posets $(P,\gamma)$. Comparing Corollaries \[sv-products-cor\] and \[esv-products-cor\] shows that this is a bialgebra morphism, and hence an LC-Hopf algebra morphism. Since the $K^{(\beta)}_{\alpha}$ form a pseudobasis for $\mcoPeak$, it is also surjective.
As discussed in Remark \[product-rmk\], results of Lam and Pylyavskyy [@LamPyl §5.4] lead to product and coproduct formulas for the multifundamental quasisymmetric functions $L^{(\beta)}_\alpha$. Applying $\ttheta$ to such identities gives analogous (co)product formulas for the multipeak quasisymmetric functions $K^{(\beta)}_\alpha$.
Shifted stable Grothendieck polynomials {#shifted-stable-sect}
---------------------------------------
A primary motivation for our definition of enriched set-valued $P$-partitions comes from the families of *shifted stable Grothendieck polynomials* $\GQ_\lambda$ and $\GP_\lambda$ introduced by Ikeda and Naruse in [@IkedaNaruse]. Mirroring the situation in Section \[stable-sect\], we can recover these power series as the enriched set-valued weight enumerators of labeled posets associated to shifted Young diagrams. In this section, we will also describe skew analogues of $\GQ_\lambda$ and $\GP_\lambda$, which have not been considered previously.
Suppose $\lambda =(\lambda_1> \lambda_2 > \dots > 0)$ and $\mu = (\mu_1> \mu_2> \dots > 0)$ are strict partitions with $\mu \subseteq \lambda$. The *shifted skew diagram* of $\lambda/\mu$ is \[sd-eq\]\_[/]{} := { (i,i+j-1) : \_i < j \_i}.Let $\SD_\lambda = \SD_{\lambda/\emptyset}$. As usual, we consider $\SD_{\lambda/\mu}$ to be partially ordered with $(i,j) \leq (i',j')$ if $i\leq i'$ and $j\leq j'$. Let $n = |\lambda| - |\mu|$ and fix a bijection $\theta : \SD_{\lambda/\mu} \to [n]$ satisfying the conditions in . For example, if $\lambda =(5,4,2)$ and $\mu=(2,1)$ then the oriented Hasse diagram representing $(\SD_{\lambda/\mu},\theta)$ is $$\begin{tikzpicture}[baseline=(center.base), xscale=1, yscale=1]
\tikzset{edge/.style = {->}}
\node (center) at (0, -1) {};
\node (a) at (3,-3) {$(1,3)$};
\node (b) at (1,-1) {$(1,5)$};
\node (c) at (2,0) {$(2,5)$};
\node (d) at (2,-2) {$(1,4)$};
\node (e) at (3,-1) {$(2,4)$};
\node (g) at (4,-2) {$(2,3)$};
\node (h) at (4,0) {$(3,4)$};
\node (i) at (5,-1) {$(3,3)$};
\draw[edge] (d) -- (a);
\draw[edge] (a) -- (g);
\draw[edge] (b) -- (d);
\draw[edge] (c) -- (e);
\draw[edge] (e) -- (g);
\draw[edge] (g) -- (i);
\draw[edge] (h) -- (i);
\draw[edge] (b) -- (c);
\draw[edge] (d) -- (e);
\draw[edge] (e) -- (h);
\end{tikzpicture}$$ It may be helpful to contrast this picture with .
The elements of $\tE(\SD_{\lambda/\mu},\theta)$ may be identified with *semistandard shifted set-valued (marked) tableaux* of shape $\lambda/\mu$ as defined in [@IkedaNaruse §9.1], i.e., fillings of $\SD_{\lambda/\mu}$ by finite nonempty subsets of $\MM$ that are weakly increasing along rows and columns in the sense of the relation $\preceq$, such that no unprimed number appears twice in the same column and no primed number appears twice in the same row. We let $$\SetSSMT(\lambda/\mu) := \tE(\SD_{\lambda/\mu},\theta)$$ and define the *$K$-theoretic Schur $Q$-function* of $\lambda/\mu$ to be \[gq-eq\] \_[/]{} := ([\_[/]{},]{}) = \_[T (/)]{} \^[|T|-|/|]{}x\^T. Both definitions are independent of $\theta$.
The special case $ \bGQ_{\lambda} := \bGQ_{\lambda/\emptyset}$ coincides with Ikeda and Naruse’s definition of a $K$-theoretic Schur $Q$-function [@IkedaNaruse Thm. 9.1]. We will show in Section \[sym-sect\] that the quasisymmetric function $\bGQ_{\lambda/\mu}$ is always symmetric in the $x_i$ variables; when $\mu=\emptyset$, this follows from [@IkedaNaruse Thm. 9.1]. Setting $\beta=0$ reduces to the definition of the *skew Schur $Q$-function* $Q_{\lambda/\mu}$ described, for example, in [@Stembridge1997a App. A.1].
Recall that a semistandard set-valued tableau is *standard* if its entries are disjoint sets, not containing any consecutive integers, with union $\{1,2,\dots,N\}$ for some $N\geq n$. The set $\tL(\SD_{\lambda/\mu})$ of linear multiextensions of $\SD_{\lambda/\mu}$ is naturally identified with the set of standard set-valued (unmarked) tableaux of shifted shape $\lambda/\mu$; a sequence $(w_1,w_2,\dots,w_N) \in \tL(\SD_{\lambda/\mu})$ corresponds to the standard set-valued tableau containing $i$ in box $w_i$. Let $$\SetSYT_{\text{shifted}}(\lambda/\mu) := \tL(\SD_{\lambda/\mu})$$ and define $\PeakSet(T) := \PeakSet(T,\theta)$ for $T \in \SetSYT_{\text{shifted}}(\lambda/\mu)$; then $i \in \PeakSet(T)$ if and only if $i-1$, $i$, and $i+1$ all appear in $T$ with $i$ in a strictly greater column than $i-1$ and a strictly lesser row than $i+1$. Theorem \[ep-thm1\] implies that \[tilde-GQ-eq\] \_[/]{} = \_g\^\_[/]{} \^[|| - |/|]{} K\^[()]{}\_, where the sum is over peak compositions $\alpha$ and $g^\alpha_{\lambda/\mu}$ is the number of tableaux $T \in \SetSYT_{\text{shifted}}(\lambda/\mu)$ with $|T|=|\alpha|$ and $\PeakSet(T) = I(\alpha)$.
There is a second family of shifted stable Grothendieck polynomials discussed in [@IkedaNaruse], which arise in a similar way as enriched set-valued weight enumerators. Continue to let $\lambda$ and $\mu$ be strict partitions with $\mu \subseteq \lambda$. Let $n=|\lambda|-|\mu|$ and fix a bijection $\theta : \SD_{\lambda/\mu} \to [n]$ satisfying as above. Define $$V_{\lambda/\mu} := \{ (i,j) \in \SD_{\lambda/\mu} : i=j\} \subseteq \ValSet(\SD_{\lambda/\mu},\theta),$$ where the containment is equality if $\mu = \emptyset$. The elements of $\tE(\SD_{\lambda/\mu},\theta,V_{\lambda/\mu})$ are precisely the semistandard shifted set-valued tableaux in $\SetSSMT(\lambda/\mu)$ whose entries on the main diagonal contain only unprimed numbers. We define the *$K$-theoretic Schur $P$-function* of $\lambda/\mu$ to be \[gp-eq\] \_[/]{} := (\_[/]{},,V\_[/]{}) = \_ \^[|T|-|/|]{} x\^T. This formula is again independent of the choice of $\theta$.
The case $ \bGP_{\lambda} := \bGP_{\lambda/\emptyset} = \oOmega(\SD_{\lambda/\mu},\theta)$ is Ikeda and Naruse’s definition of a $K$-theoretic Schur $P$-function [@IkedaNaruse Thm. 9.1]. As with $\bGQ_\lambda$, Ikeda and Naruse prove that $\bGP_\lambda$ is symmetric in the $x_i$ variables; we extend this result to skew shapes below. Setting $\beta=0$ in recovers the *skew Schur $P$-function* $P_{\lambda/\mu} = 2^{\ell(\mu) - \ell(\lambda)} Q_{\lambda/\mu}$.
The essential reference for the properties of $\bGP_\lambda$ and $\bGQ_\lambda$ is [@IkedaNaruse]. For more background and various extensions, see [@NN2017; @NN2018; @Naruse2018].
We obtain a third interesting family of shifted stable Grothendieck polynomials as a special case of the preceding constructions. Let $\mu$ and $\lambda$ be arbitrary partitions (i.e., not necessarily strict) with $\mu\subseteq \lambda$. Suppose $\lambda$ has $k$ parts and let $\delta_k := (k,k-1,\dots,2,1)$. The partitions $\lambda+\delta_k$ and $\mu+\delta_k$ are then both strict, so we can set $$\bGS_{\lambda/\mu}
:= \bGP_{(\lambda+\delta_k)/(\mu+\delta_k)}
=\bGQ_{(\lambda+\delta_k)/(\mu+\delta_k)}
\quand
\bGS_\lambda := \bGS_{\lambda/\emptyset}.$$ This definition appears to be new. We refer to $\bGS_{\lambda/\mu}$ as the *$K$-theoretic Schur $S$-function* of $\lambda/\mu$. The name makes sense as setting $\beta=0$ recovers the *Schur $S$-function* $S_{\lambda/\mu}$ discussed in [@Stembridge1997a §A.4] and [@Macdonald §III.8], which is also the homogeneous component of $\bGS_{\lambda/\mu}$ of lowest degree.
Suppose $n = |\lambda|-|\mu|$. Since $\SD_{(\lambda+\delta_k)/(\mu+\delta_k)} \cong \D_{\lambda/\mu}$ as posets, we have by Corollary \[ttheta-cor\] that $$\bGS_{\lambda/\mu} = \tOmega(\SD_{(\lambda+\delta_k)/(\mu+\delta_k)}, \theta) = \tOmega(\D_{\lambda/\mu}, \theta) = \ttheta(G^{(\beta)}_{\lambda/\mu}).$$ This shows that $\ttheta$ is a $K$-theoretic analogue of the *superfication map* discussed in [@TKLamThesis §3.2].
We expect that the functions $\bGS_\lambda$ are related to the “$K$-theoretic Stanley symmetric functions” of classical types B, C, and D introduced in [@KirillovNaruse], as well as to $K$-theoretic generalizations of the main result in [@MP2019]. There is one other connection to the recent literature that we can explain more precisely. DeWitt has shown that $S_{\mu} = Q_{\nu}$ if $\mu$ and $\nu$ are the partitions \[mu-nu-eq\] = (m\^k) = (m+k-1,m+k-3,…,|m-k|+1) for some $m,k\in \PP$; moreover, this is the only possible identity of the form $S_{\lambda/\mu} = c Q_\nu$ with $c\in \ZZ$ apart from the equality $S_{(2,1)/(1)} = Q_{(1)}Q_{(1)} = 2Q_{(2)}$ [@Dewitt Thm. IV.3]. DeWitt’s result appears to generalize.
If $\mu$ and $\nu$ are as in then $\bGS_{\mu} = \bGQ_{\nu}$.
Since $\bGS_{(2,1)/(1)} = \bGQ_{(1)} \bGQ_{(1)} \neq 2\bGQ_{(2)}$ and since the $K$-theoretic Schur $Q$- and $S$-functions are homogeneous if $\beta$ has degree $-1$, this formula would describe all possible identities of the form $\bGS_{\lambda/\mu} = c \bGQ_\nu$ with $c\in \ZZ[\beta]$.
Symmetric functions {#sym-sect}
===================
The power series $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$ defined by and are quasisymmetric by construction. When $\mu = \emptyset$, it follows from [@IkedaNaruse Thm. 9.1] that these power series are actually symmetric. In this section, we prove that $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$ are symmetric for any $\mu$. As motivation, we start by discussing the connection between these power series and the $K$-theory of the Grassmannian.
$K$-theory of Grassmannians
---------------------------
Suppose $Z$ is a complex algebraic variety. The *$K$-theory ring* $K(Z)$ is the Grothendieck group of coherent sheaves on $Z$. The ring multiplication is the operation induced by the tensor product. Fix integers $0\leq k \leq n$ and let $\Gr(k,\CC^n)$ denote the Grassmannian of $k$-dimensional subspaces of $\CC^n$. For each partition $\lambda$ whose diagram fits in the rectangle $[k]\times[n-k]$, there is an associated *Schubert variety* $X_\lambda \subseteq \Gr(k,\CC^n)$; see [@Manivel §3.2]. If $\cO_{X_\lambda}$ denotes the structure sheaf of $X_\lambda$, then the ring $K(\Gr(k,\CC^n))$ is spanned by the corresponding classes $[\cO_{X_\lambda}]$.
Let $\Gamma$ be the additive group generated by the stable Grothendieck polynomials $G_\lambda := G^{(-1)}_\lambda$ as $\lambda$ ranges over all integer partitions, and let $I_{k,n-k}$ denote the subgroup of $\Gamma$ spanned by the functions $G_\lambda$ indexed by partitions $\lambda\not\subseteq [k]\times [n-k]$. Results of Buch [@Buch2002] show that $\Gamma$ is a ring in which the subgroup $I_{k,n-k}$ is an ideal.
If we set $[\cO_{X_\lambda}]=0$ when $\lambda\not\subseteq [k]\times [n-k]$, then $G_\lambda \mapsto[\cO_{X_\lambda}]$ induces a ring isomorphism $\Gamma / I_{k,n-k} \xrightarrow{\sim} K(\Gr(k,\CC^n))$.
Thus, the stable Grothendieck polynomials are “universal” $K$-theory representatives for Schubert varieties in type A Grassmannians, and their structure constants determine the structure constants of the ring $K(\Gr(k,\CC^n))$. The shifted stable Grothendieck polynomials have a similar interpretation in the context of Lagrangian and maximal orthogonal Grassmannians.
Fix a nondegenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $\CC^n$ and define $\OG(k,n)$ to be the *orthogonal Grassmannian* of $k$-dimensional subspaces $V\subseteq \CC^n$ that are isotropic, in the sense that $\langle V,V\rangle = 0$. Let $\cG_n$ be either $\OG(n,2n+1)$ or $\OG(n+1,2n+2)$. For each strict partition $\lambda$ whose shifted diagram fits in the square $[n]\times[n]$, or which equivalently has $\lambda \subseteq (n,n-1,\dots,2,1)$, there is an associated *Schubert variety* $\Omega_\lambda \subseteq \cG_n$ [@IkedaNaruse §8.1]. The classes $[\cO_{\Omega_\lambda}]$ of the corresponding structure sheaves are a basis for $K(\cG_n)$.
Let $\Gamma_P$ be the additive group generated by $\GP_\lambda := \GP_\lambda^{(-1)} $ as $\lambda$ ranges over all strict partitions. Let $I_{P,n}$ denote the subgroup of $\Gamma_P$ spanned by the functions $\GP_\lambda$ indexed by strict partitions $\lambda\not\subseteq (n,n-1,n\dots,2,1)$. Results in [@CTY; @IkedaNaruse] (see also [@HKPWZZ]) show that $\Gamma_P$ is a ring in which $I_{P,n}$ is an ideal.
If we set $[\cO_{\Omega_\lambda}]=0$ when $ \lambda\not\subseteq(n,n-1,\dots,2,1)$, then $\GP_\lambda \mapsto[\cO_{\Omega_\lambda}]$ induces a ring isomorphism $\Gamma_P / I_{P,n} \xrightarrow{\sim} K(\cG_n)$.
A similar result holds for the $K$-theoretic Schur $Q$-functions, with one technical caveat. Let $\LG(n)$ denote the *Lagrangian Grassmannian* of $n$-dimensional subspaces in $\CC^{2n}$ that are isotropic with respect to a fixed nondegenerate skew-symmetric bilinear form. For each strict partition $\lambda\subseteq (n,n-1,\dots,2,1)$, there is again an associated *Schubert variety* $\Omega'_\lambda \subseteq \LG(n)$, and the classes $[\cO_{\Omega'_\lambda}]$ are a basis for $K(\LG(n))$ [@IkedaNaruse §8.1].
Let $\Gamma_Q$ be the additive group generated by $\GQ_\lambda := \GQ_\lambda^{(-1)} $ as $\lambda$ ranges over all strict partitions, and let $I_{Q,n}$ be the subgroup spanned by the functions $\GQ_\lambda$ with $\lambda\not\subseteq (n,n-1,\dots,2,1)$. Let $\hat \Gamma_Q := \prod_{\lambda} \ZZ \GQ_\lambda\supsetneq \bigoplus_{\lambda} \ZZ \GQ_\lambda = \Gamma_Q$ and define $\hat I_{Q,n}$ as the completion of $I_{Q,n}$ relative to its basis of $\GQ_\lambda$’s. It then follows from [@IkedaNaruse Prop. 3.5] that $\hat\Gamma_Q$ is a ring in which $\hat I_{Q,n}$ is an ideal.
If we set $[\cO_{\Omega'_\lambda}]=0$ when $ \lambda\not\subseteq(n,n-1,\dots,2,1)$, then $\GQ_\lambda \mapsto[\cO_{\Omega'_\lambda}]$ induces a ring isomorphism $\hat \Gamma_Q / \hat I_{Q,n} \xrightarrow{\sim} K(\LG(n))$.
We have to state this result in terms of the completions $\hat \Gamma_Q$ and $\hat I_{Q,n}$ because it is still an open problem to show that $\Gamma_Q$ is a ring; see [@IkedaNaruse Conj. 3.2]. If this holds then $\GQ_\lambda \mapsto [\cO_{\Omega'_\lambda}]$ would also induce an isomorphism $\Gamma_Q /I_{Q,n} \xrightarrow{\sim} K(\LG(n))$. To prove Ikeda and Naruse’s conjecture, it is enough to show that $\bGQ_\lambda \bGQ_\mu$ is always a finite linear combination of $\bGQ_\nu$’s. Results of Buch and Ravikumar [@BuchRavikumar] imply that this holds at least when $\lambda$ or $\mu$ has a single part.
Fomin–Kirillov operators
------------------------
An element $f \in \ZZ[\beta][[x_1,x_2,\dots]]$ is *symmetric* if the coefficients of $x_1^{a_1}x_2^{a_2}\cdots x_{k}^{a_k}$ and $x_{i_1}^{a_1}x_{i_2}^{a_2}\cdots x_{i_k}^{a_k}$ in $f$ are equal for every choice of $a_1,a_2,\dots,a_k \in \PP$ and every choice of $k$ distinct positive integers $i_1, i_2, \dots, i_k$. Equivalently, $f$ should be invariant under the change of variables swapping $x_i$ and $x_{i+1}$ for all $i$.
Let $\mSym$ denote the $\ZZ[\beta]$-module of all symmetric power series in $\ZZ[\beta][[x_1,x_2,\dots]]$. Let $\Sym$ denote the submodule of power series in $\mSym$ of bounded degree.
The *monomial symmetric function* of a partition $\lambda$ is the power series given by the sum $m_\lambda := \sum_{\sort(\alpha)=\lambda}M_\alpha$ over all compositions $\alpha$ that sort to $\lambda$. It is well-known that $\Sym$ is a graded Hopf subalgebra of $\QSym$, which is free as a $\ZZ[\beta]$-module with a homogeneous basis given by the power series $\{m_\lambda\}$. We identify $\mSym$ with the completion of $\Sym$ relative to this basis.
The main result of this section is the following theorem, which reduces to [@IkedaNaruse Thm. 9.1] in the case when $\mu=\emptyset$.
\[skew-sym-thm\] Let $\mu$ and $\lambda$ be strict partitions with $\mu\subseteq \lambda$. The power series $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$ are elements of $\mSym$.
We delay the proof of this result until Section \[yb-sect\]. First, we need to introduce two new families of power series closely related to $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$.
Let $\SPart$ be the free $\ZZ[\beta][[x_1, x_2,\ldots]]$-module with a pseudobasis given by the set of all strict partitions, and write $\mSPart$ for the corresponding completion. Let $ \langle\cdot,\cdot\rangle : \SPart \times \mSPart \to \ZZ[\beta][[x_1, x_2,\ldots]] $ denote the associated form making the natural (pseudo)bases of strict partitions in $\SPart$ and $\mSPart$ dual to each other. In other words, $\langle\cdot,\cdot\rangle$ is the nondegenerate $ \ZZ[\beta][[x_1, x_2,\ldots]]$-bilinear form, continuous in the second coordinate, such that $\langle \mu,\nu\rangle = \delta_{\mu\nu}$ for all strict partitions $\mu$ and $\nu$.
Let $\mu$ be a strict partition of $n \in \NN$ with shifted diagram $\SD_\mu$ as in . The *$r$th diagonal* of $\mu$ is the set of positions $(i,j) \in \SD_\mu$ with $j - i =r$. The *outer corners* of $\mu$ are the positions $(i,j) \notin \SD_\mu$ such that $\SD_\mu\sqcup \{(i,j)\}$ is the shifted diagram of a strict partition of $n+1$. The *inner corners* of $\mu$ are the positions $(i,j) \in \SD_\mu$ such that $\SD_\mu \setminus\{(i,j)\}$ is the shifted diagram of a strict partition of $n-1$.
For $n \in \NN$, define $a_n : \SPart \to \SPart$ to be the continuous linear map such that if $\mu$ is a strict partition then $$a_n \mu = \begin{cases}
\beta \cdot \mu & \textrm{if $\mu$ has an inner corner on its $n$th diagonal}, \\
\nu & \textrm{if $\SD_{\nu} = \SD_\mu \sqcup\{(i,j)\}$ where $j-i=n$, and }\\
0 & \textrm{otherwise}.
\end{cases}$$ When $\beta=0$ the maps $a_i$ specialize to the *diagonal box-adding operators* considered in [@FominGreene Ex. 2.4] or [@Serrano §1.4]. For $x \in \ZZ[\beta][[x_1,x_2 \ldots]]$, let \[apq-eq\] A\_n(x) &= 1 + x a\_n,\
\_n(x) &= A\_n(x) A\_2(x) A\_1(x) A\_0(x) A\_[1]{}(x)A\_[2]{}(x) A\_[n]{}(x),\
\_n(x) &= A\_n(x) A\_2(x) A\_1(x) A\_0(x)A\_0(x) A\_[1]{}(x) A\_[2]{}(x) A\_[n]{}(x).These operators are similar to the ones which Fomin and Kirillov define in [@FominKirillov]. Fix strict partitions $\mu \subseteq \lambda$, let $n = \lambda_1 \geq \ell(\lambda)$, and define $$\bGP_{\lambda\ss \mu}
:=
\left\langle \lambda, \cdots \cP_n(x_2)\cP_n(x_1) \mu \right\rangle
\quand
\bGQ_{\lambda\ss \mu}
:=
\left\langle \lambda, \cdots \cQ_n(x_2)\cQ_n(x_1) \mu \right\rangle.$$ The first formula is well-defined because the coefficient of each fixed $x$-monomial in $\langle \lambda, \cP_n(x_N)\cdots \cP_n(x_2)\cP_n(x_1) \mu\rangle$ eventually stabilizes as $N\to\infty$. The second formula makes sense for similar reasons. (One can replace $n$ in these formulas by any integer greater than $\lambda_1$ without changing the meaning.)
From our definition, it is only clear that $\bGP_{\lambda\ss \mu}$ and $\bGQ_{\lambda\ss \mu}$ are formal power series in $\ZZ[\beta][[x_1,x_2,\dots]]$. These power series are actually quasisymmetric and related to $\bGP_{\lambda/ \mu}$ and $\bGQ_{\lambda/ \mu}$ in the following way. Let $\IC(\mu)$ be the set of inner corners of a strict partition $\mu$.
Suppose $\mu\subseteq \lambda$ are strict partitions. Then $$\bGP_{\lambda\ss \mu} = \sum_{\nu \subseteq \mu} \bGP_{\lambda/\nu}
\qquand
\bGQ_{\lambda\ss \mu} = \sum_{\nu \subseteq \mu} \bGQ_{\lambda/\nu}$$ where both sums are over all strict partitions $\nu \subseteq \mu$ with $\SD_{\mu/\nu}\subseteq \IC(\mu)$.
Thus, neither $\bGP_{\lambda\ss \lambda}$ nor $\bGQ_{\lambda \ss \lambda}$ is equal to $\bGP_{\lambda/\lambda} = \bGQ_{\lambda/\lambda} =1$.
A *vertical strip* (respectively, *horizontal strip*) is a skew shape with no two boxes in the same row (respectively, same column).
We first prove the formula for $\bGQ_{\lambda \ss\mu}$. Let $n =\lambda_1 \geq \ell(\lambda)$. From the definition of $a_i$ and $A_i(x)$, one sees that the strict partition $\lambda$ appears with nonzero coefficient in $A_0(x)A_1(x) \cdots A_n(x) \mu$ if and only if we can produce the shifted diagram of $\lambda$ by adding a vertical strip to the shifted diagram of $\mu$. Moreover, if we define $\cV_{\lambda/\mu}$ to be the collection of vertical strips $V$ such that $V \cup \SD_{\mu} = \SD_\lambda$ and $V \cap \SD_{\mu}\subseteq\IC(\mu)$, then \[coeff1-eq\] , A\_0(x) A\_1(x)A\_n(x) = \_[V \_[/]{}]{} \^[- |/|]{} (x)\^[|V|]{}. Similarly, $\lambda$ appears with nonzero coefficient in $A_n(x) \cdots A_1(x)A_0(x) \mu$ if and only if we can can produce the shifted diagram of $\lambda$ by adding a horizontal strip to the shifted diagram of $\mu$. Moreover, if we define $\cH_{\lambda/\mu}$ to be the collection of horizontal strips $H$ such that $H \cup \SD_{\mu} = \SD_\lambda$ and $H \cap \SD_{\mu}\subseteq \IC(\mu)$, then $$\left\langle \lambda, A_n(x) \cdots A_1(x)A_0(x) \mu \right\rangle = \sum_{H \in \mathcal{H}_{\lambda/\mu}} \beta^{- |\lambda/\mu|} (\beta x)^{|H|}.$$ Combining these observations, we deduce that \[bgq-ss-eq\] \_[ß]{} = \_[(V\_1, H\_1, V\_2, H\_2, …, V\_N,H\_N)]{} \^[- |/|]{} \_i (x\_i)\^[|V\_i| + |H\_i|]{} where the sum is over all tuples $(V_1,H_1,V_2,H_2,\ldots, V_N, H_N)$ such that for some sequence of strict partitions $\mu = \lambda^0\subseteq \mu^1 \subseteq \lambda^1 \subseteq \mu^2 \subseteq \lambda^2 \subseteq \dots \subseteq \mu^N \subseteq \lambda^{N} = \lambda$ it holds that $V_i \in \cV_{\mu^i/\lambda^{i-1}}$ and $H_i \in \cH_{\lambda^i/\mu^i}$.
Fix strict partitions $\mu \subseteq \lambda$. Consider the set of semistandard shifted set-valued tableaux $T$ of shape $\lambda/\nu$ where $\nu \subseteq \mu$ is a strict partition with $\SD_{\mu/\nu}\subseteq \IC(\mu)$. This set is in bijection with the sequences indexing the summands of via the map $T \mapsto (V_1,H_1,V_2,H_2,\ldots)$ where $V_i$ and $H_i$ are the sets of boxes in $T$ containing $i'$ and $i$, respectively. Moreover, under this bijection, we have $|T| = \sum_i (|V_i| + |H_i|)$ and $x^T = \prod_i x_i^{|V_i| + |H_i|}$. Thus, the desired formula for $\bGQ_{\lambda \ss \mu}$ follows by combining the definition with .
The argument needed to deduce our formula for $\bGP_{\lambda \ss \mu}$ is similar: one just needs to modify the steps above by replacing $A_0(x) A_1(x)\cdots A_n(x)$ by $ A_1(x)\cdots A_n(x)$ in and requiring all vertical strips $V \in \cV_{\lambda/\mu}$ to contain no positions on the main diagonal. We omit the details.
Applying inclusion-exclusion to the preceding result gives the following.
\[in-ex-cor\] Suppose $\mu\subseteq \lambda$ are strict partitions. Then $$\bGP_{\lambda/\mu} = \sum_{\nu \subseteq \mu} (-1)^{|\mu| - |\nu|} \bGP_{\lambda\ss \nu}
\qquand
\bGQ_{\lambda/\mu} = \sum_{\nu\subseteq \mu} (-1)^{|\mu| - |\nu|} \bGQ_{\lambda\ss \nu}$$ where both sums are over all strict partitions $\nu \subseteq \mu$.
Yang-Baxter relations {#yb-sect}
---------------------
In view of Corollary \[in-ex-cor\], to prove Theorem \[skew-sym-thm\] it suffices to show that $\bGP_{\lambda \ss \mu}$ and $\bGQ_{\lambda \ss \mu}$ are symmetric. For this it is enough to prove that the operators $\cP_n(x)$ and $\cP_n(y)$ (respectively, $\cQ_n(x)$ and $\cQ_n(y)$) commute. To show this latter fact, we follow the approach of [@FominKirillov] (see also [@FominGreene; @FominKirillov1996; @Serrano; @Yeliussiozv2019]), proving some Yang–Baxter-type equations satisfied by the factors $A_i(x)$ and $A_j(x)$.
Extending our earlier notation, for any expressions $x$ and $y$, we write x y := x + y + xy, x := , xy := x (y) = . The operators $A_i(x)$ from satisfy the following commutation relations.
\[yang-baxter-lem\] Let $i,j \in \NN$ and $x,y \in \ZZ[\beta][[x_1,x_2,\dots]]$. Then
(a) $A_i(x) A_j(y) = A_j(y) A_i(x)$ if $|i - j| > 1$,
(b) $A_i(x) A_i(y) = A_i(x \oplus y)$,
(c) $A_{i + 1}(x)A_{i}(x \oplus y) A_{i + 1}(y) = A_{i}(y)A_{i + 1}(x \oplus y) A_{i}(x)$ if $i>0$, and
(d) $A_0(x)A_1(x \oplus y)A_0(y)A_1(y \ominus x) = A_1(y \ominus x)A_0(y)A_1(x \oplus y)A_0(x)$.
The first two statements are clear from the definitions of $a_i$ and $A_i(x)$ in . For part (c), assume $i>0$ and consider $f := A_{i + 1}(x)A_{i}(z) A_{i + 1}(y)$ and $f' := A_{i}(y)A_{i + 1}(z) A_{i}(x)$; later, we will specialize $z$ to $x \oplus y$. Write $\diag_j := \{ (a,b) \in \PP\times \PP : b -a = j\}$ for the $j$th diagonal in $\PP\times \PP$. The behavior of $f$ and $f'$ on a strict partition $\mu$ depends only on local properties of its shifted diagram. There are finitely many cases to consider:
- It is not possible for the adjacent diagonals $\diag_i$ and $\diag_{i+1}$ to both contain inner corners of $\mu$ or both contain outer corners of $\mu$.
- If neither $\diag_i$ nor $\diag_{i + 1}$ contains an inner or outer corner of $\mu$, then we have $a_i \mu = a_{i + 1} \mu = 0$, so $f \mu = f' \mu = \mu$.
- Suppose $\diag_{i + 1}$ contains an outer corner of $\mu$ and $\diag_i$ does not contain an inner or outer corner. Let $\mu'$ be the strict partition whose shifted diagram is obtained from $\SD_\mu$ by adding a box in diagonal $i+1$. Then $\mu'$ has an outer corner in diagonal $i$; write $\mu''$ for the result of adding this box. Then $\mu''$ does not have an inner or outer corner in diagonal $i+1$, so we compute $f \mu = \mu + (x \oplus y) \mu' + yz \mu'' $ and $
f' \mu = \mu + z \mu' + yz \mu''.$ Thus we have $(f-f') \mu = (x\oplus y - z)\mu'$.
- If $\diag_i$ contains an outer corner of $\mu$ and $\diag_{i + 1}$ does not contain an inner or outer corner then by similar reasoning $(f-f')\mu = (z - x\oplus y)\mu'$ where $\mu'$ is the result of adding to $\mu$ its outer corner in diagonal $i$.
- If $\diag_{i + 1}$ has an inner corner and $\diag_i$ does not contain an inner or outer corner, then $f\mu = (1+\beta x)(1+\beta y)\mu = (1 + \beta \cdot x \oplus y)\mu$ and $f'\mu = (1 + \beta z)\mu$.
- If $\diag_{i }$ has an inner corner and $\diag_{i+1}$ does not contain an inner or outer corner then $f\mu = (1 + \beta z)\mu$ and $f'\mu = (1 + \beta x)(1+ \beta y) \mu = (1 + \beta \cdot x \oplus y)\mu$.
- Suppose $\diag_i$ contains an outer corner of $\mu$ and $\diag_{i + 1}$ contains an inner corner. Let $\mu'$ be the strict partition whose shifted diagram is obtained from $\SD_\mu$ by adding a box in diagonal $i+1$. Then $\mu'$ does not have an inner or outer corner in diagonal $i$, so $f\mu = (1 + \beta x)(1 + \beta y)\mu + (1 + \beta y)z \mu'$ and $f'\mu = (1 + \beta z)\mu + (x \oplus y + \beta yz)\mu'$. Thus $(f-f') \mu = (x \oplus y - z)( \beta \mu - \mu')$.
- If $\diag_i$ contains an inner corner of $\mu$ and $\diag_{i + 1}$ contains an outer corner then by similar reasoning $(f-f') \mu = (z - x \oplus y)( \beta \mu - \mu')$, where $\mu'$ is the result of adding to $\mu$ its outer corner in diagonal $i + 1$.
Comparing coefficients in each case, we see that if $y = x \oplus z$ then $f = f'$.
Our proof of part (d) is similar. Fix $w,x,y,z \in \ZZ[\beta][[x_1,x_2,\dots]]$ and consider $g := A_0(x)A_1(w)A_0(y)A_1(z)$ and $g' := A_1(z)A_0(y)A_1(w)A_0(x)$ acting on a diagram $S$. We have four cases: diagonal $0$ is always an inner or outer corner, and diagonal $1$ can be inner, outer, or neither, but never the same as $0$. We illustrate the cases with shapes $( 2 )$, $( 2, 1 )$, $( 3 )$, $( 3, 1 )$, $( 3, 2 )$, and $( 3, 2, 1 )$.
- If $\diag_0$ contains an inner corner of $\mu$ and $\diag_1$ does not contain an inner or outer corner, then the last two parts of $\mu$ must be $(2,1)$ and we have $g \mu =g'\mu = (1 + \beta x)(1 + \beta y) \mu $.
- Suppose $\diag_0$ contains an outer corner of $\mu$ and $\diag_1$ does not contain an inner or outer corner. Then the smallest part of $\mu$ must be $\geq 3$, and we can assume without loss of generality that $\mu = (3)$. Let $\mu' = (3,1)$, $\mu''=(3,2)$, and $\mu''' = (3,2,1)$. Then one can check that $$\ba
g \mu &=\mu + (x \oplus y)\mu' + wy\mu'' + wxy\mu''', \text{ and } \\
g' \mu &= \mu +(x \oplus y) \mu' +(wx + xz + yz + \beta wxz + \beta xyz) \mu'' + wxy \mu''',
\ea$$ so $(g-g')\mu = -(wx + xz + yz + \beta wxz + \beta xyz - wy)\mu''$.
- Suppose $\diag_0$ contains an outer corner of $\mu$ and $\diag_1$ contains an inner corner, so that the last part of $\mu$ must be $2$. Let $\mu'$ be the strict partition formed by concatenating $\mu$ and $(1)$. Then we have $$\ba
g \mu &= (1 + \beta w)(1 + \beta z) \mu + ( x + y + \beta wx + \beta xy)(1+\beta z)\mu',
\text{ and }
\\
g' \mu &= (1 + \beta w)(1 + \beta z)\mu + (x + y + \beta wy + \beta xy)\mu'
\ea$$ so $(g-g')\mu = \beta (wx + xz + yz + \beta wxz + \beta x y z - wy) \mu'$.
- Suppose $\diag_0$ contains an inner corner of $\mu$ and $\diag_1$ contains an outer corner. Then the last part of $\mu$ must be $1$ and the second-to-last part of $\mu$ must be $\geq 3$, and we can assume without loss of generating that $\mu = (3,1)$. Let $\mu'=(3,2)$ and $\mu'' = (3,2,1)$. Then it follows by similar calculations that $
(g-g')\mu = (wx + xz + yz + \beta wxz + \beta x y z - wy) (\beta \mu' -\mu'')
$.
It follows in each case that we have $g = g'$ provided that $$\label{yb-eq}
wy = wx + xz + yz + \beta wxz + \beta xyz,$$ and in particular this is satisfied if $w = x \oplus y$ and $z = x\ominus y$.
We can now show that $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$ are symmetric functions.
The relations in Lemma \[yang-baxter-lem\] are the same as [@FominKirillov (2.1)–(2.4)] with “$h_i$” replaced by “$A_i$” and “$+$/$-$” replaced by “$\oplus$/$\ominus$”. Making the same substitutions transforms [@FominKirillov Prop. 4.2] to the assertion that the operators $\cP_{n}(x)$ and $\cP_{n}(y)$ commute for all $x,y$, which is what we need to show to deduce that $\bGP_{\lambda\ss\mu}$ and $\bGP_{\lambda/\mu}$ are symmetric.
Fomin and Kirillov’s proof of [@FominKirillov Prop. 4.2] only depends on the fact that $+$ is a commutative group law; hence every formal consequence of the Yang–Baxter equations for their $h_i$’s also holds for our $A_i$’s, with “$+$” and “$-$” respectively replaced by “$\oplus$” and “$\ominus$”. In particular, the proof of [@FominKirillov Prop. 4.2] carries over to our context, *mutatis mutandis*, and we conclude that $\cP_{n}(x)\cP_{n}(y) = \cP_{n}(y)\cP_{n}(x)$ as desired.
We now explain how to see that the operators $\cQ_n(x)$ and $\cQ_n(y)$ likewise commute. Fomin and Kirillov’s proof of [@FominKirillov Prop. 4.2] is implicitly an inductive argument, which translates when $n=2$, for example, to the following sequence of transformations. Here, the Yang–Baxter relations from Lemma \[yang-baxter-lem\] are indicated with braces overhead: [$$\begin{aligned}
\cP_2(x) \cP_2(y) & = A_2(x)A_1(x)A_0(x)A_1(x)\overbrace{A_2(x)A_2(y)}A_1(y) A_0(y)A_1(y)A_2(y) \\
& = A_2(x)A_1(x)A_0(x)\overbrace{A_1(x)A_2(x\oplus y)A_1(y)} A_0(y)A_1(y)A_2(y) \\
& = \overbrace{A_2(x)}A_1(x)\overbrace{A_0(x)A_2(y)}A_1(x\oplus y)\overbrace{A_2(x) A_0(y)}A_1(y)\overbrace{A_2(y)} \\
& =A_2(y) \overbrace{A_2(x\ominus y) A_1(x) A_2(y)} A_0(x) A_1(x \oplus y) A_0(y) \overbrace{A_2(x) A_1(y) A_2(y \ominus x)} A_2(x) \\
& = A_2(y) A_1(y) A_2(x) A_1(x\ominus y) \overbrace{A_0(x) A_1(x \oplus y) A_0(y) A_1(y \ominus x)} A_2(y) A_1(x) A_2(x) \\
& = A_2(y) A_1(y) A_2(x) \overbrace{A_1(x\ominus y) A_1(y \ominus x)} A_0(y) A_1(x \oplus y) \overbrace{A_0(x) A_2(y)} A_1(x) A_2(x) \\
& = A_2(y) A_1(y)\overbrace{ A_2(x)A_0(y)} A_1(x \oplus y) A_2(y)A_0(x) A_1(x) A_2(x) \\
& = A_2(y) A_1(y) A_0(y) \overbrace{A_2(x) A_1(x \oplus y) A_2(y)}A_0(x) A_1(x) A_2(x) \\
& = A_2(y) A_1(y) A_0(y) A_1(y) \overbrace{A_2(x \oplus y)} A_1(x)A_0(x) A_1(x) A_2(x) \\
& = A_2(y) A_1(y) A_0(y) A_1(y) A_2(y) A_1(x) A_1(x)A_0(x) A_1(x) A_2(x) \\
& = \cP_2(y) \cP_2(x).\end{aligned}$$]{}Only minor adjustments to this inductive argument are needed to check that $\cQ_n(x)\cQ_n(y)= \cQ_n(y)\cQ_n(x)$. For example, when $n=2$ one has [$$\begin{aligned}
\cQ_2(x) \cQ_2(y) & = A_2(x)A_1(x)\overbrace{A_0(x)A_0(x)}A_1(x)\overbrace{A_2(x)A_2(y)}A_1(y) \overbrace{A_0(y)A_0(y)}A_1(y)A_2(y) \\
& = A_2(x)A_1(x)A_0(x \oplus x)\overbrace{A_1(x)A_2(x\oplus y)A_1(y)} A_0(y\oplus y)A_1(y)A_2(y) \\
& = \overbrace{A_2(x)}A_1(x)\overbrace{A_0(x\oplus x)A_2(y)}A_1(x\oplus y)\overbrace{A_2(x) A_0(y \oplus y)}A_1(y)\overbrace{A_2(y)} \\
& =A_2(y) \overbrace{A_2(x\ominus y) A_1(x) A_2(y)} A_0(x\oplus x) A_1(x \oplus y) A_0(y \oplus y) \overbrace{A_2(x) A_1(y) A_2(y \ominus x)} A_2(x) \\
& = A_2(y) A_1(y) A_2(x) A_1(x\ominus y) \overbrace{A_0(x \oplus x) A_1(x \oplus y) A_0(y \oplus y) A_1(y \ominus x)} A_2(y) A_1(x) A_2(x) \\
& = A_2(y) A_1(y) A_2(x) \overbrace{A_1(x\ominus y) A_1(y \ominus x)} A_0(y\oplus y) A_1(x \oplus y) \overbrace{A_0(x \oplus x) A_2(y)} A_1(x) A_2(x) \\
& = A_2(y) A_1(y)\overbrace{ A_2(x)A_0(y \oplus y)} A_1(x \oplus y) A_2(y)A_0(x \oplus x) A_1(x) A_2(x) \\
& = A_2(y) A_1(y) A_0(y \oplus y) \overbrace{A_2(x) A_1(x \oplus y) A_2(y)}A_0(x \oplus x) A_1(x) A_2(x) \\
& = A_2(y) A_1(y) \overbrace{A_0(y \oplus y)} A_1(y) \overbrace{A_2(x \oplus y)} A_1(x) \overbrace{A_0(x \oplus x)} A_1(x) A_2(x) \\
& = A_2(y) A_1(y) A_0(y)A_0(y) A_1(y) A_2(y) A_1(x) A_1(x)A_0(x) A_0(x) A_1(x) A_2(x) \\
& = \cQ_2(y) \cQ_2(x)\end{aligned}$$]{}where the transformation in the fifth line uses the identity A\_0(xy) A\_1(x) A\_0(x y) A\_1(y) = A\_1(y) A\_0(x y) A\_1(x)A\_0(xy) which follows from . The general case is similar and we omit the details. We conclude that the power series $\bGQ_{\lambda\ss\mu}$ and $\bGQ_{\lambda/\mu}$ are symmetric.
Symmetric subalgebras
---------------------
We have seen that $\GP_\lambda$ and $\GQ_\lambda$ are linearly independent in $\mSym$ and involve only $x$-monomials of degree at least $|\lambda|$, so the following is well-defined.
Let $\mGSym$ and $\omGSym$ denote the linearly compact $\ZZ[\beta]$-modules with the $K$-theoretic Schur $Q$- and $P$-functions $\{\bGQ_\lambda\}$ and $\{\bGP_\lambda\}$ as respective pseudobases ($\lambda$ ranging over all strict partitions).
The module $\omGSym$ may be equivalently defined by a cancellation law.
\[q-cancel-thm\] It holds that $$\omGSym = \{ f \in \mSym : f(t,\ominus t,x_3,x_4,\dots) = f(x_3,x_4,\dots)\}$$ where $t$ is an indeterminate commuting with each $x_i$.
The following is a $K$-theoretic extension of [@Stembridge1997a Thm. 3.8].
\[cap-thm\] It holds that $$\mGSym = \mSym \cap \mcoPeak
\qquand \omGSym = \mSym \cap \omcoPeak.$$ In particular, we have $\mGSym
\subseteq \omGSym$.
First, we consider the case of $\omGSym$. Fix a strict partition $\lambda$. By and Corollary \[omco-cor\], we have $\bGP_\lambda \in \omcoPeak$. Since $\bGP_\lambda$ is symmetric, $ \omGSym \subseteq \mSym \cap \omcoPeak$. On the other hand, by Lemma \[q-cancel-lem\] and Theorem \[q-cancel-thm\] we have $\omGSym \supseteq \mSym \cap \mcoPeak_{\QQ[\beta]} = \mSym \cap (\mQSym \cap \mcoPeak_{\QQ[\beta]})$, so Theorem \[inter-thm\] implies that $\omGSym \supseteq \mSym \cap \omcoPeak$.
Second, we consider the case of $\mGSym$. By , we have $\bGQ_\lambda \in \mcoPeak$ and so $\mGSym \subseteq \mSym \cap \mcoPeak$. To finish, we will check that $\mGSym \supseteq \omGSym \cap \mcoPeak$, which implies $\mGSym \supseteq \mSym \cap \mcoPeak$ by Theorem \[inter-thm\].
Let $\mGSym_{\QQ[\beta]}$ be the linearly compact $\QQ[\beta]$-module with the $K$-theoretic Schur $Q$-functions $\bGQ_\lambda$ as a pseudobasis. We claim that $\omGSym \subseteq \mGSym_{\QQ[\beta]}$. Since $\bGQ_\lambda \in \mSym \cap \mcoPeak \subseteq \omGSym$, we have $\bGQ_\lambda \in \sum_\mu \ZZ[\beta] \bGP_\mu$ where the sum is over all strict partitions. If $\beta$ has degree $0$ and each $x_i$ has degree $1$ then the nonzero homogeneous components of $\bGP_\lambda$ and $\bGQ_\lambda$ of lowest degree are $P_\lambda$ and $Q_\lambda = 2^{\ell(\lambda)} P_\lambda$, so \_2\^[()]{} \_+ \_[|| < ||]{} \_. Hence $\bGP_\lambda \in 2^{-\ell(\lambda)} \bGQ_\lambda + \sum_{|\lambda| < |\mu|} \QQ[\beta] \bGP_\mu$ and it follows that $\bGP_\lambda \in \mGSym_{\QQ[\beta]}$, so $\omGSym \subseteq \mGSym_{\QQ[\beta]}$ as claimed.
The peak quasisymmetric function $K_\lambda$ is the nonzero homogeneous component of $K^{(\beta)}_\lambda$ of lowest degree, and it is shown in the proof of [@Stembridge1997a Thm. 3.8] that $Q_\lambda \in K_\lambda + \sum_{\lambda \prec \alpha} \ZZ K_\alpha$ where $\prec$ is the partial order on compositions described in the proof of Theorem \[inter-thm\]. Since $\bGQ_\lambda \in \mcoPeak$, we must have \_K\^[()]{}\_+ \_ K\^[()]{}\_. Thus, as in the proof of Theorem \[inter-thm\], any $\QQ[\beta]$-linear combination of $\bGQ_\lambda$’s that belongs to $\mcoPeak$ must have coefficients in $\ZZ[\beta]$. By the previous paragraph, this means that $\mGSym \supseteq \mGSym_{\QQ[\beta]} \cap \mcoPeak \supseteq \omGSym \cap \mcoPeak$.
Finally, $\mGSym
\subseteq \omGSym$ holds as $\mcoPeak \subset \omcoPeak$ by Theorem \[inter-thm\].
\[vab-cor\] Suppose $(P,\gamma)$ is a labeled poset and $V\subseteq \ValSet(P,\gamma)$.
If $\tOmega(P,\gamma) \in \mSym$ then $\tOmega(P,\gamma) \in \mGSym$.
If $\tOmega(P,\gamma,V) \in \mSym$ then $\tOmega(P,\gamma,V) \in \omGSym$.
We have $\tOmega(P,\gamma) \in \mcoPeak$ by Theorem \[ep-thm1\] and $\tOmega(P,\gamma,V) \in \omcoPeak$ by Corollary \[omco-cor\], so this follows by Theorem \[cap-thm\].
We single out one especially important case. Fix strict partitions $\mu\subseteq \lambda$.
\[single-cor\] It holds that $\bGP_{\lambda/\mu} \in \omGSym$ and $\bGQ_{\lambda/\mu}\in \mGSym$.
This holds by Corollary \[vab-cor\] given , , and Theorem \[skew-sym-thm\].
Concretely, Corollary \[single-cor\] means that we have $\bGP_{\lambda/\mu} \in \sum_\nu \ZZ[\beta] \bGP_\nu$ and $\bGQ_{\lambda/\mu}\in\sum_\nu \ZZ[\beta] \bGQ_\nu$ where the sums are over all strict partitions $\nu$. We expect that these sums are actually finite with positive coefficients:
$\bGP_{\lambda/\mu} \in \bigoplus_\nu \NN[\beta]\bGP_\nu$ and $\bGQ_{\lambda/\mu}\in \bigoplus_\nu \NN[\beta]\bGQ_\nu$.
This should be a direct consequence of the geometric interpretation of $\bGP_\lambda$ and $\bGQ_\lambda$ in [@IkedaNaruse]. It would be interesting to find a combinatorial proof.
Since $\mGSym
\subseteq \omGSym$, we must have $\bGQ_\mu \in \sum_\lambda \ZZ[\beta] \bGP_\mu$. Computer calculations suggest that this expansion is actually always finite, with the following fairly simple description. As before, a *vertical strip* is a subset of $\PP\times \PP$ that contains at most one position in each row.
If $\mu$ is a strict partition then \[q-to-p-eq\] \_= 2\^[()]{} \_ (-1)\^[c(/)]{} (-/2)\^[|/|]{}\_where the sum is over the finite set of strict partitions $\lambda\supseteq \mu$ with $\ell(\mu) =\ell(\lambda)$ such that $\SD_{\lambda/\mu}$ is a vertical strip, and we define $c(\lambda/\mu)$ to be the number of distinct columns occupied by positions in $\SD_{\lambda/\mu}$.
For example, one has $\bGQ_{(3,2)} = 4 \bGP_{(3,2)} + 2\beta\cdot \bGP_{(4,2)} - \beta^2\cdot \bGP_{(4,3)}.$
The coefficients in have the form $\pm 2^i \beta^j$ for $i,j \in \{0,1,\dots,\ell(\mu)\}$, and if the coefficient of $\bGP_\lambda$ is nonzero, then its sign is $(-1)^{s(\lambda/\mu)}$ where $s(\lambda/\mu)$ is the number of nonempty columns of $\SD_{\lambda/\mu}$ with an even number of boxes.
We end this section with two results involving the power series $\bGS_\lambda$.
The map $\ttheta$ restricts to a morphism $ \mSym \to \mGSym$.
This holds since $ \ttheta(G^{(\beta)}_\lambda)=\bGS_\lambda \in \mGSym$ by Corollary \[single-cor\].
The set of $K$-theoretic Schur $S$-functions $\{\bGS_\lambda\}$, with $\lambda$ ranging over all strict partitions, is another pseudobasis for $\mGSym$.
Fix a strict partition $\lambda$. We have $\bGS_\lambda \in \mGSym$, and it follows from [@Macdonald (8.8$'$), §III.8] and [@Macdonald Ex. 7, §III.8] that $S_\lambda \in Q_\lambda + \sum_{\mu > \lambda} \ZZ Q_\mu$ where $>$ is the dominance order on strict partitions. Comparing lowest-degree terms, we deduce that $\bGS_\lambda \in \bGQ_\lambda + \sum_{\mu} \ZZ[\beta] \bGQ_\mu$ where the sum is over strict partitions $\mu$ with $\mu > \lambda$ or $|\mu|>|\lambda|$, so the corollary follows.
Antipode formulas {#antipode-sect}
=================
In this section, we show how to expand the multipeak quasisymmetric functions $\{ K^{(\beta)}_\alpha\}$ in terms of the multifundamental quasisymmetric functions $\{ L^{(\beta)}_\alpha\}$. We then derive formulas for several involutions of $\mQSym$, including the antipode.
Mirroring operators
-------------------
We start by defining an operation on posets that adds “mirror images” of certain vertices. Let $(P,\gamma)$ be a labeled poset. Assume the vertices of $P$ are all positive integers (not necessarily with the usual order on $\ZZ$) and $\gamma $ takes only positive integer values. We refer to labeled posets with these properties as *positive*. Write $\prec$ for the partial order on $P$. For each pair $(I, J)$ of subsets such that $P = I\cup J$, we define $\fkM_{IJ}(P,\gamma)$ to be the labeled poset $\fkM_{IJ}(P,\gamma) := (Q,\delta)$ with the following properties.
- As a set, we have $Q := I \cup (-J)$.
- The partial order on $Q$ has $s\prec t$ if and only if $|s| \prec |t|$ in $P$.
- We have $\delta(s) =\sign(s) \gamma(|s|)$ where $\sign(s) := s/|s| \in \{\pm 1\}$.
Thus $|Q| = |P| + |I\cap J|$, and if $s \in I \cap J$ then $-s$ and $s$ are incomparable in $Q$. If $J=\varnothing$ then $\fkM_{P \varnothing}(P,\gamma) = (P,\gamma)$, while if $I = \varnothing$ then $\fkM_{\varnothing P}(P,\gamma) = (P, -\gamma)$ reverses all arrows in the oriented Hasse diagram of $(P, \gamma)$.
Drawing labeled posets as oriented Hasse diagrams, we have $$%\fkM_{\{2,3\}\{4\}\{1\}}(P,\gamma) =
\fkM_{\{1, 2,3\}\{2, 3, 4\}}\(\
\begin{tikzpicture}[baseline=(center.base), xscale=0.7, yscale=0.5]
\tikzset{edge/.style = {<-}}
\node (center) at (0,0) {};
\node (d) at (0,3) {$4$};
\node (c) at (0,1) {$3$};
\node (b) at (0,-1) {$2$};
\node (a) at (0,-3) {$1$};
\draw[edge] (b) -- (a);
\draw[edge] (c) -- (b);
\draw[edge] (c) -- (d);
\end{tikzpicture}\ \)\ =\
\begin{tikzpicture}[baseline=(center.base), xscale=0.7, yscale=0.5]
\tikzset{edge/.style = {<-}}
\node (center) at (0,0) {};
\node (d) at (0,3) {$-4$};
\node (c) at (0,1) {$3$};
\node (c2) at (2,1) {$-3$};
\node (b) at (0,-1) {$2$};
\node (b2) at (2,-1) {$-2$};
\node (a) at (0,-3) {$1$};
\draw[edge] (b) -- (a); \draw[edge] (b2) -- (a);
\draw[edge] (c) -- (b); \draw[edge] (c2) -- (b);
\draw[edge] (b2) -- (c); \draw[edge] (b2) -- (c2);
\draw[edge] (d) -- (c); \draw[edge] (d) -- (c2);
\end{tikzpicture}
\ .
%\begin{tikzpicture}[xscale=0.6, yscale=0.5]
% \node (max) at (0,4) {$-4$};
% \node (b1) at (-1.5,2) {$-3$};
% \node (b2) at (1.5,2) {$+3$};
% \node (a1) at (-1.5,0) {$-2$};
% \node (a2) at (1.5,0) {$+2$};
% \node (min) at (0,-2) {$+1$};
% \draw (min) -- (a1) -- (b1) -- (max) -- (b2) -- (a2) -- (min);
% \draw (a1) -- (b2);
% \draw (a2) -- (b1);
%% \draw[preaction={draw=white, -,line width=6pt}] (a1) -- (b2);
%\end{tikzpicture}$$
Now, given a positive labeled poset $(P,\gamma)$, define \^[()]{}(P,) := \_[I J = P]{} \^[|I J|]{} . When $\beta=0$, we have $
\Psi^{(0)}(P,\gamma) = \sum_{\epsilon: P \to \{\pm 1\}} [(P,\epsilon \gamma)]
$ where $\epsilon \gamma$ is the map $s \mapsto \epsilon(s) \gamma(s)$; compare with [@Stembridge1997a Thm. 3.6].
Any isomorphism class of labeled posets contains at least one positive element $(P,\gamma)$, and the value of $\Psi^{(\beta)}(P,\gamma) $ does not depend on the choice of this representative. The formula for $\Psi^{(\beta)}$ therefore extends uniquely to a continuous $\ZZ[\beta]$-linear map $\mLPSet \to \mLPSet$.
Write $\Phi_<: \mLPSet \to \mQSym$ and $\Phi_{>|<} :\mLPSet^+ \to \mQSym$ for the morphisms of combinatorial LC-Hopf algebras from Theorems \[<-thm\] and \[><-thm\].
\[psi-thm\] The map $\Psi^{(\beta)} : \mLPSet \to \mLPSet$ is an LC-Hopf algebra morphism making the following diagram commute: $$\begin{tikzcd}
\mLPSet \arrow[d, "\Phi_<"] \arrow[drr, "\Phi_{>|<}"] \arrow[rr, "\Psi^{(\beta)}"] && \mLPSet \arrow[d, "\Phi_{<}"]
\\
\mQSym \arrow[rr, "\ttheta"] && \mQSym
\end{tikzcd}$$ Consequently, if $(P,\gamma)$ is a positive labeled poset then \[ijk-eq\] (P,) = \_[I J = P]{} \^[|I J|]{} (\_[IJ]{}(P,)).
The lower triangle in the diagram commutes by Theorems \[<-thm\] and \[><-thm\] and Corollary \[ttheta-cor\]. To complete the proof, it suffices by Theorem \[><-thm\] to show that $\Psi^{(\beta)}$ is an LC-Hopf algebra morphism and that $\zetaq \circ \Phi_< \circ \Psi^{(\beta)} = \uzetaLP$.
We check the second property first. Fix a positive labeled poset $(P,\gamma)$ and note that $\zetaq \circ \Phi_< = \zetaLP$. If $P = I\cup J$ then $ \zetaLP ([\fkM_{IJ}(P,\gamma)])$ is zero unless (i) $y \notin I$ whenever $y\lessdot z$ in $P$ and $\gamma(y) > \gamma(z)$ and (ii) $y \notin J$ whenever $x\lessdot y $ in $P$ and $\gamma(x) < \gamma(y)$. These conditions can only hold if $\PeakSet(P,\gamma)$ is empty (since $I\cup J=P$). In this case, if $(I, J)$ satisfies conditions (i) and (ii) then $ \zetaLP (\beta^{|I\cap J|} \cdot [\fkM_{IJ}(P,\gamma)])=t^{|P|}(\beta t)^{|I\cap J|}$. Moreover, the decompositions $I \cup J = P$ with this nonzero contribution are uniquely determined by independently assigning each element of $\ValSet(P, \gamma)$ to $I \setminus J$, to $J \setminus I$, or to $I \cap J$, and so $$\zetaq \circ \Phi_< \circ \Psi^{(\beta)}([(P,\gamma)])
= \begin{cases}
t^{|P|}(1 + 1 + \beta t)^{|\ValSet(P, \gamma)|}
&\text{if $\PeakSet(P,\gamma) = \varnothing$},
\\
0&\text{otherwise}.
\end{cases}$$ This agrees with the formula for $\uzetaLP([(P,\gamma)]) := \uzetaLP([(P,\gamma,\varnothing)])$ from Proposition \[><-prop\]. We conclude that $\zetaq \circ \Phi_< \circ \Psi^{(\beta)} = \uzetaLP$, as desired.
Finally, we must check that the map $\Psi^{(\beta)}$ is an LC-Hopf algebra morphism. It clearly commutes with the unit, counit, and product maps of $\mLPSet$, so we only need to check that $(\Psi^{(\beta)}\htimes \Psi^{(\beta)}) \circ \Delta = \Delta \circ \Psi^{(\beta)}$. Continue to let $(P,\gamma)$ be a positive labeled poset. Let $\delta : P \cup (-P) \to \ZZ$ be the map with $\delta(s) = \sign(s) \delta(|s|)$. Write $\prec$ for the partial order on $P\cup (-P)$ that has $x\prec y$ if and only if $|x| \prec |y|$ in $P$. Relative to this order, the labeled poset $(P\cup -P, \delta)$ is the same thing as $\fkM_{PP}(P,\gamma)$.
Define $\sJ(P,\gamma)$ to be the set of tuples $(I,J,S_1,S_2)$ where $I \cup J = P$ and, if $(Q,\delta) := \fkM_{IJ}(P,\gamma)$, then $S_1$ is a lower set in $Q$ and $S_2$ is an upper set in $Q$ such that $Q= S_1 \cup S_2$ and $S_1\cap S_2$ is an antichain. The value of $\Delta \circ \Psi^{(\beta)}([(P,\gamma)])$ is \[cop-eq1\] \^[|I J| + |S\_1S\_2|]{} where the sum is over all $(I,J,S_1,S_2) \in \sJ(P,\gamma)$, where each $S_i$ is partially ordered by $\prec$.
Next, define $\sK(P,\gamma)$ to be the set of tuples $(T_1,T_2,I_1,J_1,I_2,J_2)$ where $T_1$ is a lower set in $P$ and $T_2$ is an upper set in $P$ such that $P = T_1 \cup T_2$ and $T_1 \cap T_2$ is antichain, and where $T_i = I_i \cup J_i$. The value of $(\Psi^{(\beta)}\htimes \Psi^{(\beta)}) \circ \Delta([P,\gamma]) $ is \[cop-eq2\] \^[|I\_1 J\_1| + |I\_2 J\_2| + |T\_1T\_2|]{} \[\_[I\_2J\_2]{}(T\_2,)\] where the sum is over all $(T_1,T_2,I_1,J_1,I_2,J_2) \in \sK(P,\gamma)$.
We must show that and coincide. It suffices to exhibit a bijection $\sJ(P,\gamma) \xrightarrow{\sim} \sK(P,\gamma)$ such that if \[form-eq\](I,J,S\_1,S\_2) (T\_1,T\_2,I\_1,J\_1,I\_2,J\_2) then for $i\in\{1,2\}$ it holds that \[form-eq2\] (S\_i,) = \_[I\_iJ\_i]{}(T\_i, ) |I J| + |S\_1 S\_2| = |I\_1 J\_1| + |I\_2 J\_2| + |T\_1T\_2|. The desired map is as follows. Given $(I,J,S_1,S_2) \in \sJ(P,\gamma)$, let $I_i := \{s \in P : s \in S_i\},$ $J_i := \{s \in P : -s \in S_i\},$ and $T_i := \{ s \in P : \{\pm s\} \cap S_i\neq \varnothing\}$ for $i \in \{1,2\}$. Clearly $T_i = I_i \cup J_i$, $P = T_1 \cup T_2$, and $(S_i,\delta) = \fkM_{I_iJ_i}(T_i,\gamma)$ as labeled posets. Moreover, $T_1$ is a lower set in $P$ and $T_2$ is an upper set, and [$$\begin{aligned}
|I \cap J| + |S_1 \cap S_2| %& =|I \cap J| + |S_1| + |S_2| - |S_1 \cup S_2| \\
& = |I \cap J| + |S_1| + |S_2| - |Q| \\
& = |S_1| + |S_2| - |P| \\
& = (|T_1| + |I_1 \cap J_1|) + (|T_2| + |I_2 \cap J_2|) - (|T_1| + |T_2| - |T_1 \cap T_2|) \\
& = |I_1 \cap J_1| + |I_2 \cap J_2| + |T_1 \cap T_2|.\end{aligned}$$]{}We must also check that $T_1\cap T_2$ is an antichain in $P$. For this, suppose $x \in T_1 \cap T_2$ and $y \in P$ and $x\prec y$. Since $x \in T_1 \cap T_2$, both $S_1$ and $S_2$ must contain at least one of $\pm x$. From our assumptions that $S_2$ is an upper set and $S_1\cap S_2$ is an antichain,it follows that $\{ \pm y\} \cap (S_1 \cup S_2) \subseteq S_2\setminus S_1$, so $y \in T_2\setminus T_1$. We conclude that $T_1 \cap T_2$ is an antichain, so is at least a well-defined map $\sJ(P,\gamma) \to \sK(P,\gamma)$ satisfying .
To invert , suppose $(T_1,T_2,I_1,J_1,I_2,J_2) \in \sK(P,\gamma)$. Let $S_i := I_i \cup (-J_i) $ for $i \in \{1,2\}$ so that $(S_i,\delta) = \fkM_{I_iJ_i}(T_i,\gamma)$, and define $$I := \{s \in P : s \in S_1 \cup S_2 \}
\quand
J := \{s \in P : -s \in S_1 \cup S_2 \}.$$ Since every $s \in P$ has $\{\pm s\} \cap (S_1 \cup S_2) \neq \varnothing$, we have $P = I \cup J$ and $(S_1\cup S_2,\delta)= \fkM_{IJ}(P,\gamma)$, and it is clear that in this labeled poset $S_1$ is a lower set, $S_2$ is an upper set, and $S_1\cap S_2$ is an antichain. The correspondence $
(T_1,T_2,I_1,J_1,I_2,J_2)
\mapsto
(I,J,S_1,S_2)
$ defined in this way is a map $\sK(P,\gamma) \to \sJ(P,\gamma)$. This is the inverse of , so is the required bijection.
Automorphisms {#antipode-subsection}
-------------
If $w$ is a finite sequence then we write $w^\r$ for its reversal. Given a composition $\alpha \vDash n$, let $\alpha^\c$ denote the unique composition of $n$ with $I(\alpha^\c) =[n - 1] \setminus I(\alpha)$, and define $\alpha^\t := (\alpha^\c)^\r = (\alpha^\r)^\c$. If $w$ is a finite sequence of integers with no adjacent repeated entries and $\alpha \vDash \ell(w)$ has $I(\alpha) = \Des(w) := \{ i : w_i > w_{i+1}\}$, then $I(\alpha^\t) = \Des(w^\r)$.
Recall that the quasisymmetric functions $L_\alpha := L^{(0)}_\alpha = \sum_{\alpha \leq \alpha'} M_{\alpha'}$ form a homogeneous basis for $\QSym$ and a pseudobasis for $\mQSym$. Following [@LMW §3.6], we write $\omega, \psi, \rho : \QSym \to \QSym$ for the linear maps with \[omega-eq1\] (L\_) := L\_[\^]{} (L\_) := L\_[\^c]{} (L\_) := L\_[\^]{} for all compositions $\alpha$. Each of these operators is an algebra morphism; $\psi$ and $\rho$ are coalgebra anti-automorphisms; and $\omega = \psi \circ \rho = \rho \circ \psi$ is a Hopf algebra automorphism. Given a peak composition $\alpha =(\alpha_1,\alpha_2,\dots,\alpha_k)$, let \[flat-eq\] \^:= (\_k + 1 , \_[k-1]{},…, \_2,\_1 -1). If $\lambda$ is a partition, $\alpha$ is a peak composition, and $K_\alpha := K^{(0)}_{\alpha}$, then one has \[omega-eq2\] (s\_) =(s\_) &= s\_[\^T]{}\
(s\_) &= s\_ (K\_) =(K\_)&= K\_[\^]{}\
(K\_) & =K\_by [@LMW §3.6] and [@Stembridge1997a Prop. 3.5]. These maps extend uniquely by continuity to involutions of $\mQSym$ preserving $\mSym$; we denote the extensions by the same symbols.
We can evaluate $\omega$, $\psi$, and $\rho$ at other quasisymmetric functions of interest. If $f \in \ZZ[\beta][[x_1,x_2,\dots]]$ and $a,b \in \ZZ[\beta]$ then we abbreviate by writing $$f(\tfrac{ax}{1-b x}) :=f( \tfrac{ax_1}{1-b x_1}, \tfrac{ax_2}{1 -bx_2},\dots)$$ for the power series obtained by substituting $x_i \mapsto \frac{ax_i}{1-b x_i} = ax_i + abx_i^2 + \dots$ for each $i \in \PP$. If $(P,\gamma)$ is a labeled poset then let $P^*$ be the dual poset, in which all order relations are reversed, and define $\gamma^*(s) = -\gamma(s)$ for $s \in P$.
\[omega-prop\] If $\alpha$ is a composition and $(P,\gamma)$ is a labeled poset then $$\ba
\omega\( L^{(\beta)}_\alpha\) &=L^{(\beta)}_{\alpha^\t} (\tfrac{x}{1-\beta x}) \\
\psi\( L^{(\beta)}_\alpha\) &=L^{(\beta)}_{\alpha^\c} (\tfrac{x}{1-\beta x}) \\
\rho\( L^{(\beta)}_\alpha\) &=L^{(\beta)}_{\alpha^\r} \\
\ea
\qquand
\ba
\omega\(\tGamma(P,\gamma)\) &= \tGamma(P^*,\gamma)(\tfrac{x}{1-\beta x}) \\
\psi\(\tGamma(P,\gamma)\) &= \tGamma(P,\gamma^*)(\tfrac{x}{1-\beta x}) \\
\rho\(\tGamma(P,\gamma)\) &= \tGamma(P^*,\gamma^*).
\ea$$
When $\beta=1$, the top formulas are closely related to [@Patrias Prop. 38].
We use the term *word* in this proof to mean a finite sequence of positive integers. Let $W$ be the linearly compact $\ZZ[\beta]$-module with the set of all words as a pseudobasis. Define $\phi_\leq$, $\phi_<$, and $\phi_\geq$ to be the continuous linear maps $W \to \mQSym$ such that if $v$ is a word and $\alpha\vDash \ell(v)$ has $I(\alpha) = \Des(v)$, then $$\phi_\leq(v) = L_\alpha,
\qquad
\phi_<(v^\r) = L_{\alpha^\t},
\qquand
\phi_>(v) = L_{\alpha^\c}.$$ Fix a word $w$ with no adjacent repeated letters and suppose $\alpha\vDash \ell(w)$ has $I(\alpha) = \Des(w)$. Let $[[w]] \in W$ be the sum of all words that yield $w$ when adjacent repeated letters are combined, so that, for example, $[[21]] = 21 + 221 + 211 + 2221 + 2211 + 2111 + \dots$. Then [@Marberg2018 Props. 8.2 and 8.5] assert that $$\phi_\leq([[w^\r]]) = \tilde L_{\alpha^\t}(\tfrac{x}{1-x}),
\qquad
\phi_<([[w]]) = \tilde L_{\alpha},
\qquand
\phi_>([[w^\r]]) = \tilde L_{\alpha^\r},$$ where $\tilde L_\alpha := L^{(1)}_\alpha$. As $\omega\circ \phi_<(v) = \phi_\leq(v^\r)$ for any word $v$, we therefore have (L\_) = \_<(\[\[w\]\]) =\_(\[\[w\^\]\]) = L\_[\^]{}().Similarly, since $ \rho\circ \phi_<(v) = \phi_>(v^\r)$ for any word $v$, we have (L\_) = \_<(\[\[w\]\]) = \_>(\[\[w\^\]\]) = L\_[\^]{}.Substituting $x_i \mapsto \beta x_i$ and applying gives the desired expressions for $\omega( L^{(\beta)}_\alpha)$ and $\rho( L^{(\beta)}_\alpha)$. The formulas for $ \omega\(\tGamma(P,\gamma)\) $ and $\rho\(\tGamma(P,\gamma)\)$ then follow from Theorem \[p-thm1\] and Proposition \[by-def-prop\] since $\tL(P^*) = \{w^\r : w \in \tL(P)\}$ and $\tGamma(w^\r,\gamma^*) =L^{(\beta)}_{\alpha^\r}$ for $w \in \tL(P)$. Finally, we compute $\psi(L^{(\beta)}_\alpha)$ and $\psi\(\tGamma(P,\gamma)\)$ using the identity $\psi= \omega \circ \rho$.
It follows from that if $f \in \mSym$ then $\omega(f) = \psi(f)$ and $\rho(f) = f$, so there is only one nontrivial computation to make on symmetric functions.
\[omega-g-cor\] If $\mu\subseteq \lambda$ are partitions then $\omega\( G^{(\beta)}_{\lambda/\mu}\)
=G^{(\beta)}_{\lambda^T/\mu^T} (\tfrac{x}{1-\beta x})$.
When $\beta=1$ this identity is essentially [@LamPyl Prop. 9.22].
If $\theta$ is a labeling of $\D_{\lambda/\mu}$ satisfying and $\vartheta$ is an analogous labeling of $\D_{\lambda^T/\mu^T}$, then $(\D_{\lambda/\mu}, \theta^*)$ and $(\D_{\lambda^T/\mu^T},\vartheta)$ are equivalent labeled posets. Since we have $ G^{(\beta)}_{\lambda/\mu} = \tGamma(\D_{\lambda/\mu}, \theta)$ by , the result follows from Proposition \[omega-prop\].
Next, we state some formulas for the multipeak quasisymmetric functions.
\[omega-peak-prop\] If $\alpha$ is a peak composition and $(P,\gamma)$ is a labeled poset then $$\ba
\omega\( K^{(\beta)}_\alpha\)&=K^{(\beta)}_{\alpha^\flat} (\tfrac{x}{1-\beta x}) \\
\psi\( K^{(\beta)}_\alpha\) &= K^{(\beta)}_{\alpha} (\tfrac{x}{1-\beta x}) \\
\rho\( K^{(\beta)}_\alpha\) &=K^{(\beta)}_{\alpha^\flat} \\
\ea
\qquand
\ba
\omega\(\tOmega(P,\gamma)\) &=\tOmega(P^*,\gamma)(\tfrac{x}{1-\beta x}) \\
\psi\(\tOmega(P,\gamma)\)&= \tOmega(P,\gamma)(\tfrac{x}{1-\beta x}) \\
\rho\(\tOmega(P,\gamma)\)&=\tOmega(P^*,\gamma).
\ea
%\ba
%\omega\( \oK^{(\beta)}_\alpha\) &=\oK^{(\beta)}_{\alpha^\flat} (\tfrac{x}{1-\beta x}) \\
% \psi\(\oK^{(\beta)}_\alpha\) &= \oK^{(\beta)}_{\alpha} (\tfrac{x}{1-\beta x}) \\
% \rho\(\oK^{(\beta)}_\alpha\) &=\oK^{(\beta)}_{\alpha^\flat}.
% \ea$$ Moreover, the formulas on the left also hold if we replace “$K^{(\beta)}$” by “$\oK^{(\beta)}$”.
If $P = I \cup J $ and $(Q,\delta) := \fkM_{IJ}(P,\gamma)$, then $(Q,\delta^*) \cong \fkM_{JI}(P,\gamma)$ and $(Q^*,\delta^*) \cong \fkM_{JI}(P^*,\gamma)$ as labeled posets. The formulas for $\psi\(\tOmega(P,\gamma)\) $ and $\rho\(\tOmega(P,\gamma)\)$ are immediate from Theorem \[psi-thm\] and Proposition \[omega-prop\], and $\omega\(\tOmega(P,\gamma)\) = \psi \circ \rho\(\tOmega(P,\gamma)\) =
\psi \(\tOmega(P^*,\gamma)\) = \tOmega(P^*,\gamma)(\tfrac{x}{1-\beta x})$.
Given these formulas, the expressions for $\omega (K^{(\beta)}_\alpha)$, $\psi (K^{(\beta)}_\alpha)$, and $\rho (K^{(\beta)}_\alpha)$ are clear from Proposition \[k1-prop\], noting that if $w \in \tL(P)$ has $\PeakSet(w,\gamma) = I(\alpha)$ then $\PeakSet(w^\r,\gamma) = I(\alpha^\flat)$. The analogous set of identities involving $\oK^{(\beta)}_\alpha$ follow in turn by continuous linearity in view of Corollary \[equiexp-cor\].
If $\mu\subseteq \lambda$ are strict partitions then $$\omega\( \bGP_{\lambda/\mu}\)
=\bGP_{\lambda/\mu} (\tfrac{x}{1-\beta x})
\qquand
\omega\( \bGQ_{\lambda/\mu}\)
=\bGQ_{\lambda/\mu} (\tfrac{x}{1-\beta x}).$$
Since $\omega$ and $\psi$ take the same value on the symmetric functions $\bGP_{\lambda/\mu}$ and $\bGQ_{\lambda/\mu}$, this follows from , , and Proposition \[omega-peak-prop\].
We can use these formulas to prove two facts about stable Grothendieck polynomials. The following statement generalizes independent results of Ardila and Serrano [@ArdilaSerrano Thm. 4.3] and DeWitt [@Dewitt Thm. V.5] which show that $s_{\delta_n/\mu} \in \bigoplus_\nu \NN P_\nu$ for all partitions $\mu \subseteq \delta_n := (n,n-1,\dots,2,1)$.
If $n \in \NN$ and $\mu \subseteq \delta_n$ then $G^{(\beta)}_{\delta_n/\mu} \in \bigoplus_\nu \NN[\beta] \bGP_\nu$.
Say that a partition $\mu$ is *strictly contained* in $\lambda$ if $\mu_i < \lambda_i$ for $1 \leq i \leq \ell(\mu)$. It suffices to prove the proposition when $\mu$ is strictly contained in $\delta_n$, since Corollary \[sv-products-cor\] implies that in general $G^{(\beta)}_{\delta_n/\mu}$ is a product of power series of the form $G^{(\beta)}_{\delta_m/\nu}$ with $\nu$ strictly contained in $\delta_m$, and results in [@CTY; @HKPWZZ] show that products of $K$-theoretic Schur $P$-functions are finite $\NN[\beta]$-linear combinations of $K$-theoretic Schur $P$-functions.
Suppose $\mu$ is a partition strictly contained in $\delta_n$. Define $b_i = n - \mu^T_i + i$ for $i \in [n]$ and let $a_1<a_2<\dots<a_n$ be the elements of $[2n] \setminus \{b_1,b_2,\dots,b_n\}$. One has $1<b_1<b_2<\dots<b_n=2n$ and $a_i<b_i$ for each $i$. The second author’s paper [@Marberg2019a] considers certain power series $G_w$ and $\GO_y$ in $\mSym$ indexed by permutations $w \in S_n$ and $y=y^{-1} \in S_{n}$. Let $$y := (a_1,b_1)(a_2,b_2)\cdots(a_n,b_n) \in S_{2n}
\quand w:= b_1a_1 b_2 a_2 \cdots b_n a_n \in S_{2n}.$$ It follows from [@Marberg2019a Thm. 2.3 and Prop. 5.5] that $\GO_y = G_{w^{-1}}$, while [@Matsumura Thm. 3.1] implies[^1] that $G_w = G^{(\beta)}_{\delta_n/\mu^T}$. By [@Marberg2019a Lem. 5.3] and Corollary \[omega-g-cor\], we have $
\omega(\GO_y)=\omega(G_{w^{-1}}) = G_w(\frac{x}{1-\beta x}) =\omega(G^{(\beta)}_{\delta_n/\mu})$, so $G^{(\beta)}_{\delta_n/\mu} = \GO_y$. Finally, $\GO_y \in \bigoplus_\nu \NN[\beta] \bGP_\nu$ by [@Marberg2019a Thm. 1.9].
The previous result holds in a stronger form when $\mu = \emptyset$.
For a partition $\lambda$ and a strict partition $\nu$, one has that $G^{(\beta)}_{\lambda} = \bGP_{\nu}$ if and only if $\lambda =\nu= \delta_n$ for some $n \in \NN$.
For a short proof that $s_{\delta_n} = P_{\delta_n}$, see [@Macdonald Ex. 3, §III.8].
First, suppose $G^{(\beta)}_{\lambda} = \bGP_{\nu}$. The $\beta$-degree-$0$ terms of the left and right sides are respectively $s_\lambda$ and $P_\nu$, and the lowest-degree component of $P_\nu$ is $s_\nu$, so it must be the case that $\lambda = \nu$. The papers [@HMP1; @HMP4] study certain symmetric functions $\iF_z$ indexed by involutions $z \in S_\infty$. It follows from [@HMP4 Thm. 4.20] that every Schur $P$-function $P_\nu$ occurs as $\iF_z$ for some $z$, while [@HMP1 Prop. 3.34 and Thm. 3.35] assert that $s_\lambda= \iF_z$ for some $ z $ only if $\lambda =\delta_n$ for some $ n \in \NN$. Thus it must be that $\lambda = \nu = \delta_n$ for some $n \in \NN$.
Conversely, the papers [@Marberg2019a; @MP2019b] study certain symmetric functions $\GSp_z$ indexed by fixed-point-free involutions $z \in S_{2n}$. When $z = (1,n+1)\cdots(n,2n) \in S_{2n}$, [@MP2019b Thm. 4.17] shows that $\GSp_z = \bGP_{\delta_{n-1}}$ while [@Marberg2019a Thms. 2.5 and 5.2 and Prop. 5.5] imply that $\GSp_z = G^{(\beta)}_{\delta_{n-1}}$. Thus $\bGP_{\delta_n} = G^{(\beta)}_{\delta_n}$.
The antipode $\antipode$ of the Hopf algebra $\QSym$ is the linear map $\QSym \to \QSym$ with $L_\alpha \mapsto (-1)^{|\alpha|} L_{\alpha^\t}$ for all compositions $\alpha$ [@LMW §3.6]; the antipode of $\mQSym$ is the unique continuous extension of this map. One can evaluate $\antipode$ at many elements of interest in $\mQSym$ using the following observation. Let $f^{(\beta)} \in \ZZ[\beta][[x_1,x_2,\dots]]$ and define $f^{(-\beta)} := f^{(\beta)}|_{\beta\mapsto -\beta}$.
Assume $f^{(\beta)} \in \mQSym$ is homogeneous of degree $n$ when we define $\deg(\beta) = -1$ and $\deg (x_i) = 1$. Then $\antipode\(f^{(\beta)}\) = (-1)^n \omega\( f^{(-\beta)}\)$.
This hypothesis applies whenever $f^{(\beta)} \in \{ L^{(\beta)}_\alpha, %G^{(\beta)}_{\lambda/\mu},
K^{(\beta)}_\alpha, \oK^{(\beta)}_{\alpha},\bGQ_{\lambda/\mu}, \dots\}$, in particular. Taking $f^{(\beta)} = L^{(\beta)}_\alpha$ and then specializing to $\beta=1$ recovers [@Patrias Thm. 41], while taking $f^{(\beta)} = \bGP_\lambda$ gives $\bGP_\lambda$ = (-1)\^[||]{} \^[(-)]{}\_() for all strict partitions $\lambda$. Comparing with [@HKPWZZ Prop. 3.5] shows that the power series which Hamaker *et al.* denote as $\GP_\lambda$ and $K_\lambda$ may be given in our notation as $\GP_\lambda := \GP_\lambda^{(-1)}$ and $K_\lambda :=\GP_\lambda^{(1)}(\tfrac{x}{1-x})$, and are related by the identity $K_\lambda = (-1)^{|\lambda|}\antipode(\GP_\lambda)$.
Using these formulas, one can check directly that $\antipode$ preserves the cancelation laws in Lemma \[q-cancel-lem\] and Theorem \[q-cancel-thm\], as must hold for $\mGSym$ and $\omGSym$ to be LC-Hopf subalgebras of $\mSym$. By contrast, the involutions $\omega$ and $\psi$ of $\mQSym$ do not preserve $\mGSym$, $\omGSym$, $\mcoPeak$, or $\omcoPeak$.
Although $\omega(\mSym) =\mSym$, the family of stable Grothendieck polynomials $\{G^{(\beta)}_\lambda\}$ is not itself preserved by $\omega$. To correct this, Yessulizov [@Yeliussizov2017] has introduced a two-parameter generalization $G^{(\alpha,\beta)}_\lambda \in \ZZ[\alpha,\beta][[x_1,x_2,\dots]]$ of $G^{(\beta)}_\lambda =: G^{(0,\beta)}_\lambda$ that has $\omega(G^{(\alpha,\beta)}_\lambda) = G^{(\beta,\alpha)}_{\lambda^T}$. It would be interesting to describe analogous two-parameter generalizations $\GP^{(\alpha,\beta)}_\lambda$ and $\GQ^{(\alpha,\beta)}_\lambda$ of the $K$-theoretic Schur $P$- and $Q$-functions with $\omega(\GP^{(\alpha,\beta)}_\lambda) = \GP^{(\beta,\alpha)}_\lambda$ and $\omega(\GQ^{(\alpha,\beta)}_\lambda) = \GQ^{(\beta,\alpha)}_\lambda$.
[^1]: Matsumura’s result concerns certain polynomials $\fk G_\sigma(x,\xi) \in \ZZ[\beta][x_1,x_2,\dots,\xi_1,\xi_2,\dots]$ indexed by permutations $w \in S_\infty$. These are related to the power series $G_w$ by the identity $G_w = \lim_{m\to \infty} \fk G_{1^m\times w}(x,0)$, where convergence is in the sense of formal power series.
|
---
abstract: |
We consider the multihop broadcasting problem for $n$ nodes placed uniformly at random in a disk and investigate the number of hops required to transmit a signal from the central node to all other nodes under three communication models: Unit-Disk-Graph (UDG), Signal-to-Noise-Ratio (SNR), and the wave superposition model of multiple input/multiple output (MIMO).
In the MIMO model, informed nodes cooperate to produce a stronger superposed signal. We do not consider the problem of transmitting a full message nor do we consider interference with other messages. In each round, the informed senders try to deliver to other nodes the required signal strength such that the received signal can be distinguished from the noise.
We assume a sufficiently high node density $\rho= \Omega(\log n)$ in order to launch the broadcasting process. In the unit-disk graph model, broadcasting takes $\mathcal{O}(\sqrt{n/\rho})$ rounds. In the other models, we use an Expanding Disk Broadcasting Algorithm, where in a round only triggered nodes within a certain distance from the initiator node contribute to the broadcasting operation.
This algorithm achieves a broadcast in only $\mathcal{O}\left(\frac{\log n}{\log \rho}\right)$ rounds in the SNR-model. Adapted to the MIMO model, it broadcasts within $\mathcal{O}(\log \log n - \log \log \rho)$ rounds. All bounds are asymptotically tight and hold with high probability, i.e. $1- n^{-\mathcal{O}(1)}$.
author:
- |
Christian Schindelhauer\
University of Freiburg,\
Georges-Köhler-Allee 51,\
79110 Freiburg im Breisgau, Germany\
`schindel@tf.uni-freiburg.de` Aditya Oak\
Technical University of Darmstadt,\
Hochschulstra[ß]{}e 10,\
64289 Darmstadt, Germany\
`oak@st.informatik.tu-darmstadt.de` Thomas Janson\
University of Freiburg,\
Georges-K[ö]{}hler-Allee 51,\
79110 Freiburg im Breisgau, Germany\
`thomas@janson-online.de`
date: 'August 30, 2019'
title: 'Collaborative Broadcast in $\mathcal{O}(\log \log n)$ Rounds'
---
![Four rounds of repeated collaborative broadcast in the UDG, SNR, and MIMO model for 10,000 senders randomly distributed in a disk with radius 30 and wavelength $\lambda= 0.1$[]{data-label="fig:teaser"}](flooding-comparison-10000.pdf){width=".7\textwidth"}
Introduction
============
Understanding the limits of multi hop communications and broadcasting is important for the development of new technologies in the wireless communication sector. In the recent decades, ever more realistic models for communication have been considered. First, graph models have been used to describe the communication between wireless communication nodes, resulting in the Radio Broadcast model [@peleg2007time]. However, this model neglects the communication range, which has led to a geometric graph model, the Unit-Disk Graph (UDG) [@UnitDiskGraphs1990], which we also consider here. It is based on the observation that there is a path loss of the sender energy with increasing distance between the sender and the receiver. In order to distinguish the signal from noise, the signal to noise energy ratio (SNR) has to be above certain threshold, which leads to the disk shaped model for radio coverage.
However, if one carefully models the influence of the noise on the SNR, one sees that interference-free communication links inside a disk are still possible because of the polynomial nature of the energy path loss. This model is known as the SINR-model [@Goussevskaia07]. This, however, is still far from reality, where one sees a superposition of electromagnetic waves, which can be expressed by the addition of the complex Fourier coefficients. In this model, diversity gain and energy gain [@Tse_fundamentals_book] enable higher bandwidth and higher communication range. There is a trade-off between these two features and certain properties of the number of coordinated senders, receivers and the channel matrix have to be met. However, one sees that in practice this one-hop communication has already led to better networking solutions.
For our theoretical analysis, we concentrate on an open space model with no interfering communications. We want to find the theoretical limitations of a collaborative multi-hop broadcast. For this, we are interested in sending a carrier signal with no further modulated information. This signal is sent by the sender node positioned at the center of a disk in which all other nodes are randomly distributed. Thus, in the first round the first sender activates some small number of neighboring nodes. Then, in every subsequent round, all of them try to extend the set of informed nodes as far as possible, who then join in the next round, until all nodes of the disk are informed (or the process cannot reach any further nodes).
Related Work
============
Broadcasting algorithms have been widely optimized for speed, throughput, and energy consumption. A lot of algorithms apply MAC (medium access control) protocols like TDMA (Time Division Multiple Access) [@UnitDiskGraphs1990; @gandhi2008minimizing; @Halldorsson:2018:LIS:3212734.3212766; @UnitDiskSINRLotker2009], CDMA (Code Division Multiple Access) [@5779066; @Sirkeci-Mergen_First], FDMA (Frequency Division Multiple Access) [@Sirkeci-Mergen_First] to increase spatial reuse. Physical models with high path loss exponent $\alpha > 2$ are beneficial here and increase the spatial reuse with only local interference. With spatial reuse, parallel point-to-point communications are possible which either spread the same broadcast message in the network or pipeline multiple broadcast messages at the same time. The latter can achieve a constant broadcasting rate for path loss exponent $\alpha > 2$. Cooperative transmission with MISO (Multiple Input Single Output) or MIMO (Multiple Input Multiple Output) is applied to increase the transmission range and broadcast speed by a constant factor (where underlying MAC protocols still work).
Here, we focus on broadcast speed and allow as many as possible nodes cooperate in transmitting the same broadcast message with MIMO. The obvious trade-off here is broadcast speed against broadcast rate, since pipelining and spatial reuse are limited.
Broadcasting has been first considered for a graph based model, where interference prevents communication and a choice has to be made which link should be used for propagation. Since we do not consider interference and allow the usage of all links, a simple flooding algorithm achieves the optimal bound of the diameter of the network. So, these works (see [@peleg2007time] for a survey) do not apply here. However, even if interference is considered there is only a constant factor slow down in the Unit-Disk-Graph model [@gandhi2008minimizing]. Note that Unit-Disk-Graphs are connected, when the node density of the randomly placed nodes is large enough [@xue2004number].
Launched by the seminal paper of [@Gupta00thecapacity], the SNR (Signal to Noise Ratio) model has gained a lot of interest. Here, signals can be received if the energy of the sending nodes is a constant factor larger than the sum of noise energy and interference. This model leads to a smooth receiver area with near convexity properties [@AvienEmek12].
If the energy of each sender is constrained, Lebhar et al. [@UnitDiskSINRLotker2009] show that the SNR-model does not give much improvement compared to the UDG-model. So, they incorporate the unit disk model into the SINR (Signal to Interference and Noise) model. The focus of their work is finding TDMA scheduling schemes to enhance the network capacity while the path-loss exponent in the SINR model is chosen with $\alpha > 2$ such that interferences have only local effects for unsynchronized transmitters. In this context, the SNR model is used for each sender separately. So, the problem of broadcast mainly reduces to range assignment and scheduling problem, for which the number of rounds approaches the diameter [@Halldorsson:2018:LIS:3212734.3212766].
For the superposition model the problem of point-to-point communication has been considered mostly for beam-forming for senders (MISO/MIMO) or receivers (SIMO/MIMO). For MIMO (Multiple Input Multiple Output), most of the research is concerned with the energy gain and diversity gain, as well as the trade-off. For an excellent survey we refer to [@Tse_fundamentals_book]. Besides the approach, where sender antennas and receiver antennas are connected to one device and only a one hop communication is considered, a lot of work is dedicated to collaboration of independent senders and receivers, for which we now discuss some noteworthy contributions. A transmission with cooperative beamforming requires phase synchronization of the collaborating transmitters to produce a beam and sharing the data to transmit. Dong et al. [@dpp2000] present for this a two phase scheme: in phase one, the message is spread among nodes in a disk in the plane around the node holding the original message. The open-loop and closed-loop approach can be used to synchronize nodes to the destination or a known node position and time synchronization. In phase two, the synchronized nodes jointly transmit the message towards the destination.
In [@freitas2012energyWSN] a three phase scheme is presented. In order to save energy for a Wireless Sensor Network, in the first phase, a sensor sends its message via SIMO to a group of nearby nodes. In the second phase the nodes use MIMO beamforming to another group of nodes nearby of the receiver and in the final phase the last group of nodes sends the message via MISO to the recipient.
For the MIMO model in [@NGS09_Linear_Capacity_Beamforming] and [@ozgur2010linearCapacity] the authors give a recursive construction, which provides a capacity of $n$ for $n$ senders using MIMO communication using its diversity gain. Yet, in [@franceschetti2009capacity], an upper bound of $\sqrt{n}$ for such a diversity gain has been proved. These seemingly contradicting statements have been addressed in [@ozgur2013spatial], where they address the question whether distributed MIMO provides significant capacity gain over traditional multi-hop in large ad hoc networks with $n$ source-destination pairs randomly distributed over an area A. It turns out that the capacity depends on the ratio $\sqrt{A}/\lambda$, which describes the spatial degree of freedom. If it is larger than $n$ it allows $n$ degrees of freedom [@ozgur2010linearCapacity], if it is less than $\sqrt{n}$ the bound of [@franceschetti2009capacity] holds. For all regimes optimal constructions are provided in these papers. While in [@ozgur2010linearCapacity] path loss exponents $\alpha\in (2,3]$ are considered, for $\alpha>3$ the regularity of the node placement must be taken into account [@NGS09_Linear_Capacity_Beamforming].
While this research is largely concerned with the diversity gain, we study the physical limitations of the energy gain in MIMO. In [@oyman2007power; @oyman2011cooperative], a method is presented to amplify the signal by using spatially distributed nodes. They explore the trade-off between energy efficiency and spectral efficiency with respect to network size. In [@mlo13_telescopic_beamforming], a distributed algorithm is presented in which rectangular collaborative clusters of increasing size are used to produce stronger signal beams.
Janson et al. [@JS14_Beamforming_LogLog_TR] analyze the asymptotic behavior of the rounds for a unicast in great detail and prove an upper and lower bound of $\Theta(\log \log n)$ rounds. If the nodes are placed on the line it takes an exponential number of rounds [@JS13_Beamforming_Line]. The generalization of these observations for different path loss models can be found in [@diss-janson-2015]. In [@6962163] it is shown that the sum of all cooperating sender power can be reduced to the order of one sender, while maintaining a logarithmic number of rounds to send a message over an $n$ hop distance.
A practical approach already uses this technology. Glossy [@5779066] is a network architecture for time synchronization and broadcast including a network protocol for flooding, integration in network protocols of the application, and implementation in real-world sensor nodes. If multiple nodes transmit the same packet in a local area, the same symbol of the different transmitters will overlap at a receiver without inter-symbol interference if the synchronization is sufficient. The superposed signals of the same message have random phase shifts and in the expectation add up constructively. Faraway, out of sync, transmitters produce noise-like interference the influence of which is alleviated at the receiver via pseudo-noise codes. While a high node density increases interference in common network protocols, a higher density is beneficial here and increases the transmission range and reduces the number of broadcasting rounds. Glossy is the underlying technology for the so-called Low-Power Wireless Bus [@Ferrari:2012:LWB:2426656.2426658], where this multi-hop broadcast allows to flood the network with a broadcasting message. The energy efficiency was further improved in Zippy [@sutton2015zippy], which is an on-demand flooding technique providing robust wake-up in the network. Unlike Glossy, Zippy uses an asynchronous wake-up flooding. In [@kumberg2017exploiting] the problem of Rayleigh fading for synchronized identical signals is addressed by producing a low frequency wake-up signal, which results from the beat frequency of closely chosen frequencies. This allows the usage of a passive receiver technology.
Sirkeci-Mergen et al. [@Sirkeci-Mergen_First] propose a multistage cooperative broadcast algorithm similar to our work. Their nodes are also uniformly distributed in a disk. A continuum approximation is used to approximate the behavior of the disk with high node density. A minimum SNR threshold is assumed for successful reception of the message. Their algorithm works in stages, in the first stage, the node at the center of the disk transmits the message. All nodes which receive this message are considered as level one. In the next stage, level one nodes re-transmit the message, in this way set of informed nodes keeps growing in radially outward direction. Nodes belonging to same levels form concentric rings. Source node emits single block of data.
A similar problem and a similar algorithm has been considered in [@sirkeci2010broadcast]. Sirkeci-Mergen et al. consider source node transmitting a continuous message signal. Initially source node which is at the center of the disk, transmits the message signal. In the next round, level one nodes, i.e. the set of nodes that received the message in the previous round, transmit the message signal which is received by next level and the source node does not transmit message. In the following round, the source transmits the next message block. In this way, levels send and receive the message block in alternate rounds. In our work, we consider that in each round, all informed nodes send a single message cooperatively and we prove bounds on the number of rounds needed.
Jeon et al. [@jeon2007two] also consider a system model similar to our work. They use two phase opportunistic broadcasting to achieve linear increase in propagation distance. In phase one, nodes inside a disk of specific radius broadcast message with different random phases while in phase two, a node broadcasts the message to its neighboring nodes. These phases are performed repeatedly to broadcast the message. Improving on this work we obtain better bounds by coordinating the phase of the nodes, while we consider only the path loss factor of $\alpha=2$.
To our knowledge, no research so far has evaluated the asymptotic number of rounds to cover the disk using cooperative broadcast using MIMO, which is the main focus of this work. While [@5779066; @Ferrari:2012:LWB:2426656.2426658; @kumberg2017exploiting; @sutton2015zippy] use only simulation and [@jeon2007two; @sirkeci2010broadcast; @Sirkeci-Mergen_First] prove all their statements only for the expectation in the continuum limit, i.e. when the number of nodes approaches infinity. Our results are to our knowledge the first asymptotic results in MIMO that hold for a finite number of nodes $n$ with high probability, i.e. $1-n^{-\mathcal{O}(1)}$.
#### Notations
The $L_2$-norm is denoted by $|\!|p|\!|_2 = \sqrt{x^2 + y^2}$ for $p=(x,y) \in {\ensuremath{\mathbb{R}}}^2$. For representation of signal waves we use complex numbers ${\ensuremath{\mathbb{C}}}$ where the imaginary number is $i=\sqrt{-1}$. For $z= a+b i$ the complex conjugate is $z^* = a-bi$, the absolute value $|z| = \sqrt{z \cdot z^*}=
\sqrt{a^2+b^2}$ and the real part is $\Re(z) = a = \frac{z+z^*}{2}$, the imaginary part is $\Im(z) = b = \frac{z-z^*}{2}$. The exponent for the base of the Euler number $e$ gives $e^{a+b i} = e^a (\cos b + i \sin b)$. Note that $z = |z| \cdot e^{i \arg(z)}$, where $\arg(z) \in [0, 2\pi)$.
The Models
==========
We assume $n$ nodes $v_1, \ldots, v_n \in {\ensuremath{\mathbb{R}}}^2$ uniformly distributed in a disk of radius $R$ centered at origin, where the additional node $v_0$ resides. The density is denoted by $\rho= n/(\pi R^2)$. Each node knows the disk radius $R$, its location and all nodes are perfectly synchronized, see Fig. \[broadfig\].
![Randomly positioned nodes in a disk of radius $R$ with $v_1$ in the center. The first two phases of the considered collaborative Broadcast algorithms\[broadfig\]](broadcast.pdf){width="1\linewidth"}
We say that a node is triggered or informed, when it has received a signal carrying no further information. The objective is to send the broadcast signal from the center node $v_0$ to all other nodes where in each round the set of sending nodes is increased by the triggered nodes of the last round.
We concentrate on broadcasting a pure sinusoidal signal and leave the problem of broadcasting a complete message to subsequent work. The sinusoidal signal has wavelength $\lambda$ and we normalize the speed of light as $c=1$ by choosing proper units for time and space. In our theoretical framework we assume that every node knows its exact position in the plane, is synchronized (well enough in order to emit phase-coordinated signals) and is able to precisely emit the signal at a given point in time with a certain phase shift and a fixed amplitude.
We consider three communication models in our analysis: Unit-Disk-Graph (UDG), the Signal-to-Noise Ratio (SNR), and MISO/MIMO (Multiple Input—Single/Multiple Output) for coordinated senders. The difference between MIMO and MISO is whether we consider a single receiver or multiple receivers. Since, MIMO is the more general term we prefer this term throughout this paper.
The MIMO Model
--------------
The coordination of nodes refers here to synchronized signals allowing a radiation pattern containing strong beams, i.e. a beamforming gain. Many physical properties are covered in the [**Multiple Input/Multiple Output (MIMO)**]{} model based on superposition of waves. Every node can serve either as sender or as receiver. A node can demodulate a received signal $rx(t) \in {\ensuremath{\mathbb{C}}}$ if the square of the length of the Fourier coefficient over an interval of $\delta\gg \lambda$ is larger than $\beta$, i.e. $$\begin{aligned}
z &=& \frac1{\delta}\int\displaylimits_{t=t_0}^{t_0+\delta} \hbox{rx}(t) \ e^{-i 2 \pi t/ \lambda} \mathrm{d}t \ ,\nonumber \\
|z|^2 / N_0 & \geq & \beta \ .\label{eqsnr}
\end{aligned}$$ with imaginary number $i=\sqrt{-1}$ and $t$ denoting time. In this notation we normalize the energy with respect to the time period and assume $\delta$, $N_0$ and $\beta$ are constant. The bound (\[eqsnr\]) demands that the signal-to-noise energy ratio is large enough to allow a successful signal reception, i.e. $\text{SNR} \ge \beta$ for signal power $|z|^2$ and additive white noise with power $N_0$ over time $\delta$.
Each sending node $j \in \{1,\ldots, n\}$ can start sending at a designated time $t_1$ and stops at $t_2$, described by the function $$s_j(t) = \begin{cases}
a \cdot e^{i 2\pi (t-t_1)/ \lambda} \ , & t \in [t_1,t_2] \ ,\\
0 \ , & \hbox{otherwise} \ , \nonumber
\end{cases}$$ where $a\in {\ensuremath{\mathbb{C}}}$ may encode some signal information, e.g. via Quadrature Amplitude Modulation (QAM). Since we are only interested in transmitting a single signal we choose $a=1$ or $a= e^{i \varphi}$, when we use a phase shift $\varphi$. The total signal received at a node $q \in {\ensuremath{\mathbb{R}}}^2$ is modeled by $$\hbox{rx}(t) = \sum_{j=1}^n \frac{s_j(t- |\!|q - v_j|\!|_2)}{|\!|q-v_j|\!|_2}\ , \nonumber$$ which models the free space transmission model with a path loss factor of two for the logarithm of sender and receiver energy ratio. We are aware, that this equation describes only the far-field behavior, which starts at some constant numbers $c_f>1$ of wavelengths, i.e. $|\!|q-v_i|\!|_2 \geq c_f \lambda$ (Antennas have unit size and are neglected in this equation). Hence, every time $|\!|r-v_i|\!| < c_f \lambda$, we will replace the denominator $|\!|q-v_i|\!|_2$ by $c_f \lambda$ in this expression. We assume that $c_f \lambda \leq 1$ and therefore $\lambda < 1$.
Unit Disk Graph Model
---------------------
For nodes $v_1, \ldots, v_n$, the geometric [**Unit Disk Graph**]{} is defined by the set of edges $(v_i,v_j)$ where nodes have distance $|\!|v_i,v_j|\!|_2 \leq 1$. In each round a message or signal can be sent from a node to an adjacent node. So, collaborative sending is simply ignored. Yet, we also ignore the negative effect of interference. In this model messages can be sent along edge in parallel, independently from what happens somewhere else.
The following Lemma shows the strong relationship between the single sender MIMO model and the UDG model.
If only one sender $u$ sends a signal in the MIMO model with amplitude $a\in {\ensuremath{\mathbb{R}}}^+$, then a node $v$ in distance $d$ receives it if and only if $d\leq \frac{a}{\sqrt{\beta N_0}}$.
In the MIMO model the sender produces the signal $s_j(t) = a\ e^{i 2\pi t/ \lambda +i\phi}$ for $t \in [0,T]$ for some sending time $T$ and phase shift $\phi$ and otherwise $s_j(t) = 0$. For distance $d$ the received signal is $$\hbox{rx}(t) = \frac{s_j(t- d)}{d} = \frac{a\ e^{i 2\pi (t-d)/ \lambda+i\phi}}{d}
\ ,$$ if $t \in [d,T+d]$ and otherwise $\hbox{rx}(t)= 0$. Therefore for $\delta = T$: $$\begin{aligned}
z & = & \frac1{T}\int\displaylimits_{t=d}^{d+T} \hbox{rx}(t) \ e^{-i 2 \pi t/ \lambda} \mathrm{d}t \\
& = & \frac1{T}\int\displaylimits_{t=d}^{d+T}\frac{a\ e^{i 2\pi (t-d)/ \lambda+i\phi}}{d}\ e^{-i 2 \pi t/ \lambda} \\
& = & \frac1{T}\int\displaylimits_{t=d}^{d+T}\frac{a\ e^{-i 2\pi d/ \lambda +i\phi}}{d}\ \mathrm{d}t \\
& = & \frac{a}{d}\ \ e^{-i 2\pi d/ \lambda+ i \phi} \\\end{aligned}$$ Now $ |z| = a/d$ and therefore if $|z|^2 = \frac{a^2}{d^2} \geq \beta N_0$ then $u$ can receive the signal.
This Lemma implies that if $a^2= \beta N_0$, then the MIMO model is equivalent to the Unit-Disk Graph (UDG) model with sending radius 1, if only one sender is active. In order to fairly compare these two models, we fix $a=1$ and set $\beta N_0 = 1$.
The Signal-to-Noise-Ratio Model
-------------------------------
The [**Signal-to-Noise-Ratio (SNR)**]{} model adds the received signal energy of all senders, i.e. a signal is received at $q$ in the SNR model, if for sender energy $S_j := a_j^2$, where $a_j$ denotes the amplitude of sender $v_j$ the sum of the received signal energy is large enough: $$\hbox{\sl RS} := \sum_{j=1}^n \frac{S_j}{(|\!|q-v_j|\!|_2)^2} \ , \quad \hbox{where} \quad \quad \frac{\hbox{\sl RS}}{N_0} \geq \beta \ .$$ If we assume that the senders’ starting time is not coordinated but independently chosen at random, then the following Lemma shows that the MIMO model in the expectation is equivalent to the SNR model.
At the receiver $q$ the expected signal energy $S$ of senders $v_1, \ldots, v_n$ with random phase shift $\phi_i$ and amplitude $a_i$ in the MIMO model is $${\mathbb{E}\left[S\right]} = \hbox{\sl RS} = \sum_{j=1}^n \frac{a_j^2}{(|\!|q-v_j|\!|_2)^2} \ .$$
We assume that at the receiver $q$ senders have started sending such that the sending time intervals $[t_j, t'_j]$ of sender $v_j$ covers the time interval $[T,T+\delta]$ at $q$, i.e. $t_j+ |\!|q-v_j|\!|_2 \leq t$ and $t'_j |\!|q-v_j|\!|_2 \leq t+ \delta$. Each sender sends the signal $s_j(t) = a_j\ e^{i 2\pi t/ \lambda +i\phi_j}$ in interval $[t_j, t'_j]$ with random phase shift $\phi_j$. The received signal at $q$ during $t\in [T,T+\delta]$ is by definition: $$\begin{aligned}
\hbox{rx}(t) &=& \sum_{j=1}^n \frac{s_j(t- |\!|q - v_j|\!|_2)}{|\!|q-v_j|\!|_2}\\
& = & \sum_{j=1}^n \frac{a_j}{|\!|q-v_j|\!|_2} \ e^{i 2\pi (t-|\!|q-v_j|\!|_2)/ \lambda +i\phi_j}\ .\end{aligned}$$ So, the received signal $z$ is: $$\begin{aligned}
z &=& \frac1{\delta}\int\displaylimits_{t=t_0}^{t_0+\delta} \hbox{rx}(t) \ e^{-i 2 \pi t/ \lambda} \mathrm{d}t \\
&=& \frac1{\delta}\int\displaylimits_{t=t_0}^{t_0+\delta}
\sum_{j=1}^n \frac{a_j}{|\!|q-v_j|\!|_2} \ e^{i 2\pi (t-|\!|q-v_j|\!|_2)/ \lambda +i\phi_j}\ e^{-i 2 \pi t/ \lambda} \mathrm{d}t \\
&=& \frac1{\delta}\int\displaylimits_{t=t_0}^{t_0+\delta}
\sum_{j=1}^n \frac{a_j}{|\!|q-v_j|\!|_2} \ e^{-i 2\pi |\!|q-v_j|\!|_2/ \lambda +i\phi_j} \mathrm{d}t \\
&=& \
\sum_{j=1}^n \frac{a_j}{|\!|q-v_j|\!|_2} \ e^{-i 2\pi |\!|q-v_j|\!|_2/ \lambda +i\phi_j} \\
&=& \
\sum_{j=1}^n b_j \ e^{i\sigma_j} \ , \\\end{aligned}$$ where we substitute $b_j= \frac{a_j}{|\!|q-v_j|\!|_2} $ and $\sigma_j=-2\pi |\!|q-v_j|\!|_2/ \lambda +\phi_j \bmod 2 \pi$. Note that $\sigma_1, \ldots, \sigma_n$ are again independent random variables and uniform distributed over $[0,2\pi]$. Now, we observe. $$\begin{aligned}
{\mathbb{E}\left[\left| \sum_{j=1}^{n} b_j e^{i \sigma_j} \right|^2 \right]} &=&
{\mathbb{E}\left[\left(\sum_{j=1}^{n} b_j e^{i \sigma_j} \right) \cdot \left(\sum_{j=1}^{n} b_k e^{- i \sigma_j} \right) \right]} \\
&=&
{\mathbb{E}\left[\sum_{j,k \in \{1, \ldots, n\}} b_j b_k\ e^{i (\sigma_j - \sigma_k)} \right]} \\
&=&
\sum_{j,k \in \{1, \ldots, n\}} b_j b_k\ {\mathbb{E}\left[e^{i (\sigma_j-\sigma_k)} \right]} \\
&=&
\sum_{j \in \{1, \ldots, n\}} b_j^2\ {\mathbb{E}\left[e^{i (\sigma_j - \sigma_j)} \right]}
+
\sum_{j,k \in \{1, \ldots, n\}, j\neq k} b_j b_k\ {\mathbb{E}\left[e^{i \sigma_j}\right]} {\mathbb{E}\left[e^{-i \sigma_k} \right]} \\
&=&
\sum_{j,k \in \{1, \ldots, n\}} b_j^2\ ,\end{aligned}$$ where we use that $\sigma_j$ and $\sigma_k$ are independent and that $ {\mathbb{E}\left[e^{i \sigma_j} \right]} =0$ because $\sigma_j$ is uniform over $[0, 2\pi]$. Therefore ${\mathbb{E}\left[|z|^2\right]} = \sum_{j=1}^n \frac{a_j^2}{(|\!|q-v_j|\!|_2)^2}$.
This proof can also be found in [@2012RandomMIMOJanson]. Unlike in the coordinated MIMO model, in the SNR model signals are sent with random phasing which induces a more regular radiation pattern.
Under the assumption that ${\mathbb{E}\left[|z|^2\right]}/N_0 \geq \beta$ induces a successful reception, $a_j= 1$ and $\beta N_0 = 1$ we derive the Signal-to-Noise Ratio (SNR) model, where the energy of the uncorrelated received signals add up. Again, this model reduces to the UDG model if only one node is sending.
UDG Coverage {#sec:UDG}
============
Let $\rho =\frac{n}{\pi R^2}$ denote the density of nodes. The probability that $n-1$ nodes are not in a given area of size $\pi/8$ is $$\left(1-\frac{\pi/8}{\pi R^2}\right)^{n-1} =
\left(1-\frac{\rho \pi}{8 n}\right)^{n-1} \leq \exp\left(-\left( 1-\frac1n\right) \frac{\rho \pi}{8} \right)\ ,$$ which is less than $1/n^{c}$ for $\rho \geq \frac{8}{\pi} c \ln (n+1)$ for any $c>1$ and $n\geq 2$. Now, consider six equally sized sectors of a unit disk around a node; there is at least a node with probability $1 - n^{-c+1}$.
![For a disk with radius $1$ and area $\pi$ and node density $\rho = \Omega(\log n)$, each of the sectors $S_1, \ldots, S_6$ around a disk with area $\pi/8$ and sender $v_j$ in the center are not empty with high probability\[sectors\]](sectors.pdf){width="0.25\linewidth"}
From this, it follows that UDG is connected (see [@xue2004number] for a better bound) and that the diameter of the UDG is at most $8R = \mathcal{O}\left(\sqrt{n/\rho}\right)$.
For $\rho>1$ in the UDG model, broadcasting needs $\Omega(\sqrt{n/\rho})$ rounds to inform all nodes with high probability.
The probability that none of the $n-1$ non centered nodes are at a distance larger than $R-1$ from the center is for $R>1$: $$\left(1- \frac{2R-1}{R^2} \right)^{n-1} \leq e^{-\frac{n-1}{R}} = e^{-\Theta(\sqrt{n \rho})}.$$ Hence, with high probability some nodes are in this outer rim, which can be reached only after at least $R-2 = \Omega(\sqrt{n}{\rho})$ rounds.
For large enough density $\rho=\Omega(\log n)$ this bound is tight. The probability that $n-1$ nodes are not in a given area of size $\pi/8$ is less than $1/n^{c}$ for $\rho \geq \frac{8}{\pi} c \ln (n+1)$ for any $c>1$ and $n\geq 2$.
For $\rho= \Omega(\log n)$ in the UDG model, broadcasting needs $\Theta(\sqrt{n/\rho})$ rounds to inform all nodes with high probability.
Consider two nodes $v_j$ and $v_k$ with distance $d\leq R$. We have seen that each subregion around a node depicted in Fig \[sectorrouting\] contains at least a node with high probability. Now, we route starting from $v_j$ along the line $L$ connecting $v_j$ and $v_k$ by choosing a node from a sector which is closer to $r_k$ in a sector which in a corridor of width $2$ around $L$. We pick a node from this sector and observe that the messages advances by a distance of at least $\frac14$ in the direction towards $v_k$.
![Routing from $v_j$ to $v_k$ using the unit-disk graph and non-empty sectors.\[sectorrouting\]](sectorrouting.pdf){width="85.00000%"}
Hence, it takes at most $4R$ hops, where $R^2 = \frac{n}{\pi \rho}$.
A Lower Bound for SNR Collaborative Broadcasting {#sec:SNR}
================================================
The expected number of nodes $n(r)$ in a disk of radius $r$ around the origin is sharply concentrated around the expectation $\rho \pi r^2$, if it is at least logarithmic in $n$, which follows from an application of Chernoff bounds.
\[densifity\] For $n$ randomly distributed nodes in a disk of radius $R$ and a given smaller disk of radius $r$ within this disk, let $n(r)$ denote the number of nodes there within. Then we observe: $$\begin{aligned}
{\mathbb{E}\left[n(r)\right]} &=& \pi \rho r^2\ , \\
{\text{Prob}\left[n(r) \geq (1+c) {\mathbb{E}\left[n(r)\right]}\right]}& \leq & e^{-\frac13 \min\{c, c^2\} \pi \rho r^2}
\ .
\\ {\text{Prob}\left[n(r) \leq {\textstyle \frac12} {\mathbb{E}\left[n(r)\right]}\right]}& \leq & e^{-\frac18 \pi \rho r^2} \ . \label{dlower}\end{aligned}$$
We can reformulate $n(r)$ as the sum of the independent Bernoulli variables $X_i$, which denote $X_i=1$ when node $i$ falls into the smaller disk of radius $r$, and otherwise $X_i=0$. We have ${\text{Prob}\left[X_i=1\right]}= \frac{\pi r^2}{\pi R^2} = \frac{r^2}{R^2}$ and thus the expectation of $n(r)$ is the following. $${\mathbb{E}\left[n(r)\right]} = {\mathbb{E}\left[\sum_{i=1}^n X_i\right]} = n \frac{r^2}{R^2} = \frac{\pi \rho n r^2 }{n}= \pi\rho r^2\ ,$$ using $\rho = \frac{n}{\pi R^2}$. The other inequalities follow by applying Chernoff bounds to $n(r)$.
In the SNR-model for $\pi \rho \geq 1$ and $\rho =o(n)$ at least $\Omega\left(\frac{\log n}{\max\{1,\log \rho\}}\right)$ rounds are necessary to broadcast the signal to all $n$ nodes with high probability.
We start with the center node in the middle of the disk and denote by $r_j$ the maximum distance of an informed node from the center of the disk. Let $n_j$ denote the number of informed nodes in round $j$. By definition $r_0 = 0$ and $n_0=1$. Then, in round one we have $r_1=1$ by applying the SNR model for one sender.
We consider two cases.
1. Case: $\pi \rho \geq k \log n$.
Then, the expected number of nodes $n_1$ is $\pi \rho$ by Lemma \[densifity\] and for $\pi \rho \geq 1$ it is bounded as $n_1\leq 2\pi \rho$ with high probability by choosing $c=3k$. Consider a receiver in distance $d$ and assume for the lower bound argument that all nodes $n(r_j)$ in radius $r_j$ send the signal. Since $\pi \rho r_j^2 \geq k \log n$ we have $n(r_j) \geq \frac12 \pi \rho^2$ with high probability. So, for $d \geq 4 \sqrt{\rho} r$ and $\rho \geq 1$ we have $$d-r
\geq 4 \sqrt{\rho} r-r
\geq (4 \sqrt{\rho}-1)r
\geq (4 \sqrt{\rho}-\sqrt{\rho})r
\geq 3 \sqrt{\rho} r
> \sqrt{2 \pi \rho} r \ .$$ Then, the received energy is at most $\frac{n(r)}{(d-r)^2}$ where $$\frac{n(r)}{(d-r)^2} < \frac{n(r)}{2 \pi \rho r^2} \leq \frac{2 \pi \rho r^2}{2 \pi \rho r^2} \leq 1\ \ ,$$ with high probability. So, no node farther away than $r_{j+1} = 4 \sqrt{\rho} r_j$ is informed in the SNR model in round $j$.
By induction only nodes in distance of at most $r_t = \left(4 \sqrt{\rho}\right)^t$ can be informed after $t$ rounds with probability larger than $\frac1{n^{\mathcal{O}(1)}}$, which only can inform all nodes outside the disk of radius $R-1= \frac{n}{\pi \rho}-1$ if $ t\geq \Omega\left(\frac{\log n}{\log \rho}\right)$ for $\rho=o(n)$.
2. $\pi \rho\geq 1$ and $\pi \rho \leq k \log n$.
For the proof we have to overcome the difficulty that the number of nodes in the unit disk may be too small to ensure high probability. We resolve this problem by overestimating the first radius $r_1=\sqrt{\frac{k}{\pi\rho} \ln n}$. Then, the expected number of nodes in this disk is ${\mathbb{E}\left[n(r_1)\right]} = \pi \rho r_1^2 = 3 k \ln n$ and $n(r_1)\geq 2 {\mathbb{E}\left[n(r_1)\right]}$ with small probability, i.e. $1/n^k$.
Like in the first case we assume that in round $r_j$ all nodes in this radius send. So, for $d \geq 4 \sqrt{\rho} r$ and $\rho \geq 1$ the received energy is less than $1$ within a distance of at most $r_{i+1} = 4 \sqrt{\rho} r_j$.
Now the recursion is $$r_t = (4\sqrt{\rho})^{t-1} r_1 = (4\sqrt{\rho})^{t-1} \sqrt{\frac{k}{\pi\rho} \ln n} = 4 (4\sqrt{\rho})^{t-2} \sqrt{\frac{k}{\pi} \ln n} \ .$$
After $t$ rounds nodes in distance of at most $r_t$ can be informed, which can inform all nodes in the disk of radius $R= \frac{n}{\pi \rho}$ if $$4 (4\sqrt{\rho})^{t-2} \sqrt{\frac{k}{\pi} \ln n} \geq R = \frac{n}{\pi \rho}\ ,$$ yielding $$t \geq 2 + \frac{\log n - \frac12 \log \log n - \frac12\log \pi - \log \rho + \log k +2}{2+ \frac12 \log \rho} = \Omega\left(\frac{\log n}{\max\{1,\log \rho\}}\right)$$ since $\rho=o(n)$ and $\rho\geq 1$.
A Lower Bound for MIMO Collaborative Broadcasting {#sec:MISO_Lower}
=================================================
If the unit length amplitudes of all senders in a disk of range $r$ are superpositioned, in the best case this results in a received absolute amplitude proportional to the number of senders divided by the distance.
\[misolowerbound\] Assuming that randomly placed senders are in a disk of radius $r$, then the maximum distance of a node which can be activated is at most $4\pi \rho r^2$ with high probability for $\rho r^2 = \Omega(\log n)$.
The expected number of senders in a disk of radius $r$ is $\pi \rho r^2$. Using Chernoff bounds and $\rho r^2 = \Omega(\log n)$ one can show that this number does not exceed $2 \pi \rho r^2$ with high probability.
Now, in the best case, all waves at a receiver $r$ perfectly add up resulting in a received signal of at most $\left|\hbox{rx}\right|\ \leq \ \sum_{i=1}^{2 \pi \rho r^2}
\frac{1}{|\!|r-s_i|\!|_2}\ .$ We overestimate this signal by replacing the denominator with $d-r$, where $d$ is the distance of the receiver from the senders’ disk’s center. Hence, we receive a signal if $\left|\hbox{rx}\right|^2 = (2 \pi \rho r^2)^2 \geq (d-r)^2$. So, we get $d \leq r+ 2\pi \rho r^2 \leq 4 \pi \rho r^2$.
This Lemma implies the following lower bound.
Any broadcast algorithm using MIMO needs at least $\Omega(\log \log n - \log \log \rho)$ rounds to inform all $n$ nodes with high probability.
We use Lemma \[misolowerbound\] by overestimating the effect of triggered nodes which are bound to disks with radii $r_j$. We assume that we start with $r_0 = \log n$ for $\rho\geq 1$. Now, let $r_{j+1} = 4\pi \rho r_j^2$ denote the largest distance of a node in the next round.
So $r_j \leq (4 \pi \rho \log n)^{2^j}$, which reaches $R-1=\sqrt{n/(\pi \rho)}-1$ at the earliest for some $j= \Omega(\log \log n - \log \log \rho)$.
This claim also follows from the considerations in [@JS13_Beamforming_Line] and [@JS14_Beamforming_LogLog_SSS] and more extensive in [@diss-janson-2015] where a lower bound of $\Omega(\log \log n)$ rounds for the unicast problem has been shown. Here, we adapt this argument to include the density $\rho$.
Expanding Disk Broadcasting {#sec:SNR_EDB}
===========================
For the SNR model a simple flooding algorithm works as well as the algorithm we propose. A straight-forward observation is a monotony property, i.e. every increase in sending amplitude and every additional sending node increases the coverage area. For the upper bound we use Algorithm \[alg:EDB\] which is slower, yet still asymptotically tight to the lower bound and easier to analyze. We choose $r_{j+1} = \frac14 \sqrt{\rho} r_j$, starting with $r_1=1$ and prove the following Lemma.
If $\rho= \Omega(\log n)$, then in round $j\geq 1$ all nodes in distance $r_{j+1}$ from the origin have been informed with high probability.
Lemma \[densifity\] states that the expected number of nodes $n(r_j)$ in the disk of radius $r_j$ is $\rho \pi r_j^2$. Lemma \[densifity\] (\[dlower\]) shows that ${\text{Prob}\left[n(r_i)\leq \frac12 \pi \rho r_j^2\right]} \leq e^{-\frac18 \rho \pi r_j^2} \leq e^{-\frac18\rho}$, which is a small probability $1/n^c$ for $\rho= \Omega(\log n)$.
The maximum distance from any node in the disk of radius $r_{j+1}$ to a node in this disk is at most $r_j + r_{j+1} \leq 2 r_{j+1}$. Hence, the received signal has an expected SNR of at least $$\frac{n(r_j)}{(2r_{j+1})^2} \geq
\frac{\frac12 \rho \pi r_j^2}{(2r_{j+1})^2} = \frac{\frac12 \rho \pi r_j^2}{(2 \frac14 \sqrt{\rho} r_j)^2} = 2\pi > \beta = 1 \ .$$
Therefore $r_j= (\rho/16)^{(j-1)/2}$ and for $j\geq 1+ 2\frac{\log n - \log( \pi \rho)}{(\log \rho) - 4}
= \Theta(\log n/\log \rho)$ we have $r_j \geq R$ and all nodes are informed.
\[co:SNR\_log\] In the SNR-model collaborative broadcasting needs $ \mathcal{O}(\log n/\log \rho)$ rounds for $\rho > 16$, if broadcasting starts with at least $\Omega(\log n)$ nodes, or $\rho=\Omega(\log n)$.
We conjecture that the result of Corollary \[co:SNR\_log\] not only holds for our (line-of-sight, path loss exponent 2) SNR model but also holds for the model proposed in [@NGS09_Linear_Capacity_Beamforming; @ozgur2007hierarchical] where the path loss exponent is $\alpha \le 2$. Then, the channel from sender $v_j$ to receiver $v_k$ has an contribution of $s_j(t) h_{j,k}(t) $ for emitted signal $s_j(t)$ and $h_{j,k}(t) = |\!|v_k-v_j|\!|_2^{-\alpha/2} \cdot e^{i\cdot \theta_{j,k}(t)}$ with random phase shift $\theta_{j,k}(t)$ at time $t$. We discuss further conjectures about the influence of the path loss factor in the Outlook.
The transmission range grows in each round exponentially with factor $r_j / r_{j-1} = \sqrt{\rho/16}$. In order to minimize this term, we define the radii downwards. Let $p = \lceil2\log R/(\log \rho/16)\rceil$ denote the last round and define $r'_p := R = \sqrt{\frac{n}{\rho \pi}}$. Then, define $r'_{j-1} := r'_j/\sqrt{\rho/16}$. Note that $r'_j \leq r_1 =1$ and therefore $r'_2\leq r_2$ can be informed in one step.
So, the overall time $T$ is mostly determined by the speed of light: $$\begin{aligned}
T &\le& \sum_{j=1}^{p} r'_j
\ \le\ R \left( 1 + \sum_{j=1}^{p-1} \frac{1}{(\log \rho/16)^{j/2}}\right)= R \left(1 + \mathcal{O}\left(\frac{1}{\sqrt{\log \rho}}\right)\right) \ .\end{aligned}$$ Since we assume that $\rho= \Omega(\log n)$ we have in this setting the following corollary.
\[c:snrlight\] The speed of the SNR Broadcast approaches the speed of light for growing $n$.
However, the time effort for decoding and encoding at a node is for practical applications usually much larger than the time caused by the speed of light, the main factor is the number of times the signals have to be relayed, i.e. the number of rounds.
MIMO {#sec:MISO}
====
In MIMO we only analyze the expanding broadcasting algorithm, since the coverage area is far from convex nor does every additional sending node help, see Fig. \[figspikes\]. We use a start radius $r_1= c_{2}/\lambda$ and the expansion $ r_{j+1} = c_1 \rho r_j^{3/2}\lambda^{1/2}$ for a constant $c_1>0$ to be defined later.
![Randomly placed nodes reached by a cooperative broadcast in the MISO model from the set of senders randomly placed in the red disk.\[figspikes\]](broadcastRandomPlacement_500kNodes-sendingradius-x-lambda-y-size-z.pdf){width="1\linewidth"}
Informing all nodes in radius $r_1= c_{2}/\lambda$ is done by a different algorithm, since the MISO broadcast does not work here. For constant $\lambda$ this can be done with single sender broadcasts (resulting in the UDG broadcast). For $\lambda=o(1)$ the number of rounds in the first phase may have a significant impact.
In the subsequent MIMO rounds, the senders $v_k$ are synchronized with a phase shift $\varphi_\ell = - 2 \pi |\!|v_\ell - v_0|\!|_2 / \lambda$ such that the resulting signal of $v_\ell$ is $e^{i (2 \pi t / \lambda + \varphi_\ell)}$. These phases try to imitate the pattern of single sender in the center, the energy of which grows double exponentially in each round.
For the analysis we consider only the signal strength at one receiver and analyze whether MISO works for this sender. We prove that the SNR ratio of the collaborative broadcast signal at every receiver is above the threshold with high probability. So, MISO with high probability results in MIMO with high probability for all receivers in the next disk rim.
Recall that the density is defined as $\rho= \frac{n}{\pi R^2}$ and let $\rho \geq c_{3} \log n$.
\[safety\] For constant wavelength $\lambda$, density $\rho = \Omega(\log n)$ every receiver in distance $d$ can be triggered with high probability, if $ 15r_j \leq d \leq c_1 \rho r_j^{3/2}\lambda^{1/2} $, for a constant $c_1$.
We consider an arbitrary node $q$ in distance $d$ from the first sender $v_0$ in the center. We prove that this node is triggered with high probability and thus all receivers in this distance will be triggered likewise with this probability.
First we analyze the expected received signal of a receiver in distance $d$, which is given by an integral. The complex value of this integral will be asymptotically estimated using a geometric argument over the intersection of ellipses with equal phase shift impact and the sender disk.
In this proof $m+1$ denotes the number of triggered senders in a disk of radius $r$. Sender $v_0$ resides at $(0,0)$. The other $m$ senders are uniformly distributed in the disk of radius $r$. We investigate the received signal $\hbox{\sl rx}$ at a receiver $q$ outside of this disk with distance $d\geq r$. Wlog. we assume $q$ lies on the $x$-axis.
Define for $p=(p_x,p_y)$: $$\Delta_d(p) := \sqrt{p_x^2+p_y^2}+ \sqrt{(d-p_x)^2+p_y^2}-d \ .$$ Using this notion the received signal strength is given by
For $0\leq w \leq \tau+\lambda/2$ and senders $v_1, \ldots, v_n \in D(v_0,r)$ the received signal is given as $\hbox{\sl rx}(t)= \hbox{\sl rx} \cdot
e^{i 2 \pi (t- d)/\lambda}$, where $\hbox{\sl rx} =
\sum_{j=1}^n \frac{ e^{-i 2 \pi
\Delta_d(v_j)/{\lambda}} }{|\!|q-v_j|\!|_2} \ .$
We will estimate the absolute value $|\hbox{\sl rx}|$ as follows.
First, we see that there is an easy characterization by ellipses $E_{\tau}$ with focal points in $v_0$ and $q$, which characterize whether senders help or interfere, see Fig. \[oak-senders\]. The parameter $\tau$ describes the phase at which the sender’s signal arrives at the receiver $q$. The main contributor to the received signal comes from the area within $E_{\tau}$ which has an area of $\Theta(r^{3/2} \lambda^{1/2})$, which corresponds to the innermost dark ellipse in Fig. \[oak-senders\]. The other areas more or less cancel themselves out.
![Senders in a disk of radius 10, colored according to the phase difference perceived by a receiver located at point (100, 0) for wavelength 1 wavelength $\lambda=1$ [@oakmasterthesis].\[oak-senders\]](Gray_Phase_Difference.jpg){width="0.8\linewidth"}
To prove this, we give a formula which describes exactly the expected signal at a given point $t_0$. This expectation will be estimated by carefully chosen bounds. We denote by $D_r$ the disk with center $(0,0)$ and radius $r$.
Consider the ellipse $E_{\tau}$ with focal points $v_0=(0,0)$ and $q= (d,0)$: $$E_{\tau} := \{ p \in {\ensuremath{\mathbb{R}}}^2 \mid |\!|p|\!|_2 + |\!|p-q|\!|_2 = d+ \tau \} \ .$$ We will use the following notation do describe all points inside this ellipse: $$E_{\leq \tau} := \{ p \in {\ensuremath{\mathbb{R}}}^2 \mid |\!|p|\!|_2 + |\!|p-q|\!|_2 \leq d+ \tau \} \ .$$
$
{\mathbb{E}\left[\hbox{\sl rx}\right]}
= \frac{1}{d} + \frac{m}{2\pi r^2}
\iint\displaylimits_{(x,y) \in D_r}
\frac{e^{- i \Delta_d(x,y) 2\pi/\lambda} \ \mathrm{d} x \ \mathrm{d} y}{\sqrt{(x-d)^2+y^2}} \ .
$
In order to estimate this expectation we introduce the following complex valued functions for the disk $D_r = \{p \in {\ensuremath{\mathbb{R}}}^2 \mid |\!|p|\!|_2 \leq r\}$. $$\begin{aligned}
s({d,\lambda,r}) &:=&\iint\displaylimits_{(x,y) \in D_r}
\frac{e^{- i \Delta_d(x,y) 2\pi/\lambda} \ \mathrm{d} x \ \mathrm{d} y}{\sqrt{(x-d)^2+y^2}}
$$ Since the maximum phase shift in the disk of radius $r$ is at point $(-r,0)$ with $\Delta_d(-r,0) = 2r$ we have the following relationship between these functions $s({d,\lambda,r}) = h_{d,\lambda,r}(2r)$. Furthermore, there is the following linearity within $h$: $$\forall c>0: s({d,\lambda,r}) = \frac1c\ s({c d,c \lambda,c r})$$
Note that the following relationship holds. $$\begin{aligned}
{\mathbb{E}\left[\hbox{\sl rx}\right]} &=& \frac1d + \frac{m}{2\pi r^2} s(d,\lambda,r) \\
&=& \frac1d + \frac{m}{2\pi r}s(d/r,\lambda/r,1) \ .\end{aligned}$$ Now the estimation of the signal is based on the following lemma.
\[achtvier\] For $w\geq \pi$, $\lambda \leq 2$ and $d>15$: $$\begin{aligned}
\Im( s(d,\lambda,1)) &\geq& \frac{9}{2,240\sqrt2} \frac{\sqrt{\lambda}}{d+1} \ . \end{aligned}$$
We need the following definitions, where $h_{\infty,\lambda}(w)$ is used to estimate the signal energy in the far-distance case. We first estimate its size and then apply it to $s(d,\lambda,r)$. Note that $U=D_1$. $$\begin{aligned}
h_{d,\lambda}(w) &:=& \iint\displaylimits_{(x,y) \in E_{\leq w} \cap U}
e^{i \Delta_d(x,y) 2\pi/\lambda}\ \mathrm{d} x \ \mathrm{d} y\ , \\
h_{\infty,\lambda}(w) &:=& \iint\displaylimits_{(x,y) \in E_{\leq w} \cap U}
e^{i \Delta_{\infty}(x,y) 2\pi/\lambda} \ \mathrm{d}x \ \mathrm{d}y \ .\end{aligned}$$ where $
\Delta_d(x,y) := \sqrt{x^2+y^2}+ \sqrt{(d-x)^2+y^2}-d$ and $\Delta_{\infty}(x,y) = \sqrt{x^2+y^2} -x $.
We use a geometric argument and concentrate on the area of the intersection of the unit disk $U$ and the $E_{\leq w}$ described by $f(w,d)= \iint\displaylimits_{(x,y) \in E_{\leq w} \cap U} 1 \ \mathrm{d} x \ \mathrm{d} y
$. Using the following function for the area of the segment of depth $z$ of an ellipse with radii $r_1$ and $r_2$ we derive a closed form for this function, see Fig. \[segment\]. $$\begin{aligned}
g(x) & := & \arccos\left(1-x\right)-\left(1-x\right) \sqrt{1-(1-x)^2}\\
s(r_1,r_2,z) &=& r_1 r_2\ g\left(\frac{z}{r_1}\right) \end{aligned}$$
![Area $s(r_1,r_2,z)$ of a segment of an ellipse \[segment\]](ellipse.pdf){width="0.35\linewidth"}
The main notations are shown in Fig \[f:not\].
![Sender $v_0=(0,0)$, receiver $q=(d,0)$, intersections $(x_0,y_0), (x_0,-y_0)$, the center $(d/2,0)$ of the ellipse, the radii $r_1$ and $r_2$\[f:not\]](Disk-Ellipse.pdf){width=".6\linewidth"}
The cutting depth $z_0$ of the circle and the cutting depth $z_1$ for the ellipse are $$\begin{aligned}
z_0 & = & w\cdot \frac{2d-2+w}{2d}\\z_1 &=&
1- z_0 +w/2= \frac{(2-w)(d+w)}{2d}\ .
\end{aligned}$$ The radii of the ellipse are given by $$\begin{aligned}
r_1 & = & \frac{d+w}2\ , \\r_2 & = & \frac12 \sqrt{(d+w)^2 - d^2}\ =\ \frac12 \sqrt{(2d+w)w}\ .\end{aligned}$$ The other parameters are $$\begin{aligned}
x_0 & = & 1- z_0(w,d) \ = \ 1- w\cdot \frac{2d-2+w}{2d} \\
y_0 & = & \sqrt{r^2 - x_0^2} \ , \end{aligned}$$ which are the solutions to the equations $$\begin{aligned}
x_0^2 + y_0^2& = & 1 \\
(d-x_0)^2 + y_0^2 & = & (d-1+w)^2 \\
r_1^2 + \frac14 d^2&=& \frac14(d+w)^2 \ .\end{aligned}$$
This implies for $f(w,d):= |E_{\leq w} \cap U|$ (time shift $w$ and receiver in distance $d$): $$\begin{aligned}
f(w,d) &=& s(1,1,z_0) + s(r_1,r_2, z_1)\\
&=& g(z_0)+ r_1r_2 \ g\left( \frac{z_1}{r_1}\right)\\
&=& g\left( \frac{w(2d+w-2)}{2d}\right)+\frac14 (d+w) \sqrt{w(2d+w)}\
g\left( \frac{2-w}{d }\right)
$$\[AI-proof\] Define the derivatives $f'$ and $f''$ with respect to the first parameter: $$\begin{aligned}
f'(x,y)& := &\quad\frac{\mathrm{d} f(x,y)}{\mathrm{d} \ x} \label{eq:neuns} \\
f''(x,y)& := &\quad\frac{\mathrm{d}^2 f(x,y)}{\mathrm{d}^2 \ x} \label{d:fss}
\end{aligned}$$ The following observations of the derivatives are useful later on.
For $x\in [0,2]$, $y\geq 1$: $$\begin{aligned}
f'(x,y) & > & 0 \label{eq:neun} \\
\sqrt{x} \ \ < \ \ f(x,y) & < & \frac{7}3 \sqrt{x} \label{eq:fff}
\label{eq:zehn} \\
\hbox{For $y\geq 2$:}\quad \frac32 \sqrt{x} \ \ < \ \ f(x,y) \label{eq:ffff}\\
\hbox{For $y>1$, $x\leq 2$:}\quad \frac{f(x,y)}{f(x/2,y)} & > & \frac{7}{5}
\label{eq:droelf}
$$ \[la1\]
Inequality (\[eq:neun\]) follows by the definition of $f(x,y)$. The other inequalities have been verified by computer generated function tables using interval arithmetics, see also the plots in Figs. \[f1\]-\[f3\] to get an intuition.
![Plot of functions $f(x,y)$ and $f'(x,y)$ for exemplary values $y\in\{1,2,\infty\}$\[f1\]](plots-f-fs.pdf){width=".5\linewidth"}
In the automated proofs we only use rational numbers and only test if for two numbers $a<b$. The automated proofs check whether for some function $h$ the inequality $h(x) < c$ holds for $x\in [a,b]$, where $h(x)$ is finite in $[a,b]$. For this we divide the interval $[a,b]$ in smaller intervals $[a_0,a_1], [a_1,a_2], \ldots, [a_{n-1},b]$. Then, for an interval we calculate $f([a_j,a_{j+1}]) = [s_j,\ell_{j}]$ using interval arithmetics [@hickey2001interval]. So, it holds that $s_j \leq f(x) \leq \ell_{j+1}$ for all $x \in [a_j,a_{j+1}]$. Then, we check whether $\ell_j < c$ for all $i$. If this is not the case, we increase the number of intervals $n$ to have smaller intervals. By basic analysis it follows that a set of small enough intervals must exist that deliver a positive result if $h(x)<c$.
Analogously, we generate proofs for two-dimensional cases to check that $f(x,y) < c$ holds for $x \in [a,b], y\in [a',b']$, where we use input intervals $[a_j,a_{j+1}] \times [a'_k, a'_{k+1}]$.
However, a direct application is not possible before we have not resolved the following two problems: we have an open interval $y\in [1,\infty)$ and some comparisons involve functions on both side of the inequality.
For the first problem we substitute $z=\frac1{y}$ and consider functions over the interval $z \in [0,1]$. At $z=0$ we have to show that the discontinuity is removable. This way we prove Inequality (\[eq:droelf\]) noting that $\lim_{y\rightarrow \infty} \frac{f(2,y)}{f(1,y)} = \sqrt{2} \geq \frac75$.
For the second problem we divide by one side and get a removable discontinuity. In order to remove the discontinuity at $x=0$ for $f(x,y)/\sqrt{x}$, we observe that $$\lim_{y\rightarrow \infty} f(x,y) = \frac{1}{3} (x+1) \sqrt{(2-x) x}+\arccos (1-x)$$ such that we get a removable discontinuity for $\frac{f(x,y)}{\sqrt{x}}$ for $y\rightarrow \infty$. Using the substitution $z=1/y$ the automatic selection of intervals for interval arithmetics produces the proofs for inequalites (\[eq:fff\]) and (\[eq:ffff\]).
![The function $f(x,y)/f(x/2,y)$ for $y\in\{1,3/2,3,14,\infty\}$ relevant for Inequality (\[eq:droelf\]) \[f4\]](plotf2dif.pdf){width=".5\linewidth"}
For (\[eq:neuns\]), (\[eq:fff\]), (\[eq:ffff\]) $f$, $g$, the first and second derivatives of $f$ are the following. $$\begin{aligned}
f(x,y) & = & g\left( \frac{x(x+2y-2)}{2y} \right)+ \frac14 (x+y) \sqrt{x(x+2y)} \ g\left(\frac{2-x}{y}\right) \\
\frac{\mathrm{d} \ g(x)}{\mathrm{d} \ x} & = & \sqrt{1-(1-x)^2} \\
f'(x,y)&:=& \frac{df(x,y)}{dy} \\&=&
g\left(\frac{2-x}{y}\right) \frac{2 x^2+4 x y+y^2}{4 \sqrt{x (x+2 y)}} +
\ \frac{(x+y-2)\sqrt{x(2-x)(x+2 y-2)(x+2 y) }}{2 y^2} \\
f''(x,y)&:=& \frac{d^2f(x,y)}{\mathrm{d}^2 \ y} =T_1(x,y)+ T_2(x,y) + T_3(x,y) \ ,
\end{aligned}$$ where we have the following terms. $$\begin{aligned}
T_1(x,y)
& = & \frac{
-3 x^4-2 x^3 (6 y-7)-2 x^2 \left(7 y^2-21 y+10\right)
-4 x \left(y^3-8 y^2+10 y-2\right)+4 y \left(y^2-3 y+2 \right) }{2 y^2 \sqrt{(2-x) x (x+2 y-2) (x+2 y)}} \ \\[2ex]
T_2(x,y)& = & -\frac{ \sqrt{(2-x) (x+2 y-2)} \left(2 x^2+4 x y+y^2\right)}{2 y^2 \sqrt{x (x+2 y)}}\\
T_3(x,y)& = &\frac{(x+y) \left(2 x^2+4 x y-y^2\right) }{4 (x (x+2 y))^{3/2}} g\left(\frac{2-x}{y}\right)\end{aligned}$$
We can prove via interval arithmetics for $x \in [0,2], y\geq 1$ the following inequalities, see also Figures \[ff1\], \[ff2\], \[ff3\], \[ff4\].
The following inequalities hold for $y\geq1$, $x \in [0,2]$:
$$\begin{aligned}
-2\ & \leq &\ \ T_1(x,y) \sqrt{x(2-x)} \nonumber \\[-3.3ex]&&\hspace{25ex}\leq\ 2 \label{T1}\\[1ex]
-3\ & \leq &\ \ T_2(x,y) \sqrt{x} \nonumber \\[-3.3ex]&&\hspace{25ex}\leq\ 0 \ \label{T2}\\[1ex]
-1 \ & \leq &\ \ T_3(x,y) x^{3/2} \nonumber \\[-3.3ex]&&\hspace{25ex}\leq\ 1 \label{T3} \\[1ex]
1 \ &\leq &\quad \frac{g(x)}{x^{3/2}} \nonumber\\[-3.9ex]&&\hspace{25ex}\leq\ 2 \label{g32} \ .\end{aligned}$$
For $x\in \left(0, \frac1{100}\right], y\geq 1$ $$T_3(x,y) x^{3/2} \ \leq \ -\frac15 \ . \label{T33215}$$ where discontinuities at $x=0$ and $x=2$ are removed.
For $x\in \left[\frac1{100}, 2-\frac1{100}\right]$, $y\geq 1$ $$\label{fss18}
f''(x,y)\ \leq\ -\frac18\ .$$
Possible discontinuities at $x=0$ and $x=2$ of the functions $T_1(x,y) \sqrt{x(2-x)}$, $T_2(x,y) \sqrt{x}$, and $T_3(x,y) x^{3/2}$ can trivially be removed by algebraic transformation. The function $\frac{g(x)}{x^{3/2}}$ is also finite with a removable discontinuity at $x=0$, since $$\lim_{x\rightarrow 0} \frac{g(x)}{x^{3/2}} = \frac43 \sqrt2 \ ,$$ which can be seen by the Taylor series. For the automated proofs we again replace $z= \frac1y$ and prove the claims for the interval $z\in [0,1]$ with interval arithmetics.
Since $f''$ is finite over this domain, we can apply an automated proof with interval arithmetics for $f''(x,1/z) < -\frac18$ for $x \in [1/100,1-1/100]$ , removing the discontinuity for $z=0$.
![The function $T_1(x,y) \sqrt{x(2-x)}$ for $y\in \{1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.8, 2.5, 3.2, 5, 7, \infty\}$, relevant for Inequality (\[T1\]) \[ff1\]](plots-T1x2x.pdf){width=".5\linewidth"}
![The function $T_2(x,y) \sqrt{x}$ for $y\in \{1, 1.1, 1.3, 1.8, 2.8, 4.5, 10,\infty\}$, relevant for Inequality (\[T2\]) \[ff2\]](plots-T2.pdf){width=".5\linewidth"}
![The function $T_3(x,y) x^{3/2}$ for $y\in \{1, 1.3, 2, 5,\infty\}$ , relevant for Inequality (\[T3\]) \[ff3\]](plots-T3.pdf){width=".5\linewidth"}
![The function $g(x)/x^{3/2}$, relevant for Inequality (\[g32\]) \[ff4\]](plots-gx32.pdf){width=".5\linewidth"}
\[l:fss\] For all $y>1$, $x\in (0,2)$ $$f''(x,y)<-\frac14\ .$$ For $x\in \left[ 2-\frac1{100},2\right)$, $y\geq 1$ $$f''(x,y)\ \leq\ -199\ .$$
The second derivative $f''(x)$ tends to $-\infty$ at the borders of $x\in [0,2]$. We have proved (\[T33215\]) via interval arithmetics for $x\leq \frac{1}{100}$: $x^{3/2} T_3(x,y) \leq -1/5$. Then, it follows for $x \in [0, \frac1{100}]$:
![The function $f''(x,y)$ for $y\in\{1,1.2,2,\infty\}$ of Lemma \[l:fss\] \[f:fss\]](plots-fss.pdf){width=".5\linewidth"}
$$\begin{aligned}
f''(x,y) & \leq & T_3(x,y)+ \frac{2}{\sqrt{x(2-x)} } \\
& \leq & -\frac{1}{5 x^{3/2}} + \frac{2\sqrt{x}}{\sqrt{(2-x)} } \\
& = & \frac{1}{\sqrt{x}} \left(-\frac{1}{5 x} + \frac{2 x}{\sqrt{(2-x)} }\right) \\
& < & -199\end{aligned}$$
For $x\geq 2-\frac1{100}$ we get $$\begin{aligned}
f''(x,y) & \leq & T_1(x,y) + T_2(x,y) + T_3(x,y) \\
& \leq & -\frac12 \sqrt{x(2-x)} + x^{3/2} \\
& \leq & -\frac{\sqrt{100}}{2 \sqrt{2-\frac{1}{100}}} + (2-1/100)^{3/2} \\
& < & -7/5\end{aligned}$$
These lemmas imply:
\[l:fsss\] For $x\in (0,2), y\geq 1$: $$f''(x,y)\leq -\frac18\ .$$
![The function $f(x,y)/\sqrt{x}$ for $y\in\{1,2,\infty\}$ relevant for Inequalites (\[eq:fff\]) and (\[eq:ffff\]) \[f3\]](plots-fsqrt.pdf){width=".5\linewidth"}
Using $f$ we can describe $h_{\infty}$ as follows.
For $w\in [0,2]$: $$h_{d, \lambda}(w) = \int_{x = 0}^{w} e^{i 2\pi x/ \lambda} f'(x,d)
\ \mathrm{d}x\ .$$
From the above considerations we know that the area of $E_{\leq w} \cap U$ is described by $ f(w, d)\ .$ Hence, the differential area with points with phase $w$ is given by $$\frac{\mathrm{d}\ f(w,d)}{\mathrm{d}\ w} = f'(w,d)\ .$$
For the orientation of $h_{d, \lambda}(w)$ we get the following Lemma.
For all $w \geq 0$: $$\arg(h_{d, \lambda}(w)) \in (0,\pi)\ .$$
We extend $f'(x,y)$ as $f'(x,y)= 0$ for $y\geq 2$. We consider two consecutive intervals of distance $\lambda$: $$\begin{aligned}
h_{d, \lambda}(w)&=&\sum_{j=0}^{\lceil 2/\lambda \rceil} \int_{x = 0}^{\lambda} e^{i 2\pi x/ \lambda} f'\left(x+j \lambda,{d}\right)
\ \mathrm{d}x\ \\
&=& \sum_{j=0}^{\lceil 2/\lambda \rceil} \int_{x = 0}^{\lambda/2}e^{i 2\pi x/ \lambda}\left(f'\left({x+j \lambda},{d}\right) f'\left({x+j \lambda + \frac12 \lambda},{d}
\right)\right)
\ \mathrm{d}x\ \\
&=& \int_{x = 0}^{\lambda/2} e^{i 2\pi x/ \lambda} \sum_{j=0}^{\lceil 2/\lambda \rceil} \left(f'\left({x+j \lambda},{d}\right)-
f'\left({x+j \lambda + \frac12 \lambda},{d}
\right)\right)
\ \mathrm{d}x\ \\\end{aligned}$$ Since $f''(x,y) \leq -\frac18$ we have $$f'\left({x+j \lambda},{d}\right)-f'\left({x+j \lambda + \frac12 \lambda},{d}
\right)\geq \frac1{16} \lambda$$
Because every difference is positive and it follows that $\arg(h_{d, \lambda}(w)) \in (0,\pi)$.
From this consideration one can only imply that $\Im(h_{d, \lambda}(w)) = \Omega(\lambda)$ for $w\geq \lambda$. To achieve a better bound we concentrate on the first term of the sum.
For $x\leq \lambda/2$: $$f'\left({x},{d}\right) \ \geq \ \frac{7}{5} f'\left({x + \frac12 \lambda},{d}
\right)$$
Since $x\leq \lambda/2$ we have $2x \leq x+\lambda$ and by Lemma \[la1\] and the monotony of $f'$ it follows: $$f'({x},{d}) \ \geq \ \frac{7}{5} f'({2 x},{d})\ \geq \ \frac{7}{5}\ f'\left({x + \frac12 \lambda},{d}
\right)\ .$$
For $w \geq \lambda/2$: $$\Im(h_{d, \lambda}(w)) \geq \int_{x = 0}^{\lambda} \sin(2\pi i\ x/ \lambda) \ f'\left({x},{d}\right)
\ \mathrm{d}x\ .$$
Because $f''(x,y)<0$ the other sum terms for $j>0$ may only increase the imaginary terms.
$$\int_{x = 0}^{\lambda/2} \sin(2\pi i\ x/ \lambda) \ f'\left({x},{d}\right)
\ \mathrm{d}x\ \geq
\frac27
\int_{x = 0}^{\lambda} \sin(2\pi i\ x/ \lambda) \ f'\left({x},{d}\right)
\ \mathrm{d}x$$
For $x\leq \lambda$ by the claim above we have: $$f'\left({x},{d}\right) \geq \frac75 \
f'({x+\lambda/2},{d})$$ Hence $$f'(x,d) -f'\left(x+\lambda/2,d\right) \ \ \geq \ \ \frac27 \ f'(x,d) \ .$$
For $d\geq 2$ $$\int_{x = 0}^{\lambda/2} \sin(2\pi i\ x/ \lambda) \ f'\left({x},{d}\right)
\ \mathrm{d}x\ \ \geq \ \frac1{16\sqrt2} \sqrt{\lambda}$$
We use that $\frac32 \sqrt{x}< f(x,y) < \frac73 \sqrt{x}$ from (\[q73\]) and that $f'(x,y)$ decreases with $x$ (Lemma \[l:fsss\]). $$\begin{aligned}
\int_{x = 0}^{\lambda/2} \sin(2\pi i\ x/ \lambda) \ f'\left(x,d\right)
\ \mathrm{d}x\
& \geq &
\int_{x = \lambda/4}^{\frac34 \lambda} \sin(2\pi i\ x/ \lambda) \ f'\left(x,d\right)
\ \mathrm{d}x\ \\
& \geq &
\int_{x = \lambda/4}^{\frac34 \lambda} \frac1{\sqrt2} \ f'\left(x,d\right)
\ \mathrm{d}x\ \\
& \geq &
\int_{x = \lambda/4}^{\frac34 \lambda} \frac1{\sqrt2} \ f'\left(x,d\right)
\ \mathrm{d}x\ \\
& \geq &
\frac{1}{2\sqrt2} \ \left( f\left({\frac34 \lambda},d\right)-
f\left({\frac14 \lambda},d\right)\right)\\
& \geq & \frac{1}{2\sqrt2} \left(\frac32 \sqrt{\frac34 \lambda}-
\frac73 \sqrt{\frac14 \lambda}\right)\\
& \geq & \frac{\frac32\sqrt3-\frac73}{4\sqrt2}\
\lambda^{1/2} \label{q73} \\
& \geq & \frac{1}{16\sqrt2}\
\lambda^{1/2} \end{aligned}$$
This implies the following Lemma:
\[lA10\]For $w \geq \lambda/2$, $d \geq 2r$: $$\Im(h_{d, \lambda}(w))) \geq \frac{1}{56\sqrt2} \sqrt{\lambda} \ .$$
Follows by the combining the above claims.
We now estimate for $d\geq c_f \lambda +1$: $$u_{d,\lambda}(w) :=\iint\displaylimits_{(x,y) \in E_{\leq w} \cap U}
\frac{e^{2\pi
i \Delta_d(x,y)/\lambda}}{\sqrt{(x-d)^2+y^2}} \ \mathrm{d} x \ \mathrm{d} y\ ,$$ which is of interest since $u_{d,\lambda}(2) = s(d,\lambda,1)$. Note that $\Delta_d(p):= |\!|p|\!|_2 + |\!|p-q|\!|_2- d\ .$ We change coordinates from $(x,y)$ to $(z, \ell)$, where $z$ is the $x$-coordinate of the Ellipse $E_{w}$ and $\ell$ is the distance from the point to the receiver. Note that $$\left(
\begin{matrix}
x(\ell,z) \\
y(\ell,z)
\end{matrix}\right) =
\left(
\begin{matrix}\displaystyle
\frac{2 d^2-2 d \ell+2 d z-2 \ell z+z^2}{2 d}\\[2ex]
\displaystyle
\frac{\sqrt{z} \sqrt{2 \ell-z} \sqrt{4 d^2-4 d \ell+4 d z-2 \ell z+z^2}}{2 d}\end{matrix}\right)$$
For $0 \leq w \leq 2$: $$\begin{aligned}
u_{d,\lambda}(w)
&=& \int\displaylimits_{z=0}^{w}
\int\displaylimits_{\ell=d-1+z}^{d+z/2}
2\frac{e^{2\pi i z/\lambda}}{\ell}
|\det(\phi(\ell,z))| \mathrm{d}\ell \ \mathrm{d}z\ \end{aligned}$$ where for $\ell\geq z$: $$|\det(\phi(\ell,z))| = \frac{2 \ell (d-\ell+z)}{\sqrt{z} \sqrt{2 \ell-z} \sqrt{(2 d+z) (2 d-2 \ell+z)}}$$
where $\phi(\ell,z)$ is the Jacobian of $\left(
\begin{matrix}
x(\ell,z) \\
y(\ell,z)
\end{matrix}\right)$.
$$f'(z,d) =
\int\displaylimits_{\ell=d-r+z}^{d+z/2}
2 |\det(\phi(\ell,z))| \ \mathrm{d}\ell$$
Define $$t(d,z, \ell) := \int\displaylimits_{\ell=d-1+z}^{d+z/2}
\frac{2}{\ell}
|\det(\phi(\ell,z))| \mathrm{d}\ell \ .$$
\[A13\] For $d > 1$ and $\ell \in [d-1,d+1]$ for all $w \in [0,2]$: $$\frac{f'(z,d)}{d+1} \ \ \leq \ \ t(d,z, \ell)
\ \ \leq \ \
\frac{f'(z,d)}{d+1} \left(1+ \frac{2}{d-1} \right)$$
Note that $0 < d-1 \leq \ell \leq d+1$. Hence, $$\frac{1}{d+1} \ \ \leq \ \ \frac{1}{\ell} \leq \frac{1}{d-1} = \frac1{d+1}
\left(1+\frac{2}{d-1}\right)$$
We choose $d \geq 15$ and get $\frac{2}{d-1} \leq \frac17$.
For $w \geq \lambda/2$, $d \geq 15$: $$\Im(s(d,\lambda,1)) \geq \frac{1}{112\sqrt2 (d+1)}\sqrt{\lambda} \ .$$
We consider two consecutive intervals of distance $\lambda$: $$\begin{aligned}
u_{d,\lambda}(w) &=&\sum_{j=0}^{\lceil 2/\lambda \rceil}
\int\displaylimits_{z = 0}^{\lambda}
\int\displaylimits_{\ell=d-1+z+j \lambda}^{d+(z+j\lambda)/2}
2\frac{e^{2\pi i z/\lambda}}{\ell}
|\det(\phi(\ell,z))| \ \mathrm{d}\ell \ \mathrm{d}z\
\\
&=& \sum_{j=0}^{\lceil 2/\lambda \rceil}
\int\displaylimits_{z = 0}^{\lambda/2}\frac{e^{2\pi i z/\lambda}}{\ell}
\left( t(d,z+j \lambda,\ell) -t(d,z+j\lambda+\lambda/2) \right)
\ \mathrm{d}z
\end{aligned}$$ Now for $z\le \lambda/2$, $d\geq 15$: $$\begin{aligned}
t(d,z+\lambda/2, \ell(x,y)) &\leq& \frac87\frac{1}{d+1}
f'(z+\lambda/2,d) \label{ineq89}\\
&\leq& \frac87 \cdot\frac57 \cdot \frac1{d+1} f'(z/2 + \lambda/4,d)\nonumber \\
&\leq& \frac87 \cdot\frac57 \cdot \frac1{d+1} f'(z,d) \label{eq5787}\\
&\leq&
\frac{40}{49} t(d,z, \ell(x,y)) \nonumber\end{aligned}$$ The first line (\[ineq89\]) follows from Lemma \[A13\]. Line (\[eq5787\]) follows from $z/2 + \lambda/4 \geq z$ and that $f'$ is monotone decreasing because $f''(x,y)< -\frac18$ by Lemma \[l:fsss\].
Hence $t(d,z, \ell(x,y)) - t(d,z+\lambda/2, \ell(x,y)) \geq \frac{9}{40} t(d,z, \ell(x,y)$ which we use in line \[l940\]: $$\begin{aligned}
\Im(s(d,\lambda,1)) &=& \Im(u_{d,\lambda}(2)) \\
&=& \sum_{j=0}^{\lceil 2/\lambda \rceil}
\int\displaylimits_{z = 0}^{\lambda}
\frac{\sin(2\pi i\ z/ \lambda)}{\ell} t(d,z+j\lambda, \ell(x,y))
\ \mathrm{d}\ell\ \ \mathrm{d}z \nonumber\\
& \geq &
\sum_{j=0}^{\lceil 2/\lambda \rceil}
\int\displaylimits_{z = 0}^{\lambda/2}
\frac{9}{40} \frac{\sin(2\pi i\ z/ \lambda)}{\ell}
t(r,d,z+j\lambda, \ell(x,y))
\ \mathrm{d}\ell\ \mathrm{d}z \label{l940}\\
& \geq &
\sum_{j=0}^{\lceil 2/\lambda \rceil}
\int\displaylimits_{z = 0}^{\lambda/2}
\frac{9}{40} \frac{\sin(2\pi i\ z/ \lambda)}{d+1} f'(z+j \lambda,d) \ \mathrm{d}z \nonumber\\
& \geq & \frac{9}{40} \frac{1}{d+1}
\sum_{j=0}^{\lceil 2/\lambda \rceil}
\int\displaylimits_{z = 0}^{\lambda}
\sin(2\pi i\ z/ \lambda) f'(z+j \lambda,d) \ \mathrm{d}z \nonumber\\
& = & \frac{9}{40}\frac{1}{d+1} \Im(h_{d,\lambda}(2))\label{hdef}\\
& \geq &
\frac{1}{56\sqrt2}\frac{9}{40}\frac{1}{d+1} \sqrt{\lambda} \nonumber\\
& \geq &
\frac{9}{2,240\sqrt2}\frac{ \sqrt{\lambda}}{d+1} \nonumber
\end{aligned}$$
In line (\[hdef\]) we use the definition of $h_{d,\lambda}(2)$ and then apply Lemma \[lA10\].
For $d>15r$: $${\mathbb{E}\left[\Im[\hbox{\sl rx}]\right]} \geq
\frac{9}{4,480\pi\sqrt2} \frac{m\sqrt{\lambda}}{d\sqrt{r}}
\ .$$
$
{\mathbb{E}\left[\Im[\hbox{\sl rx}]\right]} \geq
\frac{m}{2\pi r}
\frac{9}{2,240\sqrt2} \frac{\sqrt{\lambda/r}}{d/r+1} \ \ \geq \ \
\frac{9}{4,480\pi\sqrt2} \frac{m\sqrt{\lambda}}{d\sqrt{r}}
$.
We apply the Hoeffding bound (Theorem 2 of [@hoeffding1963probability]), which states for $n$ independently distributed random variables $X_j\in {\ensuremath{\mathbb{R}}}$ strictly bounded by the intervals $[a_j, b_j]$: $${\text{Prob}\left[\left|\overline{X} - {\mathbb{E}\left[\overline{X}\right]}\right| \geq t\right]} \leq
2 \exp\left(- \frac{2 n^2 t^2}{\sum_{j=1}^n (b_j-a_j)^2} \right),\
\hbox{where $\overline{X} = \frac1n \sum_{j=1}^n X_j$.}$$ We will use $X_j = \Im\left[
\frac{ e^{2 \pi
i \Delta_d(v_j)/{\lambda}} }{|\!|q-v_j|\!|_2}\right]
\in [-\frac{1}{d-r},\frac{1}{d-r}] =: [a_j,b_j]$ denoting the signal produced by each of the $m$ non central senders. Furthermore, we set $t = \frac12 {\mathbb{E}\left[\overline{X}\right]}$. Hence, $a_j-b_j=\frac{2}{d-r} \geq \frac{2}{d-\frac1{15} d}\geq \frac{30}{14 d} $.
Note that ${\mathbb{E}\left[\overline{X}\right]} \geq c_{6} \sqrt{\frac{\lambda}{r}} \frac{1}d \geq c_{6}\sqrt{\frac{\lambda}{r}} \frac{14}{30} (a-b) $, where $c_{6}=\frac{9}{4,480\pi\sqrt2}$. Therefore $$\frac{t^2}{(b-a)^2} = \frac{\frac14{\mathbb{E}\left[\overline{X}\right]}^2 }{(b-a)^2} \geq
c_{6}^2 \left(\frac{14}{30}\right)^2 \frac{\lambda}{r}$$ This implies the following Lemma.
For $d\geq 15r$ and $c_4=\frac12 c_{6}$ and $c_{7} = 2c_{6}^2 \left(\frac{14}{30}\right)^2$ $${\text{Prob}\left[\Im[\hbox{\sl rx}] \leq c_4\frac{m\sqrt{\lambda}}{d\sqrt{r}}\right]} \leq
2 \exp\left(- c_{7} \frac{\lambda m}{r} \right)\ .$$
This implies the following Lemma.
\[lakjfd\] For $r \geq r_1$ and $d>15 r$: $${\text{Prob}\left[|rx|^2 \leq \left(c_4 \frac{m\sqrt{\lambda}}{d\sqrt{r}}\right)^2\right]} \leq \frac{1}{n^{c_5}}\ . \label{rxright}$$
Note that $\Im(z) \geq a$ implies $|z|^2 = \Im(z)^2 + \Re(z)^2 \geq a^2$. It remains to show that $ \exp\left(- c_{7} \frac{\lambda m}{r}\right) \leq \frac{1}{n^{c_5}}$. Using that $m \geq \frac12 \rho \pi r^2$ from Lemma \[densifity\] (\[dlower\]) with probability $1-e^{-\frac18 \rho \pi r^2}$. So, the error probability is bounded for $n\geq 2 $ as $\frac{1}{n^{c_{5}}}$, where $c_5 =\frac1{2 \ln 2} c_{2} c_{3} c_{7} \pi + 1$. Using $r\geq r_1 = c_{2}/\lambda$, $\rho \geq \frac{c_{3}}{\ln 2}\ln n$ we bound the error probability of Lemma \[lakjfd\] as follows. $$\begin{aligned}
2 e^{- c_{7} \frac{\lambda m}{r} }
& \leq & e^{ - \frac12 c_{7} \lambda \rho \pi r + \ln 2}\\
& \leq & e^{ - \frac12 c_{2} c_{7} \rho \pi + \ln 2}\\
& \leq & e^{ - \frac1{2 \ln 2} c_{2} c_{3} c_{7} \pi \ln n + \ln 2}\\
& \leq & n^{ - \frac1{2 \ln 2} c_{2} c_{3} c_{7} \pi + 1}\\
& \leq & n^{ - c_5}\\
$$
The right side inside the probability of (\[rxright\]) is larger than the SNR threshold $\beta=1$ if $d \leq c_4 m r^{1/2}\lambda^{1/2} $. This implies that all nodes in distance $d \leq \frac14 c_4 \rho r_{j}^{3/2}\lambda^{1/2} =c_1 \rho r_{j}^{3/2}\lambda^{1/2} $ for $c_{1}=\frac14c_{4}$ can be informed in round $j$ with high probability. This completes the proof of Theorem \[safety\].
For constant wavelength $\lambda$ MISO broadcasting takes $ \mathcal{O}(\log \log n - \log \log \rho)$ rounds to broadcast the signal.
The algorithm works in two phases. In the first phase, we inform all nodes in radius $r_1=\mathcal{O}(1)$ using the UDG Broadcast algorithm with single senders. This takes at most $\mathcal{O}(1)$ rounds. In the second phase we use the phase shift $\varphi_\ell = |\!|v_\ell-v_0|\!|_2$ for all senders $v_\ell$. Now the radii increase double exponentially with $r_{j+1} = c_1 \rho r_j^{3/2} \lambda^{\frac12}$. Note that $r_{j+1} \geq 15 r_j$ if $ r_j \geq \frac{225}{c_1^2 \rho^2\lambda} \geq \frac{225}{c_1^2 c_3^2 \lambda\log^2 n}\geq \frac{c_{2}}{\lambda} = r_1$ which holds for large enough number of nodes $n$. After $j= \mathcal{O}(\log \log n - \log \log \rho)$ rounds we have reached $$\begin{aligned}
r_j &=& r_1^{(\frac32)^{j}} \left(c_1\rho\lambda^{\frac12}\right)^{1+\frac32 +\ldots + (\frac32)^{j-1}} \\
&=& r_t^{(\frac32)^{j}} \left(c_1\rho\lambda^{\frac12}\right)^{2(\frac32)^{j}-2}\\
&\geq & R = \sqrt{\frac{n}{\pi \rho}}\, .
\end{aligned}$$ Note that in the first round $15 r_1$ nodes are already informed. So, all nodes in distance $r_2$ can be informed and the minimum distance of $15 r_1$ from the senders is kept. In every next round we decrease $r_j$ by a factor of $1/15$ to ensure that this minimum distance of $15 r_j$. The above recursion changes only to the extent that $c_1$ is replaced by $c'=c_1/15$ without changing the asymptotic behavior.
Because of the super-exponential growth of $r_i$ the same observation as in Corollary \[c:snrlight\] can be made for the MISO broadcast.
\[c:misolight\] The speed of the MIMO Broadcast approaches the speed of light for growing $n$.
Conclusions
===========
We have compared the number of rounds of collaborative broadcasting in three communication models. All of them are derived from the far-field superposition MISO/MIMO model where the signal-to-noise ratio allows a communication range of one unit.
The first is the Unit-Disk-Graph model. Typically, parallel communication is seen here as a problem, resulting in the Radio Broadcasting model. For the Unit Disk Graph such interference results only in an extra overhead of a constant factor [@gandhi2008minimizing]. The delimiting factor is the diameter of the graph, proportional to $\sqrt{n/\rho}$. For the SNR-model one can achieve broadcasting in a logarithmic number of rounds which comes from the addition of the senders’ signal energy. This allows to extend the disk of informed nodes by a factor of $\Theta(\sqrt{\rho})$, where $\rho$ is the sender density. This result is not surprising, since the area grows quadratically in the diameter and the path loss is to the power of two as well, which fits well to the law of conservation of energy.
For the MIMO model it is already known that beamforming increases the energy beyond the SNR model. It is possible to achieve logarithmic number of rounds for unicast on the line [@JS13_Beamforming_Line] and $\mathcal{O}(\log \log n)$ for the plane [@JS14_Beamforming_LogLog_TR]. The used beams are very narrow and for a growing number of sender nodes the ratio between beam range and angle decreases. So, the extended range results from the focus of the energy on a smaller beam, while the signal energy is reduced elsewhere.
This leads to the question, how it is possible that a broadcast with high signal strength can take place. The answer is, that this panoramic beam draws its increased energy from a signal reduction from the three-dimensional space into the two-dimensional space, where the receivers happen to be located.
Outlook
=======
We have focused on broadcasting only a single sinusoidal signal and not a message consisting of many different signals. There, one faces inter-signal interference and inter-symbol interference. For inter-signal interference, note that the received signal has a constant phase shift. We bounded the interference of non-synchronized signals only by a constant factor with respect to the main signal. So, tighter bounds are necessary. For the inter-symbol interference, a special encoding may reduce the interference caused by signals used for other parts of the message.
In [@JS14_Beamforming_LogLog_SSS] we have described a $\mathcal{O}(\log \log n)$ unicast algorithm without the need of perfect synchronization. There is some hope that a similar technique works here, too. Furthermore, the question remains open whether flooding in MISO is as efficient as the broadcasting algorithm relying on the Expanding Disk Algorithm.
A drawback of the MISO broadcasting is that we assume the wavelength is constant, while the density grows logarithmic. So, the average distance of two nodes is smaller than the wavelength. Since, we use far-distant sending ranges this does not affect the validity of the claims. But a broadcasting time bound for constant density and very small wavelength seems desirable. For a constant density, we have to overcome the problem that the first sender might have no neighbors in the unit disk range. For this, on can focus only on situations where the broadcasting is possible in the UDG model. For small wavelength the solution seems to be more difficult. One approach might be to use the SNR broadcast for small ranges and then switch to MIMO broadcast. However, the SNR model equals the MISO/MIMO model for random phases only in the expectation and not with high probability. So, one has to analyze a SNR broadcast algorithm in the MISO/MIMO model where not all nodes might be triggered (a realistic problem known as Rayleigh fading). If this approach works out, then an additional number of $\mathcal{O}(\log (1/\lambda))$ steps are needed in MIMO broadcast, which is asymptotically better than the SNR broadcast if $1/\lambda$ is smaller than any polynomial.
Another interesting question concerns the influence of the path loss exponent $\alpha$, which we choose as $\alpha=2$. It has no influence to the UDG model. As an anonymous reviewer pointed out in the SNR model one expects for $\alpha<2$ a bound of $\mathcal{O}(\log \log n)$ for broadcasting, for $\alpha=2$ we have proved a bound of $\Theta(\log n)$ and for $\alpha>2$ a bound of $\mathcal{O}(n^{1/2})$ like in the UDG model is to be expected.
We conjecture for MIMO that our results can be generalized for $\alpha < 3$ because of area size of around $\Theta(r^{3/2} \lambda^{1/2})$ of nearly synchronous senders. For larger path loss the asymptotic number of rounds increase. For $\alpha=3$ we expect a logarithmic bound and for $\alpha>3$ the same behavior as in the Unit Disk Graph. Finally, the communication model is still very simple. Instead of a constant signal-to-noise ratio one might consider Gaussian noise. It is also unclear how obstacles influence the algorithm, or for which other path loss exponents, the double logarithmic number of rounds can be guaranteed.
Acknowledgments
===============
We like to thank the organizers of the Dagstuhl Seminar 17271, July 2 - 7, 2017, Foundations of Wireless Networking, where this research has begun and first results have been found. We would like to thank Alexander Leibold, who performed and checked the automated proofs and anonymous reviewers of a previous version for their detailed and valuable input. We would also like to thank Tigrun Tonoyan, Magnus Halldorrson and Zvi Lotker for many fruitful discussions.
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---
abstract: 'Co-evolution exhibited by a network system, involving the intricate interplay between the dynamics of the network itself and the subsystems connected by it, is a key concept for understanding the self-organized, flexible nature of real-world network systems. We propose a simple model of such co-evolving network dynamics, in which the diffusion of a resource over a weighted network and the resource-driven evolution of the link weights occur simultaneously. We demonstrate that, under feasible conditions, the network robustly acquires scale-free characteristics in the asymptotic state. Interestingly, in the case that the system includes dissipation, it asymptotically realizes a dynamical phase characterized by an organized scale-free network, in which the ranking of each node with respect to the quantity of the resource possessed thereby changes ceaselessly. Our model offers a unified framework for understanding some real-world diffusion-driven network systems of diverse types.'
author:
- Takaaki Aoki
- Toshio Aoyagi
date: '2011-07-06'
title: 'Scale-Free Structure Emerging from Co-Evolution of a Network and the Distribution of a Diffusive Resource on It'
---
Today, the term “network” is common in our everyday lives, in which it often refers to large-scale, complexly structured, conglomerations of interactions in real-world systems. Typically, these networks are not static, but change continuously in response to the activity of the subsystems that they connect. For example, traffic networks among cities supporting the transportation of people and products are frequently reformed to meet the current needs as cities develop or decay, and conversely, this reformation of the networks influences the growth or decay of the cities. A similar process takes place in the case of communication networks [@Kalapala:2006ie; @*Barrat:2004bk]. In social networks, human behavior is strongly influenced by social relationships, and at the same time, the relationships among people change continually as a result of their behavior. This intricate interplay between individuals and their relationships creates the complex structures of human societies [@Palla:2007if; @*Kossinets:2006je]. The essence of such real-world systems resides in the co-evolving dynamics of the individual subsystems and the networks of interactions through which they are connected. To understand the mechanisms governing such dynamical network organization, we have to consider the interplay between the dynamics both on and of a network.
In the last decade, there have been two major trends in the investigation of the type of co-evolving dynamics described above. One trend is to focus on the topology of the network. It is well known that real-world networks possess some common topological features [@Watts:1998db; @Barabasi:1999p178; @Newman:2003p839]. One such feature is a scale-free structure, in which the “node degree” (the total number of links connected to a node) exhibits a power-law distribution [@PRICE:1965vs; @Barabasi:1999p178]. In the attempt to explain these features, many models describing the evolution of the topology have been proposed. The other trend is to focus on the collective behavior of the dynamical subsystems interacting on complex but static networks [@Boccaletti:2006p403]. The aim of such studies is to investigate how various observed topological features of the network influence the nature of the dynamical systems coupled through the links of the network.
Most studies employing the approaches described above focus on only one of the two aspects of co-evolving dynamics, evolution of the network topology or evolution of the dynamical states of the nodes on a static network. However, co-evolving dynamics, in which the network topology and the nodal states evolve simultaneously and interdependently, is an interdisciplinary subject of growing interest [@Gross:2008p274; @*Bohme:2011gw; @*Vazquez:2008dh; @*Nardini:2008jn; @*Holme:2006wi; @*Aoki:2009p4; @*Perc:2010p822; @*PhysRevE.84.036101]. To facilitate further systematic studies of such systems, a general mathematical framework for modeling co-evolving real-world networks is needed. As a first step toward this goal, in this Letter, we propose a simple model of co-evolving weighted networks. This model is schematically depicted in Fig.1(a). The basic concept of our model is as follows. We assign a dynamical variable, $x_i$, to each node. The dynamics of these variables are governed by a reaction-diffusion equation in which nodes are coupled through the weighted links of the network. This dynamical variable at each node can be regarded as the quantity of the “resource” at that node. Additionally, we assume a physically reasonable resource-dependent dynamics for the link weights. We systematically investigate the collective behavior that emerges asymptotically through the interplay between these dynamics both on and of the network.
![(color online). (a) A schematic illustration of co-evolving dynamics of a network and the distribution of a diffusive resource on it. (b, c, d) Typical features of the network that emerge through the co-evolving dynamics. The cumulative distributions of the quantity of the resource, $x_{i}$, the weights, $w_{ij}$, and the strengths, $s_{i}$, converge to power-law forms. The topology of the network, represented by the adjacency matrix $a_{ij}$, is chosen as an Erdös-Rényi random graph with $N$ = 16384 and $\langle k \rangle$ = 10. The initial distribution of the resource and the initial weights were generated according to a normal distribution with mean $\mu$ = 1 and standard deviation $\sigma$ = 0.1. Other parameter values are as follows: $\kappa$ = 0.05, $D$ = 0.34, $\epsilon$ = 0.01. []{data-label="fig:CoEv"}](fig1.eps)
We now describe the model. First, let us consider a weighted network with $N$ nodes. The link structure of the network is defined by the adjacency matrix $a_{ij}$, in which $a_{ij}$=1 if a link exists between the nodes $i$ and $j$ and $a_{ij}$=0 otherwise. We assign a time-dependent symmetric weight $w_{ij}(t)$ ($=w_{ji}(t))$ to each existing link. This weight represents the strength of the interaction. Here we study a system in which it is these strengths of the interactions that change in time. More precisely, we consider a system in which there exists a fixed set of connections among nodes, with each connection characterized by a weight, $w_{ij}(t)$.
We consider the reaction-diffusion dynamics of a single quantity on this network. We refer to this quantity as the “resource", which may be, for example, molecules, cells, people or money. The value of this quantity at the $i$th node at time $t$ is represented by $x_i(t)$. In general, the evolution of $x_i(t)$ is assumed to be described by an equation of the following form: $$\Delta x_i (t)= F(x_i(t) )+ \text{diffusion process via weighted links},$$ where $\Delta x_i(t) \equiv x_i(t+1) - x_i(t)$. Here $F(x)$ represents a reaction process undergone by the resource. The weights $w_{ij}(t)$ control the diffusion process as follows. This process can be understood as consisting of the combined motion of many random walkers, in which the walkers at the node $i$ move to the node $j$ in a single time step with the time-dependent probability $D w_{ji}(t) / s_i(t)$. Here, $s_i(t)$ is the strength of the node $i$, defined by $s_i (t)\equiv \sum_{j \in {\cal N}_{i}} w_{ji}(t) = \sum_j a_{ij}w_{ij}(t)$, where ${\cal N}_{i}$ is the set of nodes connected to the node $i$. The master equation for the resource is thus given by $$\begin{aligned}
\Delta x_{i}(t) = F(x_{i}(t) ) + D \sum_{j \in {\mathcal N}_{i} } \left( \frac{w_{ij}(t)}{s_{j}(t)} x_j(t) - \frac{w_{ji}(t)}{s_{i}(t)} x_i(t) \right),\end{aligned}$$ where the second and third terms are the inward and outward currents of the resource at the $i$-th node, respectively. For the reaction process, we employ simple dissipation with equilibrium state $x=1$, described by $$F(x) = -\kappa(x -1).$$
Next, we describe the evolution of the structure of the network. It is reasonable to assume that the evolution of a weight $w_{ij}(t)$ depends on the quantities of the resource at the corresponding nodes, $x_i(t)$ and $x_j(t)$. As a first step, we assume a linear dependence on each, with the weight $w_{ij}(t)$ merely relaxing to $x_i(t) x_j(t)$, appealing to the law of mass action. Then, employing the simplest form, we stipulate the dynamics of the weights to be described by $$w_{ij}(t+1) - w_{ij}(t) = \epsilon \left[ x_i(t) x_j(t) - w_{ij}(t) \right], \eqno(2)$$ where the parameter $\epsilon^{-1}$ represents the relaxation time scale of the weight dynamics. It should be noted that in these dynamics it is possible for a link to be effectively eliminated, because weights can become vanishingly small.
The co-evolving dynamics of the entire system are described by the simple equations (1) and (2). Despite their simplicity, however, we have found that the interplay between the two types of dynamics that they describe can yield power-law distributions of the resource and the weights, even when the underlying topology of the network is not scale-free. Figures \[fig:CoEv\](b)-(d) display a typical result of the numerical simulations, where an Erdös-Rényi (ER) random graph was used for $a_{ij}$. As displayed in Fig. \[fig:CoEv\](b), the cumulative distribution of the resource takes a power-law form with exponent $\gamma \sim -1$ in the asymptotic state. This result is consistent with an empirical law found to characterize many physical and social phenomena, including word frequencies in natural languages, populations of cities, statistics of Web access, and company sizes [@Miller:1958p1240; @*Gabaix:1999p1317; @*Huberman:1998p1328; @*Aoyama:2000p1361; @*Dragulescu:2001p1511; @*PhysRevLett.108.168701], namely, Zipf’s law [@zipforig]. As seen in Figs. \[fig:CoEv\](c) and (d), the weights $w_{ij}$ and strengths $s_{i}$ also exhibit power-law distributions in the asymptotic state with different exponents. For details of the network dynamics, see Figure S1 and the movie included in the Supplemental Material.
![(color). (a) Dependence of the exponent of the cumulative distribution of the resource on the decay constant, $\kappa$, and the diffusion constant, $D$. The other parameter values are the same as in Fig. \[fig:CoEv\]. The points labeled “(b)” and “(c)” indicate the parameter values used in the corresponding cases. (b) Time evolution of the quantities of the resource at the 20 most resource-rich nodes in the asymptotic state for a system with dissipation ($\kappa$ = 0.05). In this case with nonzero $\kappa$, the distribution of the resource over the network realizes a steady state asymptotically, but, as seen here, the quantity of the resource at each node and the rankings of the nodes continue to change indefinitely. (c) Same as in (b), but with no dissipation ($\kappa$ = 0). In this case, the quantities of the resource at the top 20 nodes remain fixed.[]{data-label="fig:Stationary"}](fig3.eps)
In the following, we focus on the extreme case $\kappa$ = 0, which can be readily treated analytically. In this case, the resource merely diffuses over the network without dissipation, and the total quantity of the resource is conserved. Furthermore, to obtain insight into the interplay between the diffusive resource and the weight dynamics, here we investigate the situation in which the resource dependence of the weight dynamics takes the generalized form $$w_{ij}(t+1) - w_{ij}(t) = \epsilon \left[ x_i(t)^\alpha x_j(t)^\alpha - w_{ij}(t) \right]. \eqno(3)$$ We determine the types of organized network structure realized as a function of the parameter $\alpha$, which controls the non-linearity of the resource dependence. Figure \[fig:DiffusionOnly\] summarizes the numerical results. As seen there, two distinct types of network structure appear through the dynamics, as determined by the value of $\alpha$. In the case $\alpha \ge 1$, there emerges a power-law resource distribution in which a few very resource-rich nodes (hubs) and many resource-poor nodes coexist. Contrastingly, in the case $\alpha <1$, the resource is distributed almost evenly among all nodes, and no prominent hubs appear.
We now analyze the asymptotic behavior of the system, making some simplifying approximations that allow us to extract meaningful results. Let us first consider separately the dynamics of the resource distribution and set of link weights in the artificial cases that the other is held fixed. First, if the link weights were static, the quantities $x_{i}$ would converge to the equilibrium solution $x_{i}^{*} = s_{i} / (\sum_{k} s_{k}/N)$. Second, if the resource distribution were held fixed, the weights would relax to the solution $w_{ij}^{*} = (x_i x_j)^{\alpha}$. With these considerations, we conjecture that the asymptotic dynamics can be approximated by updating the variables $x_{i}$ and $w_{ij}$ in an alternating manner using two maps, $x_{i}(n+1)= s_{i}(n) / (\sum_{k} s_{k}(n)/N)$ and then $w_{ij}(n+2)=(x_i(n+1) x_j(n+1))^{\alpha}$, where $n$ denotes the number of the iteration. Although this approximation might be too crude, the results it produces exhibit reasonable agreement with the numerical simulations of the original system, as shown below. Next, using the first of the above maps, we can eliminate $x_i(n)$ in the second. Then, using the update rule for the strength $s_{i}$, we obtain $$s_{i}(n+1) = c_{n} s_{i}(n)^{\alpha} \sum_{j=1}^{N} a_{ij} s_{j}(n)^{\alpha}, \eqno (4)$$ where $c_{n} = 1/ (\sum_{k} s_{k}(n)/N)^{2\alpha}$. In addition, we consider the case in which the network topology $a_{ij}$ is chosen as a regular random graph, which means that each $a_{ij}$ is selected randomly, but $\sum_{j} a_{ij}$ is a predetermined constant, $k_{0}$. Then, using a mean-field approximation, we finally obtain the following relation for the strength after $n$ iterations with the initial value $s_{i}(0)$: $$s_{i}(n)\propto s_{i}(0)^{\alpha^{n}}. \eqno (5)$$ The asymptotic behavior resulting from the above maps can be classified into three cases, corresponding to three types for the value of $\alpha$: $\alpha>1$, $\alpha<1$ and $\alpha=1$. We now consider these individually.
In the case $\alpha >1$, for a given initial strength distribution $P^{0}(s)$, we have $
P^{n}(s) \propto P^{0} (s^{\alpha^{-n}} ) s^{-1 +\alpha^{-n}}.
$ In the asymptotic limit $n \to\infty$, the distribution $P^{n}(s)$ behaves as $
P^{n}(s) \to s^{-1}.
$ In Fig. \[fig:DiffusionOnly\](b), it is seen that this power-law strength distribution appears not only in the case of a regular random graph but also in the case of an ER random graph. This suggests that the above analytic result holds for a more general network topology. As shown in the top graph of Fig. \[fig:DiffusionOnly\](b), an increase in the link weight between two nodes tends to become larger when the quantities of the resource at these nodes increase. This leads to the emergence of hub nodes. This scaling-up effect represents a kind of “rich get richer” or “economies of scale” behavior.
In the opposite case, $\alpha < 1$, $s_{i}(n)$ in Eq. (5) converges to a uniform constant in the limit $n \to \infty$; i.e., Eq. (5) becomes $s_{i}(n) \propto s(0)^{0}$. The validity of this theoretical prediction has been confirmed numerically for regular random graphs, as shown in Fig. \[fig:DiffusionOnly\](b). For ER random graphs, owing to the variability of the degree, the strengths do not converge to identical values but, rather, to some distribution with finite variance. Note that the strength distributions for both types of random graphs in the case $\alpha <1$ possess a characteristic scale, which implies a finite mean strength.
In the critical case, $\alpha$ = 1, the situation is very delicate. Theoretically, according to Eq (5), the strength after $n$ iterations should be given by $s(n)=s(0)$, and thus the strength distribution should not change. However, this prediction is inconsistent with the numerical results, as we have already seen that a power-law strength distribution is realized for both regular random graphs and ER random graphs (see Fig. \[fig:DiffusionOnly\]). For complete graphs, however, the situation is different. In this case, the strength distribution remains unchanged, as predicted by the analytic treatment (data not shown). These results lead us to conclude that a more accurate approximation of the asymptotic dynamics is required [^1].
In the case with no dissipation ($\kappa = 0$), the exponent of the power-law distribution is always $-1$, provided that $\alpha \ge 1$. This type of strength distribution has been reported for the global cargo shipping network [@Kaluza19012010]. In the general case with dissipation ($\kappa \neq 0$), the exponent of the resource power-law distribution generally depends on the parameter values, such as the decay constant, $\kappa$, and the diffusion constant, $D$. Figure \[fig:Stationary\](a) plots the exponent of the cumulative distribution of the resource in ($\kappa$,$D$) space. These results were obtained by fitting the numerically generated distributions to the form $x^{\gamma}$ with a least-squares fit. As seen there, the exponent decreases to about -1.5 with increasing $\kappa$ and decreasing $D$, and eventually the resource distribution ceases to be of a power-law type. We thus see that the resource disparity among the nodes is an increasing function of $D$ and a decreasing function of $\kappa$.
Interestingly, even in the case that the system has realized its asymptotic, stationary power-law distribution, the resource ranking among the nodes continues to change through the co-evolving dynamics. In Fig. \[fig:Stationary\](b), it is seen how the quantities of the resource possessed by the top 20 nodes (according to the ranking at $t$ = 36000) gradually change in time from $t$ = 36000 to $t$ = 40000, with the rankings occasionally being exchanged. The link weights similarly continue to evolve over the entire network, even in the asymptotic regime. This kind of dynamical phase of an organized scale-free network is observed generally in the case $\kappa \neq 0$. Note that this dynamical system described by Eqs. (1) and (2) includes no random noise. Contrastingly, in the case $\kappa$ = 0, the resource ranking of the nodes becomes fixed asymptotically (see Fig. \[fig:Stationary\](c)). Therefore, in addition to elucidating the statistical characteristics of the network emerging asymptotically, our model is able to describe the microscopic dynamics exhibited by a dynamical state in which both the link weights and the resource distribution continue to change indefinitely (see Fig. S2 in the Supplemental Material). This suggests that our model may be applicable to the investigation of the vicissitudes of social phenomena, including the dynamics of business activity, webpage access rankings and city populations [@Gautreau02062009].
In conclusion, we have proposed a simple model of a co-evolving weighted network exhibiting a dissipative diffusion process of a resource over a weighted network and resource-dependent evolution of the network link weights. We have demonstrated numerically and analytically that both the resource and weight distributions exhibit power-law forms in the asymptotic state as a result of the interplay between these two types of dynamics. We believe that the most important finding of this paper is the existence of a dynamical phase of the organized scale-free network for a system with dissipation. From this, we conclude that our model can treat the dynamical formation of microscopic structure in a weighted network. We believe that our model provides a useful basic framework for the modeling of co-evolving weighted networks and that through the application of various generalizations, it should be useful for investigating real-world systems of many kinds. For example, generalizing the model to include reaction dynamics with multiple resources, it would be readily applicable to a wide range of actual physical and social systems, including systems driven by chemical reaction-diffusion dynamics, predator-prey dynamics and population dynamics.
We thank N. Masuda for fruitful discussions. This work was supported by KAKENHI (24120708, 24740266, 21120002, 23115511).
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12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} [****, ()](\doibase 10.1073/pnas.0811113106)
[^1]: The direct numerical simulation of Eq (4), without the mean-field approximation, yields results consistent with those obtained from the original equations, (1) and (3).
|
---
abstract: |
The autocorrelation and the linear complexity of a key stream sequence in a stream cipher are important cryptographic properties. Many sequences with these good properties have interleaved structure, three classes of binary sequences of period $4N$ with optimal autocorrelation values have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period $N$. In this paper, we use the interleaving technique to construct a binary sequence with the optimal autocorrelation of period $2N$, then we calculate its autocorrelation values and its distribution, and give a lower bound of linear complexity. Results show that these sequences have low autocorrelation and the linear complexity satisfies the requirements of cryptography. \
[*Keywords: Linear complexity, minimal polynomial, interleaved sequences, the autocorrelation value*]{}
author:
- |
Shidong Zhang$^{1,2}$, Tongjiang Yan$^{1,2}$\
\
[College of Science, China University of Petroleum, Qingdao, Shandong, China, 266580$^1$]{}\
[Key Laboratory of Network Security and Cryptology, Fujian Normal University, Fuzhou, Fujian, China, 350117 $^2$]{}\
[(Email: yantoji@163.com)]{}\
title: '**Linear Complexity and Autocorrelation of two Classes of New Interleaved Sequences of Period $2N$** '
---
Introduction
============
A sequence $\textbf{u}=\{u_i\}_{i=0}^{\infty}$, if $\textbf{u}$ satisfies $u_{i+N}=u_i$, where $u_i\in\{0,1\}$, is called a binary sequence of period $\emph{N}$. The set $U=\{0\leq i <N:u_i=1\}$ is called the*characteristic set* of $\textbf{u}$. If $|\emph{U}|=N/2$ for even N or $|\emph{U}|=(N\pm 1)/2$ for odd $\emph{N}$, where $|\emph{U}|$ denotes the cardinality of $\emph{U}$, then such a sequence **u** is called a balanced sequence. Let $\textbf{u}=\{u_i\}_{i=0}^{\infty}$ and $\textbf{v}=\{v_i\}_{i=0}^{\infty}$ be two sequences of period $N$. The periodic correlation between them is defined by $$\label{eq01}
R_{\textbf{u}, \textbf{v}}(\tau) = \sum_{i = 0}^{N-1}(-1)^{\textbf{u}(i)+\textbf{v}(i+\tau)}, 0\leq\tau<N,$$ where the addition $t+\tau$ is performed modulo $N$. $R_{\textbf{u}, \textbf{v}}(\tau)$ is called the (periodic) cross-correlation function of $\textbf{u}$ and $\textbf{v}$. If $\textbf{u} = \textbf{v}$, $R_{\textbf{u}, \textbf{v}}(\tau)$ is called the (period) autocorrelation function of $\textbf{u}$, denoted by $R_\textbf{u}(\tau)$ for short [@Ref3].
According to the remainder of $N$ modulo $4$, the optimal values of out-of-phase autocorrelations of binary sequences are classified into four types as follows: (1) $R_\textbf{a}(\tau)=-1$ if $N\equiv3 \bmod 4$; (2) $R_\textbf{a}(\tau)\in\{-2, 2\}$ if $N\equiv2 \bmod 4$; (3) $R_\textbf{a}(\tau)\in\{1, -3\}$ if $N\equiv1 \bmod 4$; (4) $R_\textbf{a}(\tau)\in\{0, -4,4\}$ if $N\equiv0 \bmod 4$, where $0<\tau<N$. For the second case, it is exactly the autocorrelation of the sequence we constructed. Binary sequences with low correlation have very significant applications in communication systems, radar and cryptography [@Ref2; @Ref4]. Sequences should have low autocorrelation to eliminate the effect of multipath, and low cross-correlation to extract the desired user’s signal from the rest of users.
In cryptographic applications, the linear complexity of a sequence is considered as the most important property. Generally speaking, a sequence with large linear complexity is favorable for cryptography to resist the well-known Berlekamp-Massey algorithm [@Ref6; @Ref7], and the sequence can be recovered easily if the linear complexity is less than half the period [@Ref8]. It is worth to know the linear complexity of a sequence before applying them in any applications.
Many sequences with these good properties have interleaved structure, interleaved technique is widely used to analyse and design sequences [@Ref6]. In $2008$, based on the interleaved structure, Yu and Gong [@Ref18] presented a family of binary sequences of period $4N$ with optimal autocorrelation magnitude, i.e., $R_{\emph{\textbf{s}}}(\tau)\in \{0,\pm4\}$ for all $0<\tau<n$, and they also determined the linear complexity of the proposed sequence. Later, Tang and Gong [@Ref15] generalized the sequences in [@Ref18] and obtained more binary sequences of period $4N$ with optiaml autocorrelation value.
In this paper, using the interleaved technique, We construct the binary sequence with the otimal autocorrelation of period $2N$, and we calculate its autocorrelation value and distribution, and give a lower bound of linear complexity. Results show that these sequences have low autocorrelation and the linear complexity satisfies the requirements of cryptography.
This paper is organized as follows. Section $2$ introduces some related definitions and lemmas which would be used later. In Section $3$, We first give the interleaved structure, then calculate the autocorrelation value and its distribution of the sequences. In Section $4$, we give a lower bound of linear complexity. In Section $5$, we give a Remark. Conclusions are given in Section $6$.
Preliminaries
=============
Interleaved Sequence
--------------------
[@Ref8] Let $\{a_0,a_1,\cdots,a_{T-1}\}$ be a set of $T$ sequences of period $N$. An $N \times T$ matrix $U$ is formed by placing the sequence $a_i$ on the $i$th column, where $0 \leq i \leq T-1$. Then one can obtain an interleaved sequence $\textbf{u}$ of period $NT$ by concatenating the successive rows of the matrix $U$. For simplicity, the interleaved sequence $\textbf{u}$ can be written as $$\textbf{u}=\mathbf{I}(a_0,a_1,\cdots,a_{T-1}),$$ where $\mathbf{I}$ denotes the interleaved operator.
Let $\textbf{\emph{s}} = (\textbf{\emph{s}}(i))_{i = 0}^\infty$ be a sequence over a field $\mathbb{F}_2$. A polynomial of the form $$f(x)=1+c_1x+c_2x^2+ \cdots +c_rx^r\in \mathbb{F}[x]$$ is called the characteristic polynomial of the sequence $\textbf{\emph{s}}$ if $$\textbf{\emph{s}}(i)=c_1\textbf{\emph{s}}(i-1)+c_2\textbf{\emph{s}}(i-2)+ \cdots +c_r\textbf{\emph{s}}(i-r), \forall i \geq r.$$
Among all the characteristic polynomials of $\textbf{\emph{s}}$, the monic polynomial $m_\textbf{\emph{s}}(x)$ with the lowest degree is called its minimal polynomial. The linear complexity of $\textbf{\emph{s}}$ is defined as the degree of $m_\textbf{\emph{s}}(x)$, which is described as LC($\textbf{\emph{s}}$).
Let $\textbf{\emph{s}} = (\textbf{\emph{s}}(0),\textbf{\emph{s}}(1), \cdots , \textbf{\emph{s}}(n-1))$ be a binary sequence of period $n$ and define the sequence polynomial $$\label{eq02}
S(x) = \textbf{\emph{s}}(0)+\textbf{\emph{s}}(1)x+ \cdots +\textbf{\emph{s}}(n-1)x^{n-1}.$$
Then, its minimal polynomial and linear complexity can be determined by Lemma $1$.
[@Ref15] Assume a sequence ***s*** of period $n$ with sequence polynomial S(x) is defined by Equation $\mathrm{(\ref{eq02})}$. Then
1. the minimal polynomial is $m_\textbf{\emph{s}}(x)= \frac{x^n-1}{\mathrm{gcd}(x^n-1,\textbf{\emph{s}}(x))};$
2. the linear complexity is $\mathrm{LC}(\textbf{\emph{s}})=n-\mathrm{deg}(\mathrm{gcd}(x^n-1,S(x))),$
where $\mathrm{gcd}(x^n-1,S(x))$ denotes the greatest common divisor of $x^n-1$ and $S(x)$.
[@Ref22] Let **a** be a binary sequence of period $n$, and $\textbf{\emph{s}}_\textbf{a}(x)$ be its sequence polynomial. Then
1. $S_\textbf{b}(x) = x^{n-\tau}S_\textbf{a}(x), \; \textrm{if} \; \textbf{b} = L^\tau(\textbf{a})$;
2. $S_\textbf{b}(x) = S_\textbf{a}(x)+\displaystyle\frac{x^n-1}{x-1}, \; \textrm{if} \; \textbf{b} \; is \; the \; complement \;\\ sequence \; of \; \textbf{a}$;
3. $S_\textbf{u}(x) = S_\textbf{a}(x^2)+xS_\textbf{b}(x^2), \; \textrm{if} \; \textbf{u} = \mathbf{I}(\textbf{a},\textbf{b})$.
[@Ref17] Let m be an integer. Correlation of sequences satisfies the following properties:
1. $R_{L^m(\textbf a)\textbf b}(\tau)=R_{\textbf {ab}}(\tau-m)$;
2. $R_{\textbf aL^m(\textbf b)(\tau)}=R_{\textbf {ab}}(\tau+m)$;
3. $R_{\textbf{ab}}(\tau)=R_{\textbf{ab}}(\tau+N)=R_{\textbf{ba}}(N-\tau)$;
4. $R_{\textbf{ab}}(\tau)+R_{\textbf{a} \bar{\textbf{b}}}(\tau)=R_{\textbf{ab}}(\tau)+R_{\bar{\textbf{a}}\textbf{b}}(\tau)=0$.
The autocorrelation of $\textbf{u}=\mathbf{I}(\textbf{a},\textbf{b})$ $$R_\textbf{u}(\tau)=
\begin{cases}
R_\textbf{a}(\tau/2)+R_\textbf{b}(\tau/2) & \mbox{if }\tau \mbox{ is even}, \\
R_{\textbf{ab}}(\frac{\tau-1}{2})+R_{\textbf{ba}}(\frac{\tau+1}{2}) & \mbox{if }\tau \mbox{ is odd}.
\end{cases}$$
**Proof** For the case $\tau$ is even, we can know the location of sequence $\textbf{a}$ is replaced by $L^{\frac{\tau}{2}}(\textbf{a})$, and the location of sequence $\textbf{b}$ is replaced by $L^{\frac{\tau}{2}}(\textbf{b})$, in other words, $\mathbf{I}(\textbf{a},\textbf{b})$ becomes $\mathbf{I}(L^{\frac{\tau}{2}}(\textbf{a}),L^{\frac{\tau}{2}}(\textbf{b}))$, so by the definition, we have $R_\textbf{u}(\tau)=R_\textbf{a}(\frac{\tau}{2})+R_\textbf{b}(\frac{\tau}{2})$.
For the case $\tau$ is odd, we can know the location of sequence $\textbf{a}$ is replaced by $L^{\frac{\tau-1}{2}}(\textbf{b})$, and the location of sequence $\textbf{b}$ is replaced by $L^{\frac{\tau+1}{2}}(\textbf{a})$, in other words, $\mathbf{I}(\textbf{a},\textbf{b})$ becomes $\mathbf{I}(L^{\frac{\tau-1}{2}}(\textbf{b}),L^{\frac{\tau+1}{2}}(\textbf{a}))$, so by the definition we have $R_\textbf{u}(\tau)=R_{\textbf{ab}}(\frac{\tau-1}{2})+R_{\textbf{ba}}(\frac{\tau+1}{2})$. Hence, we have completed the proof of Lemma $4$.
Let $N$ be a prime and $\beta$ a primitive element of the integer residue ring $Z_N=\{0,1,\cdots,N-1\}$, such that for any $j\in Z_N \backslash \{0\}$, there exists an integer $k$ satisfying $j=\beta^k$. Denote by $D_0$ a multiplicative subgroup generated by $\beta^4$, i.e., $D_0=\{\beta^{4k}:0\leq k<f\}$. Then, $Z_N^*=Z_N \backslash \{0\}$ can be decomposed as $Z_N^*=\cup_{j=0}^3{D_j}$, where $D_j=\{\beta^{4k+j}:0\leq k<f\}$ is called the cyclotomic class $j$ of order $4$.
[@Ref3] Let $\textbf{u}$ and $\textbf{v}$ be two binary sequences with characteristic sets $D_0\cup D_1$ and $D_1\cup D_2$ respectively, and $\textbf{u}(0)=0$ and $\textbf{v}(0)=0$. Then we have $$\begin{aligned}
R_\textbf{u}(\tau)=\left\{ \begin{array}{lll}
N& \tau=0,\\
-1-2y& \tau \in D_0\cup D_2,\\
-1+2y& \tau \in D_1\cup D_3,
\end{array} \right.
\end{aligned}$$ $$\begin{aligned}
R_{\textbf{v}}(\tau)= \left\{ \begin{array}{lll}
N& \tau=0,\\
-1+2y& \tau \in D_0\cup D_2,\\
-1-2y& \tau \in D_1\cup D_3,
\end{array} \right.
\end{aligned}$$ and $$\begin{aligned}
R_{\textbf{u} \textbf{v}}(\tau)=\left\{ \begin{array}{ll}
-3& \tau \in D_2,\\
1& otherwise.
\end{array} \right.
\end{aligned}$$
[@Ref21] Let $S_{\textbf{u}}(x)$ and $S_{\textbf{v}}(x)$ be the sequence polynomials of sequences $\textbf{u}$ and $\textbf{v}$, $\alpha$ a primitive $N$th root of unity over the field $GF(2^m)$, that is the splitting field of $x^N-1$. Then $$\begin{aligned}
S_{\textbf{u}}(\alpha^j)= \left\{ \begin{array}{lll}
S_{\textbf{u}}(\alpha)& j\in D_0,\\
S_{\textbf{v}}(\alpha)& j\in D_1,\\
1+S_{\textbf{u}}(\alpha)& j\in D_2,\\
1+S_{\textbf{v}}(\alpha)& j\in D_3,\\
\frac{N-1}{2}\pmod 2& j=0,
\end{array} \right.
\end{aligned}$$ $$\begin{aligned}
S_{\textbf{v}}(\alpha^j)= \left\{ \begin{array}{lll}
S_{\textbf{v}}(\alpha)& j\in D_0,\\
1+S_{\textbf{u}}(\alpha)& j\in D_1,\\
1+S_{\textbf{v}}(\alpha)& j\in D_2,\\
S_{\textbf{u}}(\alpha)& j\in D_3,\\
\frac{N-1}{2}\pmod 2& j=0,
\end{array} \right.\end{aligned}$$
where
$S_{\textbf{u}}(x)=\sum_{i\in D_0\cup D_1} x^{i},$\
$S_{\textbf{v}}(x)=\sum_{i\in D_1\cup D_2} x^{i}.$
[@Ref21] $S_{\textbf{u}}(\alpha)\in \{0,1\}$ if and only if $2\in D_0$.
$S_{\textbf{v}}(\alpha)\in \{0,1\}$ if and only if $2\in D_0$.
**Proof** Since $(D_0,\cdot)$ is a group, we have $qD_0=D_0$ and $q^{-1}\in D_0$ for any $q\in D_0$. Hence $$\begin{aligned}
& &S_{\textbf{v}}(\alpha^q)\\
&=&\sum_{i\in D_1\cup D_2}{\alpha^{qi}}\\
&=&\sum_{y\in D_1\cup D_2}{\alpha^y}\\
&=&S_{\textbf{v}}(\alpha).\end{aligned}$$ Since the characteristic of the field $GF(2^m)$ is $2$, it follows that $(S_{\textbf{v}}(\alpha))^2=S_{\textbf{v}}({\alpha}^2)$. From the above, we have $S_{\textbf{v}}(\alpha^2)=S_{\textbf{v}}(\alpha)$ if and only if $2\in D_0$, that is to say $S_{\textbf{v}}(\alpha)\in \{0,1\}$ if and only if $2\in D_0$. So we have completed the proof of Lemma $8$.
New Construction Method and the Autocorrelation Values and Distribution
========================================================================
In this section, assume that $\textbf{u}$ and $\textbf{v}$ are two binary sequences with characteristic sets $D_i\cup D_j$ and $D_j\cup D_l$ respectively, and we define $$\begin{aligned}
\textbf{u}'(t)=\left\{ \begin{array}{lll}
u(t)& t\neq 0,\\
1& t=0.
\end{array} \right.
\end{aligned}$$ $$\begin{aligned}
\textbf{v}'(t)=\left\{ \begin{array}{lll}
v(t)& t\neq 0,\\
1& t=0.
\end{array} \right.
\end{aligned}$$ we proposes one new way to construct sequences, then we calculate its autocorrelation values and distribution.
New Construction Method and Sequence Correspondence
---------------------------------------------------
Let $N=4f+1$, where $f=y^2$ is odd and $N\geq 5$, and sequences $\textbf{u}$ and $\textbf{v}$ are the same as before, new construction as the following: $$\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v}),$$ $$\textbf{\emph{s}}'=\mathbf{I}(\textbf{u}',L^{(N+1)/2}\textbf{v}).$$
Compare the above two constructions, we can know that when $\textbf{\emph{s}}'=\mathbf{I}(\textbf{u}',L^{(N+1)/2}\textbf{v})$, the sequence is a balanced sequence, then we give the autocorrelation value for different characteristic sets for $\textbf{\emph{s}}'$, which is similarly demonstrated when $\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v})$. Below we calculate their autocorrelation of $\textbf{\emph{s}}'$ in different situations.
the Autocorrelation Values and Distribution
-------------------------------------------
$\mathbf{Case \; 1.}$ $(i,j,l)=(0,1,2), and (i,j,l)=(2,1,0)$.
**Proof** When $(i,j,l)=(0,1,2)$, we know that the characteristic sets of $\textbf{u}'$ and $\textbf{v}$ are $D_0\cup D_1$ and $D_1\cup D_2$ respectively. Let $\beta$ be a primitive element of the integer residue ring $Z_N=\{0,1,\cdots,N-1\}$, such that for any $j\in Z_N \backslash \{0\}$, there exists an integer $k$ satisfying $j=\beta^k$. We have $1=\beta^{4f}$, since $ord(\beta)=N-1$ and $f$ is odd, $f$ can be empressed by $f=2t+1$, $t$ is an integer, so $(\beta)^{4t+2}=-1$, we have $-1\in D_2$. For the sequences $\textbf{u}$ and $\textbf{v}$, since $-1\in D_2$, by Lemmas $3(3)$ and $5$, we know: $$\begin{aligned}
R_{\textbf{v} \textbf{u}}(\tau)=\left\{ \begin{array}{lll}
-3& \tau \in D_0,\\
1& otherwise.
\end{array} \right.
\end{aligned}$$
For the autocorrelation values of sequences $\textbf{u}'$ and $\textbf{v}'$, since the only difference between sequences $\textbf{u}$ and $\textbf{u}'$ is $\textbf{u}'(0)=1$, we only need to know the values of $\textbf{u}'(\tau)$ and $\textbf{u}'(N-\tau)$, since $-1\in{D_2}$, $\textbf{u}'(\tau)$ and $\textbf{u}'(N-\tau)$ take different values of $0$ and $1$, we have $(-1)^{\textbf{u}'(0)+\textbf{u}'(\tau)}+(-1)^{\textbf{u}'(0)+\textbf{u}'(N-\tau)}=0$, so we have $R_{\textbf{u}}(\tau)=R_{\textbf{u}'}(\tau)$, similarly we know $R_{\textbf{v}}(\tau)=R_{\textbf{v}'}(\tau)$, for the cross-correlation value of sequences $\textbf{u}$ and $\textbf{v}'$, we only to know $\textbf{v}(\tau)$ and $\textbf{v}(N-\tau)$, then by Lemma $5$ and Equation $(6)$, we have: $$\begin{aligned}
R_{\textbf{u}'\textbf{v}}(\tau)=\left\{ \begin{array}{ll}
3& \tau \in D_1,\\
-1& otherwise.
\end{array} \right.
\end{aligned}$$ $$\begin{aligned}
R_{\textbf{v}\textbf{u}'}(\tau)=\left\{ \begin{array}{ll}
3& \tau \in D_3,\\
-1& otherwise.
\end{array} \right.
\end{aligned}$$
By Lemma $4$, we know: $$R_\textbf{\emph{s}}(\tau)=
\begin{cases}
R_\textbf{u}(\tau/2)+R_{\textbf{v}'}(\tau/2)& \tau \mbox{ is even},\\
R_{\textbf{u}'\textbf{v}}(\frac{\tau+N}{2})+R_{\textbf{v}\textbf{u}'}(\frac{\tau+N}{2})& \tau \mbox{ is odd}.
\end{cases}$$
Then by Lemma $5$, we have $$\begin{aligned}
R_\textbf{\emph{s}}(\tau)=\left\{ \begin{array}{lll}
2N& \tau=0,\\
-2& \tau \mbox{ is even},\tau \neq 0,\\
-2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_0,\\
2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_1 ,\\
-2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_2,\\
2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_3,\\
-2& \tau=N.\\
\end{array} \right.
\end{aligned}$$
When $(i,j,l)=(2,1,0)$, compared to $(i,j,l)=(0,1,2)$, their difference is that the position of the base sequence has changed, so the autocorrelation values and cross-correlation values of the base sequences are unchanged, then by Equation $(7)$, the autocorrelation values of the sequence $\textbf{\emph{s}}$ is the same as above.\
$\mathbf{Case \; 2.}$ $(i,j,l)=(1,2,3),(i,j,l)=(3,2,1)$.
**Proof** According to the proof of case $1$, we can know that when the characteristic sets (i,j,l)=(1,2,3) and (i,j,l)=(3,2,1), the autocorrelation values of the sequences are equal, and we can know the autocorrelation values of the base sequence: $$\begin{aligned}
R_{\textbf{u}'}(\tau)=\left\{ \begin{array}{lll}
N& \tau=0,\\
-1+2y& \tau \in D_0\cup D_2,\\
-1-2y& \tau \in D_1\cup D_3,
\end{array} \right.
\end{aligned}$$ $$\begin{aligned}
R_{\textbf{v}}(\tau)= \left\{ \begin{array}{lll}
N& \tau=0,\\
-1-2y& \tau \in D_0\cup D_2,\\
-1+2y& \tau \in D_1\cup D_3,
\end{array} \right.
\end{aligned}$$
The cross-correlation values of the base sequence: $$\begin{aligned}
R_{\textbf{u}'\textbf{v}}(\tau)=\left\{ \begin{array}{ll}
3& \tau \in D_2,\\
-1& otherwise.
\end{array} \right.
\end{aligned}$$ $$\begin{aligned}
R_{\textbf{v}\textbf{u}'}(\tau)=\left\{ \begin{array}{ll}
3& \tau \in D_0,\\
-1& otherwise.
\end{array} \right.
\end{aligned}$$
Then by Equation $(7)$, we have $$\begin{aligned}
R_\textbf{\emph{s}}(\tau)=\left\{ \begin{array}{lll}
2N& \tau=0,\\
-2& \tau \mbox{ is even},\tau \neq 0,\\
2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_0,\\
-2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_1 ,\\
2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_2,\\
-2& \tau \mbox{ is odd}, \frac{\tau+N}{2} \in D_3,\\
-2& \tau=N.\\
\end{array} \right.
\end{aligned}$$
The above is the autocorrelation function of sequences $\textbf{\emph{s}}'$, the sequences $\textbf{\emph{s}}$ can be similarly demonstrated, in the case of the same characteristic set, except for $\tau=N$, the autocorrelation values are opposite to each other, in other cases, the autocorrelation values are the same, and its proof is omitted for simplicity. Next we give the autocorrelation distribution of the sequence $\textbf{\emph{s}}$.
The autocorrelation value distribution of $\textbf{\emph{s}}=\mathbf{I}(\textbf{u}',L^{(N+1)/2}\textbf{v})$: $$\begin{aligned}
R_\textbf{\emph{s}}(\tau)=\left\{ \begin{array}{lll}
2N& 1\,\,time,\\
-2& \frac{7N-3}{4}\,\,time,\\
2& \frac{N-1}{4}\,\,times.
\end{array} \right.
\end{aligned}$$ When $\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v})$: $$\begin{aligned}
R_\textbf{\emph{s}}(\tau)=\left\{ \begin{array}{lll}
2N& 1\,\,time,\\
-2& \frac{7N-7}{4}\,\,time,\\
2& \frac{N+3}{4}\,\,times.
\end{array} \right.
\end{aligned}$$
**Proof** Let $Q$ denote the quadratic residual set of modulo $N$, $P$ be a quadratic non-residual set for modulo $N$, then $Q=D_0\cup D_2$, $P=D_1\cup D_3$, since $-1\in D_2$, and $N$ is odd, if $a\in Q$ is odd(even), then $N-a\in Q$, and $N-a$ is even(odd). Thus half elements in $Q$ are even(odd). So is $P$. Then by Equations $(8)$ and $(9)$, the Equation $(10)$ is proved. When $\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v})$ similarly demonstrated. So Theorem $1$ is proved.
A Lower Bound of Linear Complexity
==================================
Since $N=4f+1$, $2\notin D_0 \cup D_2$, then we only consider the case $2\notin D_0$, $\textbf{u}'$ and $\textbf{v}$ are two binary sequences with characteristic sets $D_0\cup D_1$ and $D_1\cup D_2$ respectively, then we have:
The minimal polynominls of $\textbf{u}'$ and $\textbf{v}$:
1. $m_{\textbf{u}'}(x)=x^N-1$;
2. $m_{\textbf{v}}(x)=\displaystyle\frac{x^N-1}{x-1}$.
**Proof** Let $\alpha$ be a primitive $N$th root over $GF(2^m)$ as before. Since $N=4f+1$, $2\notin D_0$, by Lemmas $7$ and $9$, $S_{\textbf{u}'}(\alpha) \notin \{0,1\}$ and $S_{\textbf{v}}(\alpha) \notin \{0,1\}$, so $\{\alpha^j: 0<j\leq {N-1}\}$ are not the roots of $S_{\textbf{u}'}(\alpha)$ and $S_{\textbf{v}}(\alpha)$. Since $S_{\textbf{u}'}(\alpha^0)=(\frac{N-1}{2}+1)=1 \pmod 2$, $S_{\textbf{v}}(\alpha^0)=\frac{N-1}{2}=0 \pmod 2$, we have $\textrm{gcd}(S_{\textbf{u}'}(x),x^{N}-1)=1$, $\textrm{gcd}(S_{\textbf{v}}(x),x^{N}-1)=x-1$, then by the Lemma $1$, we have completed the proof.
When $2\in D_1$, we have $S_{\textbf{v}}(x^2)=S_{\textbf{u}'}(x)$, and $2\in D_3$, we have $S_{\textbf{u}'}(x^2)=S_{\textbf{v}}(x)$.
**Proof** When $2\in D_1$, $$\begin{aligned}
& &S_{\textbf{v}}(x^2)\\
&=&\sum_{i\in D_1\cup D_2} x^{2i}\\
&=&\sum_{g\in D_2\cup D_3} x^{g}\\
&=&S_{\textbf{u}}(x);\end{aligned}$$ When $2\in D_3$, $$\begin{aligned}
& &S_{\textbf{u}}(x^2)\\
&=&1+\sum_{i\in D_0\cup D_1} x^{2i}\\
&=&1+\sum_{h\in D_0\cup D_3} x^{h}\\
&=&S_{\textbf{v}}(x).\end{aligned}$$ We have completed the proof.
Let $\textbf{\emph{s}}'$ be the interleaved sequence of period $2N$ as before and $S_{\textbf{\emph{s}}'}(x)$ is the sequence polynomial of $\textbf{\emph{s}}'$, $\textbf{\emph{s}}'=\mathbf{I}(\textbf{u}',L^{(N+1)/2}\textbf{v})$. Then the linear complexity $LC(\textbf{\emph{s}}')$ is bounded by $LC(\textbf{\emph{s}}') \geq 2N-4$, when $\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v})$, the linear complexity $LC(\textbf{\emph{s}})$ is bounded by $LC(\textbf{\emph{s}}) \geq 2N-5$.
**Proof** Let $S_{\textbf{u}'}(x)$ and $S_{\textbf{v}}(x)$ be the sequence polynomials of $\textbf{u}'$ and $\textbf{v}$, the characteristic sets $(i,j,l)=(0,1,2),(1,2,3),(2,1,0)$ or $(3,2,1)$, $\alpha$ be a primitive $N$th root of unity over the field $GF(2^m)$. Let us take the characteristic set (i,j,l)=(0,1,2) as an example, and other cases can be similarly proved. By Lemma $2(3)$, we have $$\begin{aligned}
& &\textrm{gcd}(S_{\textbf{\emph{s}}'}(x),x^{2N}-1)\\
&=&\textrm{gcd}((S_{\textbf{u}'}(x^2)+x^{\frac{N+1}{2}}S_{\textbf{v}}(x^2),x^{2N}-1).\end{aligned}$$
Since $S_{\textbf{\emph{s}}'}(\alpha^0)=1$, then $\textrm{gcd}(S_{\textbf{\emph{s}}'}(\alpha^0),((\alpha^0)^{2N}-1))=1$, so $x-{\alpha^0}$ is not a common factor. Then we consider whether $\{\alpha^j:j\in Z_N^*\}$ is the common root of $S_{\textbf{\emph{s}}'}(x)$ and $x^{2N}-1)$.
When $2\in D_1$, by Corollary $3$, we have $$\begin{aligned}
& &\textrm{gcd}(S_{\textbf{u}'}(x^2)+x^{\frac{N+1}{2}}S_{\textbf{v}}(x^2), x^{2N}-1)\\
&=&\textrm{gcd}(S_{\textbf{u}'}(x)(S_{\textbf{u}'}(x)+x^{\frac{N+1}{2}}),x^{2N}-1).\end{aligned}$$
Since $\textrm{gcd}(S_{\textbf{u}'}(x),x^{N}-1)=1$, so we only consider $\textrm{gcd}(S_{\textbf{u}'}(x)+x^{\frac{N+1}{2}},x^{2N}-1)$, for $\{\alpha^j:j\in D_0\}$, $(\alpha)^{\frac{j.(N+1)}{2}}$ are not equal to each other, then by Lemma $6$,for $\{\alpha^j:j\in D_0\}$, $S_{\textbf{u}'}(x)$ is the same value, we can know in the set $\{\alpha^j:j\in D_0\}$, only one number may be the root of the equation $S_{\textbf{u}'}(x)+x^{\frac{N+1}{2}}=0$. Similarly, there may be only three roots for the other three cases, so the linear complexity is bounded by $LC(\textbf{\emph{s}}') \geq 2N-4$.
When $2\in D_3$, then by Corollary $3$, we have $$\begin{aligned}
& &\textrm{gcd}(S_{\textbf{u}'}(x^2)+x^{\frac{N+1}{2}}S_{\textbf{v}}(x^2), x^{2N}-1)\\
&=&\textrm{gcd}(S_{\textbf{v}}(x)(1+x^{\frac{N+1}{2}}S_{\textbf{v}}(x)),x^{2N}-1).\end{aligned}$$
Since $\textrm{gcd}(S_{\textbf{v}}(x),x^{N}-1)=1$, we only consider $\textrm{gcd}(1+x^{\frac{N+1}{2}}S_{\textbf{v}}(x),x^{2N}-1)$, according to the above proof, for $\{\alpha^j:j\in D_0\}$, $S_{\textbf{v}}(x)$ is the same value, we can know in the set $\{\alpha^j:j\in D_0\}$, only one number may be the root of the equation $1+x^{{\frac{N+1}{2}}}S_{\textbf{v}}(x)$. Similarly, there may be only three roots for the other three cases, so the linear complexity is bounded by $LC(\textbf{\emph{s}}') \geq 2N-4$. Hence, we have completed the proof of Theorem $2$.
When $\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v})$, the proof of linear complexity is the same as for $\textbf{\emph{s}}'=\mathbf{I}(\textbf{u}',L^{(N+1)/2}\textbf{v})$, the only difference between $\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v})$ and $\textbf{\emph{s}}'=\mathbf{I}(\textbf{u}',L^{(N+1)/2}\textbf{v})$ is that $1$ is the root of $\textbf{\emph{s}}=\mathbf{I}(\textbf{u},L^{(N+1)/2}\textbf{v})$, so the linear complexity is bounded by $LC(\textbf{\emph{s}}) \geq 2N-5$.
Remark
======
$\textbf{\emph{s}}$ and $\textbf{\emph{s}}'$ possess optiaml autocorrelation if and only if its characteristic set is an almost difference set, sequences $\textbf{\emph{s}}$ and $\textbf{\emph{s}}'$ can also be obtained by the results in [@Ref1].
Conclusion
==========
In this paper, we use the interleaving technique to construct a binary sequence with the optimal autocorrelation of period $2N$. From the section $3$, we can conclude that this sequence is optimal, the sequence has low autocorrelation values, and we give the distribution of the autocorrelation values. Especially, in section $4$, based on the discussion of roots of the sequence polynomials over the field $\textrm{GF}(2^m)$, we give a lower bound of linear complexity, the linear complexity can be reached $2N-4$, far larger than half of a period. Results show that these sequences have low autocorrelation and the linear complexity satisfies the requirements of cryptography.
Acknowledgments {#acknowledgments .unnumbered}
===============
The project is supported by the open fund of Fujian Provincial Key Laboratory of Network Security and Cryptology Research Fund (Fujian Normal University) (No.15002), the Natural Science Fund of Shandong Province (No.ZR2014FQ005), the National Natural Science Foundations of China(No.61170319)and the Fundamental Research Funds for the Central Universities (No.11CX04056A, 15CX08011A,15CX05060A).
[10]{}
K. T. Arasu, Cunsheng Ding, Tor Helleseth, P. Vijay Kumar, Halvard Martinsen, “Almost difference sets and their sequences with optimal autocorrelation,” [*IEEE Transactions on Information Theory*]{}, vol. 47, no. 7, pp. 2934–2943, 2001.
P. Z. Fan and M. Darnell, *Sequence Design for Communications Applications*. London, U.K.: Wiley. Press 1996.
Yang Yang, Xiaohu Tang and Zhengchun Zhou, “The Autocorrelation Magnitude of Balanced Binary Sequence Pairs of Prime Period $N\equiv 1\pmod{4}$ With Optimal Cross-Correlation,” [*IEEE Communications Letters*]{}, vol. 19, no. 4, pp. 585–588, 2015.
S. W. Golomb and Guang Gong, *Signal Design for Good Correlation-for Wireless Communication*, Cryptography and Radar. Cambridge, U.K.: Cambridge Univ. Press, 2005.
Xiaohu Tang and Cunsheng Ding, “New classes of balanced quaternary and almost balanced binary sequences with optimal autocorrelation value,” [*IEEE Transactions on Information Theory*]{}, vol. 56, no. 12, pp. 6398–6405, 2010.
James L. Massey, “Shift-register synthesis and BCH decoding,” [*IEEE Transactions on Information Theory*]{}, vol. 15, no. 1, pp. 122–127, 1969.
Jianqin Zhou and Wei Xiong, “An algorithm for computing m-tight error linear complexity of sequences over GF(pm) with period pm,” [*International Journal of Network Security*]{}, vol. 15, no. 1, pp. 59-63, 2013.
Nian Li and Xiaohu Tang, “On the linear complexity of binary sequences of period 4N with optimal autocorrelation value/magnitude,” [*IEEE Transactions on Information Theory*]{}, vol. 57, no. 11, pp. 7597–7604, 2011.
T Helleseth £¬ K Yang, “On Binary Sequences of Period $n = p^m-1$ with Optimal Autocorrelation,”Springer London. Press, pp. 209–217, 2002.
Ying Cai and Cunsheng Ding, “Binary sequences with optimal autocorrelation,” [*Theoretical Computer Science*]{}, vol. 410, no. 24/25, pp. 2316–2322, 2009.
H. D. L¨¹ke, H. D. Schotten, and H. Hadinejad-Mahram, “Binary and quadriphase sequence with optimal autocorrelation: A survey,” [*IEEE Transactions on Information Theory*]{}, vol. 49, no. 12, pp. 3271–3282, 2003.
Xiaohu Tang and Guang Gong, “New constructions of binary sequences with optimal autocorrelation value/magnitude,” [*IEEE Transactions on Information Theory*]{}, vol. 56, no. 3, pp. 1278–1286, 2010.
Jong-Seon No, Habong Chung, Hong-Yeop Song, Kyeongcheol Yang, Jung-Do Lee, Tor Helleseth, “New construction for binary sequences of period $p^m-1$ with Optimal autocorrelation using $(z+1)^d+az^d+b$,” [*IEEE Transactions on Information Theory*]{}, vol. 47, no. 4, pp. 1638–1644, 2001.
Gohar M. M. Kyureghyan, Alexander Pott, “On the Linear Complexity of the Sidelnikov-Lempel-Cohn-Eastman Sequences,” [*Des. Codes Cryptography*]{}, vol. 29, no. 1-3, pp. 149–164, 2003.
Rudolf Lidl and Harald Niederreiter, *Finite Fields (Encyclopedia of Mathematics and Its Applications )*. New York: Cambridge University Press, 1997.
V.M.Sidelnikov, “Some k-valued pseudo-random sequences and nearly equidistant codes,” [*Problems of Information Transmission*]{}, vol. 5, no. 1, pp. 12–16, 1969.
Tongjiang Yan and Guang Gong, “Some Notes on Constructions of Binary Sequences with Optimal Autocorrelation,” http://arxiv.org/abs/1411.4340, 2014.
Abraham Lempel, Martin Cohn, Willard L. Eastman, “A class of balanced binary sequences with optimal autocorrelation properties,” [*IEEE Transactions on Information Theory*]{}, vol. 23, no. 1, pp. 38–42, 1977.
Xuan Zhang, Qiaoyan Wen, and Jing Qin, “Constructions of sequences with almost optimal autocorrelation magnitude,” [*Journal of Electronics and Information Technology*]{}, vol. 33, no. 8, pp. 1908–1912, 2011.
Wilfried Meidl, Arne Winterhof, “Some Notes on the Linear Complexity of Sidel’nikov-Lempel-Cohn-Eastman Sequences,” [*Des. Codes Cryptography*]{}, vol. 38, no. 2, pp. 159–178, 2006.
Cunsheng Ding, Tor Helleseth and Kwok-Yan Lam, “Several Classes of Binary Sequences with Three-Level Autocorrelation,” [*IEEE Transactions on Information Theory*]{}, vol. 45, no. 7, pp. 2606–2612, 1999.
Qi Wang and Xiaoni Du, “The linear complexity of binary sequences with optimal autocorrelation,” [*IEEE Transactions on Information Theory*]{}, vol. 56, no. 12, pp. 6388–6397, 2010.
[**Shidong Zhang**]{} biography. Shidong Zhang was born in 1992 in Shandong Province of China. He was graduated from Jining University . He will study for a postgraduate degree at China University of Petroleum in 2016. And his tutor is Tongjiang Yan. Email:zhangshdo1992@163.com\
[**Tongjiang Yan**]{} biography. Tongjiang Yan was born in 1973 in Shandong Province of China. He was graduated from the Department of Mathematics, Huaibei Coal-industry Teachers College, China, in 1996. In 1999, he received the M.S. degree in mathematics from the Northeast Normal University, Lanzhou, China. In 2007, he received the Ph.D. degree in Xidian University. He is now a professor of China University of Petroleum. His research interests include cryptography and algebra. Email:yantoji@163.com\
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abstract: 'We experimentally demonstrate the efficient channeling of fluorescence photons from single q-dots on optical nanofiber into the guided modes, by measuring the photon-count rates through the guided and radiation modes simultaneously. We obtain the maximum channeling efficiency to be $22.0$ ($\pm4.8$)% at fiber diameter of $350$ nm for the emission wavelength of $780$ nm. The results may open new possibilities in quantum information technologies for generating single photons into single-mode optical-fibers.'
author:
- 'Ramachandrarao Yalla,$^{1}$ Fam Le Kien,$^{1}$ M. Morinaga,$^{2}$ and K. Hakuta$^{1}$'
title: Efficient channeling of fluorescence photons from single quantum dots into guided modes of optical nanofiber
---
Efficient collection of fluorescence photons from a single emitter into a single-mode fiber is a major challenge in the context of quantum information science. For that purpose various novel techniques have been proposed so far. The examples would include micropillar cavities [@yamamoto], photonic crystal cavities [@vuckovic], solid immersion lens [@sil], and plasmonic metal nano-wires [@lukin]. However, in these techniques the subsequent coupling of fluorescence photons into a single-mode fiber may reduce the actual collection efficiency. In the view of the ability to directly couple fluorescence photons into a single-mode fiber, tapered optical fibers with sub-micron diameter, termed as optical nanofibers, would be particularly promising. It has been theoretically predicted that, by positioning the emitter on the nanofiber surface, one can channel the fluorescence photons into the nanofiber guided-modes with an efficiency higher than $20\%$ [@ducloy; @spem], and moreover, fibers can be tapered adiabatically to keep the light transmission into the single-mode fiber higher than $90\%$ [@Nayak1; @Arno1].
In the last decade, optical nanofibers have been attracting considerable attention in the field of quantum optics. Many works have been reported so far using laser-cooled atoms. Channeling of fluorescence photons into the guided modes has been demonstrated [@Nayak1], and photon correlations from single atoms have been measured systematically through nanofiber guided modes [@Nayak2; @Nayak3]. Fluorescence emission spectrum has been measured for few atoms through the guided mode by combining optical-heterodyne and photon-correlation methods [@Das]. Various schemes have been proposed for trapping atoms around the nanofiber [@trap0; @trap; @trap-Arno], and the trapping has been experimentally demonstrated [@Arno2] using dipole-trapping method via two-color laser-fields [@trap]. However, regarding the channeling efficiency of fluorescence photons into the guided modes, it has not been measured yet, although the works so far imply a reasonable correspondence to the theoretical predictions [@Nayak1]. One reason would be due to a fact that atoms are not on the nanofiber surface and the atom-surface distance could not be estimated accurately.
Recently, two groups have reported the photon-counting measurements from semiconductor q-dots deposited on nanofibers [@nano; @opex]. They absolutely measured the photon-count rates into the guided modes for one q-dot. They discussed the channeling efficiency of fluorescence photons into the guided modes based on the measured results. However, as pointed out in Ref. [@opex], the value which can be obtained from such measurements is not the channeling efficiency itself, but is a product of the channeling efficiency and the quantum efficiency of the q-dot. Therefore, the channeling efficiency cannot be determined from the measurements without accurate information on the quantum efficiency for the one q-dot which is measured.
In the present Letter, we experimentally determine the channeling efficiency of fluorescence photons from single q-dots on optical nanofiber into the guided modes. We measure the photon-count rates through the guided and radiation modes simultaneously for various diameters of nanofiber. The measured results completely reproduce the theoretical predictions [@ducloy; @spem] within the experimental errors. The maximum channeling efficiency is obtained to be $22.0$ ($\pm4.8$)% at the fiber diameter of $350$ nm for the emission wavelength of $780$ nm.
Figure 1 illustrates the schematic diagram of the experimental setup. Main part of the setup consists of inverted microscope (Nikon, Eclipse Ti-U) with a computer controlled $\it{x}$-$\it{y}$ stage, optical nanofiber, and sub-pico-liter needle-dispenser (Applied Micro Systems, ND-2000). Optical nanofiber is placed on the $\it{x}$-$\it{y}$ stage to precisely control the nanofiber position to the focus point of the microscope. Optical nanofibers are produced by adiabatically tapering commercial single-mode optical-fibers (SMF1, cut-off wavelength: $1.3$ $\mu$m) using a heat and pull technique. The diameter of nanofiber is measured using a scanning electron microscope (SEM) prior and after the optical experiments. The thinnest diameter of the nanofiber is $300$-$400$ nm, and the nanofiber diameter varies along the fiber axis by $100$ nm/$1$ mm. The transmission through the optical nanofiber is measured to be $90\%$ using a fiber-coupled superluminescence light emitting diode (SLED) at $800$ nm.
We use core-shell type colloidal CdSeTe (ZnS) q-dots having emission wavelength at $790$ nm (Invitrogen, Q21371MP). We use the sub-pico-liter needle-dispenser to deposit q-dots on nanofibers. The dispenser consists of a taper glass-tube which contains diluted q-dot solution and a needle having a tip of diameter $17$ $\mu $m. The needle axis is adjusted to coincide with the axis of the microscope, and the needle-tip position is computer-controlled along the axis. Once the needle tip passes through the taper glass-tube, it carries a small amount of q-dot solution at its edge. In order to deposit q-dots on nanofiber with minimum scattering loss, the needle-tip position is adjusted so that the q-dot solution at its tip just touches the nanofiber surface. Note that this method could deposit q-dots only on the $\it{upper}$ $\it{surface}$ $\it{of}$ $\it{nanofiber}$. The deposition is done for several positions along the fiber axis corresponding to the fiber diameter of $300$-$800$ nm. The transmission through the optical nanofibers is dropped to $81\%$ after the depositions. The depositions are done for three optical nanofiber samples, and the following measurements are carried out for all the deposited positions.
{width="8cm"}
The q-dots are excited using cw laser-diode LD at a wavelength of $640$ nm. The excitation beam is focused to the nanofiber by the microscope objective lens OL ($40$X, NA= $0.6$). Regarding the fluorescence photons channeled into the guided modes, in order to guarantee the observation through the fundamental-mode ($HE_{11}$), SMF1 is spliced to another single-mode fiber SMF2 (cut-off wavelength: $557$ nm) at both ends marked as $S_{1}$, $S_{2}$ in Fig.1. The fluorescence light beam from each end of SMF2 is filtered from the scattered excitation laser light with a color glass filter FL1 (FL2) (HOYA, R72) and re-coupled into a multi-mode fiber. At one end of the multi-mode fiber, fluorescence photons are detected with a fiber-coupled avalanche-photodiode APD1 (Perkin Elmer, SPCM-AQR/FC). At the other end of multi-mode fiber, fluorescence emission spectrum is measured using an optical-multichannel-analyzer OMA (Andor, DV420A-OE).
Regarding the radiation modes, fluorescence photons are collected by OL, coupled into a multi-mode fiber by FC3, and detected by a fiber-coupled avalanche-photodiode APD2. A set of two filters FL3 (HOYA, R70/R72) is used to reject the scattered laser light from the focus point. Characteristics of APD1 and APD2 are the same, and signals from APD1 and APD2 are accumulated and recorded using photon-counting system (Hamamatsu, M8784). Photon-counting measurements for both guided and radiation modes and spectrum measurements are carried out for each deposited position simultaneously. Additionally, we perform photon-correlation measurements through the guided modes for all deposited positions [@opex]. All the above fluorescence measurements are carried out for the three nanofiber samples by keeping the excitation laser intensity at a low value of $50$ W/cm$^{2}$ so that q-dots may not deteriorate [@degrade; @blink].
The channeling efficiency $\eta_{c}$ into the nanofiber guided modes can be expressed as following, $$\eta_{c}= \frac{n_{g}}{n_{g}+n_{r}}= \frac{1}{1+n_{r}/n_{g}}$$ where $n_{g}$ and $n_{r}$ are photon emission rates into the guided and radiation modes, respectively. Observable photon-count rates by APD1 and APD2 are expressed as followings, $$n_{g}^{(obs)}=\frac{1}{2}\eta_{APD1}\kappa_{g}n_{g},\
n_{r}^{(obs)}= \eta_{APD2}\kappa_{r}\eta _{r}n_{r}$$ where $\kappa_{g}$ and $\kappa_{r}$ are light-transmission factors for the paths of guided and radiation modes, respectively. Factor $1/2$ for $n_{g}^{(obs)}$ corresponds to a fact that fluorescence photons into the guided modes are detected only for one direction of the nanofiber. $\eta_{APD1}$ and $\eta_{APD2}$ are quantum efficiency of APD1 and APD2, respectively, and are assumed to be the same. $\eta_{r}$ is an effective collection efficiency for the radiation modes. Thus, the ratio ${n_{r}}/{n_{g}}$ can be written as following, $$\frac{n_{r}}{n_{g}}= \frac{n_{r}^{(obs)}}{n_{g}^{(obs)}}\times{\frac{\kappa_{g}}{2\kappa_{r}\eta _{r}}}= \frac{n_{r}^{(obs)}}{n_{g}^{(obs)}}\times C$$ where $C$= ${\kappa_{g}}/{{2\kappa_{r}\eta _{r}}}$.
$\kappa_{g}$-value was measured to be $49.6$ ($\pm2.1$)%. The measurement procedure is following. The SLED output is spliced to SMF2 at the FL1-end, and the output power is measured at the APD1-position. Input power to the optical nanofiber is measured by cleaving the SMF1 before entering into the optical nanofiber. The measured value is consistent with a value calculated as a product of transmission factors of optical nanofiber ($81\%$), splicing point between SMF1 and SMF2 ($81\%$), FL2 ($83\%$), and coupling efficiency into the multi-mode fiber at FC2 ($90\%$).
$\kappa_{r}$-value was obtained to be $23.5$ ($\pm1.3$)% as a product of transmission factors of all optical components in the path and coupling efficiency into multi-mode fiber. Transmission factors are measured for OL ($74\%$), BS ($63\%$), FM ($83\%$), and FL3 ($75\%$), using the SLED light. The coupling efficiency into multi-mode fiber at FC3 was obtained to be $81\%$ by the following procedure. First, SLED light is introduced from the LD-port and is focused to the nanofiber. The scattered light from the focused spot is collected through the OL, and its power is measured both at FC3-position and at APD2-position through multi-mode fiber.
Regarding the radiation modes, the effective collection efficiency $\eta_{r}$ consists of two factors. One is from numerical aperture (NA) of the OL. The collection efficiency of the OL is estimated to be $10\%$ from the NA-value of $0.6$. The other factor arises from the nanofiber itself. The q-dots are deposited on the upper surface of nanofiber and the OL collects the fluorescence photons from the down side of nanofiber. Therefore, the nanofiber acts as a cylindrical lens and the collection efficiency of the OL may be enhanced by the lens effect of nanofiber. We calculated the enhancement factor based on the formalism developed in Ref. [@Stratton], and estimated the average enhancement factor by assuming random azimuthal distribution of q-dots on the upper surface of nanofiber. It was found that the average enhancement factor could be assumed to be constant with a value of $1.48$ ($\pm0.03$) for the fiber diameters of the present measurements. We use this average enhancement factor to obtain the effective collection efficiency $\eta_{r}$. Thus, we obtain the $\eta_{r}$-value to be $14.8$ ($\pm0.3$)% and consequently the $C$-value to be $7.13$ ($\pm0.84$) by combining the values of $\kappa_{g}$, $\kappa_{r}$, and $\eta_{r}$.
{width="8cm"}
Figure 2 shows the fluorescence photon-count rate measured for a deposited nanofiber by scanning the focusing point along the nanofiber. The scanning speed is $6$ $\mu $m/s, and signals are measured through the guided mode by APD1. One can clearly see eight sharp peaks along the nanofiber with a typical separation of $0.5$ mm. Origin of the horizontal axis corresponds to the center of the nanofiber. Nanofiber diameter varies from 400 nm at the origin to 750 nm at the position $3.5$ mm. Inset shows an expanded profile of a peak marked by arrow. The width should be limited by the focused spot size on nanofiber and is about $1.5$ $\mu $m FWHM.
{width="8cm"}
Figure 3(a) shows the typical fluorescence photon-count rates from q-dots on nanofiber at fiber diameter of 400 nm. Black and red traces correspond to the photon-count rates through the guided and radiation modes, respectively. Measurement time is $5$ min with a time bin size of $100$ ms. One can readily see that the two traces exactly match by each other. Photon-count rates show a clear single step blinking behavior, revealing that the number of deposited q-dots is one. This single q-dot deposition could be further confirmed by measuring the anti-bunching dip in the normalized photon-correlations, and the dip-value was measured to be $0.035$ $\textless\textless1$. Above measurements were performed for all the depositions of the three nanofiber samples, and the number of q-dots at each deposition was measured to be one or two, similarly as in Ref. [@opex]. Regarding the fluorescence spectrum, the center wavelength distributes over the range of $80$ nm from $740$ to $820$ nm with a typical FWHM of 52 nm.
Figure 3(b) shows the photon-count rate histograms for the black and red traces for the whole measurement time with a counting interval of $1$ kcps. By fitting the histograms with Gaussian profiles [@blink], we obtain ${n_{g}^{(obs)}}$ and ${n_{r}^{(obs)}}$ to be $44.3$ ($\pm5.4$) and $24.8$ ($\pm3.7$) kcps, respectively. Using the relation of Eq. (3), the ratio ${n_{r}}/{n_{g}}$ is obtained to be $3.99$ ($\pm1.55$). Thus, we obtain using the Eq. (1) the $\eta_{c}$-value to be $20.0$ ($\pm6.2$)%. Using the same procedure, the $\eta_{c}$-values were obtained at various fiber diameters for the three nanofiber samples. We obtained average value of $\eta_{c}$ at each fiber diameter for the three nanofiber samples. Regarding the fiber diameter of $400$ nm, the average $\eta_{c}$-value was $21.5$ ($\pm2.4$)%.
{width="8cm"}
Figure 4 shows the channeling efficiency $\eta_{c}$ as a function of fiber size parameter ($k_{0}$a= $2$$\pi$$a$/$\lambda$). The size parameter is calculated for each deposited position by using the measured fiber-diameter $2a$ and the observed emission-wavelength $\lambda$. The red curve exhibits the theoretical prediction for the channeling efficiency into the $HE_{11}$-mode assuming the nanofiber refractive-index of 1.45. Experimental values are shown by black squares with error bars. The error bars in horizontal axis are due to the variation of emission wavelength at each deposited position. The error bars in vertical axis are due to the fluctuation of the measured values. The fluctuation would mainly be due to the measurement ambiguity, but another cause may happen from the ambiguity of the enhancement factor due to the nanofiber lens effect. For the experimental analysis, we used the average enhancement factor assuming random azimuthal distribution of deposition, but the enhancement factor for each deposited position would be different from the average value. Such ambiguity should induce the fluctuation in the measured results. Although ambiguity of $\pm20$% still exists, attention must be specially paid that the measured results have completely reproduced the theoretical prediction. The channeling efficiency into the guided modes reaches to a maximum value of $22.0$ ($\pm4.8$)% at the fiber size parameter of $1.43$, which corresponds to the fiber diameter of $350$ nm for the emission wavelength of $780$ nm.
In summary, we have experimentally demonstrated the efficient channeling of fluorescence photons from single q-dots on optical nanofiber into the guided modes, by measuring the photon-count rates through the guided and radiation modes simultaneously. We have obtained the maximum channeling efficiency to be $22.0$ ($\pm4.8$)% at the fiber diameter of $350$ nm for the emission wavelength of $780$ nm. The present results may open new possibilities in quantum information technologies for generating single photons into single-mode optical-fibers.
We thank Kali Nayak for helpful discussions. This work was carried out as a part of the Strategic Innovation Project by Japan Science and Technology Agency.
[99]{}
G. Solomon, M. Pelton, and Y. Yamamoto, Phys. Rev. Lett. **86**, 3903 (2001). G. Shambat, J Provine, K. Rivoire, T. Sarmiento, J. Harris, and J. Vuckovic, Appl. Phys. Lett. **99**, 191102 (2011). T. Schroder, F. Gadeke, M. J. Banholzer, and O. Benson, New J.Phys. **13**, 055017 (2011). A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, Nature **450**, 402-406 (2007). V. V. Klimov and M. Ducloy, Phys. Rev. A **69**, 013812 (2004). F. L. Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, Phys. Rev. A **72**, 032509 (2005). K. P. Nayak, P. N. Melentiev, M. Morinaga, Fam Le Kien, V. I. Balykin, and K. Hakuta, Opt. Express **15**, 5431 (2007). A. Stiebeiner, R. Garcia-Fernandez, and A. Rauschenbeutel, Opt. Express **18**, 22677-22685 (2010). K. P. Nayak and K. Hakuta, New J. Phys. **10**, 053003 (2008). K. P. Nayak, Fam Le Kien, M. Morinaga, and K. Hakuta, Phys. Rev. A **79**, 021801(R) (2009). M. Das, A. Shirasaki, K. P. Nayak, M. Morinaga, Fam Le Kien, and K. Hakuta, Opt. Express **18**, 17154 (2010). V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, and M. Morinaga, Phys. Rev. A **70**, 011401(R) (2004). F. L. Kien, V. I. Balykin, and K. Hakuta, Phys. Rev. A **70**, 063403 (2004). G. Sagu, A. Baade, and A. Rauschenbeutel, New J. Phys. **10**, 113008 (2008). E. Vetsch, D. Reitz, G. Sague, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, Phys. Rev. Lett. **104**, 203603 (2010). M. Fujiwara, K. Toubaru, T. Noda, H. Q. Zhao, and S. Takeuchi, Nano Lett. **11**, 4362 (2011). R. R. Yalla, K. P. Nayak, and K. Hakuta, Opt. Express **20**, 2932 (2012); arXiv:1112.0624v1 (2011). T. Ota, K. Maehashi, H. Nakashima, K. Oto, and K. Murase, Phys. Status Solidi B **224**, 169 (2001). K. Goushi, T. Yamada, and A. Otomo, J. Phys. Chem. C **113**, 20161 (2009). J. A. Stratton, $\it{Electromagnetic Theory}$ (McGraw-Hill, New York, 1941), Chap. 6; Details will be reported elsewhere.
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abstract: 'A two cocycle is associated to any action of a Lie group on a symplectic manifold. This allows to enlarge the concept of anomaly in classical dynamical systems considered by F Toppan in \[[*J. Nonlinear Math. Phys.*]{} [**8**]{}, Nr. 3 (2001), 518–533\] so as to encompass some extensions of Lie algebras related to noncanonical actions.'
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\[solomin-firstpage\]
Introduction
============
The concept of anomalies in classical dynamical systems was considered by F Toppan in a recent paper [@t]. This concept allowed him to establish interesting re-interpretations of some relevant results.
The general situation he discusses is the following: the momentum mappings associated to a canonical action of a Lie group $G$ on a classical system (the Noether charges) yield a nontrivial central extension of ${\rm Lie} (G)$, the Lie algebra of $G$.
The aim of this paper is to generalize this kind of ideas in order to encompass some noncanonical actions of Lie Groups on classical systems that give rise to representations of extensions of ${\rm Lie} (G)$. These extensions are in general noncentral.
Our approach is based on the introduction and analysis of a two cocycle on ${\rm Lie} (G)$ associated to any action of $G$ on a symplectic manifold $(M, \omega)$. This cocycle takes values in $C^{\infty}(M)$, but, if the action of $G$ is symplectic, its evaluation at any point in $M$ yields a real valued two cocycle which is cohomologous to the cocycle defining the central extension mentioned above (see, for instance, [@am] or [@w]).
The generalization we propose will allow us to consider in our context an example involving the Mickelsson–Fadeev cocycle by means of classical objects. This cocycle is associated to the quantum anomalous commutator of the constraints of the Gauss-law in a $(3+1)$-dimensional Yang–Mills theory interacting with Weyl fermions and it was first computed by L Fadeev by using path-integral techniques [@fs] (and also by cohomological ones [@f]). Almost simultaneously, J Mickelsson constructed a representation of the extension of the current algebra associated to this cocycle by using topological techniques.
In fact, we shall see that the Mickelsson–Fadeev cocycle is cohomologous, modulo constants, to the cocycle associated to the action of the gauge group on the classical system defined by the effective lagrangian.
The canonical two-cocycle and the momentum maps
===============================================
In this section we consider a symplectic manifold $(M,\omega)$ and an action of a Lie group $G$ on $M$ which is not assumed to be symplectic. This action induces a natural action of $G$ on the space $C^\infty(M)$: $$(g\cdot f)(x) = f\left( g^{-1}
\cdot x\right),\ \ \ \forall g \in G.$$
The derivative of this action produces a nontrivial action of ${\rm
Lie}(G)$ on $C^{\infty}(M)$ by $$a\cdot f = -L_{{\tilde{X}_a}}f,$$with ${\tilde{X}_a}$ the infinitesimal generator associated to $a\in {\rm Lie} (G)$ and $L$ the Lie derivative.
Under such action $C^{\infty}(M)$ becomes a ${\rm Lie}(G)$-module. In general, ${\rm
Lie}(G)$ acts on all differential forms on $M$ in the same way: $a \cdot \alpha =
-L_{{\tilde{X}_a}} \alpha$.
In order to define the cohomology of ${\rm Lie}(G)$ with coefficients in $C^{\infty}(M)$, the standard coboundary operator is introduced: $$\begin{gathered}
(\delta \alpha)(a_1,\ldots,a_{n+1}) = \sum_{i=0}^{n+1} (-1)^{(i+1)}
a_i \cdot \alpha (a_1,\ldots,\hat{a_i},\ldots,a_{n+1}) \\ \phantom{(\delta
\alpha)(a_1,\ldots,a_{n+1}) =}{}+ \sum_{i < j} \alpha ([a_i,a_j],a_1, \ldots,
\hat{a_i},\ldots,\hat{a_j},\ldots,a_{n+1}) \end{gathered}$$ where $\alpha$ is one $n$ cochain on ${\rm Lie}(G)$ with values in $C^{\infty}(M)$ (i.e. an alternate multilineal map), $a_1,a_2,\ldots,a_{n+1} \in {\rm Lie}(G)$, the symbol ‘\^’ meaning that the variable under it has been deleted and the symbol ‘$\cdot$’ denoting the action of ${\rm Lie}(G)$ on $C^{\infty}(M)$.
The space $Z^n({\rm Lie}(G), C^{\infty}(M))$ of $n$-cocycles consists of $n$ cochain $\alpha$ with $\delta \alpha = 0$ and the space $B^n({\rm Lie}(G), C^{\infty}(M))$ of $n$-coboundaries consists of $n$ cochain such that exists some $(n-1)$ cochain $\beta$ with $\alpha
= \delta \beta$.
The cohomology groups are defined as $H^n({\rm Lie}(G),C^{\infty}) =
\displaystyle{\frac {Z^n({\rm Lie}(G), C^{\infty}(M)} {B^n({\rm Lie}(G),
C^{\infty}(M))}}$.
It is well known that the second cohomology group $H^2({\rm Lie}(G),C^{\infty}(M))$ is related to the extensions of ${\rm Lie}(G)$ by $C^{\infty}(M)$.
The semidirect sum of ${\rm Lie}(G)$ and $C^{\infty}(M)$ consists of pairs $(a,f) \in
{\rm Lie}(G) \times C^{\infty}(M)$ with the Lie commutator $$[(a,f),(b,g)] =
([a,b],\; a\cdot f - b\cdot g).$$
Let $\alpha \in H^2({\rm Lie}(G),C^{\infty}(M))$. We can try to define a modified commutator by $$[(a,f),(b,g)]_{\alpha} = ([a,b],\; a \cdot f - b \cdot g + \alpha
(a,b)].$$
(The Jacobi identity for the modifed commutator is easily seen to be equivalent to the cocycle condition $\delta \alpha =0$.)
So, each $\alpha \in H^2({\rm Lie}(G),C^{\infty}(M))$ defines a new Lie algebra.
Let $\alpha_1$ and $\alpha_2 \in Z^2({\rm Lie}(G),C^{\infty}( M)).$ The Lie algebras formed from these cocycles are isomorphic through a mapping of the type $\phi (a,f) = (a, f + \beta(a))$, where $\beta \in H^1({\rm Lie}(G), C^{\infty}(M))$.
The condition $$[\phi(a,f),\phi(b,g)]_{\alpha_1} = \phi ([(a,f),(b,g)]_{\alpha_2})$$ is the same as $\alpha_1 - \alpha_2 = \delta \beta$ with $\beta \in H^1({\rm Lie}(G),C^{\infty}(M))$ (i.e. $\alpha_1 $ and $\alpha_2$ are cohomologous: $\alpha_1 \simeq \alpha_2$).
Thus, up to an isomorphism of the above type the Lie algebra extensions are parametrized by elements of $H^2({\rm Lie}(G),C^{\infty}(M))$.
Real valued cocycles and coboundaries can be recovered from the previous construction just by considerig real valued cochains as constant functions.
Now, we introduce the $C^\infty (M)$-valued two-cocycle on ${\rm Lie} (G)$ canonically associated to the action of $G$ mentioned above.
Let $\Omega(a,b)(x) = \omega({\tilde{X}_a},{\tilde{X}_b})(x)$ $\forall \; a, b\in {\rm Lie}(G)$ and $x \in M$. Then, $\Omega$ is a $C^\infty
(M)$-valued two-cocycle on ${\rm Lie}(G)$.
$$\begin{gathered}
(\delta \Omega)(a,b,c) = a \cdot \Omega(b,c) - b \cdot
\Omega(a,c) + c \cdot \Omega(a,b)\\
\phantom{(\delta \Omega)(a,b,c) =}{} - \Omega([a,b],c) + \Omega ([a,c],b) - \Omega ([b,c],a)\\
\phantom{(\delta \Omega)(a,b,c)}{}= - L_{a} \omega ({\tilde{X}_b},{\tilde{X}_c})
+ L_{{\tilde{X}_b}} \omega ({\tilde{X}_a},{\tilde{X}_c}) - L_{{\tilde{X}_c}} \omega ({\tilde{X}_a},{\tilde{X}_b})\\
\phantom{(\delta \Omega)(a,b,c) =}{}- \omega ({\tilde{X}_c}, \tilde{X}_{[a,b]}) + \omega ({\tilde{X}_b}, \tilde{X}_{[a,c]}) -
\omega ({\tilde{X}_a}, \tilde{X}_{[b,c]})\\
\phantom{(\delta \Omega)(a,b,c)}{} = - {\tilde{X}_a} \omega({\tilde{X}_b},{\tilde{X}_c}) + \omega ([{\tilde{X}_a},{\tilde{X}_b}],{\tilde{X}_c}) + \omega ({\tilde{X}_b},[{\tilde{X}_a},{\tilde{X}_c}])\\
\phantom{(\delta \Omega)(a,b,c) =}{}+ {\tilde{X}_b} \omega({\tilde{X}_a},{\tilde{X}_c}) - \omega ([{\tilde{X}_b},{\tilde{X}_a}],{\tilde{X}_c}) - \omega ({\tilde{X}_a},[{\tilde{X}_b},{\tilde{X}_c}])\\
\phantom{(\delta \Omega)(a,b,c) =}{}- {\tilde{X}_c} \omega({\tilde{X}_a},{\tilde{X}_b}) + \omega ([{\tilde{X}_c},{\tilde{X}_a}],{\tilde{X}_b}) + \omega ({\tilde{X}_a},[{\tilde{X}_c},{\tilde{X}_b}])\\
\phantom{(\delta \Omega)(a,b,c) =}{} - \omega([{\tilde{X}_a},{\tilde{X}_b}],{\tilde{X}_c}) + \omega([{\tilde{X}_a},{\tilde{X}_c}],{\tilde{X}_b}) - \omega([{\tilde{X}_b},{\tilde{X}_c}],{\tilde{X}_a})\\
\phantom{(\delta \Omega)(a,b,c)}{}= - {\tilde{X}_a} \omega({\tilde{X}_b},{\tilde{X}_c}) + {\tilde{X}_b} \omega({\tilde{X}_a},{\tilde{X}_c}) - {\tilde{X}_c} \omega({\tilde{X}_a},{\tilde{X}_b})\\
\phantom{(\delta \Omega)(a,b,c) =}{}+ \omega ([{\tilde{X}_a},{\tilde{X}_b}],{\tilde{X}_c}) - \omega ([{\tilde{X}_a},{\tilde{X}_c}],{\tilde{X}_b}) + \omega ([{\tilde{X}_b},{\tilde{X}_c}],{\tilde{X}_a})\\
\phantom{(\delta \Omega)(a,b,c)}{} = -3 (d \omega) ({\tilde{X}_a},{\tilde{X}_b},{\tilde{X}_c}) = 0,
\end{gathered}$$
where $d$ denotes the usual exterior differential operator.
The cocycle $$\Omega(a,b)(x) = \omega ({\tilde{X}_a},{\tilde{X}_b})(x)$$ will be called the [**canonical cocycle**]{} associated to the action of $G$ on $(M,\omega)$.
Thus, each action of $G$ on $M$ yields an extension of ${\rm Lie} (G)$ by $C^\infty (M)$.
When the action of $G$ is symplectic, $\Omega (a,b)(x)$ and $\Omega (a,b)(x_0)$ are cohomologous as real valued two cocycles on ${\rm Lie} (G)$ for every $x, x_0 \in M$. Then, in this case, the element $ [\Omega (a,b)(x)]$ in $H^2({\rm Lie}
(G), {\mathbb R})$ is independent of $x$. We shall denote it by $\tilde{\Omega} (a,b)$. In this way, a central extension of ${\rm Lie}(G)$ can be defined.
On the other hand, assuming, as henceforth we shall do for the symplectic actions to be considered, that the action of $G$ admits momentum mappings $J_a$ (that is, for each $ a \in
{\rm Lie}(G)$ there exists $J_a \in C^{\infty}(M)$ such that $d J_a =
i_{{\tilde{X}_a}} \omega$), $\forall \; a,b \in {\rm Lie}(G)$ the function $$\Sigma (a,b)(x):= \{J_a,J_b\}(x) - J_{[a,b]}(x),$$ with $\{\;,\; \}$ the Poisson bracket associated to $\omega$, turns out to be constant on $M$ and, as a function of $(a,b)$, it is a real valued two cocycle on ${\rm Lie} (G)$ [@am; @w]. This cocycle will be denoted by $\tilde {\Sigma} (a,b).$
Since for any $x_0 \in M$, $J_{[a,b]}(x_0)$ is trivial in $H^{2}({\rm Lie} (G), {\mathbb R})$, then $\tilde {\Sigma} (a,b)$ is equivalent to $\{J_a,J_b\}(x_0)= \Omega
(a,b)(x_0)$ in this cohomology. Hence, as real two cocycles on ${\rm Lie}(G)$, $$\{J_a,J_b\}(x) - J_{[a,b]}(x) \simeq \tilde{\Omega} (a,b).$$
Thus, for symplectic actions, the momenta give rise to a representation of the central extension of ${\rm Lie}(G)$ determined by $\tilde{\Omega} (a,b)$.
According to F Toppan [@t], an anomaly appears in a classical dynamical system when the central extension associated to a symplectic action of $G$ on $(M,\omega)$ is nontrivial. Let us recall that a necessary condition for it to occur is that symplectic potentials are not preserved by the action [@am].
Now, for a not necessarily symplectic action of $G$, we shall show that, under some additional hypotheses, the extension of ${\rm Lie}( G)$ defined by $\Omega$ can still be represented by the Poisson brackets of some functions associated to the action of $G$.
More precisely, we shall see that if $\omega$ can be written as $$\omega = \omega _{i} + \Delta \omega$$ with $\omega_i$ $G$-invariant and $\Delta \omega$ a closed form (not necessarily non-degenerate) on $M$ such that $L_{{\tilde{X}_a}} \omega = L_{{\tilde{X}_a}} \Delta \omega$ and if $J_a$ are momentum maps corresponding to the symplectic action of $G$ on $(M,\omega_i)$, we have
Let $\Delta X_a$ denote the vector field ${\tilde{X}_a} - X_{J_a}$. Under the additional assumption $$\omega (\Delta X_a, \Delta X_b)=0 \quad \forall \; a, b \in {\rm Lie}(G)$$ it holds $$\{J_a,J_b\} - J_{[a,b]} \simeq \Omega (b,a) \quad \mbox{as} \quad
C^{\infty}(M)\mbox{-valued cocycle on}\quad {\rm Lie}(G).$$
$$\begin{gathered}
\{J_a,J_b\} = \omega(X_{J_a},X_{J_b}) = \omega ({\tilde{X}_a},{\tilde{X}_b}) -
\omega({\tilde{X}_a},\Delta X_b) - \omega(\Delta X_a,{\tilde{X}_b}) \\
\phantom{\{J_a,J_b\}}{} = \omega({\tilde{X}_a},{\tilde{X}_b}) + \omega(\Delta X_b,{\tilde{X}_a}) - \omega(\Delta X_a,{\tilde{X}_b}) \\
\phantom{\{J_a,J_b\}}{} = \omega({\tilde{X}_a},{\tilde{X}_b}) + \Delta \omega({\tilde{X}_b},{\tilde{X}_a}) - \Delta \omega({\tilde{X}_a},{\tilde{X}_b}) \\
\phantom{\{J_a,J_b\}}{} = \omega({\tilde{X}_a},{\tilde{X}_b}) - 2\Delta \omega({\tilde{X}_a},{\tilde{X}_b}). \end{gathered}$$
On the other hand, $$\begin{gathered}
(\delta J)(a,b) = a\cdot J_b - b\cdot J_a - J_{[a,b]} =
-L_{\tilde{X}_a} J_b + L_{\tilde{X}_b} J_a - J_{[a,b]} \\
\phantom{(\delta J)(a,b)}{} = -dJ_b({\tilde{X}_a}) + dJ_a({\tilde{X}_b}) - J_{[a,b]}\\
\phantom{(\delta J)(a,b)}{} = -\omega_i({\tilde{X}_b},{\tilde{X}_a}) + \omega_i({\tilde{X}_a},{\tilde{X}_b}) - J_{[a,b]} = 2\omega_i({\tilde{X}_a},{\tilde{X}_b}) - J_{[a,b]}. \end{gathered}$$ So, $J_{[a,b]} = 2\omega_i({\tilde{X}_a},{\tilde{X}_b}) - (\delta J)(a,b)$. Then, we conclude that: $$\begin{gathered}
\{J_a,J_b\} -J_{[a,b]} = \omega({\tilde{X}_a},{\tilde{X}_b}) - 2\Delta
\omega({\tilde{X}_a},{\tilde{X}_b}) - 2\omega_i({\tilde{X}_a},{\tilde{X}_b}) + (\delta J)(a,b) \\
\phantom{\{J_a,J_b\} -J_{[a,b]} }{}= \Omega(a,b) +(\delta J)(a,b) \end{gathered}$$ as we wanted.
Thus, under some additional hypothesis, noncentral extensions associated to nonsymplectic actions can also be represented by physically relevant functions and then considered as anomalies of classical dynamical systems.
The additional hypothesis in the previous proposition is obviously fulfilled by any symplectic action since, in this case, $\Delta X_a= 0$ $\forall \; a
\in {\rm Lie}(G)$. Then, as a $C^\infty (M)$-valued two cocycle on ${\rm Lie}\, (G)$, $$\Sigma (a,b)(x):= \{J_a,J_b\}(x) - J_{[a,b]}(x) \simeq \Omega(b,a)(x).$$
On the other hand, as mentioned above, $\Sigma (a,b)(x)$ is constant on $M$ and, as real valued cocycle, $$\begin{gathered}
\tilde{\Sigma} (a,b)= [\Sigma (a,b)(x)] \simeq \Omega(b,a)(x),\\
-\tilde{\Sigma} (a,b) = -[\Sigma (a,b)(x)]. \end{gathered}$$
Nevertheless, the opposite signs do not yield a contradiction. In fact, from the proof of Proposition 2 we have $$\begin{gathered}
\tilde{\Sigma} (a,b)= \{J_a,J_b\}(x_0) - J_{[a,b]}(x_0)=
\Omega(a,b)(x_0) + (\delta J)(a,b)(x_0) \\
\phantom{\tilde{\Sigma} (a,b)}{}=\Omega(a,b)(x_0) + 2\omega_i({\tilde{X}_a},{\tilde{X}_b})(x_0) - J_{[a,b]}(x_0)=
\Omega(a,b)(x_0) - J_{[a,b]}(x_0). \end{gathered}$$
To wit, $$\tilde {\Sigma} (a,b)- \omega_i({\tilde{X}_a},{\tilde{X}_b})(x_0) =-J_{[a,b]}(x_0).$$
So, in this case, it follows from the Proposition 2 that, as expected, $$\tilde {\Sigma} (a,b) \simeq \tilde {\Omega}(a,b)$$ as real valued cocycles.
It is shown in [@ci] that, under the same hypothesis as in the previous proposition, for any $C^{\infty}$-valued two cochain $h(a,b)$ on ${\rm Lie}
(G)$ such that $dh(a,b) = (\delta \alpha) (a,b)$, $\alpha (a)$ being a local symplectic potential of $-L_{\tilde X_a}\omega$, the following equality holds: $$d \{J_a,J_b\} - d J_{[a,b]} = d h(a,b),$$ and then, $$\{J_a,J_b\} - J_{[a,b]} = h(a,b) + c(a,b),$$ with $c(a,b)$ a real valued two cocycle on ${\rm Lie}(G)$.
Proposition 2 tells us that, by taking $h(a,b) = \Omega(a,b)+
(\delta J)(a,b)$, we get rid of $c(a,b)$. It is worth to notice that no explicit expression for $h(a,b)$ nor for $c(a,b)$ is given in [@ci].
[**A simple example.**]{} Let $G={\mathbb R}^2$ acting by translations on $Q={\mathbb R}^2$ and lift the action to $TQ \approx {\mathbb R}^4$.
Let the lagrangian $L:T {\mathbb R}^2 \simeq {\mathbb R}^4 \rightarrow {\mathbb R}$ be defined as $$L\big(q^1,q^2,\dot{q^1},\dot{q^2}\big)=\frac {1}{2} \big(\big(\dot{q^1}\big)^2+\big(\dot{q^2}
\big)^2\big) +{\big(q^1\big)}^2\dot{q^2}.$$
It is clear that $L$ can be written as $L=L^i + \Delta L$ with $L^i=\frac {1}{2}
\big(\big(\dot{q^1}\big)^2+\big(\dot{q^2}\big)^2\big)$ invariant under the action of ${\mathbb R}^2$ and $\Delta L= {\big(q^1\big)}^2\dot{q^2}$.
The Legendre transform is given by $$FL\big(q^1,q^2,\dot{q^1},\dot{q^2}\big)=\big(q^1,q^2,\dot{q^1},\dot{q^1} +
{(q^1)}^2\big).$$
The lagrangian two form $\omega_L= FL^*\big( dq^1 \wedge dp_1 + dq^2
\wedge dp_2\big)$ turns out to be $$\begin{gathered}
\omega_L=\frac {{\partial}^2L}{\partial \dot{q^1} \partial q^2}\,
d{q^1} \wedge d{q^2} + \frac {\partial^2L}{\partial \dot{q^1}
\partial \dot{q^2}} \, d{q^1} \wedge d{\dot{q^2}}+ \frac {{\partial}^2L}{\partial \dot{q^2} \partial q^1}\,
d{\dot{q^1}} \wedge d{q^1} \\
\phantom{\omega_L=}{}+ \frac {{\partial}^2L}{\partial
\dot{q^2} \partial \dot{q^1}} \, d{q^2} \wedge d{\dot{q^1}}
+ \frac {{\partial}^2L}{\partial \dot{q^1} \partial \dot{q^1}}\,
d{{q^1}} \wedge d{\dot{q^1}} + \frac
{{\partial}^2L}{{\partial}\dot{q^2} \partial \dot{q^2}}\, d{q^2}
\wedge d{\dot{q^2}}. \end{gathered}$$
So, $$\omega_L = d{q^1} \wedge d{\dot{q^1}} + d{q^2} \wedge
d{\dot{q^2}} - 2q^1 d{q^1} \wedge d{q^2}.$$
For $a=(a_1,a_2) \in {\mathbb R}^2$, the infinitesimal generator is $${\tilde{X}_a}=(a_1,a_2,0,0)=a_1 \frac {{\partial}}{{\partial}{q^1}} +
a_2 \frac {{\partial}}{{\partial}{q^2}}.$$
The momentum mapping $J_a:{\mathbb R}^2 \rightarrow {\mathbb R}$ must satisfy $$dJ_a=i_{{\tilde{X}_a}}\omega_L^i = i_{{\tilde{X}_a}}\omega_L^i=a_1 d\dot{q^1} + a_2
d\dot{q^2}$$ and $$dJ_a= \frac {\partial J_a}{\partial q^1}\,
dq^1 + \frac {\partial J_a}{\partial q^2} \, dq^2 + \frac
{{\partial}J_a}{{\partial}\dot{q^1}} \, d \dot{q^1} + \frac
{{\partial}J_a}{{\partial}\dot{q^2}} \, d \dot{q^2}.$$
We can take $$J_a=a_1 \dot{q^1} + a_2 \dot{q^2}.$$
The hamiltonian vector associated to the function $J_a$ by $\omega_L$ is $$X_{J_a}={\tilde{X}_a} - 2q^1 a_2 \frac {{\partial}}{{\partial}\dot{q^1}} - 2q^1 a_1
\frac {{\partial}}{{\partial}\dot{q^2}}.$$
Then, $$\Delta X_a=-2q^1\left(a_2 \frac {{\partial}}{{\partial}\dot{q^1}} +
a_1 \frac {{\partial}}{{\partial}\dot{q^2}}\right).$$
Now, it is easy to see that, for this example, it holds $$\omega_L(\Delta X_a,\Delta
X_b)=0 \quad \forall \; a, b \in {\rm Lie}\, ({\mathbb R}^2).$$
For $a=(a_1,a_2)$ and $b=(b_1,b_2) \in {\mathbb R}^2$, the canonical cocycle turns out to be $$\Omega(b,a)=\omega_L({\tilde{X}_b},{\tilde{X}_a})=-2q_1a_1b_2 + 2q^1b_1a_2.$$
Thus, $$\Omega((1,0),(0,1))=-2q^1.$$
On the other hand, a direct computation yields $$(\delta J)(a,b) = -L_{{\tilde{X}_a}} {\tilde{X}_b} +L_{{\tilde{X}_b}} {\tilde{X}_a} = -dJ_a ({\tilde{X}_b}) + dJ_b({\tilde{X}_a}) = 0.$$ So, $$\{J_a,J_b\}_{\omega _L} = \omega_L (X_{J_a},X_{J_b}) =
-2q_1a_1b_2 + 2q^1b_1a_2.$$
An extension involving the Mickelsson–Faddev cocycle
====================================================
The Mickelsson–Fadeev extension of ${\rm Map}\big(S^3,SU(3)\big)$ is a non central one of interest in Quantum Field Theory. It is related to the anomalous commutator of the constraints of the Gauss-law appearing in a $(3+1)$-dimensional Yang–Mills theory interacting with Weyl fermions.
Fifteen years ago, Faddeev [@f] computed this anomalous commutator by means of functional integration techniques in the following way:
The lagrangian of the theory is given by $${\mathcal L} = tr F^2 + i \bar \psi {\cal D}_A \psi,$$ where $\psi $ is a Weyl fermion, $A \in {\mathcal A}$, the space of connections on a trivial bundle over the manifold $S^4$ with structure group $SU(3)$, $F=dA$ is it associated curvture and ${\mathcal D}_A $ is the covariant operator Dirac coupled to $A$.
The quantization of the theory is carried out in stages. First, the fermionic path integral is computed and then, once the effective lagrangian is obtained, the bosonic integration is performed.
The first step yields a result which is not gauge invariant: $$W[A]= \int D \psi D \bar {\psi}\exp(i \bar \psi {\mathcal D}_A \psi)= \det {\mathcal D}_A .$$ In fact, since ${\mathcal D}_A $ is an unbounded operator, its determinant is divergent, a regularizing method must be applied, and this procedure gives rise to a not gauge invariant result.
Let us notice that the variation of $W[A]$ coincides with the variation of the Wess–Zumino–Witten lagrangian and is the integrated anomaly of the theory: $$W[A^g]
= \exp(i {\alpha}_{1}(A;g)) W[A],$$ where $g \in {\mathcal G}$, the gauge group, and ${\alpha}_{1}(A;g)$ is a 1-cocycle on ${\mathcal G}$ with values in $C^{\infty}
(\mathcal A)$.
Now, for the second step, the lagrangian that must be considered is the effective one: $${\mathcal L}_{E} (A) = \frac {1}{2}\,{\rm tr}\, F_{\mu \nu}F^{\mu \nu} + W[A].$$
Now, the constraints of the Gauss-law are just the momenta (i.e. the Noether charges) associated to the action of the gauge group with respect to the original lagrangian.
By using the Johnson–Bjrken–Low method [@fsb], Fadeev obtained the noncentral extension of ${\rm Lie}(G)$ represented by the equal-time commutators of the operators $\tilde {\boldsymbol{G}}_a$, the quantum representants of the momenta ${\boldsymbol{G}}_a$: $$[\tilde {\boldsymbol{G}}_a(x),\tilde {\boldsymbol{G}}_b(y)] [A] =
C^c_{ab} \tilde {\boldsymbol{G}}_c(y) \delta (x-y)
+ S_{(a,b)}[A](x,y),$$ where $ S_{(a,b)}[A](x,y) = \delta
(x-y) \cdot {\rm MF}_{(a,b)}[A]$ .
The two cocycle ${\rm MF}_{(a,b)}[A] = \displaystyle{\frac {1}{24 \pi} \int_{S^3}
d^{3}x \,{\rm tr}\,(A[da,db])}$ is [**the Mickelsson–Faddeev cocycle**]{}.
In order to relate the Mickelsson–Fadeev cocycle to a canonical one, we consider the symplectic manifold $(T
{\mathcal A},\omega_{{\mathcal L}_E})$, where $\omega_{{\mathcal L}_E}$ is the lagrangian form associated to the effective lagrangian ${\mathcal L}_E$.
Notice that $\omega _{{\mathcal L}_E}$ can be written as $\omega _{{\mathcal L}_E} = \omega
_{\rm can} + {\Delta }{{\omega}_E}$, where $\omega _{\rm can}$ is the symplectic and gauge-invariant structure associated to ${{\mathcal
L}_{\rm YM}}= \frac {1}{2} \,{\rm tr}\, F_{\mu \nu}F^{\mu \nu} $.
Let us consider the momentum map associated to the lift of the action of the gauge group ${\mathcal G}$ on ${\cal A}$ $$g \cdot A = A^g = g^{-1}Ag + g^{-1}dg$$ to $(T{\mathcal A},\omega _{\rm can})$.
These maps ${\boldsymbol{G}}_a: T{\mathcal A} \rightarrow {\mathbb R}$ are given by $d{\boldsymbol{G}}_a = i_{{{\tilde{X}_a}}}\omega _{\rm can}$ $\forall \; a \in {\rm Lie}({\mathcal G})$ where ${\tilde{X}_a}$ is the infinitesimal generator associated to $a\in {\rm Lie}
(\mathcal G)$.
As mentioned above, these maps are the constraints of the Gauss-law [@fs].
Let us consider the cocycle $\Omega _{\omega _{{\mathcal L}_E}}$ defined on ${\rm Lie} ({\mathcal G})$ with values in the ${\rm Lie}({\mathcal G})$-module $C^{\infty}(T{\mathcal A})$ canonically associated to the action de ${\mathcal G}$ on ${\mathcal A}$.
The following lemma shows that the additional hypothesis is fulfilled in this case.
If $\Delta X_a = {\tilde{X}_a} - X_{{\boldsymbol{G}}_a}$, where $X_{{\boldsymbol{G}}_a}$ is the Hamiltonian vector field of the momentum map ${\boldsymbol{G}}_a$ corresponding to $\omega _{{\mathcal L}_E}$, $$\omega_{{\mathcal L}_E} (\Delta X_a,\Delta X_b) = 0
\quad \forall \; a , b \in {\rm Lie}({\mathcal G}).$$
For any form $\omega$ as in Proposition 2 we have $$i_{\Delta X_a} \omega (\cdot) = i_{{\tilde{X}_a} - X_{{\boldsymbol{G}}_a }}\omega (\cdot) =
i_{{\tilde{X}_a}} \omega (\cdot) - i_{X_{{\boldsymbol{G}}_a}} \omega (\cdot) =
i_{{\tilde{X}_a}} \omega (\cdot) - i_{{\tilde{X}_a}} \omega _{\rm can} (\cdot) =
i_{\tilde{X}_a} \Delta \omega(\cdot).$$
Since in the example we are considering , $i_{\Delta X_a} \Delta
\omega _E= 0$ $\forall \; a \in {\rm Lie}({\mathcal G})$ (formula 4.37 in [@i]), then $$\begin{gathered}
\omega_{{\mathcal L}_E} (\Delta X_a,\Delta X_b) = \Delta \omega
_E({\tilde{X}_a},\Delta X_b) = 0 \quad \forall \; a, b \in {\rm Lie}({\mathcal G}).\tag*{\qed}
\end{gathered}$$
Now, it follows from Proposition 2 that $$\{J_a,J_b\} - J_{[a,b]} \simeq \Omega (b,a).$$
On the other hand, as shown in [@i], $$\{J_a,J_b\} - J_{[a,b]} \simeq {\rm MF}_{ (a,b)}+ c(a,b).$$ Then, modulo constants, ${\rm MF}_{(a,b)}$ turns out to be cohomologous to the canonical cocycle $\Omega (b,a)$.
[99]{}
Abraham R and Marsden J E, Foundations of Mechanics, Benjamim Cummings Reading, 1978.
Cariñena J F and Ibort L A, Noncanonical Groups of Transformations, Anomalies and Cohomology, [*J. Math. Phys.*]{} [**29**]{}, Nr. 3 (1988), 541–545.
Faddev L D, Operator Anomaly for the Gauss-Law, [*Phys. Lett.*]{} [ **B145**]{}, Nr. 1–2 (1984), 81–84.
Faddev L D and Shatashvili S L, Realization of Schwinger Term in the Gauss Law and the Possibility of Correct Quantization of a Theory with Anomalies, [*Phys. Lett.*]{} [**B167**]{}, Nr. 2 (1986), 225–228.
Faddev L D and Slavnov A A, Gauge Fields: Introduction to Quantum Theory, Addisson-Wesley Publishing Company, 1986.
Inamoto T, Symplectic Structures in the Chirally Gauged Wess– Zumino–Witten Model, [*Phys. Rev.*]{} [**45**]{}, Nr. 4 (1992), 1276–1290.
Toppan F, On Anomalies in Classical Dynamical Systems, [*J. Nonlinear Math. Phys.*]{} [**8**]{}, Nr. 3 (2001), 518–533.
Woodhouse N M J, Geometric Quantization, Claredon Press Oxford, 1992.
\[solomin-lastpage\]
|
---
abstract: 'Batch normalization (BN) has become a de facto standard for training deep convolutional networks. However, BN accounts for a significant fraction of training run-time and is difficult to accelerate, since it is memory-bandwidth bounded. Such a drawback of BN motivates us to explore recently proposed weight normalization algorithms (WN algorithms), i.e. weight normalization, normalization propagation, and weight normalization with translated ReLU. These algorithms don’t slow-down training iterations and were experimentally shown to outperform BN on relatively small networks and datasets. However, it is not clear if these algorithms could replace BN in large-scale applications. We answer this question by providing a detailed comparison of BN and WN algorithms using ResNet-50 network trained on ImageNet. We found that although WN achieves better [*training*]{} accuracy, the final [*test*]{} accuracy is significantly lower ($\approx 6\%$) than that of BN. This result demonstrates the surprising strength of the BN regularization effect which we were unable to compensate using standard regularization techniques like dropout and weight decay. We also found that training with WN algorithms is significantly less stable compared to BN, limiting their practical applications.'
author:
- |
Igor Gitman[^1]\
Carnegie Mellon University\
Pittsburgh, PA\
`igitman@andrew.cmu.edu`\
Boris Ginsburg\
NVIDIA\
Santa Clara, CA\
`bginsburg@nvidia.com`\
bibliography:
- 'main.bib'
title: 'Comparison of Batch Normalization and Weight Normalization Algorithms for the Large-scale Image Classification'
---
Introduction
============
Batch normalization (BN) ([@ioffe2015batch]) is used in most modern convolutional neural network architectures including ResNet networks ([@he2016deep], [@he2016identity], [@xie2016aggregated], [@zagoruyko2016wide]) and the latest Inception networks ([@ioffe2015batch], [@szegedy2016rethinking], [@szegedy2017inception]). It is generally observed that BN speeds-up training by improving the conditioning of the problem and easing the back-propagation of the gradients ([@ioffe2015batch], [@xie2017all]). Thus, the total number of iterations needed for convergence is decreased. On the other hand, even though BN is not computationally intensive, the per-iteration time[^2] could be noticeably increased. This is because BN is memory-bandwidth limited, since it requires two passes through the input data: first to compute the batch statistics and then to normalize the output of the layer. For example, BN takes about $\sfrac{1}{4}$ of the total training time of the ResNet-50 network ([@he2016deep]) on the ImageNet classification problem ([@ILSVRC15]) using Titan X Pascal GPU (see figure \[fig:bntime\]). Moving to the new Volta GPUs, BN is going to take an even bigger proportion of run-time, because convolutions are easier to optimize for.
There are many alternative normalization algorithms that could be used instead of BN for training deep neural networks. These algorithms can be roughly divided into three groups. The first group is based on the idea of extending the normalization to different dimensions of the output. This group includes Layer Normalization ([@ba2016layer]) which uses channel dimension instead of a batch dimension to perform normalization, Instance Normalization () which only normalizes over the spatial dimensions of the output and Divisive Normalization () which is applied over channel dimension as well as over a local spatial window around each neuron. The second group consists of direct modifications to the original batch normalization algorithm. This group includes such methods as Virtual BN () which proposes to use a separate and fixed batch for each example in order to perform the normalization, Ghost BN ([@hoffer2017train]) in which normalization is performed independently across different splits of the batch, and Batch Renormalization ([@ioffe2017batch]) or Streaming Normalization ([@liao2016streaming]) which both modify the original algorithm to use global averaged statistics instead of the current batch statistics. The final group includes algorithms based on the idea of normalizing weights instead of activations. This group consists of Weight Normalization ([@salimans2016weight]), Normalization Propagation ([@arpit2016normalization]) and Weight Normalization with Translated ReLU (). These algorithms are all based on the idea of dividing weights by their $l_2$ norm and differ only in minor details; we commonly refer to them as weight normalization algorithms (WN algorithms).
In this paper we focus our analysis on WN algorithms, since they are not memory-bandwidth limited, and thus don’t slow-down training iterations. It is known that WN algorithms perform as well as or better than BN for relatively small networks and datasets like CIFAR-10 ([@krizhevsky2009learning]). However, there has not been much work on comparing them on the type of large-scale problems encountered in practice. In this paper we provide such a comparison by training ResNet-50 on the ImageNet dataset. This comparison demonstrates that it is possible to get better [*training*]{} curves with WN algorithms than using BN. However, the final [*test*]{} accuracy of WN is 6% less than that of a BN-based network. This reveals the surprising strength of the BN regularization effect which we were not able to replicate using traditional regularization techniques such as weight decay and dropout ([@srivastava2014dropout]). We also observed that when training very deep networks (i.e. ResNet-50), WN algorithms only partially normalize activations and thus the norm of the output increases from layer to layer. This instability grows after gradient updates, sometimes causing networks to diverge in the middle of training. We give an explanation of this instability and argue that it is an inherent property of WN algorithms which limits their practical applications.
![Distribution of training computation time for ResNet-50 on ImageNet using Titan X Pascal GPU. Batch normalization takes about $\sfrac{1}{4}$ of the total runtime. This graph is a courtesy of M. Milakov, NVIDIA.[]{data-label="fig:bntime"}](images/bntime.png){width="40.00000%"}
Related work
============
[@salimans2016weight] show that WN can achieve better accuracy than BN on the problems of image classification, generative modeling, and reinforcement learning. For the case of image classification, the algorithms were compared using a 12-layer ConvPool-CNN-C network ([@springenberg2014striving]) on the CIFAR-10 dataset. [@arpit2016normalization] provides a more extensive comparison of BN with their normalization propagation algorithm. They demonstrate superior results on , CIFAR-100 and SVHN ([@netzer2011reading]) datasets using 12-layer Network in Network ([@lin2013network]) architecture. However, in both cases the comparison for image classification is limited to relatively shallow networks and small datasets. Thus the provided results don’t reveal how well these algorithms scale to more practical networks and datasets, which is highlighted in our work.
Weight normalization with translated ReLU was introduced by [@xiang2017effects] in the context of generative adversarial networks ([@goodfellow2014generative]) where it was shown to achieve superior results to BN. However no comparison was provided for the case of image classification problem.
[@shang2017exploring] provides a comparison of normalization propagation and weight normalization with BN for deep residual networks trained both on CIFAR-10, CIFAR-100 datasets as well as on ImageNet. They also find that WN algorithms can’t replace BN since they have at least $3\%$ gap in test top-1 accuracy. However, they don’t explore an overfitting issue and don’t experiment with weight normalization with translated ReLU. Overall, the work of Shang et al is very similar to ours, but we were not aware of their results at the time of conducting our experiments.
Normalization Algorithms
========================
In this section we give a description of all normalization algorithms explored in the paper in the context of convolutional neural networks. We highlight the underlying theoretical assumptions as well as practical issues related to each algorithm.
Batch Normalization
-------------------
Batch normalization was introduced by [@ioffe2015batch] as a technique for accelerating neural network training by standardizing the distribution of the inputs for each layer. The purpose of the algorithm is two-fold: first it helps to reduce variation of the input distribution to each layer, which the authors refer to as internal covariate shift. Second, BN partially solves the problem of vanishing and exploding gradients ([@pascanu2013difficulty]) which is especially severe for very deep neural networks ([@xie2017all]). During training, BN layer performs the following operation for each channel (feature map) $j=1 \dots C$: $$o_j = \gamma_j\frac{x_j - {\mathbb{E}}_B{\left[x_j\right]}}{\sqrt{{\text{Var}}_B{\left[x_j\right]} + \epsilon}} + \beta_j$$ Here $o$ is the output and $x$ is the input of the BN layer, ${\mathbb{E}}_B{\left[x_j\right]}$ and ${\text{Var}}_B{\left[x_j\right]}$ are the mean and variance of the $j^{\text{th}}$ channel with respect to all pixels in the current mini-batch: $${\mathbb{E}}_B{\left[x_j\right]} = \frac{1}{mHW}\sum_{i=1}^m\sum_{p=1}^H\sum_{q=1}^W x^i_{jpq}$$ where $m$ is the number of samples in a mini-batch, $H$ and $W$ are the height and width of the channel. ${\text{Var}}_B{\left[x_j\right]}$ is defined analogously. $\epsilon$ is a small constant used for numerical stability. Scale $\gamma_j$ and bias $\beta_j$ are optional parameters, usually used in the case when the BN layer is applied after convolution and before non-linearity. If we denote the total layer dimension with $D := mHWC$ then the norm of the BN output is always equal to $\sqrt{D}$, preventing the signal from vanishing or exploding during the forward pass. This in turn helps to reduce the problem of vanishing or exploding gradients. It should be noted that BN only works if $m$ is large enough, since ${\mathbb{E}}_B{\left[x_j\right]}$ and ${\text{Var}}_B{\left[x_j\right]}$ must approximate the true population statistics. Thus the algorithm is not well suited for cases when the mini-batch size is small or when training in an online setting.
Weight Normalization Algorithms
-------------------------------
**Weight Normalization** (WN) was introduced by [@salimans2016weight] as an alternative to batch normalization. The idea of WN is to decouple the direction of the weights from their norm and thereby improve the conditioning of the optimization problem. For example, for the convolutional layer, weights have to be normalized and multiplied by a learned scaling parameter: $$o_j = \gamma_j\frac{W_j * x}{{\left\|W_j\right\|}_F + \epsilon} + \beta_j$$ Here $x, o, \gamma, \beta$ and $\epsilon$ are defined as in BN, $W$ are the layer weights, ${\left\|W_j\right\|}_F$ is the Frobenius norm of weights for output channel $j$, and $*$ denotes convolution.
[@arpit2016normalization] showed that when $W$ is close to orthogonal[^3] and the input is normalized then $${\mathbb{E}}_B{\left[o\right]} \approx {\mathbb{E}}_B{\left[x\right]} = 0, {\text{Var}}_B{\left[o\right]} \approx {\text{Var}}_B{\left[x\right]} = I \Rightarrow {\left\|o_j\right\|}_2 \approx \sqrt{\frac{D_{l+1}}{D_l}}{\left\|x\right\|}_2$$ for each layer $l$. When $D_l=D_{l+1} \Rightarrow {\left\|o_j\right\|}_2={\left\|x\right\|}_2$ meaning that WN will propagate the normalization through convolutional layers. Which implies that WN can be considered as an alternative to BN. However, the above analysis doesn’t take into account the non-linear layers of the network. For example, consider the case when ReLU ([@nair2010rectified]) is applied after convolution. Assume that the output $o$ of the convolutional layer is normalized: ${\mathbb{E}}_B{\left[o\right]} \approx 0$, $ {\left\|o\right\|}_2 \approx \sqrt{D}$. Then the output of ReLU would be shifted: ${\mathbb{E}}_B{\left[\text{ReLU}(o)\right]} > 0$, and its norm would be decreased: ${\left\|\text{ReLU}(o)\right\|}_2 \approx \sqrt{\frac{D}{2}}$.
[@arpit2016normalization] suggested the **Normalization Propagation** (NP) algorithm which can be thought of as a modification of WN to account for the ReLU non-linearity. Assuming the input data follows a Gaussian distribution, the output of the ReLU follows a Rectified Gaussian distribution, making it possible to analytically perform re-normalization of the ReLU output. The update rule for the combined convolutional and ReLU layers becomes the following: $$o_j = \frac{1}{\sqrt{\frac{1}{2}{\left(1 - \frac{1}{\pi}\right)}}}{\left[\text{ReLU}{\left(\gamma_j\frac{W_j * x}{{\left\|W_j\right\|}_F + \epsilon} + \beta_j\right)} - \sqrt{\frac{1}{2\pi}}\right]}$$ The above equation was derived under assumption that $\gamma_j = 1$ and $\beta_j = 0$. Since $\gamma$ and $\beta$ change during training, [@xiang2017effects] propose to simplify the NP update rule resulting in the **Weight Normalization with Translated ReLU** (TReLU WN) algorithm: $$o_j = \text{TReLU}_{\alpha_j}{\left(\frac{W_j * x}{{\left\|W_j\right\|}_F + \epsilon}\right)}:=\text{ReLU}{\left(\frac{W_j * x}{{\left\|W_j\right\|}_F + \epsilon} - \alpha_j\right)} + \alpha_j$$ Bias $\beta$ and scale $\gamma$ are applied only to the output of the last layer to restore the representational power of the network: $$o^{\text{last}}_j = \gamma_jx^{\text{last}}_j + \beta_j$$
Batch Norm vs Weight Norm
-------------------------
It is clear that WN algorithms serve the same goal as BN: they normalize the layers activations throughout the network. However, BN performs explicit normalization by requiring the norm of the output to be exactly equal to a fixed number. WN, on the other hand, uses implicit normalization which results in the norm of the output being approximately the same as the norm of the input. In practice this difference is crucial, since it means that for WN algorithms the normalization errors might be exponentially increased throughout the network. We confirm this observation experimentally and discuss it’s practical significance in section \[sect:analysis\]. A high-level comparison between batch and weight normalization is presented in table \[tab:bn\_vs\_wn\].
Experiments
===========
CIFAR-10
--------
Our first experiments use a relatively small CIFAR-10 dataset ([@krizhevsky2009cifar10]). We compare the performance of batch normalization with three weight normalization algorithms (where weight normalization was applied after each layer) using the *cifar10-nv* architecture, a simple 12-layer network that achieves close to state-of-the-art performance in less than 1 hour of training time. The complete network architecture is presented in table \[tab:cifar10-nv\]. As a baseline we use cifar10-nv without normalization.
We trained all networks using Stochastic Gradient Descent (SGD) with momentum of 0.9 for 256 epochs with batch size of 128. We chose the best learning rate schedule and weight decay for each algorithm[^4]. Results are presented in figure \[fig:cifar10\].
![Batch normalization vs Weight normalization: training of cifar10-nv network for CIFAR-10 dataset.[]{data-label="fig:cifar10"}](images/cifar_plot.pdf){width="100.00000%"}
With these results we again confirm that WN algorithms outperform BN for a moderately small networks and datasets. We can see that WN algorithms have both better training curves (they converge faster) as well as higher final test accuracy (with the exception of the original weight normalization). However, we again emphasize that this result is not of a big practical significance since with this network size even if training is done with no normalization the final accuracy is comparable to that of a normalized network. Thus, in order to truly assess the power of different normalization techniques it is important to conduct experiments on large datasets and very deep networks which are difficult to train without some kind of normalization applied.
ResNet-50 on ImageNet
---------------------
For the large-scale experiments we trained a ResNet-50 ([@he2016deep]) network on the ImageNet. We evaluated ResNet-50 with batch normalization and weight normalization algorithms[^5].
For the original WN algorithm, the best top-1 accuracy was achieved using the same configuration as for BN: initial learning rate of 0.1, which was decreased to 0 using polynomial with power 2 decay schedule, and weight decay of 0.0001. The final comparison results are presented in figure \[fig:imagenet\].
[0.49]{} ![Batch normalization vs Weight normalization: training of ResNet-50 on ImageNet dataset. Weight norm is able to better stabilize training and helps network to converge faster to higher training accuracy. However, the test accuracy is significantly lower for WN ($\approx 67\%$ vs $\approx 73\%$ for BN). This figure reveals a surprising strength of the regularization effect of batch normalization.[]{data-label="fig:imagenet"}](images/imagenet-train-acc.pdf "fig:"){width="100.00000%"}
[0.49]{} ![Batch normalization vs Weight normalization: training of ResNet-50 on ImageNet dataset. Weight norm is able to better stabilize training and helps network to converge faster to higher training accuracy. However, the test accuracy is significantly lower for WN ($\approx 67\%$ vs $\approx 73\%$ for BN). This figure reveals a surprising strength of the regularization effect of batch normalization.[]{data-label="fig:imagenet"}](images/imagenet-test-acc.pdf "fig:"){width="100.00000%"}
[0.49]{} ![Batch normalization vs Weight normalization: training of ResNet-50 on ImageNet dataset. Weight norm is able to better stabilize training and helps network to converge faster to higher training accuracy. However, the test accuracy is significantly lower for WN ($\approx 67\%$ vs $\approx 73\%$ for BN). This figure reveals a surprising strength of the regularization effect of batch normalization.[]{data-label="fig:imagenet"}](images/imagenet-train-loss.pdf "fig:"){width="100.00000%"}
[0.49]{} ![Batch normalization vs Weight normalization: training of ResNet-50 on ImageNet dataset. Weight norm is able to better stabilize training and helps network to converge faster to higher training accuracy. However, the test accuracy is significantly lower for WN ($\approx 67\%$ vs $\approx 73\%$ for BN). This figure reveals a surprising strength of the regularization effect of batch normalization.[]{data-label="fig:imagenet"}](images/imagenet-test-loss.pdf "fig:"){width="100.00000%"}
Interestingly, weight norm outperforms batch norm in terms of convergence speed and final [*training*]{} accuracy (figure \[fig:imagenet\] (a), (c)). However, final [*test*]{} accuracy of WN is significantly lower: accuracy gap[^6] (figure \[fig:imagenet\] (b), (d)). Clearly, weight normalization suffers from overfitting which we were not able to decrease by using dropout or increasing weight decay. This result reveals the surprising strength of the regularization effect of batch normalization. It is generally observed that BN helps to prevent overfitting and reduces the need for other regularization methods, like dropout or weight decay. However, in this situation we want to emphasize the size of this regularization effect ($6\%$ for the final test accuracy) and the fact that it was not possible to achieve the same performance using dropout or weight decay for WN. Understanding the precise reason for such strong generalization is a direction for the future research.
We found the training of Resnet-50 with other flavors of WN to be problematic. Training with normalization propagation required a modification of the original algorithm to work with residual connections to ensure that output of each layer stays normal under the normal distribution assumption. And even with that modification network either converged to a poor local minimum or diverged after a few iterations if the initial learning rate was large. Training proceeded further using weight normalization with translated ReLU but tended to diverge suddenly after training for as much as 50 epochs. We discuss the reason for this instability in section \[sect:analysis\].
Weight Normalization analysis {#sect:analysis}
=============================
Theoretically, weight normalization should ensure that the outputs of the network layers are approximately normalized, assuming that the input data is normalized and weight matrices are close to orthogonal. While this assumption holds for random Gaussian initialization, it might be sufficiently violated later during the training. In practice we observed that since the weights change together, each gradient update increases correlations between different neurons, thus violating the orthogonality assumption. Moreover, if each layer increases the norm of its output even by a small fraction, the error will be exponentially magnified throughout the network. This might explain the sudden divergence of TReLU WN algorithms in the middle of the training. Figure \[fig:imagenet-outputs\] shows the norm of the outputs of the first and last layers of ResNet-50 during training of the usual weight normalization algorithm. As one can see, WN does not ensure complete normalization of the network. It should be noted that this effect is an inherent property of weight normalization itself and not specific to residual connections, activation function, bias term or convolutional operation.
![Norm of the output for the first and last layers of ResNet-50 with WN training. This figure demonstrates that weight normalization fails to fix the output norm throughout network layers since weight matrices start to violate orthogonality assumptions during training.[]{data-label="fig:imagenet-outputs"}](images/imagenet_outs.pdf){width="100.00000%"}
To verify this hypothesis we trained a simple [*linear*]{} (with no bias) 50-layer fully-connected network on ImageNet. Each layer except for the first and last was $64$ neurons. With a wide range of learning rates the network completely diverged in less than a hundred iterations due to the broken orthogonality assumptions. This experiment demonstrates the limitations of weight normalization approach for training of very deep networks.
Conclusion
==========
In this paper we provide the comparison of batch normalization and weight normalization algorithms for the large-scale image classification problem (i.e. ResNet-50 on ImageNet). We found that while having better training curves, WN algorithms show about $6\%$ lower test top-1 accuracy which cannot be restored by using dropout or increasing weight decay. This demonstrates that BN has a much stronger regularization effect than previously observed. We also demonstrated that WN algorithms are significantly unstable when applied to deep networks and can’t completely normalize activations. We therefore conclude that WN algorithms are limited in practical application to relatively shallow networks and cannot replace batch normalization for large-scale problems.
[^1]: The work was done during an internship at NVIDIA
[^2]: time required to run one back-propagation update
[^3]: For fully-connected layer with weights $W \in R^{n\times m}$ coherence of the rows: $\max_{i\neq j} \frac{{\left|w_i^Tw_j\right|}}{{\left\|w_i\right\|}_2{\left\|w_j\right\|}_2}$ has to be small which is satisfied for standard random initializations. Analogous result holds for convolutional layers.
[^4]: During training we sampled a random crop of size $28 \times 28$ and performed random horizontal flips of the images. During testing only the central $28 \times 28$ crop was used. Input data was normalized by subtracting mean and dividing by standard deviation independently for each pixel. Weights were initialized using Xavier algorithm ([@glorot2010understanding]). For all algorithms we used SGD with momentum with initial learning rate (lr) of 0.01. For NP and TReLU WN learning rate was decreased to $10^{-5}$ using a polynomial decay with power 2. For other methods we used a linear lr decay rule. For WN the final lr was $10^{-4}$ and for BN and network with no normalization final lr was $10^{-5}$ .We didn’t use weight decay (wd) for WN and for network with no normalization. For NP and TReLU WN wd=0.001 and for BN wd=0.002.
[^5]: For all experiments we used a workstation with 4 Tesla P100 GPUs. Networks were trained for 120 epochs. Batch size was 256 with 64 samples per GPU. Batch normalization was applied separately for each GPUs local chunk of a mini-batch. Training images were rescaled to the size of $256 \times 256$ pixels and then randomly cropped to $224 \times 224$. We applied random color distortions and horizontal flips and finally normalized images to $[-1, 1]$ range. During testing images were rescaled to $256 \times 256$ and the central crop of size $224 \times 224$ was used. Weights were initialized using Xavier initialization.
[^6]: It should be noted that our final test top-1 accuracy of ResNet-50 with BN is lower then that of [@he2016deep]. This is likely because we used simpler data preprocessing and only one central crop during testing.
|
---
abstract: 'We prove sharp $\ell^{p}L^{p}$ decoupling inequalities for $2$ quadratic forms in $4$ variables. We also recover several previous results [@MR3736493; @MR3447712; @MR3848437; @arxiv:1609.04107] in a unified way.'
address:
- |
Department of Mathematics\
University of Wisconsin Madison\
USA
- |
Mathematical Institute\
University of Bonn\
Germany
author:
- Shaoming Guo
- 'Pavel Zorin-Kranich'
bibliography:
- 'decoupling.bib'
title: |
Decoupling for certain quadratic surfaces\
of low co-dimensions
---
=1
Introduction
============
Let $1\le n\le d$ be integers and $P_1, \dotsc, P_n$ real quadratic forms on $\R^{d}$. We study decoupling inequalities associated to the quadratic surface $$\calS = \calS_{d, n}:=\Set{(t, P_1(t), \dotsc, P_{n}(t)) \given t\in [0, 1]^d}.$$ For a subset $R\subset [0, 1]^d$ define an extension operator $$\label{extension_operator}
E_R g(x):=\int_R g(t)e^{2 \pi i(t_1 x_1+\dots+t_d x_d+P_1(t)x_{d+1}+\dots P_n(t)x_{d+n})}dt,
\quad
x\in\R^{d+n}.$$ For a ball $B=B(c_B, r_B)\subset \R^{d+n}$ and $E>0$, define an associated weight $$w_{B, E}(x):=\Bigl( 1+\frac{\abs{x-c_B}}{r_B} \Bigr)^{-E}.$$ Typically $E$ is a fixed number that is much bigger than $(d+n)$, and will be omitted from the notation $w_{B, E}$. All implicit constants are allowed to depend on $E$. For $\delta \in 2^{-\N}$ we will denote by $\Part{\delta}$ the set of all dyadic cubes with side length $\delta$ in $[0,1]^{d}$.
\[thm:main:extension\] Let $1\le n\le 3$ and $$\label{d_n_constraint}
\begin{cases}
d\ge 1 & \text{ if } n=1,\\
2 \leq d \leq 4 & \text{ if } n=2,\\
d=3 & \text{ if } n=3.
\end{cases}$$ Assume that for every choice of linearly independent vectors $\vec{w}_{1}, \dotsc, \vec{w}_{d-n}\in \R^d$, $$\label{bl_assumption}
\det [\nabla P_1(t); \dots ; \nabla P_n(t); \vec{w}_{1}; \dots; \vec{w}_{d-n} ]
\not\equiv 0$$ when viewed as a polynomial of $t$. Moreover, assume that for every hyperplane $H\subset \R^d$, $$\label{low_dim_assumption}
\rank \Big(\big(\lambda_1 P_1+\dotsb+\lambda_n P_n\big)|_H\Big)\ge d-2$$ for some $\lambda_{1}, \dotsc, \lambda_n\in \R$.
Let $2\le p\le 2+\frac{4n}{d}$, $\epsilon>0$, and $E>0$. Then for every locally integrable function $g$, every dyadic number $\delta \in 2^{-\N}$, and every ball $B\subset \R^{d+n}$ of radius $\delta^{-2}$ we have $$\label{eq:main:extension}
\norm{E_{[0, 1]^d}g}_{L^p(w_{B,E})}
\lesssim_{\epsilon,E}
\delta^{-d(\frac{1}{2}-\frac{1}{p})-\epsilon}
\Bigl( \sum_{\Delta \in \Part{\delta}} \norm{ E_{\Delta}g }_{L^p(w_{B,E})}^p \Bigr)^{1/p}.$$
Relation to previous work
-------------------------
Theorem \[thm:main:extension\] unifies several previous results summarized in the table below and provides a new result when $n=2$ and $d=4$, which is a sharp decoupling for a large class of four dimensional quadratic surfaces in $\R^6$.
$n$ $d$ Reference
----- ---------- ---------------------
$1$ $\geq 1$ [@MR3736493]
$2$ $2$ [@MR3447712]
$2$ $3$ [@arxiv:1609.04107]
$3$ $3$ [@MR3848437]
The arguments in the above listed papers are quite different from each other. This point will be elaborated in Section \[section:overview\]. Let us first be more precise about how these results can be recovered. Notice that we use cubes with side length $\delta$, while many articles use cubes with side length $\delta^{1/2}$.
When $n=1$, Bourgain and Demeter in [@MR3374964] and [@MR3736493] proved for every (possibly hyperbolic) paraboloid $\Set{(t, P_1(t)) \given t\in [0, 1]^d}$ with non-vanishing Gaussian curvature. In this case the implication $$\label{eq:codim1-hypotheses}
\rank (P_{1}) = d
\implies
\eqref{bl_assumption} \text{ and } \eqref{low_dim_assumption}$$ can be easily verified, see [@MR3736493 Lemma 2.6].
When $n=2$ and $d=2$, Bourgain and Demeter [@MR3447712] proved for quadratic forms $$P_1(t_1, t_2)= A_1 t_1^2+2A_2 t_1 t_2+ A_3 t_2^2,
\quad
P_2(t_1, t_2)= B_1 t_1^2+2B_2 t_1 t_2+ B_3 t_2^2$$ under the assumption $$\label{181212e1.9}
\rank
\begin{bmatrix}
A_1, & A_2, & A_3\\
B_1, & B_2, & B_3
\end{bmatrix}
=2.$$ Checking amounts to checking $$\det
\begin{bmatrix}
A_1 t_1+ A_2 t_2, & A_2 t_1+A_3 t_2\\
B_1 t_1+ B_2 t_2, & B_2 t_1+B_3 t_2
\end{bmatrix}
\not\equiv 0,$$ which follows immediately from . The condition in this case is trivial as $d-2=0$.
When $d=3$ and $n=2$, Demeter, Shi, and the first author [@arxiv:1609.04107] proved for two quadratic forms $P_1$ and $P_2$ under the assumption and the assumption that they do not share any common real factor. Under these assumptions, to verify , we just need to notice that $P_1|_H$ and $P_2|_H$ can not be simultaneously zero. Let us also mention here that the method used in the current paper significantly simplifies the proof in [@arxiv:1609.04107]. To see the major differences, we refer to Section \[section:overview\].
When $d=n=3$, Oh [@MR3848437] proved under the assumption of and the assumption that $P_1|_H, P_2|_H$ and $P_3|_H$ do not vanish simultaneously for any hyperplane $H$. In this case, our assumption is just a coordinate-invariant version of Oh’s assumptions.
We would also like to point out that a few other sharp decoupling inequalities for quadratic surfaces not covered by Theorem \[thm:main:extension\] are proved in [@MR3614930; @arxiv:1804.02488; @arxiv:1811.02207]. In many of those cases $n>d$, so that our assumption would not make sense there.
Necessity of hypotheses
-----------------------
Next, let us explain the assumptions and in the case $d=4$ and $n=2$. The assumption is a necessary condition for the desired sharp decoupling inequality. If it is not satisfied, then there exists a hyperplane $H$ such that for every $\lambda_1$ and $\lambda_2$, we have $$\rank \Big(\big(\lambda_1 P_1+\lambda_2 P_2\big)|_H\Big)\le 1.$$ This further implies that after changes of variables in $\R^{d}$ and in $\R^{n}$ one can parameterize the restriction of the surface $\calS_{d, n}$ to the plane $H$ as $$(t_1, t_2, t_3, A t_1^2, 0).$$ The $\ell^{p}L^{p}$ decoupling exponent for the parabola $(t_{1},A t_{1})$ is at least $(\frac{1}{2}-\frac{1}{p})$, while the $\ell^{p}L^{p}$ decoupling exponents for the lines $(t_{2})$ and $(t_{3})$ are at least $2(\frac{1}{2}-\frac{1}{p})$, see . Using tensor products of corresponding examples we are able to pick a function $g$ that is supported near $H$, such that $$\norm{E_{[0, 1]^4}g}_{L^p(w_{B^2})} \gtrsim \delta^{-5(\frac 1 2-\frac 1 p)}
\Bigl( \sum_{\Delta \in \Part{\delta}} \norm{E_{\Delta}g}_{L^p(w_{B^2})}^p \Bigr)^{1/p}.$$ This violates the desired decoupling inequality .
We do not know whether the assumption is necessary for the decoupling inequality to hold. However, our proof seems to suggest that it is a necessary condition to run the multilinear approach of Bourgain [@MR3038558] and Bourgain and Demeter [@MR3374964]. This is indeed the case when $d=3$ and $n=2$, which is the case considered in [@arxiv:1609.04107]. More precisely, it is proven there that if the condition fails, then no matter how many and which points we pick on the surface, they will never be “transverse” in the sense of Definition \[transversality\].
Theorem \[thm:main:extension\] also includes an important class of pairs of quadratic forms, namely those pairs of quadratic forms that are simultaneously diagonalizable.
\[lem:simul-diag\] Let $2\le d\le 4$. Write $t=(t_1, \dots, t_d)$. Take two quadratic forms $$P(t)=\sum_{j=1}^d a_j t_j^2 \text{ and } Q(t)=\sum_{j=1}^d b_j t_j^2.$$ Assume that $$\label{180713e1.9}
\rank \begin{bmatrix}
a_i & a_j\\
b_i & b_j
\end{bmatrix}
= 2
\qquad\text{ for every } i\neq j.$$ Then $P(t)$ and $Q(t)$ satisfy the assumptions of Theorem \[main\_theorem\] (with $n=2$).
The same non-degeneracy condition also appeared in a recent work [@MR3652248] by Heath-Brown and Pierce, see Page 95 there. Lemma \[lem:simul-diag\] is proved in Appendix \[sec:simul-diag\].
Overview of the proof {#section:overview}
---------------------
The proof of Theorem \[eq:main:extension\] follows the multilinear approach introduced in [@MR2860188] and further developed in [@MR3374964; @MR3548534; @MR3736493; @MR3709122; @MR3848437; @arxiv:1804.02488; @arxiv:1811.02207]. In Section \[sec:gen\] we formulate this argument for general quadratic surfaces under a lower-dimensional inductive assumption (Hypothesis \[hyp:lower-dim\]) and a transversality assumption (Hypothesis \[hyp:transverse\]). In Section \[sec:spec\] we show that these assumptions are satisfied under the conditions and .
In the multilinear approach, one uses the Bourgain–Guth argument from [@MR2860188] to split the quantity that is to be estimated in a lower-dimensional and a transversely multilinear part, see Section \[sec:gen:multilinear\] and Section \[sec:gen:bourgain-guth\]. The appropriate notion of transversality was introduced in [@MR3548534; @MR3614930; @MR3709122] and is explained in Section \[sec:gen:transverse\]. In Section \[sec:ball-inflation\] and Section \[sec:bourgain-demeter\] we run a version of the Bourgain–Demeter iteration argument from [@MR3374964] to complete the proof conditionally on Hypothesis \[hyp:lower-dim\] and Hypothesis \[hyp:transverse\].
The main new idea in Section \[sec:gen\] is the way how lower dimensional contributions are controlled in Section \[sec:gen:lower-dim\]. Specifically, if there is no significant transverse contribution to $E_{[0,1]^{d}}g(x)$, then the main contribution comes from a $1/K$ neighborhood of a low degree subvariety. In the simplest case when this subvariety is a hyperplane, previous work relied on showing that its $1/K$ neighborhood lies in the $1/K^{2}$ neighborhood of a certain cylinder, see for instance [@MR3736493] and [@arxiv:1609.04107]. This step, if possible, usually involves a large amount of linear algebra calculation, see for instance [@arxiv:1609.04107 Section 4]. In the current paper, we show that the step of fitting the $1/K$ neighborhood into the $1/K^{2}$ neighborhood of a cylinder is no longer necessary. This is the content of Theorem \[thm:near-hyperplane\]. The result for hyperplanes can be extended to graphs with controlled first and second order derivatives by an argument essentially due to Oh [@MR3848437], see Theorem \[thm:near-graph\].
We extend this result to arbitrary subvarieties in Theorem \[thm:dim-reduction-variety\]. In the case of hypersurfaces $\calS$ generated by monomials this was previously done in [@arxiv:1804.02488; @arxiv:1811.02207]. However, the projection argument in those articles seems to be specific to monomials. A major difficulty in the general case is how to treat singular points of the subvariety (or, more generally, regions where the curvature is high). To this end we cover a neighborhood of the subvariety by neighborhoods of a “small” number of graphs (with controlled first and second order derivatives), see Lemma \[lem:sublevel-decomposition-of-variety\]. It is not difficult to imagine that different scales of neighborhoods have to be involved, in order not to use too many graphs. These scales are called $K_1\ll\dots\ll K_{D+1}$ in Lemma \[lem:sublevel-decomposition-of-variety\]. It is a very interesting phenomenon that in our proof we require $\log K_i\approx_{d, n, \epsilon} \log K_j$ for every $i\neq j$. In particular, we are not allowed to pick, say $K_2=2^{K_1}$. This is important in the iteration in Section \[sec:bourgain-demeter\].
In the end of the overview, let us make a few comments on the differences among proofs in [@MR3374964; @MR3736493; @MR3447712; @arxiv:1609.04107; @MR3848437]. The proofs in [@MR3374964; @MR3736493] use a $(d+1)$-linear argument, based on the $(d+1)$-linear Loomis-Whitney inequality, while the proofs in [@MR3447712; @MR3848437] use a bilinear argument, based on certain change of variables, and the proof in [@arxiv:1609.04107] uses an $M$-linear argument with $M$ ranging in an interval of integers (the same as the current paper). The bilinear argument of [@MR3447712; @MR3848437] is specific to the case $d=n$, that is the dimension of the surface of a half of the dimension of the total space.
In terms of the Brascamp-Lieb data that are involved: In [@arxiv:1609.04107] the Brascamp-Lieb data that are used are always simple, in the sense that a strict inequality can be achieved for every proper linear subspace $V$. The Brascamp-Lieb data that appear in [@MR3374964; @MR3736493] are non-simple. However, as shown in the current paper (in particular Lemma \[lem:BCCT:n=1\]), one only needs to use simple data for (hyperbolic) paraboloids. On a more technical level, this is because the first alternative in Hypothesis \[hyp:transverse\] only occurs for the trivial subspace and the full space. In contrast, in the case $d=n$, one always needs to invoke non-simple Brascamp-Lieb data.
Decoupling theorems are sometimes formulated for functions with Fourier support in $\calS$. However, in order to use a lower-dimensional inductive assumption such as Hypothesis \[hyp:lower-dim\] one needs a version of the decoupling inequality that holds for functions with Fourier support in a $\delta^{2}$-neighborhood of the surface $\calS$. We find it convenient to use such a more general version throughout, thus avoiding some technical computations as e.g. in [@MR3592159 Section 5]. This more general form of the decoupling inequality is explained in Section \[sec:fourier-support\].
Relaxed Fourier support restriction {#sec:fourier-support}
-----------------------------------
In this section we formulate a decoupling inequality for functions with Fourier support in a neighborhood of the surface $\calS$. The boxes in the figure below show how the Fourier supports will look like at different scales in the case $d=n=1$.
[cc]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
&
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
[(-1.5,-1.5) rectangle (1.5,1.5);]{}
\
Scales $2^{0}$, $2^{-1}$, $2^{-2}$ & $2^{0}$, $2^{-3}$
We proceed with a formal definition. For $\theta = a + \delta [0,1]^{d} \in \Part{\delta}$ we will denote by $f_{\theta}$ an arbitrary function of the form $M_{\theta}f$, where $f$ is an arbitrary function on $\R^{d+n}$ with $\supp \widehat{f} \subset [-C,C]^{d+n}$, $C\geq 1$, and $$M_{\theta}f(x,y) = e(a\cdot x + P(a) \cdot y) (f \circ L_{\theta})(x,y),
\quad
x\in\R^{d}, y\in\R^{n},$$ where $$L_{\theta} =
\begin{pmatrix}
\delta I_{d} & 0\\
0 & \delta^{2} I_{n}
\end{pmatrix}
\begin{pmatrix}
I_{d} & \nabla P(a)\\
0 & I_{n}
\end{pmatrix},
\quad
\nabla P(a)
=
\begin{pmatrix}
\partial_{1} P_{1}(a) & \dots & \partial_{1} P_{n}(a)\\
\vdots & & \vdots \\
\partial_{d} P_{1}(a) & \dots & \partial_{d} P_{n}(a)
\end{pmatrix},$$ and $I_{d}$ denotes the identity $d\times d$ matrix. Here $x, y$ and $(x, y)$ are treated as row vectors. In other words, we assume that $\supp \widehat{f_{\theta}}$ is contained in a parallelepiped $L_{\theta}^{*}([-C,C]^{d+n})$ whose projection onto $\R^{d}$ is a cube containing $\theta$ and that contains the graph of $P$ restricted to $\theta$.
For each $\delta>0$, let the *decoupling constant* $\Dec^{p}(\delta)$ be the smallest constant such that $$\label{eq:dec-const}
\norm[\big]{\sum_{\theta \in \Part{\delta}} f_{\theta}}_{L^p}
\le
\Dec^{p}(\delta) (\sum_{\theta} \norm{f_{\theta}}_{L^p}^p)^{1/p}.$$ The decoupling constant depends on $d,n$ and $P_{1},\dotsc,P_{n}$, and we will sometimes indicate this dependence by subscripts when several different decoupling constants are involved. We will also omit the exponent $p$ when there is only one such exponent involved.
The decoupling constant also depends, in a monotonically increasing way, on the Fourier support parameter $C$. This dependence is entirely harmless, as for dyadic $C$ the decoupling constant at scale $\delta$ with parameter $C$ can be easily controlled by the decoupling constant at scale $C\delta$ with parameter $2$. The only important thing about the parameter $C$ is that it has to be kept constant throughout various inductive procedures.
\[rem:parabolic-scaling\] The operators $M_{\theta}$ come from an action of the group of transformations generated by translations and dilations of $\R^{d}$. This makes parabolic scaling easy, and we obtain $$\norm[\big]{\sum_{\theta \in \Part[Q]{\delta}} f_{\theta}}_{L^p}
\le
\Dec^{p}(\delta/\sigma) (\sum_{\theta} \norm{f_{\theta}}_{L^p}^p)^{1/p}$$ for any dyadic numbers $0<\delta\leq\sigma\leq 1$ and any $Q \in \Part{\sigma}$.
\[rem:global-to-local\] Let $\eta$ be a positive Schwartz function on $\R^{d+n}$ such that $\supp \hat{\eta} \subset B(0,c)$ and $\eta \geq 1$ on $B(0,1)$. Let $B \subset \R^{d+n}$ be a ball of radius $\delta^{-2}$. Then applying with the Fourier support parameter $C$ replaced by $C+c$ to functions $f_{\theta}\eta_{B}$, where $\eta_{B} := \eta(\delta^{2}(\cdot-c(B)))$, we obtain $$\norm[\big]{\sum_{\theta \in \Part{\delta}} f_{\theta}}_{L^p(B)}
\leq
\Dec^{p}(\delta) (\sum_{\theta} \norm{\eta_{B} f_{\theta}}_{L^p}^p)^{1/p}.$$ By [@MR3592159 Section 4] this implies the localized estimate $$\norm[\big]{\sum_{\theta \in \Part{\delta}} f_{\theta}}_{L^p(w_{B})}
\lesssim
\Dec^{p}(\delta) (\sum_{\theta} \norm{f_{\theta}}_{L^p(w_{B})}^p)^{1/p}.$$ Similarly we can localize the rescaled decoupling inequality in Remark \[rem:parabolic-scaling\]. In fact we can localize that inequality further to ellipsoids of dimensions $\delta^{-2}\sigma$ ($d$ times) $\times \delta^{-2}$ ($n$ times), but this will not be necessary.
By Remark \[rem:global-to-local\] Theorem \[thm:main:extension\] will follow from the next result.
\[main\_theorem\] Let $1\le n\le 3$. Assume , , and . Then $$\label{main_estimate}
\Dec^{p}(\delta)\lesssim_{\epsilon} \delta^{-d(\frac{1}{2}-\frac 1 p)-\epsilon}$$ for every $2\le p\le 2+\frac{4n}{d}$ and every $\epsilon>0$.
Theorem \[main\_theorem\] will in turn follow directly from Theorem \[thm:dec:general\] once its hypotheses are verified in Section \[sec:spec\].
Sharpness of the exponents {#sec:sharpness}
--------------------------
We recall standard examples that show that for $2 \leq p,q < \infty$ the $\ell^{q}L^{p}$ decoupling inequality $$\norm[\big]{\sum_{\theta \in \Part{\delta}} f_{\theta}}_{L^p}
\lesssim
\delta^{-\Lambda} (\sum_{\theta} \norm{f_{\theta}}_{L^p}^q)^{1/q}$$ can only hold if $$\label{eq:dec-exponent-lower-bd}
\Lambda \geq \max \Bigl( d-\frac{d}{q}-\frac{d+2n}{p}, d\bigl(\frac12-\frac1q\bigr) \Bigr).$$ Consider first $f_{\theta} = M_{\theta} f$, where $f$ is a fixed Schwartz function with $\hat{f}$ positive and compactly supported. Then by scaling $\norm{f_{\theta}}_{p} \sim \delta^{-(d+2n)/p}$. On the other hand, $\sum_{\theta} f_{\theta} \gtrsim \delta^{-d}$ on a fixed neighborhood of $0$. It follows that $\Lambda \geq d-\frac{d}{q}-\frac{d+2n}{p}$.
Consider next $f_{\theta}(x)=\eta(\delta^{2} x) e^{2 \pi i c_{\theta} \cdot x}$, where $\eta$ is a Schwartz function with $\hat\eta$ compactly supported and $c_{\theta}$ is a point on the surface $\calS$ over $\theta$. Then $\norm{f_{\theta}}_{p} \sim \delta^{-2(d+n)/p}$ and by Hölder’s inequality and orthogonality $$\begin{gathered}
\delta^{-2(d+n)(\frac12-\frac1p)} \norm{\sum_{\theta} f_{\theta}}_{p}
\sim
\norm{\eta(\delta^{2}\cdot)}_{\frac{1}{1/2-1/p}} \norm{\sum_{\theta} f_{\theta}}_{p}
\geq
\norm{\sum_{\theta} \eta(\delta^{2}\cdot) f_{\theta}}_{2}\\
\gtrsim
\bigl( \sum_{\theta} \norm{\eta(\delta^{2}\cdot) f_{\theta}}_{2}^{2} \bigr)^{1/2}
\sim
\delta^{-d/2}\delta^{-2(d+n)/2}.\end{gathered}$$ It follows that $\Lambda \geq d\bigl(\frac12-\frac1q\bigr)$.
It is known from [@MR1209299 p. 118] that the $\epsilon$ loss in cannot be completely removed in general.
Notation {#notation .unnumbered}
--------
$\Part[Q]{\delta}$ is the partition of a dyadic cube $Q$ into dyadic cubes with side length $\delta$. We omit $Q$ if $Q=[0,1]^{d}$.
We use $C$ to denote a large constant that is allowed to change from line to line. Its precise value is of no relevance.
For a sequence of real numbers $\{A_i\}_{i=1}^M$, we abbreviate $\avprod A_{i} := \bigl(\prod_{i=1}^{M} A_{i}\bigr)^{1/M}$. Also, we define averaged integrals: $$\norm{f}_{\avL^{p}(B)}:=(\frac{1}{|B|}\int_B |f|^p )^{1/p} \text{ and } \norm{f}_{\avL^{p}(w_B)}:=(\frac{1}{|B|}\int |f|^p w_B)^{1/p}$$
For $\sigma>0$ and $E\subset \R^d$, we will use $N_{\sigma}(E)$ to denote the $\sigma$-neighborhood of the set $E$.
For a non-negative number $a$, we will $\lfloor a\rfloor$ to denote the greatest integer less than or equal to $a$, and $\lceil a\rceil$ to denote the least integer greater than or equal to $a$.
Acknowledgment {#acknowledgment .unnumbered}
--------------
S.G. was supported in part by a direct grant for research from the Chinese University of Hong Kong (4053295). P.Z. is partially supported by the Hausdorff Center for Mathematics in Bonn. He would also like to thank Po-Lam Yung for inviting him to the Chinese University of Hong Kong, where part of this work was conducted.
General surfaces {#sec:gen}
================
Lower dimensional decoupling {#sec:gen:lower-dim}
----------------------------
Let $\calH\subset \R^d$ be a hyper-plane that intersects $[0, 1]^d$. Without loss of generality, we write it as a graph $$\label{181212e4.2}
t_d=\calL(t') \text{ where } t'=(t_1, \dots, t_{d-1}),$$ and $\calL(t')$ is a linear form of $t'$ with $\abs{\nabla \calL}\lesssim 1$. Consider the new quadratic forms $P_{j}'(t') := P_{j}(t',\calL(t'))$, $j=1,\dotsc,n$, and define the associated decoupling constant $\Dec_{\calH}^{p}(\delta)$ analogously to . Moreover, the index $p$ will be dropped from the notation $\Dec_{\calH}^{p}(\delta)$ whenever it is clear from the context which $p$ we are using.
\[thm:near-hyperplane\] Suppose that $\Dec_{\calH}^{p}(\delta) \lesssim \delta^{-\Lambda}$. Then $$\label{eq:hyperplane-decoupling}
\norm[\Big]{\sum_{\substack{\Box \in \Part{\delta},\\ C\Box\cap \calH\neq \emptyset}} f_{\Box} }_{p}
\lesssim
\abs{\log \delta}^{C} \delta^{-\Lambda}
\Big( \sum_{\Box} \norm{f_{\Box}}^p_{p}\Big)^{1/p}.$$
In previous work of Bourgain and Demeter [@MR3736493 Section 2] and of Demeter, Shi, and the first author [@arxiv:1609.04107 Section 4] similar results were obtained in certain special cases ($d$ arbitrary, $n=1$ and $d=3,n=2$, respectively). In the language of the proof below the idea was to make a projection after which the Fourier support fits into an $O(\delta^{2})$ neighborhood of some lower dimensional surface. In this situation one can obtain by applying the lower-dimensional decoupling fiberwise at scale $\delta$. Our proof shows that with an additional induction on scales can be obtained without investigating the “geometry” of the graph of $P$.
By change of variable we may assume that $\calH = \Set{ t \given t_{d}=0 }$. Let $P'(t'):=P(t',0)$. Considering shifts of $\calH$ (which have comparable decoupling constant) we may consider only $\Box$ with $\Box \cap \calH \neq \emptyset$.
Let $A(\delta)$ be the smallest constant for which the inequality $$\norm[\Big]{\sum_{\substack{\Box \in \Part{\delta},\\ \Box\cap \calH\neq \emptyset}} f_{\Box} }_{L^p}
\leq A(\delta)
\Big( \sum_{\Box} \norm{f_{\Box}}^p_{L^p}\Big)^{1/p}$$ holds. It is easy to see that $A(\delta) \lesssim \delta^{-C}$.
Let $\Box \in \Part{\delta}$ with $\Box \cap \calH \neq \emptyset$ and $\xi = (\xi',\xi_{d}) \in \Box$. Then, since $\abs{\nabla P} \lesssim 1$ on $B(0,C)$, the projection of $L_{\Box}^{*}([-C,C]^{d+n})$ onto $\R^{d-1}\times \R^{n}$ is contained in a $O(\delta)$-neighborhood of $L_{\Box'}^{*}([-C,C]^{d-1+n})$, where $\Box'$ is the projection of $\Box$ onto $\R^{d-1}$.
For each fixed $x_{d}$ this gives a restriction on the fiberwise Fourier support restriction $\widehat{f_{\Box}(\cdot,x_{d},\cdot)}$ that is not strong enough to apply decoupling at scale $\delta$, but is sufficient to apply decoupling at scale $\delta^{1/2}$. Hence we obtain $$\begin{gathered}
\norm[\Big]{\sum_{\substack{\Box \in \Part{\delta},\\ \Box\cap \calH\neq \emptyset}} f_{\Box} }_{L^p(\R^{d-1} \times \Set{x_{d}} \times \R^{n})}\\
\lesssim
\Dec_{\calH}(c\delta^{1/2})
\Big( \sum_{\substack{\Box'\in\Part{\delta^{1/2}}, \\ \Box'\cap \calH\neq \emptyset}}\norm{ \sum_{\substack{\Box \in \Part[\Box']{\delta},\\ \Box\cap \calH\neq \emptyset}} f_{\Box} }^p_{L^p(\R^{d-1} \times \Set{x_{d}} \times \R^{n})}\Big)^{1/p}\end{gathered}$$ for every $x_{d}$. Integrating in $x_{d}$ we obtain $$\norm[\Big]{\sum_{\substack{\Box \in \Part{\delta},\\ \Box\cap C\calH\neq \emptyset}} f_{\Box} }_{p}
\lesssim
\delta^{-\Lambda/2}
\Big( \sum_{\substack{\Box'\in\Part{\delta^{1/2}}, \\ \Box'\cap \calH\neq \emptyset}}\norm{ \sum_{\substack{\Box \in \Part[\Box']{\delta},\\ \Box\cap \calH\neq \emptyset}} f_{\Box} }^p_{p}\Big)^{1/p}.$$ By scaling we have $$\norm{ \sum_{\substack{\Box \in \Part[\Box']{\delta},\\ \Box\cap \calH\neq \emptyset}} f_{\Box} }_{p}
\leq A(\delta^{1/2})
\Big( \sum_{\substack{\Box \in \Part[\Box']{\delta},\\ \Box\cap \calH\neq \emptyset}} \norm{ f_{\Box} }_{p}^{p} \Big)^{1/p}.$$ It follows that $$A(\delta) \lesssim \delta^{-\Lambda/2} A(\delta^{1/2}).$$ Iterating this inequality approximately $\log\log \frac{1}{\delta}$ times we obtain the claim.
Next, we will prove a version of Theorem \[thm:near-hyperplane\] for curved hypersurfaces.
\[hyp:lower-dim\] Suppose that for every hyperplane $\calH \subset \R^{d}$ passing through $0$ we have $\Dec_{\calH}^{p}(\delta) \lesssim \delta^{-\Lambda}$, uniformly in $\calH$.
In the situation of Theorem \[main\_theorem\] Hypothesis \[hyp:lower-dim\] will be verified in Section \[sec:spec:lower-dim\] for an appropriate exponent $\Lambda$, depending on $d$ and $n$.
\[thm:near-graph\] Let $2\leq p < \infty$ and assume Hypothesis \[hyp:lower-dim\]. Then for every $\epsilon>0$ and every hypersurface $\widetilde{\calH}\subset [0,1]^d$ that can be written as a graph $$\label{181126e4.9}
t_d=\calL(t') \text{ with }
\norm{\calL}_{C^{2}} \leq C$$ we have $$\label{181209e4.11}
\norm[\Big]{\sum_{\substack{\Box\in\Part{\delta},\\ \Box\cap \widetilde{\calH}\neq \emptyset}}f_{\Box} }_{p}
\lesssim_{\epsilon}
\delta^{-\Lambda-\epsilon}
\Big( \sum_{\Box}\norm{f_{\Box} }^p_{p}\Big)^{1/p}$$ for every $0<\delta\le 1$. The implicit constant in may depend on the constant in , but not otherwise on $\widetilde{\calH}$.
The proof is via an iteration argument, essentially due to Oh [@MR3848437]. It is also closely related to the iteration argument of Pramanik and Seeger [@MR2288738].
Let $C_{0}$ be $2^{10}$ times the constant $C$ in . For $\kappa \leq 2^{-10}$ let $A(\kappa,\delta)$ be the smallest constant such that the inequality $$\norm[\Big]{\sum_{\substack{\Box\in\Part{\delta},\\ \Box\cap \widetilde{\calH}\neq \emptyset}}f_{\Box} }_{p}
\leq
A(\kappa,\delta)
\Big( \sum_{\Box}\norm{f_{\Box} }^p_{p}\Big)^{1/p}$$ holds for all hypersurfaces $\widetilde{\calH}$ that are parameterized by functions $\calL$ with $\abs{\nabla\calL} \leq C_{0}$ and $\abs{\nabla^{2}\calL} \leq C_{0}\kappa$.
Let $\calH$ be the tangent plane at some point of $\widetilde{\calH}$. Then $\widetilde{\calH}$ is contained in the $O(\kappa)$-neighborhood of $\calH$. If $\kappa \leq \delta$, then we can apply Theorem \[thm:near-hyperplane\] and obtain $A(\kappa,\delta) \lesssim \abs{\log \delta}^{C} \delta^{-\Lambda}$.
If $\kappa > \delta$, then we can instead apply Theorem \[thm:near-hyperplane\] at scale $\kappa$. This gives $$\norm[\Big]{\sum_{\substack{\Box\in\Part{\delta},\\ \Box\cap \widetilde{\calH}\neq \emptyset}}f_{\Box} }_{p}
\lesssim
\abs{\log\kappa}^{C} \kappa^{-\Lambda}
\Big( \sum_{\substack{\Box' \in \Part{\kappa},\\ 2\Box' \cap \calH \neq \emptyset}} \norm{ \sum_{\Box \subset \Box'} f_{\Box} }^p_{p}\Big)^{1/p}.$$ The crucial observation now is that after rescaling any of the $\Box'$ to unit scale the surface $\widetilde{\calH} \cap \Box'$ becomes parameterized by a function with first derivative still bounded by $C_{0}$ and the second derivative bounded by $C_{0} \kappa^{2}$. By scaling it follows that $$\norm{ \sum_{\substack{\Box\in\Part[\Box']{\delta},\\ \Box\cap \widetilde{\calH}\neq \emptyset}} f_{\Box} }_{p}
\leq
A(\kappa^{2},\delta/\kappa)
\Big( \sum_{\Box} \norm{ f_{\Box} }^p_{p}\Big)^{1/p},$$ so that $$A(\kappa,\delta)
\lesssim
\abs{\log\kappa}^{C} \kappa^{-\Lambda} A(\kappa^{2},\delta/\kappa).$$ Applying this inequality at most approximately $\log \log \frac{1}{\delta}$ times we arrive in the situation $\kappa \leq \delta$.
In the Bourgain–Guth iteration scheme that is used to prove the equivalence between linear and multilinear decouplings we have to apply lower dimensional decoupling to families of functions with Fourier support close to a subvariety. In order to apply Theorem \[thm:near-graph\] we will cover a neighborhood of the subvariety by neighborhoods of hypersurfaces with controlled curvature. To this end the following fact will be useful.
\[lem:sublevel-f-vs-Df\] Let $\Omega \subset \R^{2}$ be open and $f \in C^{2}(\Omega) \cap C(\bar\Omega)$ with $\abs{\nabla^{2}f} \leq 1$. Then for all $\sigma,\eta>0$ we have $$\label{eq:sublevel-f-vs-Df}
\Set{ \abs{f} \leq \sigma\eta}
\subset
\Set{ \abs{\nabla f} \leq \sigma+\eta }
\cup
N_{\sigma} \bigl( \partial\Omega \cup ( \Set{f=0} \cap \Set{\abs{\nabla f} > \eta} ) \bigr).$$
By Taylor’s formula and the intermediate value theorem for every $x\in\Omega$ and $t>0$ the inequality $$\abs{f(x)} - t \abs{\nabla f(x)} + t^{2} \norm{\nabla^{2}f}_{\infty}/2 \leq 0$$ implies $\dist(x, \partial\Omega \cup \Set{f = 0}) \leq t$. In the case $\abs{f(x)} \leq \sigma\eta$ and $\abs{\nabla f(x)} > \sigma+\eta$ the above inequality holds with $t=\sigma$. Moreover, if $B(x,\sigma) \subset \Omega$, then $\abs{\nabla f}>\eta$ on that ball.
\[lem:sublevel-decomposition-of-variety\] For every natural numbers $n,A \geq 1$, every $D\geq 0$ and every sufficiently large $K>1$ there exist $$K_1\leq K_2\leq \dotsb \leq K_{D+1} = K
\text{ with }
K_{j+1} \sim_{n,j} K_{j}^{A+1}$$ such that for every normalized polynomial $P$ of degree $D$ in $n$ variables with real coefficients there exists an increasing sequence of multiindices $\alpha_{D} , \alpha_{D-1}, \dotsb, \alpha_{1}$ with $\abs{\alpha_{j}} = D-j$ such that $$\label{eq:sublevel-decomposition-of-variety}
\Set{ \abs{P} < 1/K_{D+1} } \cap B(0,1)
\subseteq
\bigcup_{j=1}^{D} N_{1/K_{j}^{A}} \Bigl( Z_{\partial^{\alpha_{j}}P} \cap \Set{\abs{\nabla \partial^{\alpha_{j}} P} \geq 1/K_{j}} \Bigr).$$ Here we say that $P$ is a normalized polynomial if $\norm{P}=1$. In other words, the $\ell^1$ sum of the coefficient of $P$ is equal to one. Also, $Z_{P} = \Set{ x \given P(x)=0 }$.
By induction on $D$. In the case $D=0$ the left hand side of is empty provided $K>1$ since $P\equiv 1$.
Suppose now that $D \geq 1$ and the conclusion is already known with $D$ replaced by $D-1$.
If $\norm{\partial^{\alpha} P} \ll 1$ for all $\abs{\alpha}=1$, then since $P$ is normalized, the left hand side of is empty provided that $K_{D+1}$ is large enough. Hence we may assume $\norm{\partial^{\alpha_{D}} P} \geq c_{1}$ for some $c_{1} = c_{1}(n,D) > 0$ and some $\alpha_{D}$.
Since $P$ is normalized we have $c_{0}\abs{\nabla^{2} P(x)} \leq 1$ for some $c_{0} = c_{0}(n,D) > 0$ and all $x\in \Omega := B(0,2)$. By Lemma \[lem:sublevel-f-vs-Df\] we obtain $$\begin{gathered}
\Set{ \abs{P} < 1/K_{D+1}}
\subset
\Set{ \abs{c_{0} P} < c_{0}/K_{D}^{A+1} }\\
\subset
\Set{ \abs{c_{0}\nabla P} < c_{0}/K_{D}+1/K_{D}^{A} }
\cup
N_{1/K_{D}^{A}} \bigl( \partial\Omega \cup (Z_{P} \cap \Set{\abs{c_{0} \nabla P} > c_{0}/K_{D}}) \bigr)\end{gathered}$$ provided $c_{0}/K_{D+1} \leq c_{0}/K_{D}^{A+1}$.
Since $K_{D}\geq 1$, the above neighborhood of $\partial\Omega$ does not intersect $B(0,1)$, and we obtain $$\begin{gathered}
\Set{ \abs{P} < 1/K_{D+1}} \cap B(0,1)
\subset
\Set{ \abs{c_{0}\nabla P} < c_{0}/K_{D}+1/K_{D}^{A} }\\
\cup
N_{1/K_{D}^{A}} (Z_{P} \cap \Set{\abs{\nabla P} > 1/K_{D}}).\end{gathered}$$ The second term is of the required form. In the first term we apply the inductive hypothesis with $D$ replaced by $D-1$, $P$ replaced by $\|\partial^{\alpha_{D}} P\|^{-1} \partial^{\alpha_{D}} P$, and $K$ replaced by $\tilde{K}$ satisfying $1/K_{D}+1/(c_{0}K_{D}^{A}) = c_{1}/\tilde{K}$.
The next result extends [@MR3709122 Claim 5.10] and [@MR3848437 Proposition 4.1].
\[thm:dim-reduction-variety\] Assume Hypothesis \[hyp:lower-dim\]. For every $D\geq 1$ and $A> 1$, for every sufficiently large $K$ there exist $$K_1\leq K_2\leq \dotsb \leq K_{D+1} = K
\text{ with }
K_{j+1} \sim_{n,j} K_{j}^{A+1}$$ such that for every non-zero polynomial $P$ of degree $D$ there exist collections of pairwise disjoint cubes $\calG_{j} \subset \Part{1/K_{j}^{A}}$, $j=1,\dotsc,D$, such that $$N_{1/K}(Z_{P}) \cap [0,1]^{d}
\subset
\bigcup_{j=1}^{D} \bigcup_{\Box \in \calG_{j}} \Box$$ and $$\norm[\Big]{ \sum_{\Box \in \calG_{j}} f_{\Box} }_{p}
\lesssim_{D,\epsilon}
K_{j}^{d}(K_{j}^{A-1})^{\Lambda+\epsilon} \Big( \sum_{\Box \in \calG_{j}} \norm{f_{\Box}}^p_{p}\Big)^{1/p}.$$
Let $P_{j} = \partial^{\alpha_{j}}P$ be as in Lemma \[lem:sublevel-decomposition-of-variety\] and $$Z_{j} = Z_{P_{j}} \cap \Set{\abs{\nabla P_{j}} \geq 1/K_{j}}.$$ Let $$\calG_{j} := \Set{\Box \in \Part{1/K_{j}^{A}} \given C\Box \cap Z_{j} \neq \emptyset} \setminus \bigcup_{j'<j} \bigcup_{\Box' \in \calG_{j'}} \Part[\Box']{1/K_{j}^{A}}.$$ Using trivial decoupling (Minkowski’s inequality) at scale $1/(C K_{j})$ it suffices to show $$\norm[\Big]{ \sum_{\Box \in \calG_{j} : \Box \subset Q} f_{\Box} }_{p}
\lesssim_{\epsilon, D}
(K_{j}^{A-1})^{\Lambda+\epsilon} \Big( \sum_{\Box} \norm{f_{\Box}}^p_{p}\Big)^{1/p}$$ for every $Q \in \Part{1/(CK_{j})}$. But if there exists $\Box \in \calG_{j}$ with $\Box \subset Q$, then $\abs{\nabla P_{j}} \gtrsim 1/K_{j}$ on $CQ$, so by the implicit function theorem $Z_{j} \cap CQ$ is a hypersurface with curvature $\lesssim K_{j}$. After scaling $Q$ to the unit scale the set $Z_{j} \cap CQ$ becomes a graph with curvature $\lesssim 1$, and the claim follows by Theorem \[thm:near-graph\].
\[cor:dim-reduction-variety\] For every $D\geq 1$ and $\epsilon>0$ there exists $c=c(D,\epsilon)>0$ such that for every sufficiently large $K$ there exist $$K^{c} \leq \tilde{K}_1\leq \tilde{K}_2\leq \dotsb \leq \tilde{K}_{D} \leq \sqrt{K}$$ such that for every non-zero polynomial $P$ of degree $D$ there exist collections of pairwise disjoint cubes $\calG_{j} \subset \Part{1/\tilde{K}_{j}}$, $j=1,\dotsc,D$, such that $$N_{1/K}(Z_{P}) \cap [0,1]^{d}
\subset
\bigcup_{j=1}^{D} \bigcup_{\Box \in \calG_{j}} \Box$$ and $$\norm[\Big]{ \sum_{\Box \in \calG_{j}} f_{\Box} }_{p}
\lesssim_{D,\epsilon}
\tilde{K}_{j}^{\Lambda+\epsilon} \Big( \sum_{\Box \in \calG_{j}} \norm{f_{\Box}}^p_{p}\Big)^{1/p}.$$
It appears somewhat unfortunate that the constant $c$ in Corollary \[cor:dim-reduction-variety\] depends also on $\epsilon$. This could make quantification of the $C_{\epsilon}\delta^{-\epsilon}$ loss in Theorem \[main\_theorem\] in the way of [@arxiv:1711.01202] less convenient. However, currently the main obstacle in that direction is the unquantified transversality in Lemma \[lem:transverse\].
Transversality {#sec:gen:transverse}
--------------
To introduce the multilinear decoupling inequality, we first need to introduce the notion of transversality. Let $M$ be a large positive integer. For $1\le j\le M$, let $V_j \subset \R^{n+d}$ be a linear subspace of dimension $d$. Let $\pi_j: \R^{n+d}\to V_j$ denote the orthogonal projection onto $V_j$. The *Brascamp–Lieb constant* $BL((V_{j})_{j=1}^{M})$ is the smallest constant (possibly $\infty$) such that the inequality $$\label{eq:BL-const-def}
\int_{\R^{n+d}} \prod_{j=1}^M f_j (\pi_j (x))^{\frac{n+d}{d M}} d x
\leq
BL((V_{j})_{j=1}^{M}) \prod_{j=1}^M \bigl( \int_{V_{j}} f_j (y) d y \bigr)^{\frac{n+d}{ d M}}$$ holds for all non-negative measurable functions $f_j: V_j\to \R$.
\[transversality\] Let $M$ be a positive integer and $\nu>0$. A collection of $M$ sets $R_1, \dotsc, R_M\subset [0, 1]^d$ is called *$\nu$-transverse* if for every choice $t_j\in R_j$ we have $$BL((V(t_j))_{j=1}^M)\le \nu^{-1},$$ where $V(t)$ denotes the tangent space of the surface $\calS_{d, n}$ at $t$.
The notion of transversality in Definition \[transversality\] goes back to [@MR3548534]. In the case $d=n$, $M=2$ it specializes to the notions used in [@MR3447712; @MR3848437], where transversality means that $V(t_{1}),V(t_{2})$ do not share common directions. In the case $n=1$, $P_{1}$ positive definite, $M=d+1$ it specializes to the notion used in [@MR3374964; @MR3592159], because the associated Brascamp–Lieb inequality is the Loomis–Whitney inequality, and the best constant in that inequality is the reciprocal of the volume of the parallelepiped spanned by the normal directions of $V_{j}$’s.
One of the main results of Bennett, Carbery, Christ, and Tao [@MR2661170] says that $$BL((V(t_j))_{j=1}^M)<\infty$$ if and only if the spaces $(V(t_j))_{j=1}^M$ satisfy the condition $$\label{eq:BCCT-condition}
\dim(V)\le \frac{d+n}{d M}\sum_{j=1}^M \dim(\pi_{t_j}(V))$$ for every linear subspace $V\subset \R^{d+n}$, where $\pi_{t}$ denotes the orthogonal projection onto $V(t)$. Moreover, from [@MR3723636] we know that the function $(t_j)_{j=1}^M \mapsto BL((V(t_j))_{j=1}^M)$ is continuous (with values in $[0,\infty]$). Indeed, it is even Hölder continuous [@arxiv:1811.11052].
In order to ensure existence of transverse sets we have to make some assumptions on the surface $\calS$.
\[hyp:transverse\] Suppose that for every subspace $V \subset \R^{d+n}$ one of the following holds.
1. $\dim \pi_{t}(V) \geq \frac{d}{d+n} \dim V$ for *every* $t\in \R^{d}$, or
2. $\dim \pi_{t}(V) > \frac{d}{d+n} \dim V$ for *some* $t\in \R^{d}$.
In the cases of Theorem \[main\_theorem\] Hypothesis \[hyp:transverse\] will be verified in Section \[sec:spec:transverse\].
\[lem:transverse\] Assuming Hypothesis \[hyp:transverse\], there exists $\theta>0$ such that the following holds. For every $K$ there exists $\nu_{K}>0$ such that for every subcollection $\calR \subset \Part{1/K}$ one of the following alternatives holds.
1. $\calR$ is $\nu_{K}$-transverse, or
2. there exists a subvariety $Z$ of degree at most $d$ such that $$\abs{\Set{R \in \calR \given 2R \cap Z\neq \emptyset}} > \theta \abs{\calR}.$$
Analogues of Lemma \[lem:transverse\] were also used in [@MR3614930; @MR3709122; @arxiv:1804.02488; @arxiv:1811.02207].
The bound $d$ on the degree is not optimal in many situations. For instance in the case $n=1$ as in [@MR3374964; @MR3736493] we can use a subvariety of degree $1$, that is, a hyperplane. This follows from Lemma \[lem:BCCT:n=1\].
In the case $d=n$ as in [@MR3447712; @MR3848437] it might have previously seemed important that only certain specific varieties can obstruct transversality. Thanks to Corollary \[cor:dim-reduction-variety\] we can afford not to keep track of which varieties may or may not arise here.
Let $V \subset \R^{d+n}$ be a subspace. If the first alternative in Hypothesis \[hyp:transverse\] holds, then the BCCT condition holds for that subspace with any choice of $t_{j}$.
Suppose now that the second alternative in Hypothesis \[hyp:transverse\] holds. The restriction of the projection operator $\pi_{t}$ to $V$ can be written in coordinates as a $d \times \dim V$ matrix whose entries are linear polynomials in $t$. By the hypothesis some minor determinant of that matrix of order $>\frac{d}{d+n} \dim V$ does not vanish for some $t$. Hence that minor determinant is a non-trivial polynomial of degree at most $d$, and the dimension of the projection is $>\frac{d}{d+n} \dim V$ outside its zero set $Z$.
In particular the BCCT condition for $(t_{j})_{j=1}^{M}$ holds for $V$ provided that $$\dim(V) \leq \frac{d+n}{dM} \sum_{j : t_{j} \not\in Z} \lfloor \frac{d}{d+n} \dim (V) + 1 \rfloor,$$ which can be equivalently written as $$\abs{\Set{j \given t_{j} \not\in Z}}/M \geq
\dim(V) \frac{d}{d+n} / \lfloor \frac{d}{d+n} \dim (V) + 1 \rfloor.$$ The number on the right-hand side is $<1$ and can take only finitely many values since $\dim(V)$ is a natural number $\leq d+n$. Let $\theta$ be $1$ minus the maximum of the right-hand side over $V$. Then the BCCT condition follows from $$\abs{\Set{j \given t_{j} \in Z}} \leq \theta M.$$ This clearly holds for $t_{j}\in R_{j}$, where $(R_{j})_{j=1}^{M}$ is an enumeration of $\calR$, unless the second alternative of the Lemma holds.
Finally, if the second alternative of the lemma does not hold, then the set of tuples $(t_{j})$ with $t_{j}\in R_{j}$ is a compact subset of the set of tuples for which the BCCT condition holds. Hence by continuity of the Brascamp–Lieb constant there exists a lower bound $\nu_{K}$ on the transversality of the tuple $\calR$.
The use of a compactness argument makes the transversality bound $\nu_{K}$ ineffective.
Multilinear decoupling {#sec:gen:multilinear}
----------------------
We use a version of the Bourgain–Guth scheme [@MR2860188] that goes back to an article of Bourgain, Demeter, and the first author [@MR3709122]. In this version the degree of multi-linearity is allowed to range in an interval depending on $K$.
For a positive integer $K$ and $0 < \delta < K^{-1}$ the *multilinear decoupling constant* $\MulDec^{p}(\delta, K)$ is the smallest constant such that the inequality $$\begin{gathered}
\label{eq:multilin-Dec}
\Bigl( \int_{\R^{d+n}} \bigl( \avprod \norm{ f_{R_{i}} }_{\avL^{p}(B(x,K))} \bigr)^{p} \dif x \Bigr)^{1/p}\\
\le \MulDec^{p}(\delta, K)
\avprod \Bigl( \sum_{J \in \Part[R_i]{\delta}} \norm{f_J}_{L^p(\R^{d+n})}^{p} \Bigr)^{\frac{1}{p}}\end{gathered}$$ holds for every $\nu_{K}$-transverse tuple $R_{1},\dotsc,R_{M} \in \Part{K^{-1}}$ with $1\leq M \leq K^{d}$, where $\nu_{K}>0$ is as in Lemma \[lem:transverse\]. Given $f_{J}$, $J\in\Part{\delta}$, we write here and later $f_{\alpha} := \sum_{J \in \Part[\alpha]{\delta}} f_{J}$ for dyadic cubes $\alpha$ of scale $\geq \delta$.
The quantity on the left-hand side of is equivalent to $$\bigl( \sum_{B' \in \Cov{K}} \avprod \norm{ f_{R_{i}} }_{L^{p}(B')}^{p} \bigr)^{1/p},$$ where $\Cov{K}$ denotes a finitely overlapping cover of $\R^{d+n}$ by balls of radius $K$.
LHS of can be thought of as morally equivalent to $\norm{ \avprod \abs{f_{R_{i}}}}_{p}$, since by the uncertainty principle the functions $f_{R_{i}}$ are morally constant at scale $K$. Following [@MR3592159], we use the formally larger averaged quantity because it can be more easily obtained in the Bourgain–Guth argument.
As for $\Dec^{p}$, we will omit the exponent $p$ in $\MulDec^{p}$ when it is clear from context.
Bourgain–Guth argument {#sec:gen:bourgain-guth}
----------------------
From Hölder’s inequality, it follows that $$\label{180713e3.4}
\MulDec^{p}(\delta, K)
\lesssim
\Dec^{p}(\delta).$$ The Bourgain–Guth argument shows that the converse inequality also holds up to some lower-dimensional terms. To be precise, we will prove
\[181209prop5.5\] Let $2 \leq p < \infty$. Assume Hypothesis \[hyp:transverse\] and Hypothesis \[hyp:lower-dim\]. Then for each $\epsilon>0$ there exists $K$ such that $$\Dec^{p}(\delta)
\lesssim_{\epsilon}
\delta^{-\Lambda-\epsilon}
+ \delta^{-\epsilon} \max_{\delta\le \delta'\le 1; \delta' \text{dyadic}} \Big[\big(\frac{\delta'}{\delta} \big)^{\Lambda} \MulDec^{p}(\delta', K)\Big].$$
It is not difficult to see that Proposition \[181209prop5.5\] can be proven by iterating the following result $O(\frac{\abs{\log\delta}}{\log K})$ many times. It is important to choose $K$ large enough depending on $\epsilon$ since we lose a constant $C_{\epsilon}$ in every step of the iteration.
\[prop:bourgain-guth-arg\] Let $1 \leq p < \infty$. Assume Hypothesis \[hyp:transverse\] and Hypothesis \[hyp:lower-dim\]. Then for every $K\ge 2$ and $0<\delta<1/K$ we have $$\label{eq:BG-arg}
\Dec^{p}(\delta)
\leq
C_{\epsilon} \sup_{K^{c(d,\epsilon)} \leq \tilde{K} \leq K} \tilde{K}^{\Lambda+\epsilon} \Dec^{p}(\delta\tilde{K})
+ C_{K} \MulDec^{p}(\delta, K).$$
Fix $f_{J}$, $J\in\Part{\delta}$.
Let $B' \in \Cov{K}$ and initialize $$\label{eq:initial-stock}
\calS_{0}(B') := \Set{ \alpha \in \Part{K^{-1}} \given \norm{f_\alpha}_{L^{p}(B')} \ge K^{-d} \max_{\alpha' \in \Part{K^{-1}}} \norm{f_{\alpha'}}_{L^{p}(B')} }.$$ We repeat the following algorithm.
Let $m\geq 0$. If $\calS_{m}(B') = \emptyset$ or $\calS_{m}(B')$ is $\nu_{K}$-transverse, then we set $$\calT(B') := \calS_{m}(B').$$ Otherwise by Lemma \[lem:transverse\] there exists a subvariety $Z$ of degree $d$ such that $$\label{eq:stock-near-variety}
\abs{\Set{ \alpha\in \calS_{m}(B') \given 2\alpha \cap Z \neq \emptyset}}
\geq
\theta \abs{\calS_{m}(B')}.$$ Let $\calG_{m,j}(B') := \calG_{j}$ be given by Corollary \[cor:dim-reduction-variety\].
Repeat the algorithm with $$\begin{aligned}
\calS_{m+1}(B')
&:=
\calS_{m}(B') \setminus \bigcup_{j=1}^{D} \bigcup_{\Box \in \calG_{m,j}(B')} \Part{\Box,1/K}.\end{aligned}$$ Since in each step we remove at least a fixed proportion $\theta$ of $\calS_{m}(B')$, this algorithm terminates after $O(\log K)$ steps.
To avoid multiple counting, we introduce $$\label{new_boxes}
\widetilde{\calG}_{m,j}(B') := \Bigl( \calG_{m,j}(B') \setminus \bigcup_{0\le m' < m} \calG_{m',j}(B') \Bigr) \setminus \bigcup_{1\le j' < j} \bigcup_{m'} \bigcup_{\Box\in \calG_{m', j'}(B')} \Part{\Box, 1/\tilde{K_j}}.$$ We estimate $$\begin{aligned}
\norm{ f }_{L^{p}(B')}
\label{eq:BG':small}&\leq
\sum_{\alpha \in \Part{K^{-1}} \setminus \calS_{0}(B')} \norm{ f_{\alpha} }_{L^{p}(B')}\\
\label{eq:BG':variety}&+
\sum_{m \lesssim \log K} \sum_{j=1}^{D} \norm{ \sum_{\beta \in \widetilde{\calG}_{m,j}(B')} f_{\beta} }_{L^{p}(B')}\\
\label{eq:BG':transverse}&+
\sum_{\alpha \in \calT(B')} \norm{ \ED_{\alpha} g }_{L^{p}(B')}\end{aligned}$$
By definition of $\calS_{0}(B')$ we obtain $$\eqref{eq:BG':small}
\lesssim
\max_{\alpha' \in \Part{1/K}}
\norm{ f_{\alpha'} }_{L^{p}(B')}.$$ By Corollary \[cor:dim-reduction-variety\] and a simple localization argument as in Remark \[rem:global-to-local\], we have $$\begin{gathered}
\eqref{eq:BG':variety}
\lesssim_{\epsilon}
\sum_{m\lesssim \log K} \sum_{j=1}^{D} \tilde{K}_{j}^{\Lambda+\epsilon} \Bigl( \sum_{\beta \in \widetilde{\calG}_{m,j}(B')} \norm{ f_{\beta} }_{L^{p}(w_{B'})}^{p} \Bigr)^{1/p}\\
\lesssim
(\log K) \sum_{j=1}^{D} \tilde{K}_{j}^{\Lambda+\epsilon} \Bigl( \sum_{\beta \in \Part{1/\tilde{K}_{j}}} \norm{ f_{\beta} }_{L^{p}(w_{B'})}^{p} \Bigr)^{1/p}\end{gathered}$$ If $\calT(B') \neq \emptyset$, then by definition of $\calS_{0}(B')$ we obtain $$\eqref{eq:BG':transverse}
\lesssim
K^{C}
\min_{\alpha' \in \calT(B')} \norm{ f_{\alpha'} }_{L^{p}(B')}
\leq
K^{C} \max_{1 \leq M \leq K^{d}} \max_{\substack{\alpha_{1},\dotsc,\alpha_{M} \in \Part{K^{-1}}\\ \nu_{K}-\text{transverse}}} \avprod \norm{ f_{\alpha_{i}} }_{L^{p}(B')}.$$ Next we sum over all balls $B'\subset \R^{d+n}$ and obtain $$\begin{aligned}
\notag
\norm{f}_{L^{p}(\R^{d+n})}
&\leq
\bigl( \sum_{B' \in \Cov{K}} \norm{f}_{L^{p}(B')}^{p} \bigr)^{1/p}\\
\label{eq:BG:small} & \lesssim
\bigl( \sum_{B' \in \Cov{K}} \max_{\alpha \in \Part{K^{-1}}} \norm{f_{\alpha}}_{L^{p}(B')}^{p} \bigr)^{1/p}\\
\label{eq:BG:variety}&+
(\log K) C_{\epsilon} \sum_{j=1}^{D} \tilde{K}_{j}^{\Lambda+\epsilon}
\Bigl( \sum_{\beta \in \Part{1/\tilde{K}_{j}}} \norm{ f_{\beta} }^p_{L^{p}(\R^{d+n})}\Bigr)^{1/p}\\
\label{eq:BG:transverse}&+
K^{C} \bigl( \sum_{B' \in \Cov{K}} \max_{1 \leq M \leq K^{d}} \max_{\substack{\alpha_{1},\dotsc,\alpha_{M} \in \Part{K^{-1}}\\ \nu_{K}-\text{transverse}}} \avprod \norm{ f_{\alpha_{i}} }_{L^{p}(B')}^{p} \bigr)^{1/p}\end{aligned}$$ Let us pause and remark that it is in this step that we require $\log \tilde{K_j}\approx_{d, n, \epsilon} \log K$. We will absorb the factor $\log K$ by $\tilde{K_j}^{\epsilon}$.
In the term , we bound $\max_{\alpha}$ by $\ell^p_{\alpha}$ and obtain $$\bigl(\sum_{\alpha \in \Part{K^{-1}}} \norm{f_{\alpha}}_{L^{p}(\R^{d+n})}^{p} \bigr)^{1/p}.$$ Each term $\norm{f_{\alpha}}_{L^{p}(\R^{d+n})}$ is a rescaled version of the left hand side of the above inequality. Therefore, one can use the definition of the decoupling constant and scaling. The same argument is also applied to .
In the last term by definition of the multilinear decoupling constant we estimate $$\begin{gathered}
\eqref{eq:BG:transverse}
\lesssim_{K}
\bigl( \sum_{1 \leq M \leq K^{d}} \sum_{\substack{\alpha_{1},\dotsc,\alpha_{M} \in \Part{K^{-1}}\\ \nu_{K}-\text{transverse}}} \sum_{B' \in \Cov{K}} \avprod \norm{ f_{\alpha_{i}} }_{L^{p}(B')}^{p} \bigr)^{1/p}\\
\leq
\MulDec(\delta, K)
\bigl( \sum_{1 \leq M \leq K^{d}} \sum_{\substack{\alpha_{1},\dotsc,\alpha_{M} \in \Part{K^{-1}}\\ \nu_{K}-\text{transverse}}} \avprod \bigl( \sum_{J \in \Part[\alpha_{i}]{\delta}} \norm{ f_{J} }_{L^{p}(\R^{d+n})}^{p} \bigr) \bigr)^{1/p}\\
\leq
\MulDec(\delta, K)
\bigl( \sum_{1 \leq M \leq K^{d}} \prod_{i=1}^{M} \sum_{\alpha_{i} \in \Part{K^{-1}}} \bigl( \sum_{J \in \Part[\alpha_{i}]{\delta}} \norm{ f_{J} }_{L^{p}(\R^{d+n})}^{p} \bigr)^{\frac{1}{M}} \bigr)^{1/p}\\
\lesssim_{K}
\MulDec(\delta, K)
\bigl( \sum_{1 \leq M \leq K^{d}} \prod_{i=1}^{M} \bigl( \sum_{\alpha_{i} \in \Part{K^{-1}}} \sum_{J \in \Part[\alpha_{i}]{\delta}} \norm{ f_{J} }_{L^{p}(\R^{d+n})}^{p} \bigr)^{\frac{p}{p M}} \bigr)^{1/p}\\
\lesssim_{K}
\MulDec(\delta, K)
\bigl( \sum_{J \in \Part{\delta}} \norm{ f_{J} }_{L^{p}(\R^{d+n})}^{p} \bigr)^{1/p}.\end{gathered}$$ Since $f_{J}$ were arbitrary this concludes the proof.
Ball inflation {#sec:ball-inflation}
--------------
The following estimate relies on Kakeya–Brascamp–Lieb type inequalities. Its proof is by now standard, see e.g. [@arxiv:1811.02207 Lemma 3.7].
\[prop:ball-inflation\] Let $K\geq 1$ be a dyadic integer and $0 < \delta \leq \rho \leq 1/K$. Let $\Set{R_j}_{j=1}^M \subset \Part{1/K}$ be a $\nu$-transverse collection of cubes. Let $B\subset \R^{d+n}$ be a ball of radius $\rho^{-2}$. Then for each $1 \leq t < \infty$ we have $$\label{0727e4.14ha}
\begin{split}
& \avL^{\frac{d+n}{d} t}_{x \in B} \avprod \ell^{t}_{J_i \in \Part[R_i]{\rho}} \norm{f_{J_i}}_{\avL^{t}(w_{B(x,1/\rho)})}\\
& \lesssim \nu^{-\frac{d}{t(d+n)}}
\avprod \ell^{t}_{J_i \in \Part[R_i]{\rho}} \norm{f_{J_i}}_{\avL^{t}(w_B)}
\end{split}$$ Here $$\avL^{p}_{x \in B}(\cdot):= \bigl( \frac{1}{\abs{B}} \int_B \abs{\cdot}^p \bigl)^{1/p}.$$
Bourgain–Demeter iteration {#sec:bourgain-demeter}
--------------------------
In this section we present a version of the iteration argument of Bourgain and Demeter. Its $\ell^{2}L^{p}$ version was introduced in [@MR3374964] and the $\ell^{p}L^{p}$ version in [@MR3736493]. The simplified version below is a special case of the iteration in [@arxiv:1811.02207].
Throughout this section let $R_1,\dotsc,R_M \in \Part{1/K}$ be $\nu_{K}$-transverse cubes.
For $\rho \in 2^{-\N}$ we define the quantity $$\begin{aligned}
A_p(\rho)
&:=
L^{p}_{x} \avprod \ell^{2}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{2}(w_{B(x,1/\rho)})}.\end{aligned}$$ Here $L^p_x$ refers to taking the $L^p$ norm of a function depending on the $x$ variable. We caution the reader that the quantities denoted by $A$ in [@MR3592159] would correspond to our $A$ with $L^{p}_{x}$ replaced by $\avL^{p}_{x\in B}$ for a large ball $B$.
Let $\tilde p:=\max(2,pd/(d+n))$ and let $0\le \kappa_{p} \le 1$ satisfy $$\frac{1}{\tilde{p}}=\frac{1-\kappa_p}{2}+\frac{\kappa_p}{p}.$$ It will be important that $\kappa_{p} \leq 1/2$ if and only if $p \leq \frac{2(d+2n)}{d}$.
\[prop:iter\] We have for each $2 \leq p < \infty$, and $\kappa_{p} \leq \kappa \leq 1$ $$\label{eq:33}
A_p(\rho)
\lesssim
\nu^{-1/\tilde{p}} \rho^{-d(1/2-1/\tilde{p})} A_{p}(\rho^{2})^{1-\kappa}
\Bigl( \Dec^{p}(\delta/\rho) \avprod \ell^{p}_{J \in \Part[R_i]{\delta}} \norm{ f_{J} }_{p} \Bigr)^{\kappa}$$
Using ball inflation from scale $\rho$ to scale $\rho^{2}$ we obtain $$\begin{aligned}
A_p(\rho)
&=
L^{p}_{x} \avL^{p}_{\tilde{x} \in B(x,1/\rho^{2})} \avprod \ell^{2}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{2}(w_{B(\tilde{x},1/\rho)})}\\
\text{by H\"older } &\lesssim
\rho^{-d(1/2-1/\tilde{p})} L^{p}_{x} \avL^{\frac{\tilde{p} (d+n)}{d}}_{\tilde{x} \in B(x,1/\rho^{2})} \avprod \ell^{\tilde{p}}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{\tilde p}(w_{B(\tilde{x},1/\rho)})}\\
\text{by Prop.~\ref{prop:ball-inflation} } &\lesssim
\nu^{-1/\tilde{p}} \rho^{-d(1/2-1/\tilde{p})} L^{p}_{x} \avprod \ell^{\tilde{p}}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{\tilde p}(w_{B(x,1/\rho^{2})})}\\
\text{by H\"older } &\leq
\nu^{-1/\tilde{p}} \rho^{-d(1/2-1/\tilde{p})} \Big( L^{p}_{x} \avprod \ell^{2}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{2}(w_{B(x,1/\rho^{2})})} \Big)^{1-\kappa}\\
&\quad\cdot
\Big( L^{p}_{x} \avprod \ell^{p}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{p}(w_{B(x,1/\rho^{2})})} \Big)^{\kappa}\end{aligned}$$ By $L^{2}$ orthogonality the first bracket is $$\lesssim
L^{p}_{x} \avprod \ell^{2}_{Q\in \Part[R_i]{\rho^{2}}} \norm{f_{Q}}_{\avL^{2}(w_{B(x,1/\rho^{2})})}
=
A_{p}(\rho^{2}).$$ In the second bracket we observe that for every $2\leq p < \infty$ we have $$\label{eq:37}
\begin{split}
&
L^{p}_{x} \avprod \ell^{p}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{p}(w_{B(x,1/\rho^2)})}\\
\text{by H\"older }&\leq
\avprod L^{p}_{x} \ell^{p}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{\avL^{p}(w_{B(x,1/\rho^2)})}\\
\text{by Minkowski }&\leq
\avprod \ell^{p}_{Q\in \Part[R_i]{\rho}} L^{p}_{x} \norm{f_{Q}}_{\avL^{p}(w_{B(x,1/\rho^2)})}\\
&\lesssim
\avprod \ell^{p}_{Q\in \Part[R_i]{\rho}} \norm{f_{Q}}_{p}\\
\text{by scaling }&\lesssim
\avprod \ell^{p}_{Q\in \Part[R_i]{\rho}} \bigl( \Dec(\delta/\rho) \ell^{p}_{J\in \Part[Q]{\delta}} \norm{f_{J}}_{p} \bigr)\\
&=
\Dec(\delta/\rho) \avprod \ell^{p}_{J\in \Part[R_i]{\delta}} \norm{f_{J}}_{p}.
\qedhere
\end{split}$$
\[prop:bourgain-demeter\] Let $2 \leq p \leq \frac{2(d+2n)}{d}$ and suppose that $$\label{eq:40}
\Dec^{p}(\delta) \lesssim \delta^{-\eta}$$ for some $\eta = d(1/2-1/p) + \sigma$ with $\sigma>0$. Then for every $K$ we have $$\MulDec^{p}(\delta,K) \lesssim_{K} \delta^{-\eta(1-2^{-\lceil d/\sigma \rceil})}.$$
Choose $\nu_{K}$-transverse $R_{1},\dotsc,R_{M} \in \Part{1/K}$. Choose functions $f_{J}$ with $$\ell^{p}_{J \in \Part[R_i]{\delta}} \norm{f_J}_{p} = 1.$$ Let $m\in\N$ be chosen later. It suffices to consider $\delta$ that are powers of $2^{2^{m}}$. Let $\rho=\delta^{2^{-m}}$. Then $$\begin{aligned}
\MoveEqLeft\relax
L^{p}_{x} \avprod \norm{f_{R_i}}_{\avL^p(B(x,K))}\\
&=
L^{p}_{x} \avL^{p}_{\tilde{x} \in B(x,1/\rho)} \avprod \norm{f_{R_i}}_{\avL^p(B(\tilde{x},K))}\\
\text{by H\"older }&\leq
L^{p}_{x} \avprod \avL^{p}_{\tilde{x} \in B(x,1/\rho)} \norm{f_{R_i}}_{\avL^p(B(\tilde{x},K))}\\
&\lesssim
L^{p}_{x} \avprod \norm{f_{R_i}}_{\avL^p(B(x,1/\rho))}\\
\text{by Minkowski}&\leq
L^{p}_{x} \avprod \ell^{1}_{J\in\Part[R_i]{\rho}} \norm{f_{J}}_{\avL^p(B(x,1/\rho))} \\
\text{by H\"older }&\leq
\rho^{-d/2}
L^{p}_{x} \avprod \ell^{2}_{J\in\Part[R_i]{\rho}} \norm{f_{J}}_{\avL^p(B(x,1/\rho))}\\
\text{by Bernstein's inequality}&\lesssim
\rho^{-d/2}
L^{p}_{x} \avprod \ell^{2}_{J\in\Part[R_i]{\rho}} \norm{f_{J}}_{\avL^2(w_{B(x,1/\rho)})}\\
&=
\rho^{-d/2}
A_p(\rho)\end{aligned}$$
By the hypothesis on $p$ we have $\kappa_{p} \leq 1/2$. Iterating Proposition \[prop:iter\] with $\kappa=1/2$ starting with $\rho=\delta^{2^{-m}}$ until we get to $\rho=\delta$, at which point we use Hölder’s inequality, we get $$\begin{aligned}
A_{p}(\rho)
&\lesssim
\prod_{l=0}^{m-1} \Bigl( C\nu^{-1/\tilde{p}} \rho^{- 2^{l} d (1/2-1/\tilde{p})} \Dec(\delta/\rho^{2^{l}})^{\kappa} \Bigr)^{(1-\kappa)^{l}}.\end{aligned}$$ By the assumption on the linear decoupling constant this is $$\label{eq:45}
\lesssim_{m,\nu}
\prod_{l=0}^{m-1} \Bigl( \rho^{- 2^{l} d (1/2-1/\tilde{p})} \delta^{-\eta/2}/\rho^{-\eta 2^{l}/2} \Bigr)^{2^{-l}}
=
\delta^{-\eta (1-2^{-m})}\rho^{\eta m/2 - dm(1/2-1/\tilde{p})}.$$ By taking a supremum over all $R_i$ and $f_{J}$ as above, we deduce $$\label{eq:49}
\MulDec(\delta, K)
\lesssim_{m, \nu}
\delta^{-\eta+2^{-m-1}((m+2)\eta-d-2dm(1/2-1/\tilde{p}))}.$$ Choosing $m = \lceil d/\sigma \rceil$ we obtain the claim since $(1/2-1/p) \geq 2(1/2-1/\tilde{p})$.
\[thm:dec:general\] Let $2 \leq p \leq \frac{2(d+2n)}{d}$. Assume Hypothesis \[hyp:transverse\] and Hypothesis \[hyp:lower-dim\] with $\Lambda \geq d(1/2-1/p)$. Then for every $\epsilon>0$ we have $$\Dec^{p}(\delta) \lesssim_{\epsilon} \delta^{-\Lambda-\epsilon}.$$
If is easy to see that $\Dec(\delta) \lesssim \delta^{-\eta}$ for some $\eta = d(1/2-1/p) + \sigma$ with $\sigma>0$. If $\eta \leq \Lambda$, then we are done. Otherwise we will be able to decrease $\eta$. Substituting the conclusion of Proposition \[prop:bourgain-demeter\] into the conclusion of Proposition \[prop:bourgain-guth-arg\] gives $$\begin{aligned}
\Dec(\delta)
&\leq
C_{\epsilon} \sup_{\log \tilde{K} \approx \log K}
\tilde{K}^{\Lambda+\epsilon} \Dec(\delta \tilde{K})
+
C_{K} \MulDec(\delta, K)\\
&\leq
C_{\epsilon} \sup_{\log \tilde{K} \approx \log K}
\tilde{K}^{\Lambda+\epsilon} (\delta \tilde{K})^{-\eta}
+
C_{K} \delta^{-\eta'}\end{aligned}$$ with $\eta' = \eta (1 - 2^{-\lceil d/\sigma \rceil})$ for any $K$. Iterating this inequality $O(\frac{\log \delta^{-1}}{\log K})$ times we obtain $$\Dec(\delta)
\lesssim
C_{K} C_{\epsilon}^{C \abs{\log\delta}/\log K } \delta^{-\max(\Lambda+\epsilon,\eta')}.$$ Choosing $K$ large enough in terms of $C_{\epsilon}$ this gives $$\Dec(\delta)
\lesssim_{\epsilon}
\delta^{-\max(\Lambda,\eta')-2\epsilon}.$$ Thus we have succeeded in decreasing $\eta$. Iterating this we can make $\eta$ arbitrarily close to $\Lambda$.
Specific surfaces {#sec:spec}
=================
Lower dimensional decoupling {#sec:spec:lower-dim}
----------------------------
In this section we verify Hypothesis \[hyp:lower-dim\].
\[181212lem5.7\] Let $d$ and $n$ be as in and assume . Then for every $2 \leq p \leq 2+\frac{4n}{d}$ and every $\epsilon>0$ we have $$\label{181212e5.34}
\Dec_{\calH}^{p}(\delta)
\lesssim_{\epsilon}
\big(\frac{1}{\delta} \big)^{d(\frac 1 2-\frac 1 p)+\epsilon}$$ for every hyperplane $\calH$ given by with $\abs{\nabla \calL}\lesssim 1$.
The case $d=1$ is trivial, so we assume $d\geq 2$.
By a compactness argument the implicit constant can be made uniform in $\calH$, so we concentrate on showing for a fixed $\calH$. By Theorem \[thm:near-hyperplane\] it suffices to find a subspace $\calH' \subset \calH$ of dimension $d-2$ on which the decoupling exponent is $(d-2)(1/2-1/p)$; then we can apply $L^{2}$ decoupling in the remaining direction.
By the hypothesis there exist $\calH' \subset \calH$ of dimension $d-2$ and $\lambda$ such that $\lambda_{1}P_{1}+\dotsb+\lambda_{n}P_{n}$ has full rank on $\calH'$. By a change of variables we may assume $\calH'=\Set{(t_{1},\dotsc,t_{d-2},0,0)}$ and $\lambda = (1,0,\dotsc,0)$. But in this case the claim is given by Theorem \[main\_theorem\] for $n=1$ and $d$ replaced by $d-2$ in view of . The only thing to be verified is the restriction on the exponents $$2+\frac{4n}{d} \leq 2+\frac{4}{d-2},$$ which is equivalent to $n(d-2) \leq d$ and is satisfied in the cases listed in .
Transversality {#sec:spec:transverse}
--------------
In this section we verify Hypothesis \[hyp:transverse\]. The case $n=1$ will be verified in Lemma \[lem:BCCT:n=1\], the case $n=2, d\ge 3$ in Lemma \[180717lem5.2\], and the remaining cases in Lemma \[181203lemma6.4\].
We write $\R^{d+n}=\R^{d}\oplus \R^{n}=S_1\oplus S_2$.
\[lem:proj-dim-lower-bd:simple\] Let $V \subset \R^{d+n}$ be a subspace. Then $$\dim( \pi_{t}(V) ) \geq \dim (V \cap S_{1})$$ for all $t$.
The proof of this lemma is straightforward, and we leave it out.
\[lem:proj-dim-lower-bd\] Suppose that holds. Let $V \subset \R^{d+n}$ be a subspace and let $$0 \leq H_{1} \leq \min(\dim(V\cap S_{1}),d-n),
\quad
0 \leq H_{2} \leq \dim(V/S_{1}).$$ Then $$\dim( \pi_{t}(V) ) \geq H_{1}+H_{2}$$ for all $t$ outside the zero set of some non-trivial polynomial of degree $H_{2}$.
Since $\R^{d+n} = S_{1} \oplus S_{2}$ we have $$\dim(V) = \dim(V \cap S_{1}) + \dim(V/S_{1}).$$
By the hypotheses on $V$ we can choose a linearly independent set $(v_{i})_{1\leq i \leq H_{1}+H_{2}} \subset V$ with $v_{i}=(w_{i},z_{i})\in \R^{d}\times\R^{n}$ such that $z_{1}=\dotsb=z_{H_{1}}=0$ and $z_{H_{1}+1},\dotsc,z_{H_{1}+H_{2}}$ are linearly independent.
Consider the vectors $n_j(t) := (e_{j}, \nabla P(t)\cdot e_j) \in \R^{d} \times \R^{n}$, $j=1,\dotsc,d$, which form a basis of the tangent space of the surface $\calS_{d, n}$ at the point $t\in \R^d$. Then $$\label{181212e5.9}
\dim(\pi_t(V))
=
\rank \big(\langle v_i, n_j(t) \rangle \big)_{\substack{1\le i\le H_{1}+H_{2}\\1\le j\le d}}.$$ The matrix on the right-hand side of can be written as $$\begin{gathered}
\big(\langle v_i, n_j(t) \rangle \big)_{\substack{1\le i\le H_{1}+H_{2}\\1\le j\le d}}
=
\big(\langle w_i, e_j \rangle + \langle z_{i}, \nabla P(t)\cdot e_j \rangle \big)_{\substack{1\le i\le H_{1}+H_{2}\\1\le j\le d}}\\
=
\big(w_i + \nabla P(t) \cdot z_{i}\big)_{1\leq i \leq H_{1}+H_{2}}.\end{gathered}$$ Here $w_i, e_j, z_i$ are all treated as column vectors. Denote $H=H_1+H_2$. Since $\nabla P(t)$ is linear in $t$, each $H\times H$ minor determinant of this matrix is a polynomial of degree at most $H_{2}$ in $t$. Suppose for a contradiction that these minor determinants vanish identically. Then also their degree $H_{2}$ homogeneous parts vanish identically, and they coincide with the corresponding $H\times H$ minor determinants of the matrix $$\big(w_{1}, \dotsc,w_{H_{1}}, \nabla P(t)\cdot z_{H_{1}+1}, \dotsc, \nabla P(t)\cdot z_{H_{1}+H_{2}} \big)$$ Therefore the latter matrix does not have full rank for any $t$. Extending $w_{1},\dots$ and $z_{H_{1}+1},\dots$ to bases of $\R^{d-n}$ and $\R^{n}$, respectively, we see that the matrix $$\big(w_{1},\dotsc,w_{d-n}, \nabla P(t)\cdot z_{H_{1}+1}, \dotsc, \nabla P(t)\cdot z_{H_{1}+n} \big)$$ does not have full rank for any $t \in \R^{d}$. But the latter matrix can be factored as $$\begin{pmatrix}
w_{1},\dotsc,w_{d-n}, \nabla P(t)
\end{pmatrix}
\begin{pmatrix}
I_{d-n} & 0\\
0 & z_{H_1+1}, \dots, z_{H_1+n}
\end{pmatrix}.$$ The latter matrix is invertible and the former is invertible for all $t$ outside a proper subvariety by the hypothesis . This contradiction finishes the proof.
\[lem:BCCT:n=1\] Let $d\ge 1$ and $n=1$. For every proper linear subspace $V\subset \R^{d+n}$, it holds that $$\Set{t \given \dim(\pi_t(V))< \dim(V)}$$ is contained in a subvariety of degree $1$.
We may assume $\dim(V)=d$. The same argument works for all other cases. Let $H_{2} := \dim(V/S_{1})$. If $H_{2} = 0$, then by Lemma \[lem:proj-dim-lower-bd:simple\] we have $$\dim(\pi_{t}(V)) \geq \dim(V \cap S_{1}) = \dim(V)$$ for all $t$. If $H_{2}=1$, then by Lemma \[lem:proj-dim-lower-bd\] with $H_{1}=\dim(V)-1$ we obtain $$\dim(\pi_{t}(V)) \geq \dim(V)$$ for all $t$ outside a subvariety of degree $1$.
\[180717lem5.2\] Let $n=2$ and $d\ge 3$. Let $V\subset \R^{d+n}$ be a non-trivial proper linear subspace.
1. If $1\le \dim(V)\le d-1$, then the set $$\Set{t \given \dim(\pi_t(V))< \dim(V)}$$ is contained in a subvariety of degree $2$.
2. If $d\le \dim(V)\le d+1$, then the set $$\Set{t \given \dim(\pi_t(V))< \dim(V)-1}$$ is contained in a subvariety of degree $2$.
Let $H_{2} := \dim(V/S_{1}) \leq 2$. If $H_{2}=0$, then by Lemma \[lem:proj-dim-lower-bd:simple\] we have $\dim(\pi_{t}(V))=\dim V$ for all $t$. Otherwise by Lemma \[lem:proj-dim-lower-bd\] with $H_{1}=\min(\dim(V)-H_{2},d-2)$ we obtain $$\dim(\pi_{t}(V)) \geq \min(\dim(V),d-2+H_{2})$$ for all $t$ outside some subvariety of degree $\leq 2$. This gives the claim unless $H_{2}=1$, $\dim(V)=d+1$. But in this case $S_{1} \subset V$, so $\dim(\pi_{t}(V)) \geq d$ for all $t$ by Lemma \[lem:proj-dim-lower-bd:simple\].
\[181203lemma6.4\] Let $d=n\ge 2$ and $V \subset \R^{d+n}$.
1. If $\dim(V)$ is odd, then $$\Set{t \given \dim(\pi_t(V)) < (\dim(V)+1)/2}$$ is contained in a subvariety of degree at most $d$.
2. If $\dim(V)$ is even, then either $$\Set{t \given \dim(\pi_t(V)) < \dim(V)/2 + 1}$$ is contained in a subvariety of degree at most $d$ or $$\dim(\pi_{t}(V)) \geq \dim(V)/2$$ for all $t$.
Let $H_{2}:=\dim(V/S_{1})$. If $H_{2} > \dim(V)/2$, then $H_{2}\geq (\dim(V)+1)/2$ for $\dim(V)$ odd and $H_{2}\geq \dim(V)/2+1$ for $\dim(V)$ even. By Lemma \[lem:proj-dim-lower-bd\] with $H_{1}=0$ we obtain the claim in this case.
If $H_{2} \leq \dim(V)/2$, then $\dim(V\cap S_{1}) \geq \dim(V)/2$, so by Lemma \[lem:proj-dim-lower-bd:simple\] we obtain $\dim(\pi_{t}(V)) \geq \dim(V)/2$ for all $t$. A case distinction between $\dim(V)$ odd and even finishes the proof.
Simultaneously diagonalizable forms {#sec:simul-diag}
===================================
Here we prove Lemma \[lem:simul-diag\]. The case $d=2$ is trivial. The case $d=3$ is contained in Demeter, Guo and Shi [@arxiv:1609.04107]. Therefore in this section we work with the case $d=4$.
Let us first verify the condition . Take two linearly independent vectors $\vec{u}, \vec{v}\in \R^4$ with $\vec{u}=(u_1, \dots, u_4)$ and $\vec{v}=(v_1, \dots, v_4)$. We need to show that $$\det\begin{bmatrix}
a_1 t_1 & a_2 t_2 & a_3 t_3 & a_4 t_4\\
b_1 t_1 & b_2 t_2 & b_3 t_3 & b_4 t_4\\
u_1 & u_2 & u_3 & u_4\\
v_1 & v_2 & v_3 & v_4
\end{bmatrix}$$ does not vanish constantly, when viewed as a polynomial of $t$. We argue by contradiction and assume that this determinant vanishes constantly. Then it is not difficult to see, via calculating this determinant directly, that $$\det \begin{bmatrix}
u_i & u_j\\
v_i & v_j
\end{bmatrix}
=0 \text{ for every } i<j.$$ This further implies that $u$ and $v$ are linearly dependent, which is a contradiction.
Next we verify the condition . We argue by contradiction and assume that there exists a hyperplane $H\subset \R^d$ such that $$\label{181208e2.2}
\max\Set{\rank (P|_H), \rank (Q|_H)}\le 1.$$ Define two diagonal matrices $$M_1:=\diag(a_1, a_2, a_3, a_4) \text{ and } M_2:=\diag(b_1, b_2, b_3, b_4).$$ The assumption implies that $$\label{181208e2.4}
\max\Set{ \rank (LM_1 L^T), \rank (LM_2 L^T)}\le 1,$$ for some $3\times 4$ matrix $L$ of a full rank. Multiplying $L$ by an invertible $3\times 3$ matrix on the left and a permutation $4\times 4$ matrix on the right and reordering $a_{j}$’s and $b_{j}$’s we may assume that $$L=
\begin{bmatrix}
1 & 0 & 0 & \lambda_1\\
0 & 1 & 0 & \lambda_2\\
0 & 0 & 1 & \lambda_3
\end{bmatrix}$$ for some $\lambda_i$. Then $$\label{180524e2.25}
L M_1 L^T=
\begin{bmatrix}
a_1 & 0 & 0\\
0 & a_2 & 0\\
0 & 0 & a_3
\end{bmatrix}
+ a_4 (\lambda_1, \lambda_2, \lambda_3)^T (\lambda_1, \lambda_2, \lambda_3)$$ and $$\label{180524e2.26}
L M_2 L^T=
\begin{bmatrix}
b_1 & 0 & 0\\
0 & b_2 & 0\\
0 & 0 & b_3
\end{bmatrix}
+ b_4 (\lambda_1, \lambda_2, \lambda_3)^T (\lambda_1, \lambda_2, \lambda_3)$$ Notice that $$\label{rank_triangle}
\rank (A+B)\le \rank (A)+\rank (B),$$ for two arbitrary matrices and $$\rank \Big((\lambda_1, \lambda_2, \lambda_3)^T (\lambda_1, \lambda_2, \lambda_3)\Big)\le 1.$$ These two facts, combined with , imply that $a_1 a_2 a_3=0$ and $b_1 b_2 b_3=0$. By at most one of $a_{1},\dotsc,a_{4}$ can be $0$, so we may assume without loss of generality $a_{3}=0$. Again by at most one of $b_{1},\dotsc,b_{4}$ can be $0$, and if $a_{3}=0$ then $b_{3} \neq 0$, so we may assume without loss of generality $b_{2}=0$. Two minor determinants of order $2\times 2$ of are $$a_1 a_4 \lambda_3^2 \text{ and } a_2 a_4 \lambda_3^2.$$ Hence we must have $\lambda_3=0$, otherwise we have a contradiction to . By a similar argument applied to $M_2$, we must have $\lambda_2=0$. So far we have obtained $$\label{180524e2.25a}
L M_1 L^T=
\begin{bmatrix}
a_1+a_4 \lambda_1^2 & 0 & 0\\
0 & a_2 & 0\\
0 & 0 & 0
\end{bmatrix}
\text{ and }
L M_2 L^T=
\begin{bmatrix}
b_1+b_4 \lambda_1^2 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & b_3
\end{bmatrix}$$ Since from we know $a_2\neq 0$ and $b_3\neq 0$, by we obtain $$a_1+a_4 \lambda_1^2=b_1+b_4\lambda^2_1=0.$$ This is a contradiction to .
|
---
author:
-
bibliography:
- 'sample.bib'
title: '[Ineffectiveness of Dictionary Coding to Infer Predictability Limits of Human Mobility]{}'
---
Abstract {#abstract .unnumbered}
========
Recently, a series of models have been proposed to predict future movements of people. Meanwhile, dictionary coding algorithms are used to estimate the predictability limit of human mobility. Although dictionary coding is optimal, it takes long time to converge. Consequently, it is ineffective to infer predictability through dictionary coding algorithms. In this report, we illustrate this ineffectiveness on the basis of human movements in urban space.
Introduction
============
Understanding and predicting human mobility is of great importance for a wide spectrum of services and applications, e.g., transportation planning, emergency management and various location-based services. Many of the related works focus on building different models to predict human mobility and the ultimate goal is to improve the accuracy of the prediction models. This leads to a fundamental measurement: *predictability*, which quantifies the degree to which human mobility is predictable. For example, a piece of very well-received work [@limits-science] estimates the predictability based on dictionary coding, and it claims that the prediction accuracy of the next location of a certain mobile phone user is at most $93\%$, no matter what prediction approach is used. In this report, we present a counterexample to that highly-cited study and illustrate that the predictability estimated based on dictionary coding is not accurate.
To be more specific, we study the predictability of human mobility in urban road networks. The advances in GPS-enabled mobile devices and pervasive computing techniques have generated massive trajectory data. For instance, Qiangsheng Taxi Company Limited, one of the largest taxi companies in Shanghai, obtained more than $3\times 10^{10}$ trajectory records in April, 2015 [@soda]. The large amount of trajectory data provide opportunities to further enhance urban transportation systems.
For example, with the help of the large collection of trajectory data and the availability of digital maps, we could adopt data-driven approach to model map-matched trajectories. Given a road network that captures the roads in a modern city and a trajectory recording the movement of a moving object in the road network, trajectory modeling is to model the likelihood of a given trajectory that passes $k$ edges sequentially. As to be detailed in Section \[sec:review\], trajectory modeling tries to compute the transition probability $P(e_{i+1}|e_1, e_2, \cdots, e_i)$, the probability that a driver who drives from edge $e_1$ to $e_i$ via edges $e_2$, $e_3$, $\cdots$ takes the edge $e_{i+1}$. As presented in [@modeling-ijcai], Recurrent Neural Networks (RNN) based models are able to predict the transition probability with $87.8\%$ accuracy.
On the other hand, we adopt dictionary coding in this paper as an alternative to model trajectory and propose multiple novel techniques to improve the coding performance. Based on the experimental study result obtained from the real dataset used by [@modeling-ijcai], the predictability limit of dictionary coding based approach is around $82\%$ which is lower than the accuracy of $87.8\%$ achieved by RNN based models proposed in [@modeling-ijcai]. It demonstrates the ineffectiveness and limitations of dictionary coding in predicting human mobility in road networks. Meanwhile, it also provides evidence for illustrating the potential mistakes in the previous dictionary coding based study on predictability limit of human mobility [@limits-science].
The rest of the report is organized as follows. First, we review related work in Section \[sec:review\]. Then, we describe several techniques to improve the coding performance in Section \[sec:methods\]. Finally, we present and discuss the experiments in Section \[sec:results\] and conclude the results in Section \[sec:conclusion\].
Literature Review {#sec:review}
=================
In this section, we review the work on three topics closely relevant to this report, including *trajectory models in road networks*, *trajectory compression*, and *predictability limits of human mobility*.
We first begin with some preliminaries about trajectories and road networks. A trajectory is the path that a moving object follows in space, as a function of time. In practice, raw trajectory data contains series of sample points, where each sample point contains a position $p_i$ and the corresponding time stamp $t_i$, as presented in Definition \[def:raw-traj\].
\[def:raw-traj\] A trajectory $T$ is a sequence of $|T|$ tuples in the form of $\langle p_{1},t_{1}\rangle$, $\langle p_{2},t_{2}\rangle$, $\cdots$, $\langle p_{|T|},t_{|T|}\rangle$, where $p_{i}$ describes the position of the moving object in terms of longitude and latitude at the time stamp $t_i$.
A road network is modeled as a directed graph $G(V, E)$, where crossroads and road segments are represented by vertices and edges respectively. Note that vehicles in urban spaces are restrained by the underlying road networks, which differ from objects that move arbitrarily in common spaces. Notwithstanding, raw trajectory data is often biased from the network because of sampling errors. Consequently, map-matching algorithms should be applied to align the sample points with the road network. The results of the map-matching process are sequences of edges in $G$. In the rest of this report, we use map-matched trajectory and edge sequence interchangeably as they both refer to a sequence of edges in the road network passed by a moving object along a trajectory.
A road network is a directed graph $G(V, E)$, in which $V$ is the set of vertices and $E$ is the set of edges.
A map-matched trajectory is a sequence of $n$ road segments, i.e., a series of edges in $G$, in the form of $e_1$, $e_2$,$\cdots$,$e_n$.
Trajectory Modeling
-------------------
In trajectory models, it is essential to figure out the probability of a given trajectory. Unfortunately, the information entropy of trajectories tends to be infinity as the precision of data grows [@compress-tods]. In other words, the uncertainty of sample points is high. As a result, the probability distribution of original trajectories, in the form of $\langle p_{i},t_{i}\rangle$ sequence as defined in Definition \[def:raw-traj\], can hardly be determined.
That being said, however, it is possible to compute the probability of a map-matched trajectory $T$ [@modeling-ijcai], i.e., $$P(T)=P(e_1)\prod_{i=1}^{n-1}P(e_{i+1}|e_1,e_2,\cdots,e_i),$$ because map-matched trajectories are integer sequences, whose information entropy is limited. Such map-matched trajectory models have been adopted to solve many problems relevant to location-based services [@recovery; @osogami; @xue; @yuan]. Two models based on Recurrent Neural Networks are designed in [@modeling-ijcai], to compute the probability of a given edge sequence. Other approaches [@zhengandni; @Ziebart] use first-order Markov chains to model map-matched trajectories, which have been proved to be insufficient [@Srivatsa].
Computing the probability of an edge sequence is equivalent to predicting road transition, i.e., computing the probability $P(e_{i+1}|e_1,e_2,\cdots,e_i)$. To the best of our knowledge, the RNN-based approaches [@modeling-ijcai] are the most effective, in terms of prediction accuracy. Nevertheless, all approaches mentioned above only give lower bounds of the prediction accuracy. The upper bounds, i.e., limits of predictability, can be bounded by information entropy [@shannon], which is introduced in Section \[subsec:limit\].
Trajectory Compression
----------------------
Data compression is to represent data with fewer bits, so that the storage cost is reduced. During the past decades, a lot of trajectory data compression techniques [@compress-tods; @press-vldb; @icde-cinct] have been proposed to meet the requirements of trajectory data storage. Here, we only review some recent works on processing map-matched trajectory data.
PRESS [@press-vldb] is the first piece of work on compressing map-matched trajectories. It implements *Shortest Path Compression* and *Frequent Path Compression* to reduce the size of edge sequences. *Minimal Entropy Labeling (MEL)* proposed in [@compress-tods] is the first method to convert original edge sequences into label sequences, which significantly reduces the alphabet size from the original $|E|$ to the max out-degree $D$ of any vertex with $D\ll |E|$. MEL allocates a unique label to each out edge of every vertex in the road network, according to frequency of edges in the data set. This labeling method is optimal for the symbol-by-symbol coding algorithms like Huffman coding, because it is able to minimize the entropy of label sequences.
Different from MEL, *Relative Movement Labeling (RML)* proposed in [@icde-cinct] takes into account the dependence between two consecutive labels. This is achieved by constructing a directed graph called ET-graph, in which a vertex $v_{ET}$ represents an edge $e_i$ in the data set, while the succeeding vertices $u_{ET}$ of $v_{ET}$ represent preceding edges $e_{i-1}$ in edge sequences. Afterwards, a label is allocated to each edge $\langle v_{ET}, u_{ET} \rangle$ in the ET-graph, according to frequency of bi-grams $\langle e_{i-1},e_i\rangle$ in the data set.
If we still apply symbol-by-symbol coding after labeling, RML does not outperform MEL. However, if arithmetic coding is applied, coding length after RML will be smaller, because arithmetic coding may encode several symbols at once. Furthermore, Lempel-Ziv algorithms converge faster after RML than after MEL, though the eventual coding length remains the same since the entropy does not change during the labeling processes.
There is a close connection between prediction and data compression. Optimal prediction is equivalent to optimal data compression. On the one hand, if one can predict probabilities of the next symbol, on the basis of the historical data, then optimal coding length is achieved by applying arithmetic coding. On the other hand, given an optimal compressor, one can use the symbol that compresses the best for the prediction. Thereby, data compression is used as a benchmark for “general intelligence”.
Predictability of Human Mobility {#subsec:limit}
--------------------------------
The first method to address the predictability of human mobility studies location data of mobile phone users [@limits-science]. The data set contains trajectories of $5\times10^4$ users, in which the length of each trajectory is truncated to $2352$.
The method used in [@limits-science] is based on information theory and coding algorithms. Information entropy is a measure of uncertainty [@shannon; @cover]. The lower the entropy is, the lower the uncertainty is, and thereby the higher the predictability is. Note that locations of mobile phone users are matched to positions of routing towers, so the number of locations and thus the entropy are limited, which makes it possible to predict future movements of users.
The authors first present the relationship between predictability and information entropy. Given a history trajectory $h_{n-1}=\{x_1,x_2,\cdots,x_{n-1}\}$ that denotes the sequence of towers at which user $i$ was observed at each consecutive hourly interval, the best guess of the next location is to choose the location $x_n$ with the highest probability, so the optimal prediction accuracy based on $h_{n-1}$ is $$\label{eq:pii}
\pi(h_{n-1}) = sup_{x_n}\{P(X_n=x_n|h_{n-1})\}.$$ Considering all the history trajectories of length $n-1$, the optimal prediction accuracy of the $n$-th location is $$\label{eq:pin}
\Pi(n)=\sum_{h_{n-1}}P(h_{n-1})\pi(h_{n-1}).$$ As the length of a historical trajectory $h_n$ tends to be infinity, the overall limit of prediction accuracy is derived in Equation (\[eq:pi\]). $$\label{eq:pi}
\Pi=\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}\Pi(i).$$ It is proven that the limit of predictability can be bounded by the entropy rate: $$\label{ieq}
H(\mathcal{X}) \leq H(\Pi) + (1-\Pi)\log_2(S-1),$$ where $H(\mathcal{X})$ is the entropy rate, $H(p) = -p\log_2p - (1-p)\log_2(1-p)$ is the binary entropy, and $S$ is the total number of the routing towers, i.e., the maximum number of different locations visited by the user.
Next, the authors use Lempel-Ziv algorithms [@lempel; @LZW] to estimate the entropy rate $H(\mathcal{X})$. According to optimality of Lempel-Ziv coding [@ziv], its average coding length per symbol is asymptotically no greater than the entropy rate. Therefore, average coding length is an approximation of the entropy rate, as the length of trajectories tends to be infinity.
However, the experimental studies presented in [@compress-tods] imply that this method has some drawbacks. On the one hand, the speed that average coding length converges to the entropy rate is very slow. One the other hand, average coding length cannot be less than the entropy, which is the limit of data compression [@shannon]. Thereby, the estimated entropy rate is usually positively biased, especially when $S$ is large or trajectories are short. If the estimated entropy rate is greater than its true value, the value of $\Pi$ will be negatively biased. Consequently, the upper bound of prediction accuracy estimated by the coding method is erroneous.
There are other works applying the coding method to determine the predictability of traffic conditions [@limits-traffic] and vehicle staying time [@limits-staying]. The two problems are similar because they both study attributes of certain areas over time, such as the average speed and the staying time.
Methodology {#sec:methods}
===========
As it is mentioned in Section \[sec:review\], the reason that average coding length of Lempel-Ziv algorithms cannot converge to the entropy rate is two-fold, i.e., the alphabet size is large and the input trajectories are short. For example, a road network of a modern city has tens or hundreds of thousands edges, and a trajectory in our dataset passes by 35 edges on average. Consequently, transformation methods that can transform trajectories into other representations are necessary before applying dictionary code. In the following, we propose two steps, namely *alphabet reduction* and *sequence construction*, as solutions to tackle the two issues stated previously.
Let $h_{n-1}$ denote the history map-matched trajectory that passes ($n-1$) edges, and the optimal prediction accuracy is $$\label{equ:opt_tra_pred}
\pi(h_{n-1}) = sup_{e}\{P(e_{n} = e|h_{n-1})\}.$$ Note Equation (\[equ:opt\_tra\_pred\]) is consistent with Equation (\[eq:pii\]), although each tower $x_i$ appearing in $h_{n-1}$ is replaced by an edge $e_i$ passed by the map-matched trajectory. In other words, the definitions of the predictability $\Pi(n)$ and $\Pi$ in the context of map-matched trajectories are similar to those in [@limits-science], i.e., in Equation (\[eq:pin\]) and Equation (\[eq:pi\]).
Alphabet Reduction
------------------
The alphabet size of inputs, i.e., the number of edges in $G$, is so large that the coding algorithm can hardly find any repeated patterns. Hence Lempel-Ziv algorithms fail to divide inputs into long distinct phrases. Consequently, the compression will be ineffective, i.e., the average coding length will not be a good approximation of the entropy rate.
In order to address the problem resulting from the large alphabet, we plan to extend the labeling methods proposed in [@compress-tods; @icde-cinct]. There is no information loss after labeling process, because all label sequences can be converted to the original edge sequences, and vice versa. In other words, there is an injective function between edge sequences and label sequences, so labeling processes do not change the entropy. Nevertheless, the alphabet size is reduced and thus the data becomes easier to compress after labeling processes. Therefore, the coding algorithms converge faster on label sequences than on edge sequences.
As it is mentioned in Section \[sec:review\], MEL considers labels to be independently distributed, while RML only takes into account dependence between two consecutive labels. The labeling method can be further improved by considering dependence among more consecutive labels, so that the coding algorithms will be more effective. More precisely, we can label edges according to frequency of longer sub-sequences in the data set. However, the sub-sequences should not be too long, otherwise their frequency will be too low because of data sparsity, conversely leading to ineffective coding.
Sequence Construction
---------------------
The second problem is that trajectories are too short. Although Lempel-Ziv algorithms are optimal, it takes long time for average coding length to converge to the entropy. In this project, every map-matched trajectory represents a single trip of a person via a car. Therefore, different from the time series studied in [@limits-traffic; @limits-staying], whose average length grows as data sets become larger, the average length of trips is relatively fixed, even though the data collection is huge. For example, the average length of map-matched trajectories in our data set is about $35$, while the data set contains more than $7\times10^7$ trips. As a result, such short inputs cannot guarantee convergence of the coding length.
In order to address this issue, our proposal is to construct a long sequence based on the trajectory data set, and then take the long sequence as the input of the coding algorithm, so that the average coding length is able to converge to the entropy. Given a set of $k$ trajectories $T_i$s with each $T_i = \{e_{i,1}, e_{i,2},\cdots,e_{i,n_i}\}$, we can convert it into a single long sequence by connecting all trajectories together, i.e., $T_{input}= \{T_1,0,T_2,0,\cdots,T_k,0\}$, where $0$ represents the end of a trajectory. Note that the entropy of the long sequence will not be smaller than that of real trajectories since this construction does not cause any information loss. However, the long sequence is not a real trajectory, so the entropy will be different. In other words, the entropy of the long sequence must be greater than the entropy of real trajectories, which will result in erroneous estimation.
We plan to design an approach to construct an arbitrarily long sequence, and the entropy of its distribution is theoretically no greater than the entropy of real trajectories. Thereby, the coding length is able to converge and the estimation of the predictability limit will be a valid upper bound. In addition, the entropy of the construction should not be too negatively biased, otherwise the estimated upper bound will be meaningless. Furthermore, the construction approach should be compatible with alphabet reduction, so that both of the problems will be solved.
In most data compression and indexing methods, trajectory data are stored in the form of $T_1$,$0$,$T_2$,$0$,$\cdots$,$T_k$,$0$, so that there is no information loss. However, this format results in an intangible entropy value. On the other hand, the entropy rate can be calculated directly, if we connect real trajectories directly without using any extra symbols, i.e., in the form of $T_1,T_2,\cdots,T_k$.
In the following, we generate a long sequence by picking up trajectories randomly from the data set, after which we build the relationship between entropy of the constructed sequence and that of the real trajectory data. First, we sample the value of trajectory length $n_i$ (i.e., $|T_i|$) from $N$, the length distribution of trajectory data. Second, we sample a trajectory whose length is exactly $n_i$ from the conditional distribution $T|N=n_i$. The sampling strategy is equivalent to sample trajectories directly from $T$.
Let $T_i = e_{i,1}, e_{i,2},\cdots,e_{i,n_i}$ denote the $i$-th trajectory in the constructed sequence. Given any sample sequence of trajectory length, i.e., $n_1$, $n_2$, $\cdots$, $n_i$, the entropy rate is written as $$\begin{aligned}
H(\mathcal{T}) &= \lim_{k \to \infty} \frac{1}{k}\sum_{k'=1}^{k}H(e_{i',j'}|e_{1,1}, e_{1,2},\cdots,e_{1,n},\cdots,e_{i',1}, e_{i',2},\cdots,e_{i',j'-1})\end{aligned}$$ and here we have $k = \sum_{i'=1}^{i-1}n_{i'} + j$ and $k' = \sum_{i''=1}^{i'-1}n_{i''} + j'$, because trajectories vary in length. Afterwards, the entropy rate is calculated as: $$\begin{aligned}
H(\mathcal{T}) &= \lim_{k \to \infty} \frac{1}{k}(\sum_{i'=1}^{i-1}\sum_{j'=1}^{n_{i'}}H(e_{i',j'}|e_{i',1}, \cdots,e_{i',j'-1})+\sum_{j'=1}^{j}H(e_{i,j'}|e_{i,1}, \cdots,e_{i,j'-1}))\\
&=\lim_{k \to \infty} \frac{1}{k}(\sum_{i'=1}^{i-1}H(e_{i',1},e_{i',2} \cdots,e_{i',n_{i'}})+H(e_{i,1},e_{i,2},\cdots,e_{i,j}))\\
&=\lim_{k \to \infty} \frac{1}{\sum_{i'=1}^{i-1}n_{i'} + j}(\sum_{i'=1}^{i-1}H(T|N=n_{i'})+H(e_{i,1}, \cdots,e_{i,j}|N=n_i))\end{aligned}$$ As $j \leq n_i$ and $H(e_{i,1}, \cdots,e_{i,j}|N=n_i)\leq H(T|N=n_i)$ for any $j$, the limit does not change if we remove them from the dominator and the numerator respectively: $$\begin{aligned}
H(\mathcal{T}) &= \lim_{k \to \infty}
\frac{\sum_{i'=1}^{i-1}H(T|N=n_{i'})}{\sum_{i'=1}^{i-1}n_{i'}}=\lim_{i \to \infty}
\frac{\sum_{i'=1}^{i}H(T|N=n_{i'})}{\sum_{i'=1}^{i}n_{i'}}\end{aligned}$$ The strong law of large numbers implies that the sample average converges almost surely to the expected value: $$\begin{aligned}
Pr[\lim_{i \to \infty} \frac{\sum_{i'=1}^{i}H(N)}{i} = E[H(N)] &=H(T|N)] = 1\\
Pr[\lim_{i \to \infty} \frac{\sum_{i'=1}^{i}N}{i}& = E[N]] = 1\end{aligned}$$ where it is noteworthy that the entropy $H(N)$ is a random variable, and $Pr[H(N)=H(T|N=n_i)]=Pr[N=n_i]$.
According to properties of limits, we have $$\begin{aligned}
Pr[\lim_{i \to \infty} \frac{\sum_{i'=1}^{i}H(N)}{\sum_{i'=1}^{i}N}& = \frac{H(T|N)}{E[N]}] = 1\end{aligned}$$ In other words, the entropy rate and thus the average coding length of Lempel-Ziv coding is $H(T|N)/E[N]$: $$\begin{aligned}
H(\mathcal{T}) &= \lim_{i \to \infty}
\frac{\sum_{i'=1}^{i}H(T|N=n_{i'})}{\sum_{i'=1}^{i}n_{i'}} = \frac{H(T|N)}{E[N]}\end{aligned}$$
The predictability of trajectory data whose length is limited can be also bounded by the entropy, by extending the results in [@limits-science]. Fano’s inequality shows that $$\begin{aligned}
H(e_i|e_1,e_2,\cdots,e_{i-1})&\leq H_F(\Pi(i))\\
\frac{1}{n}\sum_{i=1}^{n}H(e_i|e_1,e_2,\cdots,e_{i-1})&\leq \frac{1}{n}\sum_{i=1}^{n}H_F(\Pi(i))\\
\frac{H(T|N=n)}{n}&\leq \frac{1}{n}\sum_{i=1}^{n}H_F(\Pi(i))\end{aligned}$$ According to Jensen’s inequality and the fact that $H_F(p) = H(p)+(1-p)\times \log_2(S-1)$ is a convex function, the average predictability is bounded by the entropy: $$\begin{aligned}
\frac{H(T|N=n)}{n}&\leq H_F(\frac{1}{n}\sum_{i=1}^{n}\Pi(i))\\
\sum_{n}Pr[N=n]\frac{H(T|N=n)}{n}&\leq \sum_{n}Pr[N=n]H_F(\frac{1}{n}\sum_{i=1}^{n}\Pi(i))\\
\label{ieq2}
E[\frac{H(N)}{N}]&\leq H_F(\sum_{n}(\frac{Pr[N=n]}{n}\sum_{i=1}^{n}\Pi(i)) =H_F(\Pi)\end{aligned}$$ However, $E[H(N)/N]$ is not bounded by $E[H(N)]/E[N]$. Consequently, it is not proper to directly connect all trajectories of various lengths to construct the long sequence.
Now we consider the case in which all the trajectories have the same length $n$. Then the long sequence will be $e_{1,1}$, $e_{1,2}$,$\cdots$,$e_{1,n}$, $e_{2,1}$, $e_{2,2}$,$\cdots$,$e_{2,n}$, $\cdots$, $e_{i,1}$, $e_{i,2}$,$\cdots$,$e_{i,n}$, $\cdots$. After construction, we can easily separate the long sequence into original trajectories, even though the long sequence does not include extra symbols like $0$, because each trajectory is of length $n$. As a result, there is no information loss after construction, and the entropy will be no smaller than the real entropy. More precisely, the entropy rate of the long sequence is $$\begin{aligned}
H(\mathcal{T}) &= \lim_{k \to \infty} \frac{1}{k}H(e_{1,1}, e_{1,2},\cdots,e_{1,n},\cdots,e_{i,1}, e_{i,2},\cdots,e_{i,j})\\
&= \lim_{k \to \infty} \frac{1}{k}\sum_{k'=1}^{k}H(e_{i',j'}|e_{1,1}, e_{1,2},\cdots,e_{1,n},\cdots,e_{i',1}, e_{i',2},\cdots,e_{i',j'-1})\end{aligned}$$ where $k = n\times (i-1) + j$ and $k' = n\times (i'-1) + j'$. If we pick up trajectories from the data set randomly during the construction, i.e., the order of real trajectories in the long sequence is random, there will be less relevance between each part of the long sequence. In other words, we sample trajectories $T_i$ from the identical distribution $T$ independently. On the basis of this assumption, the entropy rate will be $$\begin{aligned}
H(\mathcal{T}) &= \lim_{k \to \infty} \frac{1}{k}\sum_{k'=1}^{k}H(e_{i',j'}|e_{1,1}, e_{1,2},\cdots,e_{1,n},\cdots,e_{i',1}, e_{i',2},\cdots,e_{i',j'-1})\\
&= \lim_{k \to \infty} \frac{1}{k}(\sum_{i'=1}^{i-1}\sum_{j'=1}^{n}H(e_{i',j'}|e_{i',1}, \cdots,e_{i',j'-1})+\sum_{j'=1}^{j}H(e_{i,j'}|e_{i,1}, \cdots,e_{i,j'-1}))\\
&=\lim_{k \to \infty} \frac{1}{k}(\sum_{i'=1}^{i-1}H(e_{i',1}, e_{i',2},\cdots,e_{i',n})+H(e_{i,1}, e_{i,2},\cdots,e_{i,j}))\\
&=\lim_{k \to \infty} \frac{(i-1)\times H(T)}{n\times (i-1) + j}+\lim_{k \to \infty}\frac{H(e_{i,1}, e_{i,2},\cdots,e_{i,j})}{n\times (i-1) + j}\\
&=\lim_{i \to \infty} \frac{(i-1)\times H(T)}{n\times (i-1)}+0 =\frac{H(T)}{n}\end{aligned}$$ The entropy rate and hence the average coding length will be $H(T)/n$, if we only use trajectories of length $n$ to construct the long sequence.
Now we sum up the overall approach to the problem. First, we categorize all trajectories in the data set according to their length, so that they will form many groups. Second, we construct long sequences with respect to each group and apply Lempel-Ziv coding and determine the value $H(T|N=n)/n$ for each $n$. Third, we estimate $E[H(N)/N]$ as well as $\Pi$ according to Inequation (\[ieq2\]).
Fusion of Strategies
--------------------
Sequence construction is compatible with the labeling strategy. We illustrate the idea by a simple proof. According to the fact that $$\begin{aligned}
\frac{H(e_1,e_2,\cdots,e_n)}{n}\leq H_F(\frac{1}{n}\sum_{i=1}^{n}\Pi(i))\end{aligned}$$ and $$\begin{aligned}
H(e_1,e_2,\cdots,e_n)=H(e_1,l_1,\cdots,l_{n-1}),\end{aligned}$$ we have $$\begin{aligned}
\frac{H(e_1,l_1,\cdots,l_{n-1})}{n}\leq H_F(\frac{1}{n}\sum_{i=1}^{n}\Pi(i)).\end{aligned}$$ Thereby, $$\begin{aligned}
\frac{H(l_1,\cdots,l_{n-1})}{n} \leq H_F(\frac{1}{n}\sum_{i=1}^{n}\Pi(i)).\end{aligned}$$ $\hat{H}_n=H(l_1,\cdots,l_{n-1})/(n-1)$ can be estimated through sequence construction, so we have $$\begin{aligned}
\frac{n-1}{n}\cdot \hat{H}_n \leq H_F(\frac{1}{n}\sum_{i=1}^{n}\Pi(i)) = H_F(\Pi_n)\end{aligned}$$ Note that the prediction accuracy of the first edge is nearly $0$, and we only care about the predictability of the following edges, so $$\begin{aligned}
\hat{\Pi}_n = \frac{1}{n-1}\sum_{i=2}^{n}\Pi(i) \approx \frac{1}{n-1}\sum_{i=1}^{n}\Pi(i) = \frac{n-1}{n}\cdot \Pi_n\end{aligned}$$ Finally, the predictability limit is $$\begin{aligned}
\Pi = \sum_n Pr[N=n]\cdot \hat{\Pi}_n \end{aligned}$$
Experiments {#sec:results}
===========
In this section, we present the experimental results, to prove the ineffectiveness of the approach proposed in [@limits-science]. We first introduce the data set used in the experimental study. The data set consists of real trajectory data from Qiangsheng Taxi, which contains more than $7\times10^7$ trips. The road network of Shanghai is extracted from OpenStreetMap, which contains $60,200$ edges and $28,620$ vertices. This dataset is exactly the same as the dataset used in [@modeling-ijcai], and the prediction accuracy achieved by the models proposed in [@modeling-ijcai] is 87.8%. All the algorithms are implemented with C programming language and run on a computer with Intel Core i$7$-$6820$HK CPU@$3.30$ GHz and $32$ GB memory.
Then we apply the Lempel-Ziv-Welch algorithm [@LZW], one of the most commonly used Lempel-Ziv algorithms, directly on the map-matched trajectory data. The average coding length per symbol is $15.94$ bits. Note that it only takes $\lceil\log_2 |E|\rceil$, i.e., $16$ bits to represent an edge, if we use naïve sequential coding. Consequently, the LZW algorithm can hardly compress the trajectory data, let alone does the average coding length converge to the entropy rate. This demonstrates the poor performance of the optimal coding on map-matched trajectory data.
As explained in Section \[sec:methods\], we reduce the alphabet size through the labeling methods. The average coding length drops to $1.15$ bits if MEL is applied, while that becomes $1.12$ bits if RML is applied, with both being much smaller than that of direct Lempel-Ziv coding. However, the coding length is still not satisfactory, because it leads to the estimated upper bounds of $81.9\%$ and $82.3\%$ respectively, according to Inequation \[ieq\]. Moreover, we apply both sequence construction and alphabet reduction. As the alphabet size is merely $6$ and the constructed sequences all contain at least $10^4$ labels ($5\times10^4$ on average), the performance of dictionary coding is significantly improved. Nevertheless, the estimated upper bound is $85.3\%$, which is still lower than $87.8\%$, the best prediction accuracy achieved by trajectory models so far.
Conclusion {#sec:conclusion}
==========
To draw a conclusion, the coding length of Lempel-Ziv algorithms does not converge to the entropy rate of trajectories, though the two transformation strategies are applied. Consequently, it is ineffective to infer predictability limits through the approach based on dictionary coding. In other words, the approach proposed in [@limits-science] is problematic. In the near future, we plan to design a new approach to infer predictability limits.
|
---
abstract: |
We report the final results of our study of the cosmic microwave background (CMB) with the BIMA array. Over 1000 hours of observation were dedicated to this project exploring CMB anisotropy on scales between $1{^{\prime}}$ and $2{^{\prime}}$ in eighteen $6{^{\prime}}.6$ FWHM fields. In the analysis of the CMB power spectrum, the visibility data is divided into two bins corresponding to different angular scales. Modeling the observed excess power as a flat band of average multipole $\ell_{eff} = 5237$, we find $\Delta
T_1^2=220_{-120}^{+140}\,\mu$K$^2$ at $68\%$ confidence and $\Delta
T_1^2 >0 \,\mu$K$^2$ with $94.7\%$ confidence. In a second band with average multipole of $\ell_{eff} = 8748$, we find $\Delta T_2^2$ consistent with zero, and an upper limit $880\,\mu$K$^2$ at $95\%$ confidence. An extensive series of tests and supplemental observations with the VLA provide strong evidence against systematic errors or radio point sources being the source of the observed excess power. The dominant source of anisotropy on these scales is expected to arise from the Sunyaev-Zel’dovich (SZ) effect in a population of distant galaxy clusters. If the excess power is due to the SZ effect, we can place constraints on the normalization of the matter power spectrum $\sigma_8 = 1.03^{+0.20}_{-0.29}$ at $68\%$ confidence. The distribution of pixel fluxes in the BIMA images are found to be consistent with simulated observations of the expected SZ background and rule out instrumental noise or radio sources as the source of the observed excess power with similar confidence to the detection of excess power. Follow-up optical observations to search for galaxy over-densities anti-correlated with flux in the BIMA images, as might be expected from the SZ effect, proved to be inconclusive.
author:
- |
K.S. Dawson, W.L. Holzapfel, J.E. Carlstrom,\
M. Joy, and S.J. LaRoque
title: Final Results from the BIMA CMB Anisotropy Survey and Search for Signature of the SZ effect
---
Introduction
============
The angular power spectrum of the Cosmic Microwave Background (CMB) has been measured with high signal to noise on scales from degrees to several arcminutes (e.g., Hinshaw [et al. ]{}2003, Kuo [et al. ]{}2004, Mason [et al. ]{}2003). However, observations of CMB anisotropy on arcminute scales, where secondary anisotropies such as the Sunyaev-Zel’dovich (SZ) effect (Sunyaev & Zel’dovich 1970) are expected to dominate the primary CMB anisotropy (e.g., Gnedin & Jaffe 2001), have not yet achieved comparable precision. At these finer angular scales, observations distant SZ clusters have the potential to be a powerful probe of the growth of structure in the Universe (Carlstrom, Holder, & Reese 2002). However, the power spectrum of the arcminute fluctuations reveals little information about the nature of the sources responsible for the anisotropy. As suggested by Rubino-Martin & Sunyaev (2003), higher order statistics of images can, in principle, be used to identify the unique signature of the SZ effect.
Beginning in the summer of 1998, we began a program to search for arcminute-scale CMB anisotropy using the Berkeley-Illinois-Maryland Association (BIMA[^1]) interferometer. Initial results are included in earlier publications (Holzapfel [et al. ]{}2000, Dawson [et al. ]{}2001, and Dawson [et al. ]{}2002 (hereafter D2002)). A detailed description of the BIMA analysis, results from other experiments, and a comparison with theoretical models and simulations of structure formation can be found in D2002. In this paper, we report the final results from the BIMA CMB anisotropy survey. We describe the observations of the final fields in summer 2002 with the BIMA array and the Very Large Array (VLA[^2]) in §\[sec:obs\]. The results of the power spectrum analysis, including a discussion of tests for systematic errors in the analysis are presented in §\[sec:power\]. In §\[sec:images\] we examine the BIMA image statistics in order to constrain the origin of the observed anisotropy. In §\[sec:optical\], we describe the results of follow-up optical observations used in an attempt to identify galaxy clusters in the BIMA fields. Finally, in §\[sec:con\], we summarize the results and present our conclusions.
Observations {#sec:obs}
============
The BIMA anisotropy survey consists of 18 fields that had not been previously observed for SZ galaxy clusters or CMB anisotropy at arcminute angular scales. Each field was observed with the BIMA array at a frequency of $28.5$ GHz and the VLA at a frequency of $4.8$ GHz. Analysis of ten fields observed with the BIMA array during the summers of 1998, 2000, and 2001 revealed evidence for a detection of power in excess of the instrument noise. A description of the analysis and results for the first ten fields in the survey can be found in D2002. Eight new fields were added to the survey in the summer of 2002.
BIMA Observations {#subsec:bima}
-----------------
All anisotropy observations were made using the BIMA array at Hat Creek. Nine $6.1$ meter telescopes of the array were equipped for operation at $28.5$ GHz, providing a $6.6{^{\prime}}$ FWHM field of view. The first ten fields, BDF4-BDF13, were chosen to lie in regions expected to have minimal radio point source and dust contamination. The eight fields added in 2002, BDF14-BDF21, were chosen to lie two minutes east in Right Ascension of existing fields in the survey. Observations of these fields added another 489 hours of observations to the previously published data, for a total of 1096 hours for the complete BIMA survey. Each of these new fields was observed using the same phase calibrator as its previously observed nearest neighbor and analyzed following the data reduction as described in D2002. These fields were chosen to check for possible contamination of the signal correlated with telescope position. This makes for a total of 18 independent fields in the survey, covering approximately $0.2$ square degrees. The pointing center, dates of observation, and observation time for each of the fields are given in Table \[tab:obstimes\]. We dedicated $55-65$ hours of observation with the BIMA array to each of the new fields in order to achieve a uniform RMS noise level of $<150\,\mu$Jy/beam on short baselines ($u$-$v < 1.1 \, {{\rm{k}\lambda}}$) for the entire sample. This noise level corresponds to an RMS of $15.55 \,\mu$K for a $2{^{\prime}}$ synthesized beam.
R. A. (J2000) Decl. (J2000) Observation year(s) Time (Hrs)
------- ---------------------------------------------- ---------------------------------------------------- --------------------- ------------
BDF4 $00{^{\rm h}}\,28{^{\rm m}}\,04.4{^{\rm s}}$ $+28^{\circ}\,23{^{\prime}}\,06{^{\prime \prime}}$ 98 $77.6$
HDF $12{^{\rm h}}\,36{^{\rm m}}\,49.4{^{\rm s}}$ $+62^{\circ}\,12{^{\prime}}\,58{^{\prime \prime}}$ 98, 01 $59.9$
BDF6 $18{^{\rm h}}\,21{^{\rm m}}\,00.0{^{\rm s}}$ $+59^{\circ}\,15{^{\prime}}\,00{^{\prime \prime}}$ 98, 00 $81.2$
BDF7 $06{^{\rm h}}\,58{^{\rm m}}\,45.0{^{\rm s}}$ $+55^{\circ}\,17{^{\prime}}\,00{^{\prime \prime}}$ 98, 00 $68.2$
BDF8 $00{^{\rm h}}\,17{^{\rm m}}\,30.0{^{\rm s}}$ $+29^{\circ}\,00{^{\prime}}\,00{^{\prime \prime}}$ 00, 01 $53.3$
BDF9 $12{^{\rm h}}\,50{^{\rm m}}\,15.0{^{\rm s}}$ $+56^{\circ}\,52{^{\prime}}\,30{^{\prime \prime}}$ 00, 01 $53.9$
BDF10 $18{^{\rm h}}\,12{^{\rm m}}\,37.2{^{\rm s}}$ $+58^{\circ}\,32{^{\prime}}\,00{^{\prime \prime}}$ 00, 01 $53.3$
BDF11 $06{^{\rm h}}\,58{^{\rm m}}\,00.0{^{\rm s}}$ $+54^{\circ}\,24{^{\prime}}\,00{^{\prime \prime}}$ 00, 01 $50.0$
BDF12 $06{^{\rm h}}\,57{^{\rm m}}\,38.0{^{\rm s}}$ $+55^{\circ}\,32{^{\prime}}\,00{^{\prime \prime}}$ 01 $54.8$
BDF13 $22{^{\rm h}}\,22{^{\rm m}}\,45.0{^{\rm s}}$ $+36^{\circ}\,37{^{\prime}}\,00{^{\prime \prime}}$ 01 $54.5$
BDF14 $00{^{\rm h}}\,26{^{\rm m}}\,04.4{^{\rm s}}$ $+28^{\circ}\,23{^{\prime}}\,06{^{\prime \prime}}$ 02 $62.4$
BDF15 $06{^{\rm h}}\,56{^{\rm m}}\,45.0{^{\rm s}}$ $+55^{\circ}\,17{^{\prime}}\,00{^{\prime \prime}}$ 02 $64.2$
BDF16 $12{^{\rm h}}\,34{^{\rm m}}\,49.4{^{\rm s}}$ $+62^{\circ}\,12{^{\prime}}\,58{^{\prime \prime}}$ 02 $64.5$
BDF17 $18{^{\rm h}}\,19{^{\rm m}}\,00.0{^{\rm s}}$ $+59^{\circ}\,15{^{\prime}}\,00{^{\prime \prime}}$ 02 $64.5$
BDF18 $00{^{\rm h}}\,15{^{\rm m}}\,30.0{^{\rm s}}$ $+29^{\circ}\,00{^{\prime}}\,00{^{\prime \prime}}$ 02 $57.8$
BDF19 $06{^{\rm h}}\,55{^{\rm m}}\,38.0{^{\rm s}}$ $+55^{\circ}\,32{^{\prime}}\,00{^{\prime \prime}}$ 02 $59.4$
BDF20 $12{^{\rm h}}\,48{^{\rm m}}\,15.0{^{\rm s}}$ $+56^{\circ}\,52{^{\prime}}\,30{^{\prime \prime}}$ 02 $54.7$
BDF21 $18{^{\rm h}}\,10{^{\rm m}}\,37.2{^{\rm s}}$ $+58^{\circ}\,32{^{\prime}}\,00{^{\prime \prime}}$ 02 $62.0$
VLA Observations {#sec:VLA}
----------------
To help constrain the contribution from point sources to the anisotropy measurements, we used the VLA at a frequency of $4.8\,$GHz to observe each field in the survey. With $1.5$ hours per field, these observations yielded an RMS flux of $\sim 25\,\mu$Jy/beam over a $9{^{\prime}}$ FWHM region with the same pointing center as a BIMA field. The positions of all point sources detected with significance $>6\sigma$ within $400{^{\prime \prime}}$ of the pointing center have been recorded. Measured point sources with fluxes corrected for attenuation by the primary beam at $4.8$ GHz are listed in Tables \[tab:vlasrc(1998-2001)\] and \[tab:vlasrc(2002)\].
If the spectra of the point sources are nearly flat or falling, deep observations with the VLA will identify those that lie near the noise level in the $28.5$ GHz maps. However, it is possible that a radio source with a steeply inverted spectrum may lie below the VLA detection threshold but would still contribute significantly at $28.5$ GHz. Advection dominated accretion flows are thought to be the most common inverted spectrum sources. They typically have a slowly rising spectrum, with a spectral index of $0.3$ to $0.4$ (Perna & DiMatteo, 2000) where the point source flux $S
\propto \nu^{\alpha}$. Such a shallow spectrum would only provide a factor of two increase in flux between $4.8$ GHz and $28.5$ GHz; any source not seen with the VLA would be near the noise level of the BIMA observations.
To search for point sources with more steeply inverted spectral indices, we made VLA observations at $8.0$ GHz of the five BIMA fields that most strongly indicate an excess of anisotropy power. If radio sources are the dominant contribution to the observed excess power, it is these fields that are the most likely to be contaminated. Results of these observations are found in Table \[tab:vlasrc(1998-2001)\] and Table \[tab:vlasrc(2002)\]. The $8\,$GHz observations reached a RMS flux density of $20\, \mu$Jy/beam at the center of the $5^\prime$ FWHM primary beam. Six of the 13 sources identified in the $4.8$ GHz maps were detected in the $8.0$ GHz maps with Signal-to-Noise ratio (SNR) $> 3$. The mean spectral index of the detected sources is found to be $\alpha = -0.4$. The fluxes of the other seven sources were poorly constrained because those sources were either too dim or were positioned outside of the $8.0$ GHz primary beam where the instrument is most sensitive. The $8\,$GHz images produced no additional point source detections with SNR$>6$, providing additional evidence against contamination by a population of radio sources with inverted spectra.
$\Delta$ R.A. ($^{{^{\prime \prime}}}$) $\Delta$ DEC ($^{{^{\prime \prime}}}$) 4.8 GHz ($\mu$Jy) 8 GHz ($\mu$Jy) 30 GHz ($\mu$Jy)
------- ----------------------------------------- ---------------------------------------- --------------------- --------------------- ------------------
BDF4 $-96.8$ $\phn 255.7$ $1230\pm 90.6$
BDF4 $\phn \phn 72.8$ $\phn 178.2$ $\phn 514\pm 54.9$ $468\pm313$
BDF4 $\phn \phn 99.9$ $-89.4$ $\phn 221 \pm 42.3$ $90.7\pm225$
BDF4 $-94.9$ $\phn 268.4$ $\phn 391 \pm 89.3$
HDF $-35.0$ $-85.0$ $\phn 832 \pm 56.3$ $535\pm185$
HDF $\phn 255.0$ $-89.8$ $1380 \pm 93.0$
HDF $\phn 178.3$ $-274.0$ $1520 \pm 112$
HDF $\phn 222.5$ $-86.8$ $\phn 709 \pm 64.6$
HDF $\phn \phn 69.1$ $\phn 334.1$ $1120 \pm 107$
HDF $-21.1$ $\phn \phn 66.1$ $\phn 190 \pm 36.2$
BDF6 $-136.5$ $-283.5$ $\phn 592 \pm 74.5$ $-51.3 \pm 226$
BDF7 $\phn 314.6$ $\phn \phn 47.4$ $1554 \pm 80.5$
BDF7 $\phn 173.8$ $\phn \phn 97.8$ $\phn 373 \pm 40.1$ $156\pm306$
BDF7 $\phn 253.8$ $\phn -1.1$ $\phn 284 \pm 49.5$
BDF8 $-145.9$ $-266.1$ $1381 \pm 94.1$
BDF8 $\phn \phn 27.6$ $\phn 280.9$ $\phn 611 \pm 72.8$
BDF8 $\phn 302.9$ $-79.5$ $\phn 622 \pm 86.1$
BDF9 $-221.9$ $-123.7$ $1500 \pm 78.4$
BDF9 $-192.7$ $\phn 215.8$ $1193 \pm 81.3$
BDF9 $\phn 245.2$ $-101.0$ $1039 \pm 67.8$
BDF10 $-158.5$ $-165.6$ $1670 \pm 63.9$ $1610\pm347$
BDF10 $-146.1$ $-183.9$ $\phn 320 \pm 44.4$
BDF11 $\phn \phn 87.7$ $\phn \phn 77.9$ $\phn 246 \pm 34.6$
BDF11 $\phn 342.8$ $\phn \phn \phn 8.8$ $\phn 865 \pm 101$ $387\pm179$
BDF11 $\phn \phn 42.5$ $-11.8$ $\phn 152 \pm 30.7$
BDF12 $-241.0$ $-256.7$ $1620 \pm 105$ $-97.4 \pm 368$
BDF12 $\phn 260.2$ $\phn 300.5$ $1191 \pm 129$ $-1080 \pm 905$
BDF12 $-137.4$ $-133.4$ $\phn 278 \pm 40.5$ $126 \pm 36.7$ $165\pm273$
BDF12 $\phn 170.9$ $\phn \phn 66.7$ $\phn 211 \pm 38.7$ $98.8 \pm 33.9$ $-151\pm259$
BDF13 $\phn 181.4$ $-49.5$ $\phn 721 \pm 51.2$ $\phn 531 \pm 50.8$ $935\pm271$
BDF13 $-154.0$ $\phn 299.7$ $1145 \pm 98.8$ $126 \pm 427$
BDF13 $\phn 225.1$ $-99.9$ $\phn 317 \pm 56.4$ $11.4 \pm 106$
Entries are left blank for fields not observed at $8\,$GHz. For the $30\,$GHz BIMA observations, entries are blank for those point sources which lie outside the primary beam and are not detected at $>3\sigma$ significance.
$\Delta$ R.A. ($^{{^{\prime \prime}}}$) $\Delta$ DEC. ($^{{^{\prime \prime}}}$) 4.8 GHz ($\mu$Jy) 8 GHz ($\mu$Jy) 30 GHz ($\mu$Jy)
------- ----------------------------------------- ----------------------------------------- --------------------- --------------------- ------------------
BDF14 $\phn 125.1$ $-166.8$ $1186 \pm 35.3$ $1235 \pm 63.1$
BDF14 $\phn -56.3$ $\phn \phn 30.3$ $\phn 226 \pm 24.1$ $\phn 295 \pm 22.2$ $-195\pm143$
BDF14 $\phn \phn 17.6$ $-183.4$ $\phn 578 \pm 26.5$ $\phn 367 \pm 48.9$ $110\pm244$
BDF15 $\phn \phn -6.9$ $-324.4$ $9390 \pm 78.6$ $6390\pm857$
BDF15 $-129.0$ $\phn 218.7$ $\phn 637 \pm 42.1$
BDF15 $\phn \phn 77.4$ $-282.6$ $\phn 707 \pm 51.4$
BDF16 $-157.8$ $\phn 116.7$ $1091 \pm 38.3$ $212\pm274$
BDF16 $-266.2$ $\phn 198.5$ $2553 \pm 74.8$
BDF16 $-158.4$ $\phn \phn 33.6$ $\phn 576 \pm 33.1$ $215\pm222$
BDF17 $\phn \phn 39.9$ $\phn 169.3$ $\phn 431 \pm 26.2$ $\phn 774 \pm 44.4$ $694\pm228$
BDF18 $-213.4$ $\phn 381.6$ $6902 \pm 145$
BDF18 $-106.1$ $\phn -0.7$ $\phn 465 \pm 25.7$ $145\pm168$
BDF18 $\phn 285.4$ $\phn \phn \phn 2.8$ $\phn 988 \pm 49.9$
BDF18 $\phn 137.7$ $\phn \phn 93.9$ $\phn 515 \pm 29.7$ $103\pm222$
BDF18 $\phn \phn -9.5$ $\phn 290.3$ $1636 \pm 50.5$
BDF18 $\phn 112.9$ $\phn 165.9$ $\phn 266 \pm 32.8$ $381\pm276$
BDF18 $\phn -32.2$ $\phn 105.4$ $\phn 333 \pm 25.5$ $142\pm170$
BDF19 $\phn \phn 31.9$ $\phn 169.4$ $\phn 289 \pm 32.8$ $-95\pm237$
BDF19 $-169.8$ $\phn -17.3$ $\phn 363 \pm 32.6$ $-16\pm234$
BDF19 $-128.7$ $\phn -76.5$ $\phn 403 \pm 30.4$ $167\pm210$
BDF20 $\phn 245.8$ $\phn \phn 15.9$ $3809 \pm 49.4$ $1240\pm400$
BDF20 $-131.5$ $\phn 394.5$ $4957 \pm 136$
BDF20 $\phn \phn 89.7$ $\phn 360.3$ $3272 \pm 96.6$
BDF20 $-301.9$ $\phn 149.2$ $2851 \pm 76.0$
BDF21 $\phn 277.3$ $-180.0$ $2103 \pm 57.5$
BDF21 $\phn 229.4$ $\phn \phn 74.1$ $1187 \pm 34.8$
BDF21 $-315.9$ $-224.9$ $\phn 838 \pm 82.2$
BDF21 $\phn 115.7$ $\phn 268.9$ $\phn 470 \pm 44.2$
Entries are left blank for fields not observed at $8\,$GHz. For the $30\,$GHz BIMA observations, entries are blank for those point sources which lie outside the primary beam and are not detected at $>3\sigma$ significance.
Excess Power Estimate {#sec:power}
=====================
Following the method of D2002, the excess power in the BIMA data is computed assuming that the angular power spectrum can be described by either one or two flat band powers. Point sources identified with the VLA are tabulated in §\[sec:VLA\] and are removed from the data using the constraint matrix technique described in D2002. Confidence intervals for the band powers are determined using the integrated likelihood also described in D2002. In Table \[tab:powshort\], we show the most likely $\Delta T^2$ and approximate uncertainty computed from visibilities in the $0.63-1.1\,{{\rm{k}\lambda}}$ range for each of the 18 fields in the survey. Results for the complete data set are included in Table \[tab:powall\] for the cases when the power spectrum is modeled by two bandpowers corresponding to $u$-$v$ ranges $0.63-1.1\,{{\rm{k}\lambda}}$ and $1.1-1.7\,{{\rm{k}\lambda}}$, and for a single bin covering the range $0.63-1.7\,{{\rm{k}\lambda}}$.
In the combined analysis of all 18 fields, we allow for $\Delta T^2 < 0.0$ in determining the most likely estimate of measured power and confidence intervals. Excess power corresponding to $\Delta T_1^2 > 0$, was observed with $94.7\%$ confidence in the $0.63-1.1\,{{\rm{k}\lambda}}$ bin. In the $u$-$v$ range $1.1-1.7\,{{\rm{k}\lambda}}$, the level of observed power was consistent with zero and we found an upper limit $\Delta T_2^2 < 880 \mu{\rm K}^2$ at $95\%$ confidence. When an analysis is performed combining all data into a single bin, we find an estimate of excess power excluding zero, $\Delta T^2 > 92.3\%$ confidence.
Window functions for each of these bands are produced from the noise weighted sum of the window functions for the individual visibilities. Averaged over all 18 fields, the $0.63-1.1\,{{\rm{k}\lambda}}$ band has an average value of $\ell_{eff}=5237$ with FWHM $\ell=2870$. The window function for visibilities in the $u$-$v$ range $1.1-1.7\,{{\rm{k}\lambda}}$ has an average value $\ell_{eff}=8748$ with FWHM $\ell=4150$. For the single band model covering the $u$-$v$ range $0.63-1.7\,{{\rm{k}\lambda}}$, the window function has an average value $\ell_{eff}=6864$ with FWHM $\ell=6800$.
[lcc]{} &\
& Most Likely & $\sigma$\
BDF4 & $0.0$ & $600$\
HDF & $0.0$ & $270$\
BDF6 & $380$ & $455$\
BDF7 & $300$ & $1050$\
BDF8 & $0.0$ & $390$\
BDF9 & $0.0$ & $590$\
BDF10 & $0.0$ & $360$\
BDF11 & $50$ & $920$\
BDF12 & $1590$ & $1165$\
BDF13 & $1480$ & $1260$\
BDF14 & $690$ & $920$\
BDF15 & $0.0$ & $660$\
BDF16 & $300$ & $880$\
BDF17 & $1390$ & $1400$\
BDF18 & $200$ & $990$\
BDF19 & $0.0$ & $900$\
BDF20 & $290$ & $910$\
BDF21 & $0.0$ & $310$\
\
The likelihood distribution near the maximum ($\Delta T^2_B={\overline \Delta T^2}_B$) for the data defined by bin $B$ is well described by a offset log-normal function (Bond, Jaffe, & Knox, 2000), $$\label{log-normal}
\ln{\mathcal L}({\bf \Delta T^2}) = \ln{\mathcal L}(\overline{{\bf \Delta T^2}})-\frac{1}{2}
\sum\limits_{B}\frac{(Z_B - \overline{Z}_B)^2}{\sigma_B^2}e^{2{\overline Z}_B},$$ where the offset log-normal parameters ${\bf Z}$ are defined as $$\label{lognorm}
Z_B = \ln{(\Delta T^2_B + x_B)}.$$ The likelihood functions are fit with this model to determine values for the curvature at peak $\sigma_B$, which represents the uncertainty in the measurement, and log-normal offset $x_B$ to the likelihood functions reported in Table \[tab:powall\].
[lcccccc]{} & & Likelihood\
& Most Likely & $68\%$ Confidence & $\sigma_B$ & $x_B$ & $\Delta T^2 > 0$\
$0.63-1.1\,{{\rm{k}\lambda}}$ & $220$ & $100-360$ & $130$ & $625$ & $94.7\%$\
$1.1-1.7\,{{\rm{k}\lambda}}$ & $-40$ & $<420$ & $395$ & $3040$ & $ $\
$0.63-1.7\,{{\rm{k}\lambda}}$ & $170$ & $70-290$ & $110$ & $490$ & $92.3\%$\
\
Point Source Removal {#subsec:ptsrc model}
--------------------
As discussed in §\[sec:VLA\], we have adopted a detection threshold of $6\sigma$ for identifying point sources in the VLA data. We measured the effect of eight different point source detection thresholds on the measured excess power from the combined analysis of the 18 fields. We choose point source detection limits in terms of SNR rather than flux to account for attenuation by the VLA primary beam. The noise is assumed to have an RMS of $25\,\mu$Jy/beam in all VLA fields. The results are listed in Table \[tab:ptsrc\].
------- ------------------- ----------------- ----------
Number of Sources [Most likely]{} $\sigma$
none $0$ $430$ $155$
$>40$ $7$ $350$ $150$
$>20$ $27$ $290$ $140$
$>12$ $45$ $260$ $130$
$>8$ $58$ $240$ $135$
$>6$ $62$ $220$ $130$
$>5$ $98$ $210$ $135$
$>4$ $168$ $250$ $150$
------- ------------------- ----------------- ----------
In the case for which no point sources are removed, the most likely value of $\Delta T_1^2 = 430 \, \mu{\rm K}^2$ appears significantly elevated due to contamination by radio point sources. The measured excess power drops from $430 \, \mu{\rm K}^2$ in the case of no point source constraints to a broad minimum of $\sim 200 - 250 \, \mu{\rm K}^2$ for removal of sources detected with less than $8 \sigma$ significance. As the detection threshold decreases, the removal of point sources will begin to remove a significant number of degrees of freedom from the analysis, resulting in increased uncertainty in the measurement of power. For example, a detection threshold for point sources of $4\sigma$ removes three times as many sources from the data as a detection threshold of $8-12 \sigma$. The uncertainty in the measurement begins to significantly increase with a point source detection threshold less than $6 \sigma$ while the subtraction of the additional sources have no effect on the observed excess power. Therefore, we adopt $6 \sigma$ as the point source detection threshold.
While the contribution to anisotropy from point sources is expected to scale as $\ell^2$, the estimate of excess power is consistent with zero on finer angular scales as shown in Table \[tab:powall\]. It should also be noted that no additional point sources were found in the 8 GHz VLA observations. These observations reinforce the conclusion that the contribution of point sources to the observed excess power is well constrained in the power spectrum analysis.
Systematic Tests {#subsec:jacknife}
----------------
We performed tests for systematic errors in all fields identified as having significant excess power. This analysis was limited to visibility data in the $u$-$v$ range $0.63-1.1\,{{\rm{k}\lambda}}$ described by $\Delta T_1^2$ where the most significant detection of excess power occurs. The modeled power in the second bin is fixed at $\Delta T_2^2=0$ for all tests described in this section. We searched for systematic errors by repeating the power spectrum analysis after splitting the data into a number of subsets. The first set of tests was designed to search for systematic contamination that changes with time. Relative to a source on the celestial sphere, terrestrial sources will appear to move rapidly over the course of a single observation, while the sun or moon will vary in position by many degrees over the course of a typical month long observation. We searched for such signals by analyzing the data after dividing it into subsets corresponding to the first, second, and third sections of both observation tracks and the period of observation. For the field BDF6, the only field observed in multiple years that was found to have a significant level of excess power, we also compared the results of observations taken in 1998 and 2000. Instrumental effects that manifest themselves as spurious signals on a given telescope or baseline are also a potential source of systematic error. We searched for such effects by breaking the data into subsets of four and five telescopes and looking for antenna based systematic errors. For a test of baseline based systematic errors, we created east-west and north-south baseline subsets. The different data and instrument subsets used in the search for systematic errors are listed in Table \[tab:jack-knife\]. The results for the application of these tests applied to fields BDF6, BDF12, BDF13, BDF14, and BDF17 are shown in Figure \[fig:systest\].
[Data Subset]{}
---- --------------------------------------------
1 First Four Hours UT
2 Middle Three Hours UT
3 Final Four Hours UT
4 First Third of Observation Dates
5 Middle Third of Observation Dates
6 Final Third of Observation Dates
7 Baselines from Subarray of four Telescopes
8 Baselines from Subarray of five Telescopes
9 East-West Baselines
10 North-South Baselines
11 First Year of Observation (BDF6 only)
12 Second Year of Observation (BDF6 only)
: \[tab:jack-knife\]Cuts Used in Systematic Tests
 
 
If a systematic error was associated with one of these subsets, then the level of excess power in that subset should increase. However, as can be seen in Figure \[fig:systest\], we found approximately the same level of excess power in each subset. Of the fifty-two subsets, only the final four hours of UT in BDF6 (third data point in Figure \[fig:systest\]) and the middle third of observation dates for BDF14 (fifth data point in Figure \[fig:systest\]) are found to have estimates of excess power and $68\%$ confidence intervals that lie above the $68\%$ confidence limits of the full analysis. Therefore, we consider these two subsets to be the most likely to be systematically biased. To test the possible contribution from systematic errors in these subsets to the measured excess power, we repeated the analysis on the 18 combined fields after removing these two subsets. We found an estimate of $\Delta T^2 = 180^{+140}_{-120}\,\mu$K$^2$, not significantly different than the reported value of $\Delta T^2 = 220^{+140}_{-120}\,\mu$K$^2$ for the entire survey.
Correlated Signal Between Independent Fields {#subsec:correlation}
--------------------------------------------
It is possible that a hardware malfunction could create a systematic false correlation that is constant or changes slowly with sky position. To test for this, we combined the raw visibilities from observations of all fields taken in a single summer, as if all the data came from a single pointing. If the fields contain random noise or independent sky signal, the power in the combined fields should decrease significantly when the visibilities are averaged. A false detection caused by correlations introduced in the hardware or other local effects might, depending on its stability, enhance the excess power in the anisotropy measurement when the independent observations are combined. The results of the analysis of the combined data sets, listed in Table \[tab:indep\], are consistent with instrumental noise at $68\%$ confidence, as expected for non-correlated independent observations.
To test for false correlations that vary slowly across the sky, we observed a set of fields offset by $2^{\prime}$ in RA from fields that had been previously observed. The fields BDF6, BDF12, BDF13, BDF14, and BDF17 have the most significant levels of excess power of the individual fields in the survey. We have neighboring fields for all of these except for BDF13. The raw visibilities of the neighboring fields were combined and then analyzed for excess power; the results of this analysis are shown in Table \[tab:indep\]. In all cases, the observed anisotropy power decreases as we would expect for the combination of visibilities from independent patches of sky. Therefore, we conclude that there does not appear to be any false correlation that is constant or slowly varying with sky position. In addition, there does not appear to be any correlation between excess power and proximity to the sun, none of the fields analyzed for systematic error were near the sun’s location of $11-12$ hr R.A. during the summer months. Overall, we find no evidence that our results are biased by systematic effects or astrophysical contamination.
----------------- ----------------- ----------
[Most likely]{} $\sigma$
1998 data $60$ $220$
2000 data $0$ $220$
2001 data $40$ $180$
2002 data $0$ $40$
BDF12 $1590$ $1165$
BDF19 $0.0$ $900$
BDF12/2+BDF19/2 $560$ $740$
BDF6 $380$ $1050$
BDF17 $1390$ $1400$
BDF6/2+BDF17/2 $640$ $580$
BDF4 $0$ $600$
BDF14 $690$ $920$
BDF4/2+BDF14/2 $120$ $220$
----------------- ----------------- ----------
: \[tab:indep\]Results of Combining Independent Observations
Constraints on $\sigma_8$ {#subsec:sig8}
-------------------------
As was described in the introduction, the most likely astrophysical source of the excess power observed in the survey is expected to be CMB anisotropy arising from the SZ effect in clusters of galaxies. Assuming that the observed excess power is entirely due to the SZ effect, we use a publicly available archive of N-body simulations [^3] (Schulz & White, 2003) to add the expected signal from the SZ effect to Monte Carlo noise realizations. These simulations predict the evolution of mass from $z=60$ to present in a cosmological model described by $\Omega_M=0.3$, $\Omega_{\Lambda}=0.7$, $\Omega_bh^2=0.02$, $h=0.7$, $n=1$ and $\sigma_8=1.0$. Only dark matter is included in these simulations, and the baryons contributing to the SZ signal are added assuming that the gas closely traces the dark matter. Independent regions of the N-body simulations are multiplied by the $6.6{^{\prime}}$ FWHM primary beam and transformed into the $u$-$v$ plane with the same $u$-$v$ sampling as the real BIMA data. Noise is added to each $u$-$v$ point with a variance determined from the observed visibilities. Therefore, each simulated observation has $u$-$v$ coverage and noise characteristics identical to the real BIMA observation of each field. The analysis of one hundred realizations of the BIMA survey with unique instrumental noise and simulated SZ sky resulted in an excess power of $\Delta T^2 = 216 \pm 190\,\mu$K$^2$ at $68\%$ confidence; this is remarkably close to the level of excess power found in the BIMA survey.
Komatsu and Seljak (2002) demonstrate that the amplitude of the SZ power spectrum has a strong dependence on $\sigma_8$, with $\Delta T^2 \propto \sigma_8^7$ for $\sigma_8$ near unity. We scale the simulated SZ images by $\sigma_8^{7/2}$ to produce skies corresponding to different values of $\sigma_8$. These images are transformed to the $u-v$ plane, combined with the simulated instrumental noise, and used to compute the likelihood of each value of $\sigma_8$ resulting in the observed excess power, which we approximate as $\Delta T^2_1= 180-260\mu$K$^2$. The relative likelihood that simulations with a given value of $\sigma_8$ will reproduce the observed excess power is determined from 10,000 realizations of the BIMA survey with $\sigma_8$ ranging from $0.0$ to $1.5$ The resulting relative likelihood shown in Figure \[fig:sig8\_like\] includes contributions to the uncertainty from both noise in the measurement and sample variance due the non Gaussian nature of the SZ signal and the small patch of sky surveyed. Assuming an additional $10\%$ uncertainty in the simulations (Komatsu & Seljak, 2002, Goldstein [et al. ]{}, 2003), we find $\sigma_8 = 1.03^{+0.20}_{-0.29}$ at $68\%$ confidence and $\sigma_8 = 1.03^{+0.30}_{-0.96}$ at $95\%$ confidence.
Image Analysis {#sec:images}
==============
Analysis of the complete BIMA survey results in a detection of excess power with nearly $95\%$ confidence. However, with the power spectrum analysis, we are unable to determine if the excess power is caused by sources with negative flux as would be expected from the SZ effect in galaxy clusters at 30 GHz. The statistics of the pixel flux values in the images contain additional information that can in principle constrain possible sources of the excess power.
Images are reconstructed directly from the visibility data for each field in the BIMA survey using the DIFMAP software package (Pearson et al. 1994). Positions determined from the VLA data are used to model and remove point sources ($>6\sigma$) in the $u$-$v$ plane using baselines with $u$-$v$ radius $> 1.5 \, {{\rm{k}\lambda}}$ ($\sim 50\%$ of the visibilities). BIMA fields are imaged at a resolution of $7.5{^{\prime \prime}}$ per pixel after applying a Gaussian taper with a half-power radius of $1.0 \,{{\rm{k}\lambda}}$ to the visibility data to maximize brightness sensitivity. These mapping parameters are used when presenting images of bright SZ clusters observed with the BIMA array (i.e. Laroque et al. 2003) and produce a synthesized beam which is well matched to cluster scales. A map of a typical synthesized beam can be found in Figure \[fig:beam\]. The response of the first negative sidelobe is a factor of four lower than response at the center of the beam.
![\
Typical synthesized beam created from the $u$-$v$ coverage used in the analysis of image statistics. The dotted line shows the half-power radius of the primary beam. []{data-label="fig:beam"}](f3.ps)
The typical RMS noise is $110\,\mu$Jy/beam for an image with $90{^{\prime \prime}}$ FWHM synthesized beam. The RMS for each image listed in Table \[tab:properties\] are computed directly from the noise properties of the images. The RMS temperature estimates correspond to the synthesized beamsize computed for the $u$-$v$ coverage of each experiment.
[lccc]{} & Synthesized & RMS & RMS\
& Beamsize($^{{^{\prime \prime}}}$) & [($\mu$Jy$\,$[beam]{}$^{-1}$)]{} & ($\mu$K)\
BDF4 & $90.5 \times 105.1$ & $107.1$ & $16.9$\
HDF & $91.4 \times 95.9$ & $113.7$ & $19.5$\
BDF6 & $90.7 \times 97.5$ & $92.4$ & $15.7$\
BDF7 & $91.2 \times 98.2$ & $105.4$ & $17.6$\
BDF8 & $87.9 \times 90.4$ & $109.9$ & $20.7$\
BDF9 & $88.0 \times 91.7$ & $111.4$ & $20.7$\
BDF10 & $88.0 \times 90.3$ & $109.4$ & $20.7$\
BDF11 & $89.0 \times 90.4$ & $109.8$ & $20.5$\
BDF12 & $88.7 \times 92.0$ & $112.3$ & $20.6$\
BDF13 & $89.6 \times 91.3$ & $113.1$ & $20.7$\
BDF14 & $86.6 \times 89.9$ & $106.9$ & $20.6$\
BDF15 & $87.2 \times 90.5$ & $109.1$ & $20.7$\
BDF16 & $87.4 \times 91.5$ & $108.7$ & $20.4$\
BDF17 & $86.0 \times 90.3$ & $105.2$ & $20.3$\
BDF18 & $86.3 \times 91.7$ & $107.7$ & $20.4$\
BDF19 & $88.0 \times 90.7$ & $109.5$ & $20.6$\
BDF20 & $86.8 \times 91.6$ & $108.4$ & $20.4$\
BDF21 & $86.8 \times 91.8$ & $106.7$ & $20.1$\
\
To provide a visual representation of the data used in the analysis, images of the BIMA fields are reproduced in Figure \[fig:BIMA Image statistics(1)\]. The dashed circle in each image represents the radius at which the primary beam attenuates the sky signal by a factor of two relative to the pointing center. Regions lying far outside the primary beam can be used to compute the level of instrumental noise in the map.
  
  
  
  
  
  
Image Simulations {#subsec:mc}
-----------------
Decrements caused by galaxy clusters would be expected to produce an excess of high SNR negative pixels. We attempt to detect this unique signature using higher order statistics of the image flux distribution as described in Rubino-Martin & Sunyaev, 2003. We compute asymmetry, skewness, and extrema in the observed data set. These results are compared to both simulations of instrumental noise and to simulations of SZ galaxy clusters added to instrumental noise in order to determine the significance of the observed distribution of flux.
The Fourier transform of the visibilities measured by the interferometer into the image plane introduces correlations between pixels which complicate the noise properties. In order to better understand this, we generated Monte Carlo simulations of the visibility data and transformed them into images. We created two sets of simulations, each containing 100 realizations of the full set of 18 images. These simulations are used to quantify the significance of applying the statistical tests to the BIMA images. The first set of simulations uses random complex visibilities with variances consistent with the observed noise for each image. At each point in $u$-$v$ space in the observed data, a simulated visibility is created with a real and imaginary component from a random sampling of a Gaussian distribution with variance determined from the weight of that visibility. Therefore, each simulated observation has $u$-$v$ coverage and noise characteristics identical to the real BIMA observation of each field.
In the second case, we add the expected contribution from the SZ sky to the MC noise realizations using the same simulated SZ sky images described in Section \[subsec:sig8\] with $\sigma_8$ fixed to $1.0$. Independent regions of the N-body simulations are attenuated with the $6.6{^{\prime}}$ FWHM primary beam and transformed into the $u$-$v$ plane using the exact same $u$-$v$ sampling as that in the real BIMA data. Noise is then added to the visibility data in the same way as for the noise-only simulations and the data is then transformed into the image plane for analysis.
The image statistics of the observed images are compared with those generated from the Monte Carlo simulations. For each simulation, an image is generated from the raw visibilities tapered in the $u$-$v$ plane with a $1.0 \, {{\rm{k}\lambda}}$ FWHM Gaussian. Statistics are generated directly from the images of the 100 simulations. Using a large number of Monte Carlo simulations, one can produce a probability distribution for each observable statistic. The most likely value and integrated confidence intervals can be taken from this distribution and compared to the results of the real observations. Due to the modest number of 100 simulations of the survey, the shape of the pixel distribution is strongly dependent on bin size and bin spacing, leaving the most likely value as a poorly determined quantity. Instead, the median value in the 100 simulations is taken to be the most likely estimate. The terms defining the $16$th and $84$th percentile in the distribution are taken to be the lower and upper bounds to the $68\%$ confidence intervals, respectively. The 4 nearest neighbors to the median, lower bound, and upper bound in the distribution are averaged to reduce the noise in the estimates.
We use a histogram of pixels binned in intervals of significance, $S=x_i/\sigma$, where $x_i$ is the value at pixel i and $\sigma$ is the estimated image RMS (approximately $110 \, \mu$Jy/beam). This is equivalent to weighting the pixels by the sensitivity of the observation. We can compare the pixel distributions for observed data and simulations by comparing the histogram of the images as shown in Figure \[fig:Distribution\_survey\].
 
Asymmetry of Pixel Distribution {#subsec:ass}
-------------------------------
For a given image, we can characterize the pixel flux distribution by selecting a flux interval $\Delta D$, and computing a histogram of the number of pixels with a flux between $D-\Delta D/2$ and $D+\Delta D/2$). The asymmetry of this histogram can be estimated directly as the difference in area between the positive and negative regions. A comparison of this statistic for the BIMA images and simulations is shown in Figure \[fig:Asym\_survey\].
It should be noted that the cluster simulations produce an excess of positive pixels compared to simulations of instrumental noise only. There are no sources with positive flux included in the cluster simulations, however pixels with positive flux are expected due to the sidelobes of the synthesized beam. A Gaussian taper was applied to the $u$-$v$ data to minimize the sidelobes and this effect.
It is clear from Figure \[fig:Asym\_survey\] that the observed data is consistent with the asymmetry in the simulations that include galaxy clusters and inconsistent with simulations that include only noise. For example, the excess of pixels with $S<-3$ in the observed data exceeds that in the simulations of instrumental noise at $86\%$ confidence. The number of pixels with $S<-4$ exceeds that found in the simulations of only instrumental noise at $88\%$ confidence. Therefore, the application of this statistic results in a detection of signal with the morphology of the SZ effect with a significance comparable to the significance of the detection of excess power reported in Section \[sec:power\]. In fact, the observed asymmetry in the BIMA data exceeds the mean value from the noise & SZ simulations.
As a simple exercise, a similar analysis is used to rule out positive flux (such as point sources) as the cause of excess power with slightly less confidence; we simply repeat the analysis described, but with the sign of the flux from clusters reversed. The excess of pixels with $S<-3$ in the observed data exceeds that in the simulations using positive flux at $74\%$ confidence. The excess of pixels with $S<-4$ exceeds that in simulations of instrumental noise at $76\%$ confidence. A more rigorous analysis for point sources would include a more realistic distribution of point sources and take into account the details of their removal using the full, equally weighted, set of visibilities. This expanded analysis is considered unnecessary given the results of the VLA observations, constraints on point sources, and lack of power at finer angular scales. Nonetheless, this method would be useful in cases where rejection of point sources was less certain. We again conclude that it is very unlikely that point sources are responsible for the asymmetry observed in the BIMA image pixel fluxes.
Skewness of Pixel Distribution {#subsec:skew}
------------------------------
A measure of skewness conveys information about the sign of the features producing the deviation from Gaussianity. This quantity can be estimated using the third moment of the data: $$Y = \frac{1}{N_{pix}} \sum_{i=1}^{N_{pix}} (x_i - \overline{x})^3
\label{skew}$$ The skewness of the data derived from the BIMA images is compared to the skewness determined from the Monte Carlo simulations in Figure \[fig:Skewness\_survey\].
Analysis of the simulations of instrumental noise alone results in a skewness of $0.01\pm0.08$ at $68\%$ confidence, while the simulations of clusters plus instrumental noise give a mean skewness of $-0.096^{+0.11}_{-0.15}$. The skewness in the BIMA observations was found to be $Y = -0.066$, a value that is inconsistent with the noise only simulations at $64\%$ confidence. Repeating the simple analysis described at the end of §\[subsec:ass\], the skewness is inconsistent with positive flux sources as the cause of the observed excess power at $84\%$ confidence.
Outliers in the Distribution {#subsec:out}
----------------------------
In addition to the two tests described above, we performed simulations to characterize the significance of outliers in pixel flux distributions of the individual fields. We identified the most negative pixels in each of the 18 BIMA fields and compared these with the results of the simulations. Seven BIMA fields were observed to have decrements with $S <-3\sigma$, more than occurred in $93\%$ of the survey simulations with noise only. Two BIMA fields were observed to have decrements with $S<-4\sigma$, more than occurred $99\%$ of the survey simulations including only noise. None of the observed BIMA fields had a pixel with positive flux $S >4\sigma$.
Interpretation of Image Statistics {#subsec:im_sum}
----------------------------------
In each of the statistical tests described above, the BIMA images were found to be inconsistent with those produced by instrumental noise alone at approximately the $1-2 \sigma$ level, comparable to the significance of the detection of excess power. Although the analysis does not conclusively determine the source of the excess power, the results are consistent with the signal expected from SZ galaxy clusters. The results are also inconsistent with radio point sources being the source of the observed excess power. Despite the lack of a definitive conclusion from these tests, the results are encouraging. In order to investigate what observations would be required to make a definitive measurement, we performed simulations of the 18 field survey with four times the observation time, or equivalently four times the correlation bandwidth. These simulations resulted in a detection of excess power and skewness at the $4 \sigma$ level in more than $50\%$ of the simulations. Such a detection would provide convincing evidence for SZ clusters as the source of excess power.
Optical Observations of BIMA Fields {#sec:optical}
===================================
To date, all known galaxy clusters have been discovered through optical or X-ray observations. A negative correlation of the observed fine scale CMB anisotropy with X-ray or optical emission would be a smoking gun for the discovery of a cluster through the SZ effect. To search for this correlation, we observed ten of the BIMA fields using ground-based optical telescopes. We selected the five fields with significant levels of excess power assuming that they are the most likely candidates for identifying galaxy clusters. In addition to those five fields, we observed five BIMA fields that lie at convenient RA during the nights we were awarded time. Imaging was performed using I and R filters on the LRIS instrument (Oke [et al. ]{}1995) on the 10 m Keck[^4] I telescope. Imaging in z’ was done with the MOSAIC instrument (Wolfe [et al. ]{}1998) on the 4m Kitt Peak National Observatory telescope[^5].
We performed an analysis of the optical images using a method similar to that used in the Red Cluster Survey (RCS) survey (Gladders & Yee, 2000). We first categorized objects into two redshift bins determined by color $R-I$ and $R-z'$. The first bin contains only high redshift galaxy candidates. These candidates have color $R-I > 1.0$ and $R-z' > 1.0$, implying a red sequence redshift $0.5 < z <1.0$. The second bin contains low redshift galaxy candidates. These candidates have color $0.0 < R-I < 1.0$ and $0.0 < R-z' < 1.0$ implying a redshift $z < 0.5$. All objects not satisfying these criteria are considered foreground contamination and are discarded. An exponential kernel is used to smooth the maps of detected objects in each redshift bin with scale radius of $20{^{\prime \prime}}$ to create a surface density map. We then create a product map by simply multiplying each surface density map with the corresponding BIMA map described in §\[sec:images\]. The product maps are searched for peaks and asymmetry in the distribution. Statistics are quantified with Monte Carlo simulations of random BIMA fields. The resulting analysis (described in full detail in Dawson, 2004) showed no significant correlation between overdensities of galaxies in the optical maps and decrements of flux in the BIMA $28.5$ GHz maps.
While the follow-up optical observations do not confirm the hypothesis that the observed anisotropy is caused by the SZ effect in galaxy clusters, the method of combining optical and SZ observations should prove to be a powerful technique for identifying galaxy clusters in future SZ surveys. In this first attempt at doing so, we have not modeled the expected optical cluster signature sufficiently to say what constraints this null result places on the role of SZ clusters in producing the observed excess power. There also remains the possibility that anisotropy from galaxy clusters at redshifts beyond the sensitivity of the optical data, $z>1$, could be contributing significantly to the observed signal in the BIMA survey.
Conclusion {#sec:con}
==========
In this paper, we report the final results from our search for arcminute scale CMB anisotropy using the BIMA array. Modeling the observed power spectrum with a single flat band power with average multipole of $\ell_{eff} = 6864$, we find $\Delta T^2=170^{+120}_{-100}\,\mu$K$^2$ at $68\%$ confidence and a detection of $\Delta T^2 >0$ at $92.3\%$ confidence. Dividing the data into two bins corresponding to different spatial resolutions in the power spectrum, we find $\Delta T_1^2=220^{+140}_{-120}\,\mu$K$^2$ at $68\%$ confidence for CMB flat band power described by an average multipole of $\ell_{eff} = 5237$ and $\Delta T_2^2<840\,\mu$K$^2$ at $95\%$ confidence for $\ell_{eff} = 8748$. We have used VLA observations and various cuts to test for contamination from radio point sources and systematic effects and conclude that it is unlikely that these sources are responsible for the observed signal. If we assume that the measured excess power is due to a background of distant SZ clusters, we can compare its value with that from simulations of large scale structure to place a constraint on the normalization of matter fluctuations, $\sigma_8=1.03^{+0.20}_{-0.29}$ at $68\%$ confidence.
In order to try to determine the source of the observed anisotropy power, we have performed an analysis of the BIMA image statistics. We compared the skewness, asymmetry, and outliers of the measured pixel flux distribution with simulations including noise only and noise plus SZ clusters. A statistical analysis of the BIMA survey images found that they were consistent with simulations including a background of SZ clusters, and inconsistent with simulations of instrumental noise alone or noise plus radio point sources at $1-2\sigma$. Additional Monte Carlo simulations indicate that with approximately four times the time dedicated to the survey, or equivalently four times the correlated bandwidth, the BIMA instrument would achieve the sensitivity to test the hypothesis of SZ clusters as the source of the observed excess power at greater than $99\%$ confidence. Therefore, future dedicated interferometers, such as the Sunyaev-Zel’dovich Array[^6], should be able to effectively use image statistics to determine if any observed anisotropy is due to SZ clusters. Finally, we performed a preliminary search for a correlation between red galaxy density and CMB temperature fluctuations. We are currently unable to quantify the significance of the null result, however, we expect that X-ray and optical follow-up will be essential tools for the interpretation of future SZ surveys.
We thank the entire staff of the BIMA observatory for their many contributions to this project, in particular Rick Forster and Dick Plambeck for their assistance with both the instrumentation and observations. Brian Wilhite, Josh Simon, and Steve Dawson are thanked for their early advice on optical observations and data reduction. We would also like to thank Mike Gladders for his suggestions regarding identification of clusters in the optical data and Ramon Miquel, Radek Stompor, and Chao-lin Kuo for their stimulating discussions of Bayesian statistics. We are grateful for the scheduling of time at the VLA, Keck, and KPNO observatories in support of this project. This work was supported in part by NASA LTSA grant number NAG5-7986, NSF grants AST-0096913 and PHY-0114422, and the David and Lucile Packard Foundation. The BIMA millimeter array is supported by NSF grant AST 96-13998.
Facilities: , , , .
, J.R., [Jaffe]{}, A.H., and [Knox]{}, L., 2000, , 533, 19.
, J.E., [Holder]{}, G.P., & [Reese]{}, E.D. 2002, , 40, 643
Dawson, K.S., Holzapfel, W.L., Carlstrom, J.E., Joy, M., LaRoque, S.J., & Reese, E.D. 2001, , 553, L1.
, K.S., [Holzapfel]{}, W.H., [Carlstrom]{}, J.E., [Joy]{}, M., [LaRoque]{}, S.J., & [Reese]{}, E.D., 2002 , 581, 86.
Dawson, K. 2004, Ph. D. thesis, UC Berkeley.
, J.H., [Ade]{}, A.R., [Bock]{}, J.J., [Bond]{}, J.R., [Cantaldi]{}, C., [Daub]{}, M.D., [Holzapfel]{}, W.L., [Kuo]{}, C.L., [Lange]{}, A.E., [Newcomb]{}, M., [Peterson]{}. J.B., [Pogosyan]{}, D., [Ruhl]{}, J., [Runyan]{}, M.C., and [Torbet]{}, E., 2003, , 599, 773.
, M.D. & [Yee]{}, H.K.C., 2000 , 121, 2148.
Gnedin, N.Y. & Jaffe, A.H. 2001, , 551, .
Hinshaw, G., [et al. ]{}, 2003, , 148, 195.
Holzapfel, W.L., Carlstrom, J.E., Grego, L., Holder, G., Joy, M., & Reese, E.D. 2000, , 539, 57.
Komatsu, E. & Seljak, U. 2002, , 336, 1256.
Kuo, C.L., Bock, J.J., Daub, M.D., Goldstein, J.H., Holzapfel, W.L., Lange, A.E., Newcomb, M., Peterson, J.B., Ruhl, J., Runyan, M.C., & Torbet, E., 2004, , 600, 32.
LaRoque, S.J., Joy, M., Carlstrom, J.E., Ebeling, H., Bonamente, M., Dawson, K.S., Edge, A., Holzapfel, W.L., Miller, A.D., Nagai, D., Patel, S.K., & Reese, E.D., 2003, , 583, 559.
Mason, B.S. [et al. ]{}, 2003, , 591, 540.
, J.B., [Cohen]{}, J.G., [Carr]{}, M., [Cromer]{}, J., [Dingizian]{}, A., [Harris]{}, F.H., [Labrecque]{}, S., [Lucinio]{}, R., [Schaal]{}, W., [Epps]{}, H., & [Miller]{}, J., 1995, , 107, 3750.
, T.J., [Shepherd]{}, M.C., [Taylor]{}, G.B., & [Myers]{}, S.T. 1994, , 185, 0808.
, R. & [Di Matteo]{}, T., 2000, , 542, 68.
, J.A.& [Sunyaev]{}, R.A., 2003, , 344, 155.
, A.E. & [White]{} M., 2003, , 586, 723.
, R.A. & [Zel’dovich]{}, Y.B., 1970, , 7, 3.
T., [Reed]{}, R., [Blouke]{}, M., [Boroson]{}, T., [Armandroff]{}, T., & [Jacoby]{}, G., 1998 SPIE, 487, 3355.
[^1]: The BIMA array is operated with support from the National Science Foundation
[^2]: The VLA is operated by the National Radio Astronomy Observatory, a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc.
[^3]: Data available at http://pac1.berkeley.edu/tSZ/
[^4]: Keck Observatory is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.
[^5]: KPNO is a Division of the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation.
[^6]: http://astro.uchicago.edu/sza/
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abstract: 'In this paper, we establish a baseline for object symmetry detection in complex backgrounds by presenting a new benchmark and an end-to-end deep learning approach, opening up a promising direction for symmetry detection in the wild. The new benchmark, named Sym-PASCAL, spans challenges including object diversity, multi-objects, part-invisibility, and various complex backgrounds that are far beyond those in existing datasets. The proposed symmetry detection approach, named Side-output Residual Network (SRN), leverages output Residual Units (RUs) to fit the errors between the object symmetry ground-truth and the outputs of RUs. By stacking RUs in a deep-to-shallow manner, SRN exploits the ‘flow’ of errors among multiple scales to ease the problems of fitting complex outputs with limited layers, suppressing the complex backgrounds, and effectively matching object symmetry of different scales. Experimental results validate both the benchmark and its challenging aspects related to real-world images, and the state-of-the-art performance of our symmetry detection approach. The benchmark and the code for SRN are publicly available at <https://github.com/KevinKecc/SRN> .'
author:
- 'Wei Ke[^1]'
- Jie Chen
- Jianbin Jiao
- Guoying Zhao
- 'Qixiang Ye[^2]'
title: 'SRN: Side-output Residual Network for Object Symmetry Detection in the Wild'
---
Introduction
============
Symmetry is pervasive in visual objects, both in nature creatures like trees and birds, and artificial objects like aircrafts and oil pipes in aerial images. Symmetric parts and their connections [constitute]{} a powerful part-based decomposition of shapes [@17sebastian2004recognition; @31trinh2011skeleton], providing valuable cue for the task of object recognition. With symmetry constrained, the performance of image segmentation [@18teo2015detection], foreground extraction [@19fu2014symmetry], object proposal [@21lee2015learning], and text-line detection [@27zhang2015symmetry] could be significantly improved.
The early symmetry detection, named skeleton extraction, usually involves only binary images [@37DBLP:journals/pami/LamLS92; @34saha2016survey]. In recent years, symmetry detection tends to process color images [@DBLP:conf/cvpr/LiuSZWPLRL13; @23liu2010computational], but still limited to cropped image patches with little background. This limitation is partially due to the lack of fundamental benchmarks, considering that most existing symmetry detection datasets, , SYMMAX [@14tsogkas2012learning], WH-SYMMAX [@15shen2016multiple], and SK506 [@04shen2016object], lack either object-level annotation or the in-the-wild settings, , multi-objects, part-invisibility, and various complex backgrounds.
![We propose a new benchmark, named Sym-PASCAL, for object symmetry detection in the wild. Compared with SYMMAX [@14tsogkas2012learning], WH-SYMMAX [@15shen2016multiple], and SK506 [@04shen2016object], our Sym-PASCAL spans challenges including object diversity, multi-objects, part-invisibility and various complex backgrounds. (Best viewed in color) ](figure2.pdf){width="0.99\linewidth"}
\[figure1\]
In this paper, we present a new challenging benchmark with complex backgrounds, and an end-to-end deep symmetry detection approach that processes in-the-wild images, and target at opening up a promising direction for practical applications of symmetry. The new benchmark, named Sym-PASCAL, is composed of 1453 natural images with 1742 objects derived from the PASCAL-VOC-2011 [@10pascalvoc2011] segmentation dataset. Such a benchmark is more close to practical applications with challenges far beyond those in existing datasets: (1) *diversity of objects*: multi-class objects with different illuminations and viewpoints; (2) *multi-object co-occurrence*: multiple objects exist in a single image; (3) *part-invisibility*: objects are partially occluded; and (4) *complex backgrounds*: the scenes where object located could be contextually cluttered.
For the in-the-wild symmetry detection problem, we explore the deep Side-output Residual Network (SRN) that directly outputs response image about object symmetry. SRN roots in the Holistically-nested Edge Detection (HED) network [@03xie2015holistically] but updates it by stacking multiple Residual Units (RUs) on the side-outputs. The Residual Unit (RU) is designed to fit the error between the object symmetry ground-truth and the outputs of RUs, which is computationally easier as it pursuits the minimization of residuals among scales rather than only struggles to combine multi-scale features to fit the object symmetry ground-truth. The RU we defined not only significantly improves the performance of SRN, but also solves the learning convergence problem left by the baseline HED method. By stacking multiple RUs in a deep-to-shallow manner, the receptive fields of stacked RUs could adaptively match the scale of symmetry. The contributions of this paper include:
[A new object symmetry benchmark]{} that spans challenges of diversity, multi-objects, part-invisibility, and various complex backgrounds, promoting the symmetry detection research to in-the-wild scenes.
[A Side-output Residual Network that can effectively fit the errors between ground-truth and the outputs of the stacked RUs, enforcing the modeling capability to symmetry in complex backgrounds, achieving state-of-the-art symmetry detection performance in the wild.]{}
Related Works
=============
For the applicability and beauty, symmetry has attracted much attention in the past decade. The targets of symmetry detection evolute from binary images to color object images, while the symmetry detection approaches update from hand-crafted to learning based.
**Benchmarks:** In the early research, symmetry extraction algorithms are qualitatively evaluated on quite limited binary shapes [@37DBLP:journals/pami/LamLS92]. Such shapes are selected from the MPEG-7 Shape-1 dataset for subjective observation [@11bai2007skeleton]. Later, Liu [@DBLP:conf/cvpr/LiuSZWPLRL13] use very a few real-world images to perform symmetry detection competitions. To be honest, SYMMAX [@14tsogkas2012learning] could be regarded as an authentic benchmark that contains hundreds of training/testing images with local symmetry annotation. But the local reflection symmetry it defined mainly focuses on low-level image [edges and contours]{}, missing the high-level concept of objects. WH-SYMMAX [@15shen2016multiple] and SK506 [@04shen2016object] are recently proposed benchmarks with annotation of object skeletons. Nevertheless, WH-SYMMAX is simply composed of side-view horses while SK506 consists objects with little background. Neither of them involves multiple objects in complex backgrounds, leaving a plenty of room for developing new object symmetry benchmarks.
**Methods:** Early symmetry detection methods, also named skeleton extraction [@37DBLP:journals/pami/LamLS92; @34saha2016survey], are mainly developed for the binary images by leveraging morphological image operations. When processing color images, they usually need a contour extraction or an image segmentation step as pre-processing. Considering that segmentation of in-the-wild images remains a research problem, the integration of color image segmentation and symmetry detection not only increases the complexity but also accumulates the errors.
Researchers have tried to extract symmetry in color images based on multi-scale super-pixels. One hypothesis is that the object symmetry axes are the subsets of lines connecting the center points of super-pixels [@29levinshtein2009multiscale]. Such line subsets are explored from the super-pixels using a sequence of deformable disc models extracting the symmetry pathes [@30sie2013detecting]. Their consistence and smoothness are enforced with spatial filters, e.g., a particle filter, which link local skeleton segments into continuous curves [@36widynski2014local]. Due to the lack of object prior and the learning module, however, these methods are still limited to handle the images with simple backgrounds.
More effective symmetry detection approaches root in powerful learning methods. On the SYMMAX benchmark, the Multiple Instance Learning (MIL) [@14tsogkas2012learning] is used to train a curve symmetry detector with multi-scale and multi-orientation features. To capture diversity of symmetry patterns, Teo [@18teo2015detection] employ the Structured Random Forest (SRF) and Shen [@15shen2016multiple] use subspace MIL with the same feature. Nevertheless, as the pixel-wise hand-craft feature is computationally expensive and representation limited, these methods are intractable to detect object symmetry in complex backgrounds.
Most recently, a deep learning approach, Fusing Scale-associated Deep Side-outputs (FSDS) [@04shen2016object], is shown to be capable of learning unprecedentedly effective object skeleton representations on WH-SYMMAX [@15shen2016multiple] and SK506 [@04shen2016object]. FSDS takes the architecture of HED [@03xie2015holistically] and supervises [its side-outputs]{} with scale-associated ground-truth. Despite of its state-of-the-art performance, it needs the intensive annotations of the scales for each skeleton point, which means that it uses much more human effort than other approaches when preparing the training data. Compared with FSDS, our proposed SRN can adaptively match the scales of symmetry, without using scale-level annotation.
The Sym-PASCAL Benchmark
========================
Symmetry annotation involves pixel-level fine details, and is time consuming. We thus leverage the semantic segmentation ground-truth and a skeleton generation algorithm to aid the annotation of symmetry [@38DBLP:journals/pr/ShenBHWL11].
[0.153]{} {height="10em"}
[0.163]{} {height="10em"}
[0.153]{} {height="10em"}
\[figure3\]
Categorization and Annotation
-----------------------------
Sym-PASCAL is derived from the PASCAL-VOC-2011 segmentation dataset [@10pascalvoc2011] which contains 1112 training images and 1111 testing images from 20 object classes including: person, bird, cat, cow, dog, horse, sheep, aero plane, bicycle, boat, bus, car, motorbike, train, bottle, chair, dining table, potted plant, sofa, and tv/monitor.
We categorize the 20 classes of objects into symmetry-available and symmetry-unavailable, Fig. \[figure3\]. The objects that contain lots of discontinuous parts in the segmentation masks are symmetry-unavailable, specifically potted plant, dining table, motorbike, bicycle, chair and sofa, are not selected, Fig. \[figure3a\]. The other 14 object classes are symmetry-available. Some of objects are slender and thus easy to annotate, Fig. \[figure3b\], and others with small length-width ratio or occlusion are difficult to annotate, Fig. \[figure3c\]. In total, 648/787 images are selected and annotated from the PASCAL-VOC-2011 training and testing sets. Among these images, 31.3% are with multi-object and 45.6% are with part-invisibility.
[For the images where object symmetry is obvious, , objects are composed of slender parts that are easy to annotate, we directly extract symmetry on the object segmentation masks using a skeleton extraction algorithm [@38DBLP:journals/pr/ShenBHWL11], Fig. \[figure3b\]. For such objects, the object symmetry (marked with blue curves) and their skeleton (marked with red curves) are consistent. For the images where object symmetry is not obvious, we manually extend the semantic segmentation masks and annotate symmetry on them, Fig. \[figure3c\]. For wide object as shown on the top of Fig. \[figure3c\], we extend the mask along the direction of the long axis of the object and choose the long axis as ground-truth. For occluded objects as shown at the bottom of Fig. \[figure3c\], we need to manually fill the missed parts of segmentation masks. For the pictures that contain partial objects, we empirically imagine the occluded parts to extend the segmentation masks. With these processing above, the skeleton extraction algorithm [@38DBLP:journals/pr/ShenBHWL11] is used to extract symmetry on the object segmentation masks. The object symmetry ground-truth is set as the skeleton points within the segmentation masks, shown as the blue curves in Fig. \[figure3c\].]{}
[0.24]{} ![Object-class distributions of the SK506 and Sym-PASCAL datasets.[]{data-label="figure4"}](figure4a.pdf "fig:"){width="\linewidth" height="6.5em"}
[0.23]{} ![Object-class distributions of the SK506 and Sym-PASCAL datasets.[]{data-label="figure4"}](figure4b.pdf "fig:"){width="\linewidth" height="6.5em"}
\[b\]
[L[0.15]{}|C[0.17]{}C[0.14]{}C[0.11]{}C[0.17]{}]{} & & WH- & & Sym-\
& & SYMMAX & & PASCAL\
Data & local & object & object & object\
type & symmetry & skeleton & skeleton & symmetry\
Image & in-the-wild & simple & simple & in-the-wild\
type & image & image & image & image\
\#object & – & 1 & 16 & 14\
\#training & 200 & 228 & 300 & 648\
\#testing & 100 & 100 & 206 & 787\
[0.49]{} {width="0.95\linewidth"}
[0.49]{} {width="0.95\linewidth"}
Discussion
----------
In what follows, we compare the proposed benchmark with three other representative ones, SYMMAX [@14tsogkas2012learning], WH-SYMMAX [@15shen2016multiple], and SK506 [@04shen2016object].
SYMMAX is derived from BSDS300 [@16arbelaez2011contour], which contains 200/100 training and testing images. It’s annotated with local reflection symmetry on both foreground and background. Considering that most computer vision tasks focus on the foreground, it’s more meaningful to use object symmetry instead of the symmetry about the whole image. WH-SYMMAX is developed for object skeletons, but it is made up of only cropped horse images, which are not comprehensive for general object symmetry. SK506 involves skeletons about 16 classes of objects. Nevertheless, their backgrounds are too simple to represent in-the-wild images.
As shown in Tab. \[Tab-1\]. the proposed benchmark involves more training and testing images. Particularly, these images involve complex backgrounds, multiple objects and/or occlusions. It is developed for end-to-end object symmetry in-the-wild, providing the protocol to evaluate whether or not an algorithm can detect symmetry without using additional object detectors. In Sym-PASCAL, the images for each class are more balanced than other datasets, Fig. \[figure4b\], except that the number of human objects is larger than others. In contrast, in SK506 the objects from different classes have more unbalance, Fig. \[figure4a\].
Side-output Residual Network
============================
The proposed Side-output Residual Network (SRN) roots in the well-designed output Residual Unit (RU) and a deep-to-shallow learning strategy. Given the symmetry ground-truth, the SRN is learned in an end-to-end manner.
Output Residual Unit
--------------------
Given training images, the end-to-end symmetry learning pursuits deep network parameters that best fit the symmetry ground-truth. Such a learning objective is different from that of learning a classification network [@07he2015deep]. The RU defined for output, Fig. \[figure5\], is essentially different from that in the residual network defined for features [@07he2015deep]. With the deep supervision both on the input and output of RUs, the residual of the ground-truth is computed. Formally, denoting the input of RU as $r$ and the additional mapping as ${\cal F}(y)$, the deep supervision is written as: $$\left\{ {\begin{array}{*{20}{c}}
{r \approx y}\\
{r + {\cal F}(y) \approx y}
\end{array}} \right.{\rm{ ,}}
\label{Eq1}$$ where $r$ and $r + {\cal F}(y)$ are the input and output of the RU, respectively. ${\cal F}(y)$ is regarded as the residual estimation of $y$. RUs provide shortcut connections between the ground-truth and outputs from different scales, which implies a functional module for the ‘flow’ of errors among different scales, and thus make it easier to fit complex outputs with higher adaptivity. To the extreme, if an input $r$ is optimal, it would be easier to push the residual to zero than to fit the additional mapping ${\cal F}(y)$.
{width="0.45\linewidth"}
\[figure5\]
Network Architectures
---------------------
By stacking the RUs defined, we implement a kind of new side-output deep network, named Side-output Residual Network (SRN), which incorporates the advantages of both the scale adaptability and residual learning. For SRN, the input of the first RU can be chosen as the shallowest side-output or deepest side-output, which derives two versions of SRN, Fig. \[figure6\]. In what follows, the RU is numbered as the side-output (SOP) index, and the output of the $i{\rm{ - th}}$ RU is denoted as ${\rm{RUOP}}i$, for short.
**Deep-to-shallow.** In this SRN architecture, RUs are stacked from deep to shallow, Fig. \[figure6a\]. Assume that ${s_i}$ is the $i$-th side-output, and ${r_{i + 1}}, {r_i}$ are the input and output of $i$-th RU respectively . For the first stacked RU2, the input is set as the deepest SOP3, , ${r_3} = {s_3}$. And SOP2 is used to learn the residual between RUOP3 and the ground-truth, which updates RUOP3 to RUOP2. The RUs are stacked in order until the shallowest side-output, in other words, the inputs of which are set as the output of the former one. Sigmoid is used as classifier on the output of the last stacked RU to generate the final output image.
The implementation of RU in the deep-to-shallow architecture is shown in Fig. \[figure7a\]. It’s noting that the output size of RU in this architecture is same as the side-output rather than the input image. Therefore, a Gaussian deconvolution layer is introduced to the output of RU. As the up-sampling is non-linear transformation, a weight layer is stacked to improve the scale adaptability. Instead of adding up-sampled ${r_{i + 1}}$ and $s_i$ directly, a $1 \times 1$ convolutional layer is utilized to generate ${r_i}$. The RU is formulated, $${r_i} = {w_i}^c({s_i} + w_i^r{r_{i + 1}}),
\label{Eq2}$$ where ${w_i}^c,{w_i}^r$ are the convolutional weights of concatenation layer and the up-sampled ${r_{i + 1}}$. With Eqs. (\[Eq1\]) and (\[Eq2\]), the output residual ${{\cal F}_i}(y)$ is computed, $${{\cal F}_i}(y) = w_i^c \cdot {s_i} + (w_i^rw_i^c - 1){r_{i + 1}}.
\label{Eq3}$$ When $w_i^r \cdot w_i^c$ approximates 1.0, the residual is related to only the side-output. To the extreme, along the stacking orientation of RUs, the residual ${\cal F}(y)$ approximates 0.0.
As we know, the deep layers of CNNs contain features that ignore the image details but capture high-level representations. Therefore, a deep layer SOP3 is expected to be closer to the optimal training solution. RU2 pushes the residual to zero and the response map RUOP2 is similar with the response map RUOP3. In the deep-to-shallow architecture, the deepest side-output is used as a good initialization for the ground-truth, therefore, the deep-to-shallow architecture contributes better results than the shallow-to-deep one, as shown in Sec. \[SRN-setting\].
[0.20]{} ![The implementation of the $i$-th RU.[]{data-label="figure7"}](figure7a.pdf "fig:"){width="0.9\linewidth" height="9em"}
[0.20]{} ![The implementation of the $i$-th RU.[]{data-label="figure7"}](figure7b.pdf "fig:"){width="\linewidth" height="9em"}
**Shallow-to-deep.** The architecture is shown in Fig. \[figure6b\] and the RU in Fig. \[figure7b\]. The side-outputs are up-sampled by the Gaussian deconvolution layer so that their size is consistent with the input image. Similar with Eq. (\[Eq3\]), the residual is computed, $${{\cal F}_i}(y) = w_i^sw_i^c \cdot {s_i} + (w_i^c - 1){r_{i + 1}},$$ where ${w_i}^s$ is weight parameter of the up-sampled ${s_i}$. Fig. \[figure6b\] indicates that the shallowest RUOP1 has lots of false positive pixels compare to ground-truth as SOP1 represents local structure of the input image. Along the stacking orientation, the RU3 reduces the residual so that the outputs of RU3, , RUOP3, are closer to ground-truth compared to RUOP2.
Learning
--------
Given the object symmetry detection training dataset $S = \{ ({X_n},{Y_n})\} _{n = 1}^N$ with $N$ training pairs, where ${X_n} = \{ x_j^{(n)},j = 1, \cdots ,T\}$ and ${Y_n} = \{ y_j^{(n)},j = 1, \cdots ,T\} $ are the input image and the ground-truth binary image with $T$ pixels, respectively. $y_j^{(n)} = 1$ denotes the symmetry pixel and $y_j^{(n)} = 0$ denotes non-symmetry pixel. We subsequently drop the subscript $n$ for notational simplicity, since we consider each image independently. We denote ${\bf{W}}$ as the parameters of the base network. Supposing the network has $M$ side-outputs, the $M$-th side-output is set as the basic output and $M-1$ RUs are used. We use the architecture of Fig. \[figure6a\] as example, in which $M=3$ and RUOP3 is the basic output. Fig. \[figure6b\] has similar formulation. For the basic output, the loss is computed, $$\begin{array}{l}
{{\cal L}_b}({\bf{W}},{w_b}) = - \beta \sum\limits_{j \in {Y_ + }} {\log \Pr ({y_j} = 1|X;{\bf{W}},{w_b})} \\
{\rm{ }} - (1 - \beta )\sum\limits_{j \in {Y_ - }} {\log \Pr ({y_j} = 0|X;{\bf{W}},{w_b})} ,
\end{array}$$ where ${w_b}$ is the classifier parameter for the basic output. $Y_+$ and $Y_-$ respectively denote the symmetry and non-symmetry ground-truth label sets. The loss weight $\beta = {{|{Y_ + }|} \mathord{\left/
{\vphantom {{|{Y_ + }|} {|Y|}}} \right.
\kern-\nulldelimiterspace} {|Y|}}$, and $|{Y_ + }|$ and $|{Y_ - }|$ denote the symmetry and non-symmetry pixel number, respectively. $\Pr ({y_j} = 1|X;{\bf{W}},{w_b}) \in \left[ {0,1} \right]$ is the sigmoid prediction of the basic output that measures how likely the point to be on the symmetry axis. For the $i$-th RU, $i = M-1, \cdots ,1$, the loss is computed, $$\begin{array}{l}
{{\cal L}_i}({\bf{W}},{\theta _i},{w_i}) = - \beta \sum\limits_{j \in {Y_ + }} {\log \Pr ({y_j} = 1|X;{\bf{W}},{\theta _i},{w_i})} \\
{\rm{ }} - (1 - \beta )\sum\limits_{j \in {Y_ - }} {\log \Pr ({y_j} = 0|X;{\bf{W}},{\theta _i},{w_i})}
\end{array}$$ where ${\theta _i} = (w_i^c,w_i^s)$ is the convolutional parameter of the concatenation layers and side-output layers after the $i$-th RU. ${w_i}$ is the classifier parameter for the output of $i$-th RU. The loss function for all the stacked RUs is obtained by $${\cal L}({\bf{W}},\theta ,w) = {\alpha _M}{{\cal L}_b}({\bf{W}},{w_b}) + \sum\limits_{i = {M-1}}^{1} {{\alpha _i}{{\cal L}_b}({\bf{W}},{\theta _i},{w_i})} .$$ Finally, we obtain the optimal parameters, $${({\bf{W}},\theta ,w)^*} = \arg \min {\cal L}({\bf{W}},\theta ,w).$$
In the testing phase, giving an image $X$, a symmetry prediction map is output by the last stacked RU, $$\hat Y = \Pr ({y_j} = 1|X;{{\bf{W}}^*},{\theta ^*},{w^*}).$$
Difference to Other Networks
----------------------------
The proposed SRN has significant difference with other end-to-end deep learning implementations, , HED [@03xie2015holistically], FSDS [@04shen2016object], and Laplacian Reconstruction [@05ghiasi2016laplacian]. In HED, the deep supervision is applied on side-outputs directly, while in SRN the deep supervision is applied on the outputs of RUs. According to (\[Eq2\]), each RU contains the information of two side-outputs at least, endowing SRN with the capability to smoothly model the multi-scale symmetry across deep layers. FSDS is an improvement of HED that specifies scales for side-outputs, which requires additional annotation for each scale. In contrast, SRN models the scale information with RUs, without any multi-scale annotations. SRN takes the idea of Laplacian reconstruction that uses a mask to indicate the reconstruction residual for segmentation. The difference lies in that SRN pursuits scale adaptability while the Laplacian reconstruction focuses on multi-scale error minimization.
Experimental results
====================
The proposed SRN is first evaluated and compared on the proposed Sym-PASCAL benchmark. It is then evaluated and compared with the state-of-the-art deep learning approaches on other popular datasets including SYMMAX [@14tsogkas2012learning], WH-SYMMAX [@15shen2016multiple], and SK506 [@04shen2016object].
Experimental Setup
------------------
**Implementation details.** The SRN is implemented following the parameter setting of HED [@03xie2015holistically], by fine-tuning the pre-trained 16-layer VGG net [@01simonyan2014very]. The hyper-parameters of SRN include: mini-batch size (1), learning rate (1e-8 for in-the-wild image datasets and 1e-6 for simple image datasets), loss-weight for each RU output (1), momentum (0.9), and initialization of the nested filters (0), weight decay (0.002), and maximum number of training iterations (18,000). In the testing phase, a non-maximal suppression (NMS) algorithm [@09dollar2015fast] is applied on the output map to obtain object symmetry.
**Evaluation Metrics.** The precision-recall metric with F-measure is used to evaluate the performance of symmetry detection, as introduced in [@14tsogkas2012learning]. To obtain the precision-recall curves, the detected symmetry response is first thresholded into a binary map, and then matched with the ground-truth symmetry masks. By changing the threshold value, the precision-recall curve is obtained and the best F-measure is computed.
Results on Sym-PASCAL
---------------------
### SRN setting {#SRN-setting}
Architecture Augumentation Conv1 F-measure
-------------- --------------- ------- -----------
with 0.381
w/o 0.397
with 0.371
w/o 0.396
with **0.443**
w/o **0.443**
with 0.384
w/o 0.397
: Performance of SRN under different settings on the Sym-PASCAL benchmark.
\[Tab-SRN-Setting\]
SRN is first evaluated on the new benchmark with different settings, Tab. \[Tab-SRN-Setting\]. **Architectures:** Tab. \[Tab-SRN-Setting\] shows that SRN with the deep-to-shallow architecture (F-measure 0.443) performs significantly better than the shallow-to-deep architecture (F-measure 0.397). It confirms that the deep-to-shallow architecture is easier to reduce the residual than the shallow-to-deep one as the initialization is better. **Data Augmentation:** Data augmentation can aggregate the training datasets. In this work, image rotation, flipping, up-sampling, and down-sampling (multi-scale) are used for data argumentation. For each scale, we rotate the training images every 90 degree and flip each one with different axis. The performance with/without multi-scale data argumentation is compared. Experiments show that the F-measure decreases with multi-scale augmentation, even though it produces more training data. The reason is analyzed as follows. The symmetry ground-truth is made up of curves with one-pixel thickness. The up-sampling operation produces curves that have thickness lager than one pixel, and the down-sampling operation produces discontinuous symmetry curves. **Conv1:** FSDS [@04shen2016object] doesn’t use the conv1 stage of VGG as the size of receptive field is so small (only 5) that introduces local noise of symmetry (too small to capture any symmetry response). The negative impact of small receptive field with SRN is also observed. By pairwise comparison in Tab. \[Tab-SRN-Setting\], the F-measure without conv1 is slightly better than that with conv1.
### Performance Comparison
Using the deep-to-shallow SRN with data augmentation but without conv1, we compare the performance of SRN with the state-of-the-art, as shown in Fig. \[figure8\] and Tab. \[Tab-compare-sympascal\]. All the compared results are generated by running the open source code with default parameter settings.
It’s observed that the traditional methods perform poorly and are time consuming. The best F-measure of traditional methods is 0.174, indicating the challenge of the proposed benchmark. Lindeberg [@32lindeberg1998edge] runs fastest with 5.79s per frame. Levinshtein [@29levinshtein2009multiscale], MIL [@14tsogkas2012learning], Lee [@30sie2013detecting] and Particle Filter [@36widynski2014local] need much more running time for the complex features they used.
{width="0.85\linewidth"}
The end-to-end deep learning methods perform well. HED gets the F-measure 0.369 and uses only ten milliseconds to process an image. FSDS is degenerated to HED when the scale information is not used. Its F-measure reaches 0.418 when slicing and concatenating of each side-output is used. Our proposed SRN gets the best performance with F-measure 0.443 which outperforms the baseline HED approach by 7.4%. It also outperforms the state-of-the-art method, FSDS, by 2.5%.
To show the effectiveness of the end-to-end pipeline in complex backgrounds, we compare the proposed SRN with a two-stage approach composing of semantic segmentation/object detection and skeleton extraction. We choose the best segmentation network FCN-8s [@02long2015fully] to localize objects, and the skeleton method [@38DBLP:journals/pr/ShenBHWL11] to extract symmetry, getting F-measure 0.386, Fig. \[figure8\]. We also compare the FSDS [@04shen2016object] on the detection results from the state-of-the-art object detection methods, FasterRCNN [@DBLP:conf/nips/RenHGS15] and YOLO [@DBLP:journals/corr/RedmonDGF15]. As shown in Fig. \[figure8\], the F-measures are 0.343 and 0.354, respectively. Experiments results indicate that the proposed end-to-end learning approach is a more effective and efficient way to detect object symmetry than the two-stage approaches.
The object symmetry detection results by the state-of-the-art deep leaning approaches are illustrated in Fig. \[figure9\]. From the first and second columns, it’s observed that the object symmetry obtained by our SRN approach in one-object images is more consistent with the ground-truth with/without complex background. The third and forth columns show examples that contain multiple objects, in which the proposed SRN approach achieves more accurate object symmetry detection results than other approaches. The last two columns show the results of images with occluded objects.
![Precision-recall comparison of different approaches on the Sym-PASCAL dataset.[]{data-label="figure8"}](figure8.pdf){width="0.75\linewidth"}
Methods F-measure Runtime(s)
----------------------------------------------------------------- ----------- -------------------
Partical Filter [@36widynski2014local] 0.129 25.30
Levinshtein [@29levinshtein2009multiscale] 0.134 183.87
Lee [@30sie2013detecting] 0.135 658.94
Lindeberg [@32lindeberg1998edge] 0.138 5.79
MIL [@14tsogkas2012learning] 0.174 80.35
HED (baseline) [@03xie2015holistically] 0.369 **0.10**$\dagger$
FSDS [@04shen2016object] 0.418 0.12$\dagger$
FasterRCNN [@DBLP:conf/nips/RenHGS15]+FSDS [@04shen2016object] 0.343 0.33$\dagger$
YOLO [@DBLP:journals/corr/RedmonDGF15]+FSDS [@04shen2016object] 0.354 0.12$\dagger$
FCN [@02long2015fully]+[@38DBLP:journals/pr/ShenBHWL11] 0.386 0.76$\dagger$
SRN (ours) **0.443** 0.12$\dagger$
: Performance comparison of the state-of-the-art approaches on the Sym-PASCAL dataset. $\dagger$GPU time with NVIDIA Tesla K80 []{data-label="Tab-compare-sympascal"}
[0.3]{} {width="\linewidth"}
[0.3]{} {width="\linewidth"}
[0.3]{} {width="\linewidth"}
----------- -------------------------------- ------- ------------------------ ------------------------------- ------- ------- ----------- -----------
Levinshtein Lindeberg Particle
[@29levinshtein2009multiscale] [@32lindeberg1998edge] Filter [@36widynski2014local]
SYMMAX – – 0.360 – 0.362 0.427 **0.467** 0.446
WH-SYMMAX 0.174 0.223 0.277 0.334 0.365 0.732 0.769 **0.780**
SK506 0.217 0.252 0.227 0.226 0.392 0.542 0.623 **0.632**
----------- -------------------------------- ------- ------------------------ ------------------------------- ------- ------- ----------- -----------
![The loss and F-measure comparison of HED and SRN. (Best viewed in color)[]{data-label="figure10"}](figure10.pdf){width="\linewidth"}
Results on Other Datasets
-------------------------
The performances on other three symmetry datasets are shown in Fig. \[figure11\] and Tab. \[Tab-compare-other\]. Similar with Sym-PASCAL, the deep learning based methods get significantly better performance on all the datasets, especially for the simple image datasets, WH-SYMMAX and SK506. Compared with the baseline HED, the proposed SRN improves the F-measure from 0.427 to 0.446, 0.732 to 0.780, 0.542 to 0.632 on SYMMAX, WH-SYMMAX and SK506, respectively.
Learning Convergence
--------------------
The learning convergence of the baseline HED and the proposed SRN is shown in Fig. \[figure10\]. It can be clearly seen that HED has a problem of slow convergence during learning, despite the fact that it achieves good performance on the edge and symmetry detection tasks. The reason could be that the complex backgrounds of input images seriously interrupt the end-to-end (image-to-mask) learning procedure. Benefits from the output residual fitting, the loss curve of the proposed SRN tends to converge, Fig. \[figure10\]. In addition, HED needs 12K learning iterations to get the best performance while SRN needs only 3K iterations to get the same performance.
Conclusion
==========
Symmetry detection has great applicability in computer vision yet remains not being well solved, as indicated by the low performance (often lower than 50%) of the state-of-the-art methods. In this work, we release a new object symmetry benchmark, as well as propose the Side-output Residual Network, establishing a strong baseline for object symmetry detection in the wild. The new benchmark, with challenges related to real-world images, is validated to be a good touchstone of various state-of-the-art approaches. The proposed Side-output Residual Network, with well-defined and stacked Residual Units, is validated to be more effective to perform symmetry detection in complex backgrounds. With the adaptability to object scales, the robustness to complex backgrounds, and the end-to-end learning architecture, the Side-output Residual Network has great potential to process a class of end-to-end (image-to-mask) computer vision tasks.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is partially supported by NSFC under Grant 61671427, Beijing Municipal Science and Technology Commission under Grant Z161100001616005, and Science and Technology Innovation Foundation of Chinese Academy of Sciences under Grant CXJJ-16Q218. Tekes, Academy of Finland and Infotech Oulu are also gratefully acknowledged.
[^1]: This work was supported in part by the CSC, China.
[^2]: Corresponding author
|
---
abstract: 'A general analysis of the sensitivities of neutron beta-decay experiments to manifestations of possible interaction beyond the Standard Model is carried out. In a consistent fashion, we take into account all known radiative and recoil corrections arising in the Standard Model. This provides a description of angular correlations in neutron decay in terms of one parameter, which is accurate to the level of $\sim 10^{-5}$. Based on this general expression, we present an analysis of the sensitivities to new physics for selected neutron decay experiments. We emphasize that the usual parametrization of experiments in terms of the tree level coefficients $a$, $A$ and $B$ is inadequate when the experimental sensitivities are at the same or higher level relative to the size of the corrections to the tree level description.'
author:
- 'V. Gudkov'
- 'G. L. Greene'
- 'J. R. Calarco'
title: 'General classification and analysis of neutron beta-decay experiments'
---
Introduction
============
The relative simplicity of the decay of the free neutron makes it an attractive laboratory for the study of possible extensions to the Standard Model. As is well known, measurements of the neutron lifetime and neutron decay correlations can be used to determine the weak vector coupling constant, which, in turn, can be combined with information on strange particle decay to test such notions as the universality of the weak interaction or to search for (or put a limit on) nonstandard couplings (see, for example, [@gtw2; @holsttr; @deutsch; @abele; @yeroz; @sg; @herc; @marc02] and references therein). It is less widely appreciated that precision measurements of the correlations in neutron decay can, in principle, be used as a test of the standard model without appeal to measurements in other systems. In particular, the detailed shape of the decay spectra and the energy dependence of the decay correlation are sensitive to non-standard couplings. The extraction of such information in a consistent fashion requires a rather delicate analysis, as the lowest order description of the correlation coefficients (and their energy dependencies) must be modified by a number of higher order corrections that are incorporated within the Standard Model. These include such effects as weak magnetism and radiative corrections. Recently [@eftcor] effective field theory has been used to incorporate all standard model effects in a consistent fashion in terms of one parameter with an estimated theoretical accuracy on the order of $10^{-5}$. Because this accuracy is well below that anticipated in the next generation of neutron decay experiments (see, for example, papers in [@NISTw]), this analysis provides a useful framework for the exploration of the sensitivity of various experiments to new physics.
In this paper, we extend the description of neutron beta-decay of [@eftcor] by including the most general non-standard beta-decay interactions. Our framework provides a consistent description of the modifications of the beta-decay observables at a level well below that anticipated in the next generation of experiments. Not surprisingly, we find that the different experimental observables have quite different sensitivities to the form of hypothetical non-standard couplings (i.e. vector, scalar, etc.).
Neutron $\beta$-decay beyond the Standard model.
=================================================
The most general description of neutron $\beta$-decay can be done in terms of low energy constants $C_i$ by the Hamiltonian[@ly56; @gtw1] $$\begin{aligned}
H_{int}&=&(\hat{\psi}_p\psi_n)(C_S\hat{\psi}_e\psi_{\nu}+C^\prime_S\hat{\psi}_e\gamma_5\psi_{\nu})\nonumber \\
&+&(\hat{\psi}_p\gamma_{\mu}\psi_n)(C_V\hat{\psi}_e\gamma_{\mu}\psi_{\nu}+C^\prime_V\hat{\psi}_e\gamma_{\mu}\gamma_5\psi_{\nu})\nonumber \\
&+&\frac{1}{2}(\hat{\psi}_p\sigma_{\lambda\mu}\psi_n)(C_T\hat{\psi}_e\sigma_{\lambda\mu}\psi_{\nu}+C^\prime_T\hat{\psi}_e\sigma_{\lambda\mu}\gamma_5\psi_{\nu})\nonumber \\
&-&(\hat{\psi}_p\gamma_{\mu}\gamma_5\psi_n)(C_A\hat{\psi}_e\gamma_{\mu}\gamma_5\psi_{\nu}+C^\prime_A\hat{\psi}_e\gamma_{\mu}\psi_{\nu})\nonumber \\
&+&(\hat{\psi}_p\gamma_5\psi_n)(C_P\hat{\psi}_e\gamma_5\psi_{\nu}+C^\prime_P\hat{\psi}_e\psi_{\nu}) \label{ham} \\
&+& \text{Hermitian conjugate}, \nonumber\end{aligned}$$ where the index $i=V$, $A$, $S$, $T$ and $P$ corresponds to vector, axial-vector, scalar, tensor and pseudoscalar nucleon interactions. In this presentation, the constants $C_i$ can be considered as effective constants of nucleon interactions with defined Lorentz structure, assuming that all high energy degrees of freedom (for the Standard model and any given extension of the Standard model) are integrated out. In this paper we consider only time reversal conserving interactions, therefore the constants $C_i$ can be chosen to be real. (The violation of time reversal invariance in neutron decay at the level of considered accuracy would be a clear manifestation of new physics and thus does not require an analysis of the form contained here.) Ignoring electron and proton polarizations, the given effective Hamiltonian will result in the neutron $\beta$-decay rate [@gtw1] in the tree approximation (neglecting recoil corrections and radiative corrections) in terms of the angular correlations coefficients $a$, $A$, and $B$: $$\begin{aligned}
\frac{d\Gamma ^3}{dE_ed\Omega_ed\Omega_{\nu}}= \Phi (E_e)G_F^2
|V_{ud}|^2 (1+3\lambda^2)
\hskip 2cm \nonumber \\
\times (1+b\frac{m_e}{E_e}+a\frac{\vec{p}_e\cdot
\vec{p}_{\nu}}{E_e
E_{\nu}}+A\frac{\vec{\sigma} \cdot \vec{p}_e}{E_e}
+B\frac{\vec{\sigma} \cdot \vec{p}_{\nu}}{E_{\nu}}),
\label{cor}\end{aligned}$$ Here, $\vec{\sigma}$ is the neutron spin; $m_e$ is the electron mass, $E_e$, $E_{\nu}$, $\vec{p}_e$, and $\vec{p}_{\nu}$ are the energies and momenta of the electron and antineutrino, respectively; and $G_F$ is the Fermi constant of the weak interaction (obtained from the $\mu$-decay rate). The function $\Phi (E_e)$ includes normalization constants, phase-space factors, and standard Coulomb corrections. For the Standard model the angular coefficients depend only on one parameter $\lambda = -C_A/C_V >0$, the ratio of axial-vector to vector nucleon coupling constant ($C_V=C^\prime_V$ and $C_A=C^\prime_A$): $$a=\frac{1-\lambda ^2}{1+3\lambda ^2}, \hskip 1cm A= -2\frac{\lambda
^2-{\lambda}}{1+3\lambda ^2}, \hskip 1cm B= 2\frac{\lambda
^2+{\lambda}}{1+3\lambda ^2}. \label{coef}$$ (The parameter $b$ is equal to zero for vector - axial-vector weak interactions.)
As was shown in [@gtw2] the existence of additional interactions modifies the above expressions and can lead to a non-zero value for the coefficient $b$. To explicitly see the influence of a non-standard interaction on the angular coefficients and on the decay rate of neutron one can re-write the coupling constants $C_i$ as a sum of a contribution from the standard model $C^{SM}_i$ and a possible contribution from new physics $\delta C_i$: $$\begin{aligned}
C_V &=& C^{SM}_V + \delta C_V \nonumber \\
C^\prime_V &=& C^{SM}_V + \delta C^\prime_V \nonumber \\
C_A &=& C^{SM}_A + \delta C_A \nonumber \\
C^\prime_A &=& C^{SM}_A + \delta C^\prime_A \nonumber \\
C_S &=& \delta C_S \nonumber \\
C^\prime_S &=& \delta C^\prime_S \nonumber \\
C_T &=& \delta C_T \nonumber \\
C^\prime_T &=& \delta C^\prime_T.
\label{consts}\end{aligned}$$ We neglect the pseudoscalar coupling constants since we treat[@gtw1] nucleons nonrelativistically. Defining the term proportional to the total decay rate in eq.(\[cor\]) as $\xi = (1+3\lambda^2)$ one can obtain corrections to parameters $\xi$, $a$, $b$, $A$ and $B$ due to new physics as $\delta\xi$, $\delta a$, $\delta b$, $\delta A$ and $\delta B$, correspondingly. Then, using results of [@gtw2], $$\begin{aligned}
\delta\xi &=& {C^{SM}_V}(\delta C_V+\delta C^\prime_V )+ ({\delta C_V}^2+{\delta C^\prime_V}^2+{\delta C_S}^2+{\delta C^\prime_S}^2)/2 \nonumber \\
&+& 3 [ {C^{SM}_A}(\delta C_A +\delta C^\prime_A)+ ({\delta C_A}^2+{\delta C^\prime_A}^2+{\delta C_T}^2+{\delta C^\prime_T}^2)/2], \nonumber \\
\xi \delta b &=& \sqrt{1-\alpha^2}[{C^{SM}_V}(\delta C_S+\delta C^\prime_S )+\delta C_S \delta C_V+ \delta C^\prime_S \delta C^\prime_V \nonumber \\
&+& 3({C^{SM}_A}(\delta C_T +\delta C^\prime_T)+\delta C_T \delta C_A+ \delta C^\prime_T \delta C^\prime_A )], \nonumber \\
\xi \delta a &=& {C^{SM}_V}(\delta C_V+\delta C^\prime_V )+({\delta C_V}^2+{\delta C^\prime_V}^2-{\delta C_S}^2-{\delta C^\prime_S}^2)/2 \nonumber \\
&-&{C^{SM}_A}(\delta C_A +\delta C^\prime_A)-({\delta C_A}^2+{\delta C^\prime_A}^2-{\delta C_T}^2-{\delta C^\prime_T}^2)/2, \nonumber \\
\xi \delta A &=& -2{C^{SM}_A}(\delta C_A+{\delta C^\prime_A}) + \delta C^\prime_A \delta C^\prime_A -\delta C^\prime_T \delta C^\prime_T \nonumber \\
&-& [C^{SM}_V(\delta C_A +\delta C^\prime_A)+{C^{SM}_A}(\delta C_V+\delta C^\prime_V )+\delta C_V \delta C^\prime_A +\delta C^\prime_V \delta C_A-\delta C_S \delta C^\prime_T -\delta C^\prime_S \delta C_T], \nonumber \\
\xi \delta B &=& \frac{m \sqrt{1-\alpha^2}}{E_e}[2{C^{SM}_A}(\delta C_T+\delta C^\prime_T)+{C^{SM}_A}(\delta C_S+\delta C^{\prime}_S) + {C^{SM}_V}(\delta C_T+C^\prime_T) \nonumber \\
&+& 2 \delta C_T \delta C^\prime_A +2 \delta C_A \delta C^\prime_T +\delta C_S \delta C^\prime_A +\delta C_A \delta C^\prime_S + \delta C_V \delta C^\prime_T +\delta C_T \delta C^\prime_V] \nonumber \\
&+&2{C^{SM}_A}(\delta C_A+{\delta C^\prime_A})-C^{SM}_V(\delta C_A +\delta C^\prime_A)-{C^{SM}_A}(\delta C_V+\delta C^\prime_V ) \nonumber \\
&-& \delta C_S \delta C^\prime_T - \delta C_T \delta C^\prime_S - \delta C_V \delta C^\prime_A - \delta C_A \delta C^\prime_V.
\label{nphys}\end{aligned}$$
It should be noted that we have neglected radiative corrections and recoil effects for the new physics contributions, because these are expected to be very small. However, Coulomb corrections for the new physics contributions are taken into account since they are important for a low energy part of the electron spectrum.
From the above equations one can see that contributions from possible new physics to the neutron decay distribution function is rather complicated. To be able to separate new physics from different corrections to the expression (\[cor\]), obtained in the tree level of approximation, one must describe the neutron decay process with accuracy which is better than the expected experimental accuracy. Assuming that the accuracy in future neutron decay experiments can reach a level of about $10^{-3} - 10^{-4}$, we wish to describe the neutron decay with theoretical accuracy by about $10^{-5}$ and our description must include all recoil and radiative corrections [@bilenky; @sirlin; @holstein; @sirlinnp; @sirlinrmp; @garcia; @wilkinson; @sir; @marciano]. To do this we will use recent results of calculations [@eftcor] of radiative corrections for neutron decay in the effective field theory (EFT) with some necessary modifications. The results of [@eftcor] can be used since they take into account both recoil and radiative corrections in the same framework of the EFT with estimated theoretical accuracy which is better than $10^{-5}$. However, the EFT approach does not provide all parameters but rather gives a parametrization in terms of a few (two, in the case of neutron decay) low energy constants which must be extracted from independent experiments. Therefore, the neutron $\beta$-decay distribution function is parameterized in terms of one unknown parameter (the second parameter is effectively absorbed in the axial vector coupling constant). If this parameter would be extracted from an independent experiment, it gives a model independent description of neutron beta-decay in the standard model with accuracy better than $10^{-5}$. A rough estimate of this parameter based on a “natural” size of strong interaction contribution to radiative corrections gives an accuracy for the expressions for the rate and the angular correlation coefficients which is better than $10^{-3}$ (see [@eftcor]). We vary the magnitude of this parameter in a wide range for the given numerical analysis and show that variations of the parameters in the allowed range do not significantly change our results at a level well bellow $10^{-3}$. Also, unlike [@eftcor], we use the exact Fermi function for numerical calculations to take into account all corrections due to interactions with the classical electromagnetic field. This gives us the expression for neutron decay distribution function as $$\begin{aligned}
\lefteqn{
\frac{d\Gamma ^3}{dE_ed\Omega_{\hat{p}_e}d\Omega_{\hat{p}_\nu}}
=
\frac{(G_FV_{ud})^2}{(2\pi)^5}
|\vec{p}_e|E_e(E_e^{max}-E_e)^2 F(Z,E_e)
}
\nonumber \\ && \times \left\{
f_0(E_e)
+\frac{\vec{p}_e\cdot\vec{p}_\nu}{E_eE_\nu}f_1(E_e)
+\left[\left(\frac{\vec{p}_e\cdot\vec{p}_\nu}{E_eE_\nu}\right)^2
-\frac{\beta^2}{3}
%\frac{\vec{p}_e^2}{E_e^2}
\right]f_2(E_e)
\right. \nonumber\\ && \left.
+ \frac{\vec{\sigma}\cdot\vec{p}_e}{E_e}f_3(E_e)
+ \frac{\vec{\sigma}\cdot\vec{p}_e}{E_e}
\frac{\vec{p}_e\cdot\vec{p}_\nu}{E_eE_\nu}f_4(E_e)
+ \frac{\vec{\sigma}\cdot\vec{p}_\nu}{E_\nu}f_5(E_e)
+ \frac{\vec{\sigma}\cdot\vec{p}_\nu}{E_\nu}
\frac{\vec{p}_e\cdot\vec{p}_\nu}{E_eE_\nu}f_6(E_e)
\right\},
\label{eq;theresult}\end{aligned}$$ where the energy dependent angular correlation coefficients are: $$\begin{aligned}
\lefteqn{f_0(E_e) = (1+3\lambda^2) \left( 1
%
+ \frac{\alpha}{2\pi} \delta_\alpha^{(1)}
+ \frac{\alpha}{2\pi} \; e_V^R \right) }
\nonumber \\ &&
- \frac{2}{m_N}\left[
\lambda(\mu_V+\lambda)\frac{m_e^2}{E_e}
+\lambda(\mu_V+\lambda)E_e^{max}
-(1+2\lambda\mu_V+5\lambda^2)E_e
\right] ,
\\
\lefteqn{f_1(E_e) = (1-\lambda^2)
\left( 1
%
%
+ \frac{\alpha}{2\pi}
(\delta_\alpha^{(1)}+\delta_\alpha^{(2)})
+ \frac{\alpha}{2\pi} \; e_V^R \right) }
\nonumber \\ &&
+\frac{1}{m_N}\left[
2\lambda(\mu_V+\lambda)E_e^{max}
-4\lambda(\mu_V+3\lambda)E_e
\right],
\\
\lefteqn{f_2(E_e) =
-\frac{3}{m_N}(1-\lambda^2)E_e , }
\\
\lefteqn{f_3(E_e) = (-2\lambda^2+2\lambda) \left( 1
%
%
+\frac{\alpha}{2\pi} ( \delta_\alpha^{(1)}
+\delta_\alpha^{(2)} )
+ \frac{\alpha}{2\pi} \; e_V^R \right) }
\nonumber \\ &&
+\frac{1}{m_N}\left[
(\mu_V+\lambda)(\lambda-1)E_e^{max}
+(-3\lambda\mu_V+\mu_V-5\lambda^2+7\lambda)E_e
\right],
\\
\lefteqn{f_4(E_e) =
\frac{1}{m_N}(\mu_V+5\lambda)(\lambda-1)E_e, }
\\
\lefteqn{f_5(E_e) = (2\lambda^2+2\lambda) \left(1
%
%
+\frac{\alpha}{2\pi} \delta_\alpha^{(1)}
+ \frac{\alpha}{2\pi} \; e_V^R \right) }
\nonumber \\ &&
+\frac{1}{m_N}\left[
-(\mu_V+\lambda)(\lambda+1)\frac{m_e^2}{E_e}
-2\lambda(\mu_V+\lambda)E_e^{max}
\right. \nonumber \\ && \left.
+(3\mu_V\lambda+\mu_V+7\lambda^2+5\lambda)E_e
\right] ,
\\
\lefteqn{f_6(E_e) =
\frac{1}{m_N}\left[
(\mu_V+\lambda)(\lambda+1)E_e^{max}
-(\mu_V+7\lambda)(\lambda+1)E_e
\right] \; . }\end{aligned}$$ Here $e_V^R$ is the finite renormalized low energy constant (LEC) corresponding to the “inner" radiative corrections due to the strong interactions in the standard QCD approach; $F(Z,E_e) $ is the standard Fermi function; and the functions $\delta_\alpha^{(1)}$ and $\delta_\alpha^{(2)}$ are: $$\begin{aligned}
\delta_\alpha^{(1)} &=&
\frac12
+ \frac{1+\beta^2}{\beta} {\rm ln}\left(\frac{1+\beta}{1-\beta}\right)
- \frac{1}{\beta}{\rm ln}^2\left(\frac{1+\beta}{1-\beta}\right)
+ \frac4\beta L\left(\frac{2\beta}{1+\beta}\right)
\nonumber \\ &&
+ 4 \left[\frac{1}{2\beta}{\rm ln}\left(\frac{1+\beta}{1-\beta}\right)
-1\right]
\left[{\rm ln}\left(\frac{2(E_e^{max}-E_e)}{m_e}\right)
%
+ \frac13 \left(\frac{E_e^{max}-E_e}{E_e}\right)
-\frac32
\right]
\nonumber \\ &&
+ \left(\frac{E_e^{max}-E_e}{E_e}\right)^2 \frac{1}{12\beta}
{\rm ln}\left(\frac{1+\beta}{1-\beta}\right) \, .
\\
\delta_\alpha^{(2)} &=&
\frac{1-\beta^2}{\beta}{\rm ln}\left(\frac{1+\beta}{1-\beta}\right)
+\left(\frac{E_e^{max}-E_e}{E_e}\right)
\frac{4(1-\beta^2)}{3\beta^2}
\left[\frac{1}{2\beta}{\rm ln}\left(\frac{1+\beta}{1-\beta}\right)-1
\right]
\nonumber \\ &&
+\left(\frac{E_e^{max}-E_e}{E_e}\right)^2
\frac{1}{6\beta^2}
\left[\frac{1-\beta^2}{2\beta}
{\rm ln}\left(\frac{1+\beta}{1-\beta}\right)-1
\right] \; ,\end{aligned}$$ where $\beta = p_e/E_e$. The only unknown parameter $e_V^R$ is chosen to satisfy the estimate [@sir] for an “inner” part of the radiative corrections: $\frac{\alpha}{2\pi} \; e_V^R=0.02$. In Eq.(\[eq;theresult\]) the custom of expanding the nucleon recoil correction of the three-body phase space has been used. These recoil corrections are included in the coefficients $f_i$, $i=0, 1, \cdots , 6$ defined in the partial decay rate expression, Eq.(\[eq;theresult\]). It should be noted that the expression for $f_2$ is an exclusive three-body phase space recoil correction, whereas all other $f_i$, $i= 0, 1, 3, \cdots , 6$ contain a mixture of regular recoil and phase space $(1/m_N)$ corrections.
The above expression presents all contributions from the Standard model. Therefore, the difference between this theoretical description and an experimental result can only be due to effects not accounted for by the Standard model. From the eqs.(\[nphys\]) we can see that the only contributions from new physics in neutron decay are: $$\begin{aligned}
f_0(E_e) &\longrightarrow & f_0(E_e) + \delta\xi + \frac{m}{E_e}\delta b, \nonumber \\
f_1(E_e) &\longrightarrow & f_1(E_e) + \delta a , \nonumber \\
f_3(E_e) &\longrightarrow & f_3(E_e) + \delta A , \nonumber \\
f_5(E_e) &\longrightarrow & f_5(E_e) + \delta B ,
\label{cphys}\end{aligned}$$
Since possible contributions from models beyond the Standard one are rather complicated, we have to use numerical analysis for calculations of experimental sensitivities to new physics.
The analysis of the experimental sensitivity to new physics
===========================================================
To calculate the sensitivity of an experiment with a total number of events $N$ to the parameter $q$ we use the standard technique of the minimum variance bound estimator (see, for example [@kend; @frod]). The estimated uncertainties provided by this method correspond to one sigma limits for a normal distribution. The statistical error (variance) $\sigma_q$ of parameter $q$ in the given experiment can be written as $$\label{sen1}
\sigma_q = \frac{K}{\sqrt{N}},$$ where $$\label{sen2}
K^{-2} = \frac{\int w(\vec{x})\left(\frac{1}{w(\vec{x})}\frac{\partial w(\vec{x})}{\partial q} \right)^2d\vec{x}}{\int w(\vec{x})d\vec{x}}.$$ Here $w(\vec{x})$ is a distribution function of measurable parameters $\vec{x}$. We can calculate the sensitivity of the experiment to a particular coefficient $C_i$ or to a function of these coefficients. The results for these integrated sensitivities for each type of interaction ($C_i$) and for the left-right model are given in the table \[ctab\] for the standard experiments measuring $a$, $A$ and $B$ coefficients in neutron decay, assuming that all coefficients $C_i$ have the same value of $1\cdot 10^{-3}$. The numerical test shows that results for the coefficients $K$ can be linearly re-scaled for the parameters $C_i$ in the range from $10^{-2}$ to $10^{-4}$ with an accuracy of better than $10 \%$. We can see that different experiments have different sensitivities (discovery potentials) for the possible manifestations of new physics.
Interactions $a$ $A$ $B$
-------------- ------ ------- ------
$V$ 5.26 3.60 6.95
$A$ 1.73 1.90 1.91
$T$ 2.59 7.25 1.50
$S$ 8.70 26.70 1.46
$V+A$ 2.01 1.58 3.86
: Relative statistical error ($K$) of the standard experiments to different types of interactions from new physics ($C_i$ constants) provided that these constants have the same values of $1\cdot 10^{-3}$.[]{data-label="ctab"}
The given description of neutron $\beta$-decay experiments in terms of low energy constants related to the Lorentz structure of weak interactions is general and complete. All models beyond the Standard one (new physics) contribute to the $C_i$ values in different ways. Therefore, each model can be described by a function of the $C_i$ parameters. To relate these $C$-coefficients explicitly to the possible models beyond the Standard one we can use the parametrization of reference [@herc]. It should be noted that the definitions of reference [@gtw1] used for the $C_i$ coefficients are the same as in [@herc], except for the opposite sign of $C^\prime_V$, $C^\prime_S$, $C^\prime_T$ and $C_A$. Therefore, we can re-write the relations of the $\delta C_i$, which contain contributions to the $ C_i$ from new physics, in terms of the parameters $\bar{a}_{jl}$ and $\bar{A}_{jl}$ defined in the paper [@herc] as: $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\delta C_V &=& C^{SM}_V (\bar{a}_{LL}+\bar{a}_{LR}+\bar{a}_{RL}+\bar{a}_{RR}), \nonumber \\
\delta C^\prime_V &=& -C^{SM}_V (-\bar{a}_{LL}-\bar{a}_{LR}+\bar{a}_{RL}+\bar{a}_{RR}), \nonumber \\
\delta C_A &=& -C^{SM}_A (\bar{a}_{LL}-\bar{a}_{LR}-\bar{a}_{RL}+\bar{a}_{RR}), \nonumber \\
\delta C^\prime_A &=& C^{SM}_A (-\bar{a}_{LL}+\bar{a}_{LR}-\bar{a}_{RL}+\bar{a}_{RR}) \nonumber \\
\delta C_S &=& g_S (\bar{A}_{LL}+\bar{A}_{LR}+\bar{A}_{RL}+\bar{A}_{RR}), \nonumber \\
\delta C^\prime_S &=-& g_S (-\bar{A}_{LL}-\bar{A}_{LR}+\bar{A}_{RL}+\bar{A}_{RR}), \nonumber \\
\delta C_T &=& 2 g_T (\bar{\alpha}_{LL}+\bar{\alpha}_{RR}), \nonumber \\
\delta C^\prime_T &=& -2 g_T (-\bar{\alpha}_{LL}+\bar{\alpha}_{RR}).
\label{carel}\end{aligned}$$
The parameters $\bar{a}_{jl}$, $\bar{\alpha}_{jl}$ and $\bar{A}_{jl}$ describe contributions to the low energy Hamiltonian from current-current interactions in terms of $j$-type of leptonic current and $i$-type of quark current. For example, $\bar{a}_{LR}$ is the contribution to the Hamiltonian from left-handed leptonic current and right-handed quark current normalized by the size of the Standard Model (left–left current) interactions. $g_S$ and $g_T$ are formfactors at zero-momentum transfer in the nucleon matrix element of scalar and tensor currents. For more details, see the paper [@herc]. It should be noted, that $\delta C_i + \delta C^\prime_i $ involve left-handed neutrinos and $\delta C_i - \delta C^\prime_i $ is related to right-handed neutrino contributions in corresponding lepton currents. The analysis of the three experiments under consideration ($a$, $A$ and $B$ coefficient measurements) in terms of sensitivities ($K^{-1}$) to $\bar{a}_{jl}$, $\bar{\alpha}_{jl}$ and $\bar{A}_{jl}$ parameters is presented in the table \[atab\]. For the sake of easy comparison the sensitivities in this table are calculated under assumptions that all parameters ($\bar{a}_{jl}$, $\bar{\alpha}_{jl}$ and $\bar{A}_{jl}$) have exactly the same value, $1\cdot 10^{-3}$. The expected values of these parameters vary over a wide range from $0.07$ to $10^{-6}$ (see table \[nptab\] and, paper [@herc] for the comprehensive analysis). The numerical results for the coefficients $K$ in the table can be linearly re-scaled for the parameters $\bar{a}_{ij}$, $\bar{\alpha}_{jl}$ and $\bar{A}_{ij}$ in the range from $10^{-2}$ to $10^{-4}$ with an accuracy better than $10 \%$. The relative statistical errors presented in the Table demonstrate discovery potentials of different experiments to new physics in terms of parameters $\bar{a}_{ij}$, $\bar{\alpha}_{jl}$ and $\bar{A}_{ij}$. It should be noted, that the parameter $\bar{a}_{LR}$ cannot provide sensitive information on new physics at the quark level, unless we obtain the axial-vector coupling constant $g_A$ from another experiment, since in correlations $\bar{a}_{LR}$ appears in a product with $g_A$ (see [@herc]). For discussion of significance of each of these parameters to models beyond the standard one see [@herc].
$\bar{a}_{LL}$ $\bar{a}_{LR}$ $\bar{a}_{RL}$ $\bar{a}_{RR}$ $\bar{A}_{LL}$ $\bar{A}_{LR}$ $\bar{A}_{RL}$ $\bar{A}_{RR}$ $\bar{\alpha}_{LL}$ $\bar{\alpha}_{RR}$
--- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- --------------------- ---------------------
a 0.17 0.25 135 487 1.43 1.43 283 283 0.19 79
A 1.53 0.63 423 1026 13.1 13.1 860 860 1.82 223
B 0.58 1.21 89 347 0.72 0.72 958 958 0.37 59
: Relative statistical error ($K$) of the standard experiments to different types of interactions from new physics ($\bar{a}_{ij}$ constants) provided that these constants have the same values of $1\cdot 10^{-3}$.[]{data-label="atab"}
It should be noted the results in the tables \[ctab\] and \[atab\] are calculated with the estimated value of the parameter $( \alpha /(2 \pi) \; e_V^R=0.02$. Numerical tests show that a change of this parameter by a factor two leads to changes of results in the tables by about $1\%$.
Model L-R Exotic Fermion Leptoquark Contact interactions SUSY Higgs
------------------------------ ------- ---------------- ------------------ ---------------------- --------------------- ------------------
$\bar{a}_{RL}$ 0.067 0.042
$\bar{a}_{RR}$ 0.075 0.01
$\bar{A}_{LL}+\bar{A}_{LR}$ 0.01 $7.5 \cdot 10^{-4}$ $3\cdot 10^{-6}$
$\bar{A}_{RR}+\bar{A}_{RL}$ 0.1
$-\bar{A}_{LL}+\bar{A}_{LR}$ $3\cdot 10^{-6}$
$\bar{A}_{RR}-\bar{A}_{RL}$ $4\cdot 10^{-4}$
: Possible manifestations of new physics[]{data-label="nptab"}
The calculated integral sensitivities of different experiments to a particular parameter related to new physics can be used for the estimation of the experimental sensitivity when the experimental statistics is not good enough. For the optimization of experiments it is useful to know how manifestations of new physics contribute to the energy spectrum of the measurable parameter. As an example, the contributions from $\bar{a}_{LR}$, $\bar{a}_{RL}$ and $\bar{a}_{RR}$ to the spectra for the $a$, $A$ and $B$ correlations are shown on figures (\[fig-a-aLR\]) - (\[fig-B-aRR\]). For uniform presentation all graphs on the figures are normalized by $N_f=G_F^2|V_{ud}|^2 \int f(E) dE$, where $f(E)$ is $a(E_p)$, $A(E_e)$ and $A(E_e)$, correspondingly.
![Manifestation of $a_{LR}$-type interactions on the $a$ coefficient. []{data-label="fig-a-aLR"}](aLRforSMALLa.eps)
![Manifestation of $a_{RL}$-type interactions on the $a$ coefficient. []{data-label="fig-a-aRL"}](aRLforSMALLa.eps)
![Manifestation of $a_{LR}$-type interactions on the $A$ coefficient. []{data-label="fig-A-aLR"}](aLRforLargeA.eps)
![Manifestation of $a_{RL}$-type interactions on the $A$ coefficient. []{data-label="fig-A-aRL"}](aRLforLargeA.eps)
![Manifestation of $a_{RR}$-type interactions on the $A$ coefficient. []{data-label="fig-A-aRR"}](aRRforLargeA.eps)
![Manifestation of $a_{LR}$-type interactions on the $B$ coefficient. []{data-label="fig-B-aLR"}](aLRforLargeB.eps)
![Manifestation of $a_{RL}$-type interactions on the $B$ coefficient. []{data-label="fig-B-aRL"}](aRLforLargeB.eps)
![Manifestation of $a_{RR}$-type interactions on the $B$ coefficient. []{data-label="fig-B-aRR"}](aRRforLargeB.eps)
One can see that these contributions have different shapes and positions of maxima both for different model parameters and for different angular correlations. This gives the opportunity for fine tuning in the search for particular models beyond the Standard one in neutron decays.
Using the approach developed here one can calculate the exact spectrum for a given model. For example, manifestations of the Left-Right model ($\bar{a}_{RL}= 0.067$ and $\bar{a}_{RR}=0.075$) in the measurements of the $A$ and $B$ coefficients are shown in solid lines on figures \[fig-A-asym\] and \[fig-B-asym\].
![Contributions from radiative and recoil corrections (dashed line) and from the left-right model (solid line) to the $A$ coefficient. The curves are explained in the text.[]{data-label="fig-A-asym"}](AsymforLargeA.eps)
![Contributions from radiative and recoil corrections (dashed line) and from the left-right model (solid line) to the $B$ coefficient. The curves are explained in the text.[]{data-label="fig-B-asym"}](AsymforLargeB.eps)
The dashed lines show contributions from recoil effects and radiative corrections (without Coulomb corrections) assuming that $( \alpha /(2 \pi) \; e_V^R) = 0.02$. From these plots one can see the importance of the corrections at the level of the possible manifestations of new physics.
![Contributions from radiative and recoil corrections to the $B$ coefficient for $ (\alpha /(2 \pi) \; e_V^R)=0.01$ (dashed-doted line), $( \alpha /(2 \pi) \; e_V^R)=0.02$ (dashed line), and $ (\alpha /(2 \pi) \; e_V^R)=0.03$ (solid line).[]{data-label="fig-B-corr"}](CorrforLargeB.eps)
The figure \[fig-B-corr\] shows how these corrections for the coefficient $B$ affected by the value of the parameter $( \alpha /(2 \pi) \; e_V^R)$ related to nuclear structure: dashed-doted, dashed and solid lines correspond to $0.01$, $0.02$ and $0.03$ values for the parameter.
We presented here results of analysis for only a number of parameters $\bar{a}_{ij}$ to illustrate a different level of sensitivities of experiments to the parameters. For the complete analysis of future experiments all $\bar{a}_{ij}$, $\bar{\alpha}_{ij}$ and $\bar{A}_{ij}$ parameters should be analyzed with a specific experimental conditions taken into account.
Conclusions
===========
The analysis presented here provides a general basis for comparison of different experiments of neutron $\beta$-decay from the point of view of the discovery potential for new physics. It is also demonstrates that various parameters measured in experiments have quite different sensitivities to the detailed nature of the (supposed) new physics and can, in principle be used to differentiate between different extensions to the Standard Model. Thus neutron decay can be considered as a promising tool to search for new physics, which may not only detect the manifestations of new physics but also define the source of the possible deviations from predictions of the Standard model. Our results can be used for optimization of new high precision experiments to define important directions and to complement high energy experiments. Finally we emphasize that the usual parametrization of experiments in terms of the tree level coefficients $a$, $A$ and $B$, is inadequate when experimental sensitivities are comparable or better to the size of the corrections to the tree level description. This is expected in the next generation of neutron decay experiments. Therefore, such analysis is needed for these experiments. One has to use the full expression for neutron beta-decay in terms of the coupling constants. In other words, the high precision experiments should focus on the parameters important for physics rather than on the coefficients $a$, $A$ and $B$ which are sufficient only for low-accuracy measurements.
VG thanks to P. Herczeg for helpful discussions. This work was supported by the DOE grants no. DE-FG02-03ER46043 and DE-FG02-03ER41258.
[99]{}
J. D. Jackson, S. B. Treiman and H. W. Wyld, Jr., Nucl. Phys. [**4**]{}, 206 (1957). B. R. Holstein and S. B. Treiman, Tests of spontaneous left-right-symmetry breaking,[*Phys. Rev., D*]{}[**16**]{}, 2369 (1977). J. Deutsch, in: [*Fundamental Symmetries and Nuclear Structure*]{}, eds. J. N. Ginocchio and S. P. Rosen, p.36,World Scientific, 1989. H. Abele, The Standard Model and the neutron $\beta$-decay , [*NIM, A*]{}[**440**]{}, 499 (2000). B. G. Yerozolimsky, Free neutron decay: a review of the contemporary situation, [*NIM, A*]{}[**440**]{}, 491 (2000). S. Gardner and C. Zhang, Phys.Rev.Lett. [**86**]{} 5666, (2001). P. Herczeg, Prog. in Part. Nucl. Phys. [**46**]{}, 413 (2001). W. J. Marciano, RADCOR 2002: Conclusions and Outlook, [*Nucl. Phys., B*]{} (Proc. suppl.) [**116**]{}, 437 (2003). S. Ando, H. W. Fearing, V. Gudkov, K. Kubodera, F. Myhrer, S. Nakamura and T. Sato, Phys. Lett. [**B 595**]{}, 250 (2004 ). Proceedings “Precision Measurements with Slow Neutrons”, National Institute of Standards and Technology, Gaithersburg, April 5-7, 2004. T. D. Lee and C. N. Yang, Phys. Rev. [**104**]{}, 254 (1956). J. D. Jackson, S. B. Treiman and H. W. Wyld, Jr., Phys. Rev. [**106**]{}, 517 (1957).
S. M. Bilen’kii, R. M. Ryndin, Ya. A. Saoridinskiǐ, and Ho Tso-Hsiu, Sovi. Phys. JETP [**37**]{} (1960) 1241. A. Sirlin, Phys. Rev. [**164**]{} (1967) 1767. B. R. Holstein, Rev. Mod. Phys. [**46**]{} (1974) 789; Erratum ibid [**48**]{} (1976) 673. A. Sirlin, Nucl. Phys. B [**71**]{} (1974) 29. A. Sirlin, Rev. Mod. Phys. [**50**]{} (1978) 573. A. García and M. Maya, Phys. Rev. D [**17**]{} (1978) 1376. D. E. Wilkinson, Nucl. Phys. A [**377**]{} (1982) 474. W. J. Marciano and A. Sirlin, Radiative corrections to beta decay and the possibility of a fourth generation, [*Phys. Rev. Lett.*]{} [**56**]{}, 22 (1986). W. J. Marciano and A. Sirlin, Phys. Rev. Lett. [**71**]{} (1993) 3629. A. Stuart, J. K. Ord, S. Arnold, Kendall’s Advanced Theory of Statistics: Classical Inference and and the Linear Model, v.2A, 6th eds., Arnold Publishers, 1998. A. G. Frodesen, Probability and Statistics in Particle Physics, Oxford University Press, 1979.
|
---
abstract: 'A *theory graph* is a network of *axiomatic theories* connected with meaning-preserving mappings called *theory morphisms*. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not adequately fulfill all the purposes that mathematical proofs have, and they do not exploit the structure inherent in a theory graph. We propose a new style of proof that fulfills the principal purposes of a mathematical proof as well as capitalizes on the connections provided by the theory morphisms in a theory graph. This new style of proof combines the strengths of traditional proofs with the strengths of formal proofs.'
author:
- |
William M. Farmer[^1]\
Department of Computing and Software\
McMaster University\
Hamilton, Ontario, Canada
bibliography:
- 'new-proof-style-for-math.bib'
date: 30 November 2018
title: '**A New Style of Proof for Mathematics Organized as a Network of Axiomatic Theories[^2]**'
---
[^1]: `wmfarmer@mcmaster.ca.`
[^2]: This research was supported by NSERC.
|
---
abstract: 'Over the last two decades, face alignment or localizing fiducial facial points has received increasing attention owing to its comprehensive applications in automatic face analysis. However, such a task has proven extremely challenging in unconstrained environments due to many confounding factors, such as pose, occlusions, expression and illumination. While numerous techniques have been developed to address these challenges, this problem is still far away from being solved. In this survey, we present an up-to-date critical review of the existing literatures on face alignment, focusing on those methods addressing overall difficulties and challenges of this topic under uncontrolled conditions. Specifically, we categorize existing face alignment techniques, present detailed descriptions of the prominent algorithms within each category, and discuss their advantages and disadvantages. Furthermore, we organize special discussions on the practical aspects of face alignment *in-the-wild*, towards the development of a robust face alignment system. In addition, we show performance statistics of the state of the art, and conclude this paper with several promising directions for future research.'
address:
- 'Department of Computer Science and Engineering, Nanjing University of Aeronautics and Astronautics, \#29 Yudao Street, Nanjing 210016, P.R. China'
- 'Collaborative Innovation Center of Novel Software Technology and Industrialization, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China'
author:
- Xin Jin
- Xiaoyang Tan
bibliography:
- 'face\_alignment\_survey.bib'
title: 'Face Alignment In-the-Wild: A Survey'
---
[1.25]{}
Face alignment, Active appearance model, Constrained local model, Cascaded regression, Deep convolutional neural networks.
Introduction {#sec_introduction}
============
Fiducial facial points refer to the predefined landmarks on a face graph, which are mainly located around or centered at the facial components such as eyes, mouth, nose and chin (see Fig. \[fig\_intro\_landmark\]). Localizing these facial points, which is also known as face alignment, has recently received significant attention in computer vision, especially during the last decade. At least two reasons account for this. Firstly, many important tasks, such as face recognition, face tracking, facial expression recognition, head pose estimation, can benefit from precise facial point localization. Secondly, although some level of success has been achieved in recent years, face alignment in unconstrained environments is so challenging that it remains an open problem in computer vision, and continues to attract researchers to attack it.
While face detection is generally regarded as the starting point for all face analysis tasks [@zafeiriou2015survey], face alignment can be regarded as *an important and essential intermediary step* for many subsequent face analyses that range from biometric recognition to mental state understanding. Concrete tasks may differ in the number and type of the needed facial points, as well as the way these points are used. Below we give some details on three typical tasks where face alignment plays a prominent role:
- *Face recognition:* Face alignment is widely used by face recognition algorithms to improve their robustness against pose variations. For example, in the stage of face registration, the first step is usually to locate some major facial points and use them as anchor points for affine warping, while other face recognition algorithms, such as feature-based (structural) matching [@zhao2003face; @campadelli2003feature], rely on accurate face alignment to build the correspondence among local features (e.g, eyes, nose, mouth, etc.) to be matched.
![Illustration of some example face images with 68 manually annotated points from the IBUG database [@sagonas2013300].](images/introduction/landmark_samples.pdf "fig:"){width="48.00000%"}\[fig\_intro\_landmark\]
- *Attribute computing:* Face alignment is also beneficial to facial attribute computing, since many facial attributes such as eyeglasses and nose shape are closely related to specific spatial positions of a face. In [@kumar2009attribute], six facial points are localized to compute qualitative attributes and similes that are then used for robust face verification in unconstrained conditions.
- *Expression recognition:* The configurations of facial points (typically between 20-60) are reliable indicative of the deformations caused by expressions, and the subsequent analysis will reveal the particular type of expression that may lead to such deformation. Many works [@rudovic2010coupled; @valstar2012fully; @senechal2011combining; @bailenson2008real; @li2015efficient] follow this idea and use various features extracted from these points for expression recognition.
The above-mentioned applications, as well as numerous ones yet to be conceived, urge the need for developing robust and accurate face alignment techniques in real-life scenarios.
Under constrained environments or on less challenging databases, the problem of face alignment has been well addressed, and some algorithms even achieve performance that is close to that of human beings [@belhumeur2011localizing; @dantone2012real]. Under unconstrained conditions, however, this task is extremely challenging and far from being solved, due to the high degree of facial appearance variability caused either by intrinsic dynamic features of the facial components such as eyes and mouth, or by ambient environment changes. In particular, the following factors have significant influence on facial appearance and the states of local facial features:
- *Pose:* The appearance of local facial features differ greatly between different camera-object poses (e.g., frontal, profile, upside down), and some facial components such as the one side of the face contour, can even be completely occluded in a profile face.
- *Occlusion:* For face images captured in unconstrained conditions, occlusion frequently happens and brings great challenges to face alignment. For example, the eyes may be occluded by hair, sunglasses, or myopia glasses with black frames.
- *Expression:* Some local facial features such as eyes and mouth are sensitive to the change of various expressions. For example, laughing may cause the eyes to close completely, and largely deform the shape of the mouth.
- *Illumination:* Lighting (varying in spectra, source distribution, and intensity), may significantly change the appearance of the whole face, and make the detailed textures of some facial components missing.
These challenges are illustrated in Fig. \[fig\_intro\_challenge\] by the IBUG database [@sagonas2013300]. An ideal face alignment system should be robust to these facial variations on one hand; while on the other hand, as efficient as possible to satisfy the need of practical applications (e.g., real-time face tracking).
Over the last two decades, numerous techniques have been developed for face alignment with varying degrees of success. [Ç]{}eliktutan *et al.* [@cceliktutan2013comparative] surveyed many traditional methods for face alignment of both 2D and 3D faces, but some recent state-of-the-art methods are not covered. Wang *et al.* [@wang2014facial] gave a more comprehensive survey of face alignment methods over the last two decades, but the overall difficulties and challenges in unconstrained environments have not been highlighted. More recently, Yang *et al.* [@yang2015empirical] provided an empirical study of recent face alignment methods, aiming to draw some empirical yet useful conclusions and make insightful suggestions for practical applications.
The significant contribution of this paper is to give a comprehensive and critical survey of the ad hoc face alignment methods addressing the difficulties and challenges in unconstrained environments, which we believe would be a useful complement to [@cceliktutan2013comparative; @wang2014facial; @yang2015empirical]. To be self-contained, some traditional methods for face alignment covered in [@cceliktutan2013comparative; @wang2014facial] are also included. However, contrary to the previous works, we pay special attention to study and summarize the motivation and successful experiences behind the state-of-the-art, expecting to offer some insights into the studies of this field. Furthermore, we organize special discussions on the practical aspects of constructing a face alignment system, including training data augmentation, face preprocessing, shape initialization, accuracy and efficiency tradeoffs. This in our opinion is a very important topic in practice, but is mostly ignored in previous studies. In addition, we show comparative performance statistics of the state of the art, and propose several promising directions for future research.
In the following Section \[sec\_categorization\], we briefly describe the main idea of face alignment and categorize existing methods into two main categories. Then, the prominent methods within each category are reviewed and analyzed in Section \[sec:generative\_methods\] and \[sec:discriminative\_methods\]. In Section \[sec\_development\], we investigate some practical aspects of developing of a robust face alignment system. In Section \[sec\_evaluation\], we discuss a few issues concerning performance evaluation. Finally, we conclude this paper with a discussion of several promising directions for further research in Section \[sec\_conclusion\].
Overview {#sec_categorization}
========
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
**[Approach]{} & **[Representative works]{}\
**[Generative methods]{} &\
*Active appearance models (AAMs)* &\
Regression-based fitting & Original AAM [@cootes2001active]; Boosted Appearance Model [@liu2007generic]; Nonlinear discriminative approach [@saragih2007nonlinear]; Accurate regression procedures for AMMs [@sauer2011accurate]\
Gradient descent-based fitting & Project-out inverse compositional (POIC) algorithm [@matthews2004active]; Simultaneous inverse compositional (SIC) algorithm [@gross2005generic]; Fast AAM [@tzimiropoulos2013optimization]; 2.5D AAM [@martins2013generative]; Active Orientation Models [@tzimiropoulos2014active]\
*Part-based generative deformable models* & Original Active Shape Model (ASM) [@cootes1995active]; Gauss-Newton deformable part model [@tzimiropoulos2014gauss]; Project-out cascaded regression [@tzimiropoulos2015project]; Active pictorial structures [@antonakos2015active]\
**[Discriminative methods]{} &\
*Constrained local models (CLMs)*&\
PCA shape model & Regularized landmark mean-shift [@saragih2011deformable]; Regression voting-based shape model matching [@cootes2012robust]; Robust response map fitting [@asthana2013robust]; Constrained local neural field [@baltrusaitis2013constrained]\
Exemplar shape model & Consensus of exemplar [@belhumeur2011localizing]; Exemplar-based graph matching [@zhou2013exemplar]; Robust Discriminative Hough Voting [@jin2016face]\
Other shape models & Gaussian Process Latent Variable Model [@huang2007component]; Component-based discriminative search [@liang2008face]; Deep face shape model [@wu2015discriminative]\
*Constrained local regression* & Boosted regression and graph model [@valstar2010facial]; Local evidence aggregation for regression [@martinez2013local]; Guided unsupervised learning for model specific models [@jaiswal2013guided]\
*Deformable part models (DPMs)* & Tree structured part model [@zhu2012face]; Structured output SVM [@uvrivcavr2012detector]; Optimized part model [@yu2013pose]; Regressive Tree Structured Model [@hsu2015regressive]\
*Ensemble regression-voting* & Conditional regression forests [@dantone2012real]; Privileged information-based conditional regression forest [@yang2013privileged]; Sieving regression forest votes[@yang2013sieving]; Nonparametric context modeling [@smith2014nonparametric]\
*Cascaded regression*\
Two-level boosted regression & Explicit shape regression [@cao2012face]; Robust cascaded pose regression [@burgos2013robust]; Ensemble of regression trees [@kazemi2014one]; Gaussian process regression trees [@lee2015face];\
Cascaded linear regression & Supervised descent method [@xiong2013supervised]; Multiple hypotheses-based regression [@yan2013learn]; Local binary feature [@ren2014face]; Incremental face alignment [@asthana2014incremental]; Coarse-to-fine shape search [@zhu2015face]\
*Deep neural networks*\
Deep CNNs & Deep convolutional network cascade [@sun2013deep]; Tasks-constrained deep convolutional network [@zhang2014facial]; Deep Cascaded Regression[@lai2015deep]\
Other deep networks & Coarse-to-fine Auto-encoder Networks (CFAN) [@zhang2014coarse]; Deep face shape model [@wu2015discriminative]\
********
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
Classic Constrained Local Models (CLMs) typically refer to the combination of local detector for each facial point and the parametric Point Distribution Model [@cristinacce2006feature; @wang2008enforcing; @saragih2011deformable]. Here we extend the range of CLMs by including some methods based on other shape models (i.e., exemplar-based model [@belhumeur2011localizing]). In particular, we will show that the exemplar-based method [@belhumeur2011localizing] can also be interpreted under the conventional CLM framework.
We note that some deep learning-based systems can also be placed in other categories. For instance, some systems are constructed in a cascade manner [@zhang2014coarse; @lai2015deep; @trigeorgis2016mnemonic], and hence can be naturally categorized as cascaded regression. However, to highlight the increasing important role of deep learning techniques for face alignment, we organize them together for more systematic introduction and summarization.
The problem of face alignment has a long history in computer vision, and a large number of approaches have been proposed to tackle it with varying degrees of success. From an overall perspective, face alignment can be formulated as a problem of searching over a face image for the pre-defined facial points (also called face shape), which typically starts from a coarse initial shape, and proceeds by refining the shape estimate step by step until convergence. During the search process, two different sources of information are typically used: facial appearance and shape information. The latter aims to explicitly model the spatial relations between the locations of facial points to ensure that the estimated facial points can form a valid face shape. Although some methods make no explicit use of the shape information, it is common to combine these two sources of information.
Before describing specific and prominent algorithms, a clear and high-level categorization will help to provide a holistic understanding of the commonality and differences of existing methods in using the appearance and shape information. For this, we follow the basic modeling principles in pattern recognition, and roughly divide existing methods into two categories: *generative* and *discriminative*.
- *Generative methods:* These methods build generative models for both the face shape and appearance. They typically formulate face alignment as an optimization problem to find the shape and appearance parameters that generate an appearance model instance giving best fit to the test face. Note that the facial appearance can be represented either by the whole (warped) face, or by the local image patches centered at the facial points.
- *Discriminative methods:* These methods directly infer the target location from the facial appearance. This is typically done by learning independent local detector or regressor for each facial point and employing a global shape model to regularize their predictions, or by directly learning a vectorial regression function to infer the whole face shape, during which the shape constraint is implicitly encoded.
Table \[tab\_categorization\] summarizes algorithms and representative works for face alignment, where we further divide the generative methods and discriminative methods into several subcategories. A few methods overlap category boundaries, and are discussed at the end of the section where they are introduced. Below, we discuss the motivation and general approach of each category first, and then, give the review of prominent algorithms within each category, discussing their advantages and disadvantages.
Generative methods {#sec:generative_methods}
==================
Typically, faces are modelled as deformable objects which can vary in terms of shape and appearance. Generative methods for face alignment construct parametric models for facial appearance similar to EigenFace [@turk1991face], but differ from EigenFace in that they take into account the deformation of face shape, and build appearance model in a canonical reference frame where the shape variations have been removed. Fitting a generative model aims to find the shape and appearance parameters that can generate a model instance fitting best to the test face.
According to the type of facial representation, generative methods can be further divided into two categories: Active Appearance Models (AAMs) that use the holistic representations, and part-based generative deformable models that use part-based representation.
Active appearance models {#subsec:AAMs}
------------------------
Active Appearance Models (AAMs), proposed by Cootes *et al.* [@cootes2001active], are *linear* statistical models of both the shape and the appearance of the deformable object. They are able to generate a variety of instances by a small number of model parameters, and therefore have been widely used in many computer vision tasks, such as face recognition [@lanitis1997automatic], object tracking [@stegmann2001object] and medical image analysis [@stegmann2003fame]. In the field of face alignment, AAMs are arguably the most well-known family of generative methods that have been extensively studied during the last 15 years [@cootes2001active; @matthews2004active; @gross2005generic; @tzimiropoulos2013optimization].
In the following, we first briefly introduce the basic AAM algorithm including AAM modeling and fitting, then summarize and analysis some recent advances on AAM research, and finally present some discussions about the advantages and disadvantages of AAMs.
### Basic AAM algorithm: modeling and fitting
In the section, we briefly introduce the basic AAM algorithm: modeling and fitting. Note that we do not intend to give a very comprehensive and detailed overview of the basic AAM algorithm, and refer the reader to recent surveys [@gao2010review; @wang2014facial] for more details.
***AAM modeling.*** An AAM is defined by three components, i.e., *shape model*, *appearance model*, and *motion model*. The shape model, which is coined Point Distribution Model (PDM) [@cootes1992active], is built from a collection of manually annotated facial points $\mathbf{s} = (\mathbf{x}_1^T,...,\mathbf{x}_N^T)^T$ describing the face shape, where $\mathbf{x}_i = (x_i,y_i)$ is the 2-D location of the $i$th point. To learn the shape model, the training face shapes are normalized with respect to a global similarity transform (typically using Procrustes Analysis [@gower1975generalized]) and Principal Component Analysis (PCA) is applied to obtained a set of linear shape bases. The shape model can be mathematically expressed as: $$\label{eq:PCA_shape_model}
\mathbf{s}(\mathbf{p}) = \mathbf{s_0} + \mathbf{S}\mathbf{p},$$ where $\mathbf{s}_0 \in \mathcal{R}^{\{2N,1\}}$ is the mean shape, $\mathbf{S} \in \mathcal{R}^{\{2N,n\}}$ and $\mathbf{p} \in \mathcal{R}^{n}$ is the shape eigenvectors and parameters. Furthermore, this shape model need to be composed with a 2D global similarity transform, in order to position a particular shape model instance arbitrarily on the image frame. For this, using the re-orthonormalization procedure described in [@matthews2004active], the final expression for the shape model can be compactly written using \[eq:PCA\_shape\_model\] by appending $\mathbf{S}$ with 4 similarity eigenvectors.
The appearance model is obtained by warping the training faces onto a common reference frame (typically defined by the mean shape), and applying PCA onto the warped appearances. Mathematically, the texture model is defined as follows: $$\label{eq:appearance_model}
\mathbf{A}(\mathbf{c}) = \mathbf{a_0} + \mathbf{A}\mathbf{c},$$ where $\mathbf{a}_0 \in \mathcal{R}^{\{F,1\}}$ is the mean appearance, $\mathbf{A}\in \mathcal{R}^{\{F,m\}}$ and $\mathbf{c} \in \mathcal{R}^{\{m,1\}}$ is the appearance eigenvectors and parameters respectively.
To produce the shape-free textures, the motion model plays a role as a bridge between the image frame and the canonical reference frame. Typically, it is a warp function $\mathcal{W}$ that defines how, given a shape, the image should be warped into a canonical reference frame. Popular motion models include piece-wise affine warp [@matthews2004active; @tzimiropoulos2013optimization] and Thin-Plate Splines warp [@baker2001equivalence].
***AAM fitting.*** Given an test image $\mathbf{I}$, AAM fitting aims to find the optimal parameters $\mathbf{p}$ and $\mathbf{c}$ so that the synthesized appearance model instance gives best fit to the test image in the reference frame. Formally, let $\mathbf{I}[\mathbf{p}] = \mathbf{I}(\mathcal{W}(\mathbf{p}))$ denote the vectorized version of the warped test image, then AAM fitting can be formulated as the following optimization problem, $$\label{eq:AAM_objective}
\mathrm{arg} \mathop{\mathrm{min}}_{\mathbf{p},\mathbf{c}} ||\mathbf{I}[\mathbf{p}] - \mathbf{a}_0 - \mathbf{A}\mathbf{c}||^2.$$ Solving \[eq:AAM\_objective\] is an iterative process that at each iteration an update of the current model parameters is estimated. In general, there are two main approaches for AAM fitting.
The first approach is to assume a *fixed* relationship between the residual image and the model parameter increments, and learn it via *regression*. For example, in the original AAM paper [@cootes1998active], this relationship is assumed linear and learned by linear regression, while in [@saragih2007nonlinear] a nonlinear repressor is learned via boosting. However, because the basic assumption that the regression functions are $\emph{fixed}$ is incorrect [@matthews2004active], the regression-based fitting strategies are efficient but approximate.
The second approach is to employ a standard gradient descend algorithm. But unfortunately, standard gradient descend algorithms are inefficient when applied to AAMs fitting. Matthews *et al.* [@matthews2004active] addressed this problem with a so-called project-out inverse compositional algorithm (POIC) algorithm, which decouples shape from appearance by projecting out appearance variation, and estimates the warp update in the model coordinate frame and then compose it inversely to the current warp. Although POIC is extremely fast, it is also known to have a small convergence radius, i.e., convergence is especially bad when training and testing images differ strongly [@gross2005generic]. Different from POIC that projects out the appearance variations, the simultaneous inverse compositional (SIC) algorithm [@gross2003lucas] optimizes the shape parameter and appearance parameter simultaneously under the inverse compositional framework. SIC has been shown to be more robust than POIC for generic AAM fitting, but the computational cost is much higher [@gross2005generic]. Besides SIC, another accurate AAM fitting algorithm is the Alternating Inverse Compositional (AIC) algorithm [@papandreou2008adaptive], which solves two separate minimization problems, one for the shape and one for the appearance optimal parameters, in an alternating fashion.
### Recent advances on AAMs
Recently, some extensions and improvements of AAMs have been proposed to make this classic algorithm better adapted to the task of face alignment *in-the-wild*. In general, recent advances on AAMs mainly focus on three aspects: (1) unconstrained training data [@tzimiropoulos2013optimization], (2) robust image representations [@tzimiropoulos2012generic; @antonakos2014hog] and (3) robust and fast fitting strategies [@tzimiropoulos2012generic; @tzimiropoulos2013optimization].
***Unconstrained training data.*** Although some AAM fitting algorithms (e.g., the Simultaneous Inverse Compositional (SIC) algorithm [@gross2003lucas]) are known to perform well on constrained face databases, their performance has not been assessed on in-the-wild databases until recently. Tzimiropoulos *et al.* [@tzimiropoulos2013optimization] showed that, when trained in-the-wild, AAMs perform notably well and in some cases comparably with current state-of-the-art methods, without using sophisticated shape priors, robust features or robust norms. Fig. \[fig:AAM\_in-the-wild\] shows a fitting case and the reconstruction of the image, produced by AAM built in the wild [@tzimiropoulos2013optimization].
***Robust image representations.*** Typically, AAMs use the pixel-based image representation that is sensitive to global lighting [@cootes1998active; @matthews2004active; @tzimiropoulos2013optimization], and a natural way to improve the robustness of AAMs is to use the feature-based representation. In general, robust image features, such as HOG [@dalal2005histograms], SIFT [@lowe2004distinctive] or SURF [@bay2008speeded] that describe distinctive and important image characteristics, can generalize better to unseen images. Some recent works have confirmed the robustness of the appearance model built upon feature-based representation [@tzimiropoulos2012generic; @tzimiropoulos2014active; @antonakos2015feature].
***Robust and fast fitting strategies.*** It is widely acknowledged that the Project-out Inverse Compositional (POIC) algorithm is fast but has a small convergence radius, while the Simultaneous Inverse Compositional (SIC) algorithm is accurate but very slow. Due to this, some recent advances on AAMs have focused on robust and fast fitting algorithms. For example, it was found in [@tzimiropoulos2012generic; @antonakos2015feature] that the Alternating Inverse Compositional (AIC) algorithm [@papandreou2008adaptive] performs well for generic AAM fitting. Although AIC is slower than the project-out algorithm, it is still very fast probably allowing a real-time implementation. Furthermore, by using a standard results from optimization theory, Tzimiropoulos *et al.* [@tzimiropoulos2013optimization] dramatically reduced the dominant cost for both SIC and the standard Lukas-Kanade algorithm, making both algorithms very attractive speed-wise for practical AAM systems.
![(a) A face image from the test set of LFPW [@belhumeur2011localizing], with facial points detected by the Fast-SIC algorithm proposed in [@tzimiropoulos2013optimization]. (b) Reconstruction of the image from the appearance subspace. (Fig. 2 in [@tzimiropoulos2013optimization])[]{data-label="fig:AAM_in-the-wild"}](images/representative_methods/AAM_in-the-wild.pdf){width="38.00000%"}
### Discussion
We have described the basic AAM algorithm and recent advances on AAMs. Despite the popularity and success, AMMs have been traditionally criticized for the limited representational power of their holistic representation, especially when used in wild conditions. However, recent works on AAMs [@tzimiropoulos2012generic; @antonakos2014hog; @lucey2013fourier] suggest that this limitation might have been over-stressed in the literature and that AAMs can produce highly accurate results if appropriate training data [@tzimiropoulos2013optimization], image representations [@tzimiropoulos2012generic; @antonakos2014hog] and fitting strategies [@tzimiropoulos2012generic; @tzimiropoulos2013optimization] are employed.
Despite this, AAMs are still considered to have the following drawbacks: (1) Since the holistic appearance model is used, partial occlusions cannot be easily handled. (2) For the appearance model built *in-the-wild*, the dimension of appearance parameter is very high, which makes AAMs difficult to optimize and likely to converge to undesirable local minima. One possible way to overcome these drawbacks is to use part-based representations, due to the observation that local features are generally not as sensitive as global features to lighting and occlusion. In the following section, we turn to part-based generative methods.
Part-based generative deformable models
---------------------------------------
Part-based generative methods build generative appearance models for facial parts, typically with a shape model to govern the deformations of the face shapes. In this paper, we do not distinguish the specific form of the shape model, and refer to all part-based generative methods collectively as *part-based generative deformable models*.
In general, there are two approaches to construct generative part models. The first is to construct individual appearance model for each facial part. A notable example is the well-known original Active Shape Models [@cootes1992active; @cootes1995active] that combine the generative appearance model for each facial part and the Point Distribution Model for global shapes. However, we note that a more natural and popular way is to model individual facial part is the *discriminatively* trained local detector [@cristinacce2007boosted; @saragih2011deformable; @cootes2012robust; @asthana2013robust], as adopted by a very successful family of methods coined Constrained Local Models (CLMs) [@saragih2011deformable; @asthana2013robust]. Actually, ASMs can be regarded as the predecessors of CLMs, because they are similar in both the models and the fitting process. Therefore, we refer the reader to Section \[subsubsec:CLMs\] for more details about ASMs under the CLM framework.
The second approach is to construct generative models for all facial parts simultaneously. For example, one can concatenate all facial parts (image patches) to form a part-based representation for the whole face, and then build generative appearance model for it. The Gauss-Newton Deformable Part Model (GN-DPM) [@tzimiropoulos2014gauss] has explored this idea, and build linear statistical model for both the concatenated facial parts and the shape using PCA. Benefiting from the part-based representation, the motion model of GN-DPM degenerates to similarity transformation, rather than the affine warp of AAMs. In the fitting phase, GN-DPM formulate and solve the non-linear least squares optimization problem similar to AAMs [@matthews2004active; @tzimiropoulos2013optimization]. The part-based appearance model along with a global shape model is optimized by the fast SIC algorithm [@tzimiropoulos2013optimization] in a Gauss-Newton fashion. Extensive experiments on wild face databases [@belhumeur2013localizing; @le2012interactive; @zhu2012face] demonstrate that the part-based GN-DPM outperforms AAMs by a large margin.
While GN-DPM employs the inverse compositional fitting algorithm, Tzimiropoulos *et al.* [@tzimiropoulos2015project] consider the forward algorithm for the non-linear least square optimization problem akin to that of GN-DPM. Although analytic gradient decent method is employed in [@tzimiropoulos2015project], it is only used for the derivation of the learning and fitting basis of the proposed Project-Out Cascaded Regression (PO-CR) method. In particular, PO-CR learns from data a sequence of averaged Jacobians and descent directions via regression in a subspace orthogonal to the facial appearance variation. Apart from the PCA-based appearance model in GN-DPM and PO-CR, Antonakos *et al.* [@antonakos2015active] propose to model the appearance of facial parts using multiple pairwise distributions based on the edges of a graph (GMRF), and show that this outperforms the commonly used PCA model under an inverse Gauss-Newton optimization framework.
Compared to AAMs, the part-based generative deformable models mainly have the advantages from part-based representation, i.e., more robust to global lighting and occlusion in wild conditions. As shown in [@tzimiropoulos2014gauss], part-based generative models may have the same representational power of AAMs, but are easier to optimize. That is, when the initial shape is far from the ground truth, part-based generative deformable models are more likely to get converged to a good solution, although the formulation is non-convex by nature.
Summary and discussion
----------------------
We have reviewed generative methods for face alignment in two categories, i.e., Active Appearance Models (AAMs) that use the holistic representation and the part-based generative deformable models that use the part-based representation. In general, the fitting result of a generative appearance model to a test image typically depends on two factors: (1) the representational power of the model, and (2) the difficulty in optimizing the model. As investigated in [@tzimiropoulos2014gauss], when trained in-the-wild, both AAMs and part-based generative deformable models can reconstruct the appearance of an unseen image well, but the part-based generative deformable models are considered to be easier to optimize than AAMs. Furthermore, recent results show that if unconstrained training data [@tzimiropoulos2013optimization], robust image representations [@tzimiropoulos2012generic; @antonakos2014hog] and appropriate fitting strategies [@tzimiropoulos2012generic; @tzimiropoulos2013optimization; @tzimiropoulos2014gauss; @tzimiropoulos2015project] are employed, generative methods can produce a very high degree of fitting accuracy for face alignment *in-the-wild*. These results suggest that the limitations of generative methods, especially the AAMs, might have been over-stressed in the literature. In addition, generative methods typically have the advantage of requiring fewer training examples than the discriminative methods to perform well [@antonakos2015active].
However, with recent development of unconstrained facial databases with an abundance of annotated facial data captured, the discriminative methods, which are capable of effectively leveraging large bodies of training data, are arguably now playing a more and more prominent role in face alignment *in-the-wild*. Next, we will turn to discriminative methods.
Discriminative methods {#sec:discriminative_methods}
======================
[p[0.21]{}p[0.22]{}p[0.16]{}p[0.33]{}]{} & **[Appearance model]{} & **[Shape model]{} & **[Highlights of the method]{}\
*Constrained local models* & Independently trained local detector that computes a pseudo probability of the target point occurring at a particular position. & Point Distribution Moldel; Exemplar model, etc. & The local detectors are first correlated with the image to yield a filter response for each facial point, and then shape optimization is performed over these filter responses.\
\
*Constrained local regression* & Independently trained local regressor that predicts a distance vector relating to a patch location. & Markov Random Fields to model the relations between relative positions of pairs of points. & Graph model is used to constrain the search space of local regressors by exploiting the constellations that facial points can form.\
\
*Deformable part models* & Part-based appearance model that computes the appearance evidence for placing a template for a facial part. & Tree-structured models that are easier to optimize than dense graph structures. & All parameters of the appearance model and shape model are discriminatively learned in a max-margin structured prediction framework; efficient dynamic programming algorithms can be used to find globally optimal solutions.\
\
*Ensemble regression-voting* & Image patches to cast votes for all facial points relating to the patch centers;******
Local appearance features centered at facial points. & Implicit shape constraint that is naturally encoded into the multi-output function (e.g., regression tree). & Votes from different regions are ensembled to form a robust prediction for the face shape.\
\
*Cascaded regression* & Shape-indexed feature that is related to current shape estimate (e.g., concatenated image patches centered at the facial points). & Implicit shape constraint that is naturally encoded into the regressor in a cascaded learning framework. & Cascaded regression typically starts from an initial shape (e.g., mean shape), and refines the holistic shape through sequentially trained regressors.\
\
*Deep neural networks* & Whole face region that is typically used to estimate the whole face shape jointly;
Shape-indexed feature. & Implicit shape constraint that is encoded into the networks since all facial points are predicted simultaneously. & Deep network is a good choice to model the nonlinear relationship between the facial appearance and the shape update.
Among others, deep CNNs have the capacity to learn highly discriminative features for face alignment.\
Constrained Local Models (CLMs) typically employ a parametric (PCA-based) shape model [@saragih2011deformable], but we will show that the exemplar-based method [@belhumeur2011localizing] can also be derived from the CLM framework. Furthermore, we extend the range of CLMs by including some methods that combine independently local detector and other face shape model [@huang2007component; @liang2008face; @wu2015discriminative].
Some deep network-based systems follow the cascaded regression framework, and use the shape-indexed feature [@zhang2014coarse].
Discriminative face alignment methods seek to learn a (or a set of) discriminative function that directly maps the facial appearance to the target facial points. In general, there are two main lines of research for discriminative methods. The first line is to follow the “divide and conquer" strategy by learning discriminative local appearance model (detector or regressor) for each facial point, and a shape model to impose global constraints on these local models. This line can be further subdivided into three classes: (1) *Constrained Local Models (CLMs)* that learn independent local detector for each facial point, with a shape model to regularize the detection responses of these local detectors. (2) *constrained local regression* methods that learn independent local regressor for each point and use a graph model to guide the search of these local regressors, and (3) *deformable part models* that learn the local appearance model and the tree structured shape model jointly in a discriminative framework.
The second line is to directly learn a vectorial regression function to infer the *whole* face shape, during which the shape constraint is implicitly encoded. This line can also be further subdivided into three classes: (1) *ensemble regression-voting* methods that cast votes for all facial points from local regions via regression, and ensemble the votes from different regions to form a robust prediction, (2) *cascaded regression* methods that learn a vectorial regression function in a cascade manner to estimate the face shape stage-by-stage, and (3) *deep neural networks* that employ deep convolutional networks [@sun2013deep; @zhang2014facial] or auto-encoder networks [@zhang2014coarse] to model the nonlinear relationship between the facial appearance and the shape update.
Table \[tab\_discriminative\_description\] gives a overview of the six classes of discriminative methods in our taxonomy, where the appearance model, shape model and highlights of them are listed respectively to show the differences and relations between them.
Constrained local models {#subsubsec:CLMs}
------------------------
Constrained Local Models (CLMs), which can date back to the seminal work of Active Shape Model (ASM) [@cootes1995active], are a relatively mature approach for face alignment [@cristinacce2006feature; @saragih2011deformable; @cootes2012robust; @asthana2013robust; @baltrusaitis2013constrained]. In the training phase, CLMs learn independent local detector for each facial point, and a prior shape model to characterize the deformation of face shapes. In testing, face alignment is typically formulated as an optimization problem to find the best fit of the shape model to the test image. We classify CLMs as the discriminative methods because of the discriminative nature of usual local detectors.
While the seminal work of [@saragih2011deformable] unifies various CLM approaches in a probabilistic framework, it only focuses on the CLMs using the Point Distribution Model (PDM). However, we note that some methods using other shape model (i.e., the exemplar shape model [@belhumeur2011localizing]) are also close to [@saragih2011deformable] in methodology. Hence, in this paper we refer to those methods combining independent local detector and any kind of shape model collectively as *Constrained Local Models*[^1].
In the following, we will first briefly introduce the basic Point Distribution Model (PDM) based CLM algorithm including modeling and fitting, then summarize and analysis recent advances on CLMs in handling unconstrained challenges. In particular, we will show that exemplar-based method [@belhumeur2011localizing] can also be interpreted under the conventional CLM framework [@saragih2011deformable]. Finally, we discuss the advantages and disadvantages of CLMs.
### Basic CLM algorithm: modeling and fitting
In this section, we will briefly describe the basic CLM algorithm building upon the Point Distribution Model (PDM), which has two procedures: modeling and fitting.
***CLM modeling.*** A CLM consists of two important components: *local detector* for each facial point, and the *shape model* that captures the deformations of valid face shapes. The task of local detector is to compute a pseudo probability (likelihood) that the target point occurs at a particular position. Existing local detectors can be broadly categorized into three groups.
- *Generative approach:* Generative approaches can be use to model local image patches centered at the annotated facial points. For example, [@cootes1992active; @cootes1993active] assume that the local appearance is multivariate Gaussian distributed, and use the Mahalanobis distance as the fitting response for a new image patch.
- *Discriminative classifier:* Discriminative classifier-based approach learns a binary classifier for each point with annotated image patches to discriminate whether the target point is aligned or not when testing. To cast various CLM fitting strategies in a unified probabilistic framework, the output of these classifiers are typically transformed into pseudo probabilities. Different types of classifiers have been exploited in literature, e.g., logistic regression [@saragih2011deformable], SVM [@belhumeur2011localizing; @asthana2013robust], and Local Neural Field (LNF) [@baltrusaitis2013constrained].
- *Regression-voting approach:* The regression-voting approach casts votes for the target point from a nearby region, then compute the pseudo probabilities by accumulating votes from different regions [@cristinacce2007boosted; @cootes2012robust]. The regression-voting approach has the potential to be more efficient since a locally exhaustive search is avoided.
Due to the local patch support and large variations in training, the local detectors are typically imperfect, and the correct location will not always be at the location with the highest detection response. To address this drawback, a global shape model is typically employed to regularize the detection of these local detectors. For this, conventional CLMs use the Point Distribution Model (PDM) that simply models the normalized face shapes as multivariate Gaussian and approximates them using PCA (see Equation \[eq:PCA\_shape\_model\]).
***CLM fitting.*** Overall, give an image $\mathbf{I}$, the goal of PDM-based CLMs is to find the optimal shape parameter $\mathbf{p}$ that maximizes the probability of its points corresponding to consistent locations of the facial features. By assuming that the local search of each facial point is conditionally independent, the fitting objective of PDM-based CLMs can be written as: $$\label{eq:PCA_CLM_goal}
\begin{split}
\mathbf{p^*} &= \mathrm{\mathop{arg \, max}_{\mathbf{p}}} \; p(\mathbf{p}|\{l_i\!=\!1\}_{i=1}^N, \mathbf{I}) \\
&= \mathrm{\mathop{arg \, max}_{\mathbf{p}}} \; p(\mathbf{p})\prod_{i=1}^N p(l_i\!=\!1|\mathbf{x}_i(\mathbf{p}),\mathbf{I}),
\end{split}$$ where $\mathbf{x}_i(\mathbf{p})$ is the location of the $i$th point generated by the shape model, $l_i \in \{1,-1\}$ is a discrete random variable denoting whether the $i$th facial point is aligned or not, and $p(\mathbf{p})$ is the prior distribution of $\mathbf{p}$ that can be estimated from the training data.
![ Illustration of PDM-based CLM fitting and its two components: (1) an exhaustive local search for feature locations to get the response maps $\{p(l_i = 1|\mathbf{x}, \mathbf{I})\}_{i=1}^N$ and (2) an optimization strategy to maximize the responses of the PDM constrained facial points. (Fig. 2 in [@saragih2011deformable])[]{data-label="fig:PDM_CLM_fitting"}](images/representative_methods/PDM_CLM_fitting.pdf){width="48.00000%"}
CLM fitting based on \[eq:PCA\_CLM\_goal\] is an iterative process (see Fig. \[fig:PDM\_CLM\_fitting\]) that entails (1) convolving the local detectors with the image to generate response maps, and (2) performing a global shape optimization procedure over these response maps. To make optimization efficient and numerically stable, a common choice of existing optimization strategies is to replace the true response maps with some approximate forms and then perform Guass-Newton optimization over them instead of the original response maps.
[p[0.37]{}p[0.35]{}]{} & Approximation of response map\
Isotropic Gaussian estimator [@cootes1995active] & $\mathcal{N}(\mathbf{x}_i(\mathbf{p});\bm{\mu}_i,\sigma_i^2\mathbf{I}^{(e)})$\
Anisotropic Gaussian estimator [@wang2008enforcing] & $\mathcal{N}(\mathbf{x}_i(\mathbf{p});\bm{\mu}_i,\bm{\Sigma}_i)$\
Gaussian mixture model [@gu2008generative] & $\sum_{k=1}^{K_i}\pi_{ik}\mathcal{N}(\mathbf{x}_i(\mathbf{p});\bm{\mu}_{ik},\bm{\Sigma}_{ik})$\
Gaussian kernel estimation [@saragih2011deformable] & $\sum_{\mathbf{y}_j \in \bm{\Psi}_{\mathbf{x}_i}}\pi_{\mathbf{y}_j}\mathcal{N}(\mathbf{x}_i(\mathbf{p});\mathbf{y}_j,\rho^2\mathbf{I}^{(e)})$\
The seminal framework of [@saragih2011deformable] unifies various approximation strategies for the true response maps. As listed in Table \[tab\_responsemap\_approximation\], they are (1) the isotropic Gaussian estimators used by original ASMs [@cootes1992active; @cootes1995active], where $\bm{\mu}_i$ is the the location of the maximum filter response within the $i$th response map, and $\sigma_i^{-2}$ is the detection confidence over peak response coordinate, (2) a full covariance anisotropic Gaussian estimators used in [@wang2008enforcing], where $\bm{\Sigma}_i$ is the anisotropic covariance matrix of Gaussian distribution, (3) Gaussian mixture model (GMM) used in [@gu2008generative], where $K_i$ denotes the number of modes and $\{\pi_{ik}\}_{k=1}^{K_i}$ are the mixing coefficients for the GMM of the $i$th point, and (4) a homoscedastic isotropic Gaussian kernel estimation (KDE) used by [@saragih2011deformable], where $\pi_{\mathbf{y}_j} = p(l_i=1|\mathbf{y}_j,\mathbf{I})$ denotes the likelihood that the $i$th point is aligned at location $\mathbf{y}_j$, and $\rho^2$ denotes the variance of the noise on facial point locations, $\mathbf{I}^{(e)}$ is the identity matrix. Among them, the nonparametric Gaussian kernel estimation (KDE) method [@saragih2011deformable] is considered to achieve a good tradeoff between representation power and the computational complexity. This method is known as Regularized Landmark Mean-Shift (RLMS) fitting, as the resulting update equations based on this nonparametric approximation are reminiscent of the well known mean-shift [@fukunaga1975estimation] over the facial point but with regularization imposed by the Point Distribution Model.
Due to its effectiveness and efficiency, the RLMS method [@saragih2011deformable] has been extensively investigated. For example, Baltruvsaitis *et al.* [@baltruvsaitis20123d] explored the information of depth images, and extend the RLMS [@saragih2011deformable] algorithm to a 3D vision. Unlike aforementioned approximations to response maps, [@asthana2013robust] proposes a novel discriminative regression based approach to directly estimate the parameter update, and results in significant performance improvement.
### Recent advances on CLMs {#subsec:recent_advances_CLMs}
Recently, some improvements of the conventional CLMs have been proposed to better handle various challenges in-the-wild. In general, recent advances on CLMs mainly focus on three aspects: (1) better local detectors, (2) discriminative fitting, and (3) other shape models.
***Better local detectors.*** Conventional CLMs typically use logistic regression [@saragih2011deformable] or SVM [@belhumeur2011localizing; @asthana2013robust] to train local detector, which however is plagued by the problem of ambiguity, especially on the wild databases. To mitigate this ambiguity, some advanced local detectors have been proposed, such as the Minimum Output Sum of Squared Errors (MOSSE) filters [@martins2014non] and the Local Neural Field (LNF) patch expert, which are able to capture more complex information and exploit spatial relationships between pixels, and hence can achieve better detection results.
***Discriminative fitting.*** It is widely acknowledged that the formulation based on CLMs is non-convex, and in general prone to local minima. As an alternative, Asthana *et al.* [@asthana2013robust] proposed a novel Discriminative Response Map Fitting (DRMF) method for the CLM fitting that outperforms the RLMS fitting method [@saragih2011deformable] in wild databases. We conjecture that the robustness of DRMF mainly stems from the discriminative training process, which can effectively leverage large bodies of training data.
***Other shape models.*** One problem with the Point Distribution Model (PDM) is that its the model flexibility is heuristically determined by PCA dimension. To overcome this drawback, some other shape models are proposed to combine with the local detectors for face alignment [@huang2007component; @belhumeur2011localizing; @wu2015discriminative]. In particular, we will show that the exemplar-based method [@belhumeur2011localizing] can be derived and well interpreted under the conventional CLM framework [@saragih2011deformable].
The exemplar-based method [@belhumeur2011localizing] assumes that the face shape $\mathbf{s} = (\mathbf{x}_1,...,\mathbf{x}_N)^T$ in the test image is generated by one of the transformed exemplar shapes (global models). Let $\mathbf{s}_{k,t}$ ($k=1,...,D$) denote locations of all facial points in the $k$th of the $D$ exemplars that transformed by some similarity transformation $t$, and let $\mathbf{x}_{i,k,t}$ denote location of the $i$th facial point of the transformed exemplar $\mathbf{s}_{k,t}$. By assuming that conditioned on the global model $\mathbf{s}_{k,t}$, the location of each facial point $\mathbf{x}_i$ is conditionally independent of one another, the exemplar-based shape model $p(\mathbf{s})$ can be written as follows: $$\label{eq:exemplar_shape_model}
\begin{split}
p(\mathbf{s}) & = \sum_{k=1}^D \int_{t \in{T}}p(\mathbf{s},\mathbf{s}_{k,t})dt \\
& = \sum_{k=1}^D \int_{t\in{T}} \prod_{i=1}^N p(\mathbf{x}_i|\mathbf{x}_{i,k,t})p(\mathbf{s}_{k,t})dt,
\end{split}$$ where $p(\mathbf{x}_i|\mathbf{x}_{i,k,t})$ is modeled as a Gaussian distribution centered at $\mathbf{x}_{i,k,t}$, and the prior of the global model $p(\mathbf{s}_{k,t})$ is assumed as an uniform distribution. Then, by replacing the shape model $p(\mathbf{p})$ in conventional CLM framework \[eq:PCA\_CLM\_goal\] with above exemplar-based model $p(\mathbf{s})$, we derive the objective function of [@belhumeur2011localizing] (difference in notations) as follows: $$\label{eq:exemplar_goal}
\mathbf{s}^* \! =\! \mathrm{\mathop{arg \, max}_{\mathbf{s}}} \, \sum_{k=1}^D \! \int_{t\in{T}} \prod_{i=1}^N p(\mathbf{x}_i|\mathbf{x}_{i,k,t})p(l_i\!=\!1|\mathbf{x}_i,\mathbf{I})dt.$$ This function is optimized by employing RANSAC to sample global models. Due to the use of RANSAC, the exemplar-based method [@belhumeur2011localizing] has two advantages over conventional CLMs: (1) independent of shape initialization, and (2) robust to partial occlusion, and achieves excellent performance on the wild LFPW database [@belhumeur2011localizing].
The global models in [@belhumeur2011localizing] are scored and selected by the global likelihood, i.e., multiplying the detection response of each local detector. However, as pointed by Jin *et al.* [@jin2016face], this global likelihood score function ignores the difference between local detectors, while in fact, an eye detector is typically more reliable than a chin detector. In [@jin2016face], a discriminatively trained score function is proposed to evaluate the goodness of a global model, which weighs the importance of different local detectors. Furthermore, an efficient pipeline was proposed in [@jin2016face] to alleviate the effect of inaccurate anchor points for generating global models.
### Discussion
We have reviewed the basic CLM algorithm and recent advances. In general, CLMs are considered to be more robust to partial occlusion and global lighting than the holistic approaches (e.g., AAMs) [@saragih2011deformable], due to their part-based modeling. However, the local detectors of CLMs are imperfect and have been shown to result in detection ambiguities in testing. Furthermore, since the global shape optimization is performed on the response maps, the detection ambiguities may lead to performance bottleneck, when facing various challenges in unconstrained conditions.
Another disadvantage of CLMs is that they perform an expensive locally exhaustive search for each facial point. One way to reduce the computational cost is to use a displacement expert (local regressor, i.e., estimate the relative position of the target point with respect to the given patch. We will turn to this topic in the next section.
Constrained local regression
----------------------------
Besides the CLMs, another local model-based approach is to train independent local *regressor* for each point, and employ a global shape model to restrict the search of these local regressors to anthropomorphically consistent regions [@valstar2010facial; @martinez2013local]. Since this idea is similar to CLMs, we refer to this approach as *constrained local regression*.
A representative work of this group is the Boosted Regression coupled with Markov Netwroks [@valstar2010facial] (BoRMaN) method, which iteratively uses Support Vector Regressoin (SVR) to provide an initial prediction for all points, and then applies the Markov Network to ensure that the new locations sampled to apply the local regressors are from correct point constellations. BoRMaN let each node in the graph associated to a spatial relation between two points and define pairwise relations between nodes, which allows a representation that is invariant to in-plane rotations, scale changes and translations. Essentially, BoRMaN performs an iterative sequential refinement of the estimate, where the previous target estimate becomes the test location at the next iteration. Martinez *et al.* [@martinez2013local] argue that this sequential estimation approach has a series of drawbacks, for example, sensitive to the starting point and any errors in the estimation process. To improve the robustness of BoRMaN, [@martinez2013local] propose to detect the target location by aggregating the estimates obtained from stochastically selected local appearance information into a single robust prediction, and refer to their algorithm as Local Evidence Aggregated Regression (LEAR).
The main advantage of constrained local regression approach is that combing local regressors with MRF may drastically reduce the time needed to search for point location, while its disadvantages are: (1) similar to CLMs, its performance is limited by the detection ambiguities of the independently trained local regressors, and (2) globally optimizing MRF is intractable. An alternative choice to the graph-based MRF are the tree-structured models, which are also effective to capture global elastic deformation, but easier to optimize than MRF.
Deformable part models {#subsec:DPMs}
----------------------
The tree-structured models are a natural and effective choice to model deformable objects [@yang2013articulated; @zhu2012face], which benefit from the existence of an efficient dynamic programming algorithms [@felzenszwalb2005pictorial] for finding globally optimal solutions. Actually, discriminatively trained tree-structured models have been successfully explored in many computer vision tasks, such as object detection [@felzenszwalb2010object], human pose estimation [@yang2013articulated], and recently in face alignment [@zhu2012face; @uvrivcavr2012detector; @hsu2015regressive]. We follow the nomenclature of [@felzenszwalb2010object] and refer to them collectively as *deformable part models* (DPMs).
The main challenges of applying tree-structured model for face alignment may lie in the fact that a single tree-structured pictorial structure, perhaps, is insufficient to capture various shape deformations due to viewpoint. This problem is addressed by the seminal work of Zhu *et al.* [@zhu2012face], with a unified framework for face detection, pose estimation and face alignment. They modeled every facial point as a part and used mixtures of trees to capture the global topological changes due to viewpoint; a part will only be visible in certain mixtures/views. Formally, let $T_m = (\mathcal{V}_m,\mathcal{E}_m)$ be a linearly-parameterized, tree-structured pictorial structure for the $m$th mixture. Then, given image $\mathbf{I}$ and a face shape $\mathbf{s} = (\mathbf{x}_1,...,\mathbf{x}_N)^T$, the tree structured part model of view $m$ scores $\mathbf{s}$ as: $$\label{eq:tree_structured_part_model}
\begin{split}
\mathcal{S}(\mathbf{I},\mathbf{s},m) &= \mathrm{App}_m(\mathbf{I},\mathbf{s}) + \mathrm{Shape}_m(\mathbf{s}) + \alpha^m \\
\mathrm{App}_m(\mathbf{I},\mathbf{s}) &= \sum_{i \in \mathcal{V}_m} \mathbf{w}_i^m {\bm\cdot} \phi(\mathbf{I},\mathbf{x}_i) \\
\mathrm{Shape}_m(\mathbf{s}) &= \sum_{ij \in \mathcal{E}_m} \! a_{ij}^{m}dx^2 \!+\! b_{ij}^{m}dx \!+\! c_{ij}^{m}dy^2 \!+\! d_{ij}^{m}dy,
\end{split}$$ where $\mathrm{App}_m(\mathbf{I},\mathbf{s})$ sums the appearance evidence at each part in $\mathbf{s}$, $\mathrm{Shape}_m(\mathbf{s})$ scores the mixture-specific spatial arrangement of $\mathbf{s}$, and $\alpha^m$ is a scalar bias associated with view point mixture $m$. Since parts may look consistent across some changes in viewpoint, [@zhu2012face] allows different mixtures to share part templates to reduce the computational complexity.
To learn above mixtures of tree structured part models, the Chow-Liu algorithm [@chow1968approximating] is first used to find the maximum likelihood tree structure that best explains the face shape for a given mixture. Then, for each view, all the model parameters in Eq. \[eq:tree\_structured\_part\_model\] is discriminatively learned in a max-margin structured prediction framework. In the testing phase, the input image is scored by all tree structures $T_m = (\mathcal{V}_m,\mathcal{E}_m)$ respectively, and the globally optimal shape $\mathbf{s}$ is efficiently solved with dynamic programming algorithm [@felzenszwalb2005pictorial].
Due to its simplicity and effectiveness, the tree structured part model [@zhu2012face] has been extensively investigated and improved for face alignment. U[ř]{}i[č]{}[á]{}[ř]{} *et al.* [@uvrivcavr2016multi] argue that the learning algorithm of [@zhu2012face] is a variant of a two-class Support Vector Machines, which optimizes the detection rate of resulting face detector while the facial point locations serve only as latent variables not appearing in the loss function. In contrast, U[ř]{}i[č]{}[á]{}[ř]{} *et al.* [@uvrivcavr2016multi] directly optimizes the average face alignment error with a novel objective function using the Structured Output SVMs algoirthm, which leads to a significant improvement in alignment accuracy. Yu *et al.* [@yu2013pose] presented a two-stage cascaded deformable shape model for face alignment, where a group sparse learning method is proposed to automatically select the optimized anchor points to achieve robust initialization based on the part mixture model of [@zhu2012face]. Hsu *et al.* [@hsu2015regressive] proposed to improve the run-time speed and localization accuracy of [@zhu2012face] with the Regressive Tree Structure Model (RTSM), where the tree structured model is applied on images with increasing resolution.
In general, the tree structured part model is effective at capturing global elastic deformation, while being easy to optimize unlike dense graph structure. Furthermore, it provide an unified framework to solve three tasks, namely face detection, face alignment and pose estimation, which is very appealing in automatic face analysis. However, its sluggish runtime impedes the potential for real-time facial point tracking; and perhaps due to the fact that the tree-based shape models allow for the non-face like structures to occur frequently, the performance of the tree structured part model [@zhu2012face] is reported to be slightly inferior to that of the CLMs [@saragih2011deformable; @asthana2013robust].
*A common limitation of above part-based discriminative methods (i.e., CLMs, constrained local regression, and DPMs), however, is that their performance is greatly constrained by the ambiguity of the local appearance models*. To break this bottleneck, many researchers have proposed to jointly estimate the whole face shape from the image, as described in the following sections.
Ensemble regression-voting
--------------------------
Apart from above local appearance model-based methods, another main stream of discriminative methods is to jointly estimate the whole face shape from image, during which the shape constraint is implicitly exploited. A simple way for this is to cast votes for the face shape from image patches via regression. Since voting from a single region is rather weak, a robust prediction is typically obtained by ensembling votes from different regions. We refer to these methods as *ensemble regression-voting*. In general, the choice of the regression function, which can cast accurate votes for all facial points, is the key factor of the ensemble regression-voting approach.
Regression forests [@breiman2001random] are a natural choice to perform regression-voting due to their simplicity and low computational complexity. Cootes *et al.* [@cootes2012robust] use random forest regression-voting to produce accurate response map for each facial point, which is then combined with the CLM fitting for robust prediction. Dantone *et al.* [@dantone2012real] pointed out that conventional regression forests may lead to a bias to the mean face, because a regression forest is trained with image patches on the entire training set and averages the spatial distributions over all trees in the forest. Therefore, they extended the concept of regression forests to conditional regression forests. A conditional regression forest consists of multiple forests that are trained on a subset of the training data specified by global face properties (e.g., head pose used in [@dantone2012real]). During testing, the head pose is first estimated by a specialized regression forest, then trees of the various conditional forests are selected to estimate the facial points. Due to the high efficiency of random forests, [@dantone2012real] achieves close-to-human accuracy while processing images in real-time on the Labeled Faces in the wild (LFW) database [@huang2007labeled]. After that, Yang *et al.* [@yang2013privileged] extended [@dantone2012real] by exploiting the information provided by global properties to improve the quality of decision trees, and later deployed a cascade of sieves to refine the voting map obtained from random regression forests [@yang2013sieving]. Apart from the regression forests [@dantone2012real; @yang2012face; @yang2013privileged; @yang2013sieving], Smith *et al.* [@smith2014nonparametric] used each local feature surrounding the facial point to cast a weighted vote to predict facial point locations in a nonparametric manner, where the weight is pre-computed to take into account the feature’s discriminative power.
In general, the ensemble regression-voting approach is more robust than previous local detector-based methods, and we conjecture that this robustness mainly stems from the combination of votes from different regions. However, current ensemble regression-voting approach, arguably, have not achieved a good balance between accuracy and efficiency for face alignment *in-the-wild*. The random forests approach [@dantone2012real; @yang2012face; @yang2013privileged; @yang2013sieving] is very efficient but can hardly cast precise votes for those unstable facial points (e.g., face contour), while on the other hand, the nonparametric feature voting approach based on facial part features [@smith2014nonparametric] is more accurate but suffers from very high computational burden. To pursue a face alignment algorithm that is both accurate and efficient, much research has focused on the cascaded regression approach as described in the next section.
Cascaded regression {#sec:cascaded_regression}
-------------------
Recently, cascaded regression has established itself as one of the most popular and state-of-the-art methods for face alignment, due to its high accuracy and speed [@cao2012face; @sun2013deep; @xiong2013supervised; @ren2014face; @zhu2015face]. The motivation behind this approach is that, since performing regression from image features to face shape in one step is extremely challenging, we can divide the regression process into stages, by learning a cascade of vectorial regressors.
Formally, given an image $\mathbf{I}$ and an initial shape $\mathbf{s}^0$, the face shape $\mathbf{s}$ is progressively refined by estimating a shape increment $\Delta \mathbf{s}$ stage-by-stage. In a generic form, a shape increment $\Delta \mathbf{s}$ at stage $t$ is regressed as: $$\label{eq:generic_cascaded_regression}
\Delta \mathbf{s}^{t} = \mathcal{R}^{t}\big(\Phi^{t}(\mathbf{I},\mathbf{s}^{t-1})\big),$$ where $\mathbf{s}^{t-1}$ is the shape estimated in the previous stage, $\Phi^{t}$ is the feature mapping function, and $\mathcal{R}^{t}$ is the stage regressor. Note that $\Phi^{t}(\mathbf{I},\mathbf{s}^{t-1})$ is referred to as *shape-indexed* feature [@cao2012face; @burgos2013robust] that depends on the current shape estimate, and can be either designed by hand [@xiong2013supervised; @zhu2015face] or by learning [@cao2012face; @ren2014face; @kazemi2014one]. In the training phase, the stage regressors $(\mathcal{R}^1,...,\mathcal{R}^T)$ are sequentially learnt to reduce the alignment errors on training set, during which geometric constraints among points are *implicitly* encoded.
{width="100.00000%"}
Existing cascaded regression methods mainly differ in the specific form of the stage regressor $\mathcal{R}^t$ and the feature mapping function $\Phi^t$. Here, according to the type of the stage regressor $\mathcal{R}^t$, we roughly divide existing cascaded regression methods into two categories, i.e., *two-level boosted regression*, and *cascaded linear regression*.
### Two-level boosted regression
Cascaded regression is first introduced into face alignment by Cao *et al.* [@cao2012face] in their seminal work called Explicit Shape Regression (ESR). They design a two-level boosted regression framework by again investigating boosted regression as the stage regressor $\mathcal{R}^t$. More specifically, they use a cascade of random ferns as $\mathcal{R}^t$ to regress the *fixed* shape-indexed pixel difference feature at each stage, and adopt a correlation-based feature selection strategy to learn task-specific features. This combination makes ESR a break-through face alignment method in both accuracy and efficiency, and is widely adapted ever since.
Burgos-Artizzu *et al.* [@burgos2013robust] also use the fern primitive regressor under the two-level boosted regression framework, but improve [@cao2012face] by explicitly incorporating the occlusion information into the regression target to better handle occlusions. Instead of random ferns used by [@cao2012face; @burgos2013robust], Kazemi *et al.* [@kazemi2014one] present a general framework based on gradient boosting for learning an ensemble of regression trees, achieving super-realtime performance with high quality predictions and naturally handling missing or partially labelled data. Lee *et al.* [@lee2015face] propose to use the Gaussian process regression tree (GPRT) to fit the primitive regressor under the two-level boosted regression framework, where GPRT is a Guassian process with a kernel defined by a set of trees.
### Cascaded linear regression
Although the two-level boosted regression framework has gained great success [@cao2012face; @burgos2013robust; @kazemi2014one; @lee2015face], generally speaking, any kind of stage regressor $\mathcal{R}^t$ with strong fitting capacity will be desirable. A notable example is the cascaded linear regression proposed by Xiong *et al.* [@xiong2013supervised] using strong hand-craft SIFT [@lowe2004distinctive] feature.
The primary innovation of the cascaded linear regression method [@xiong2013supervised] is a Supervised gradient Descent Method (SDM) that gives a mathematically sound explanation of the cascaded linear regression by placing it in the context of Newton optimization for non-linear least squares problem. SDM shows that a Newton update for the non-linear least squares alignment error function can be expressed as a linear combination of the facial feature differences between the one extracted at current shape and the ground truth template, resulting in a linear update function $\mathcal{R}^{t}$ at each stage, i.e., $$\label{eq:cascaded_linear_regression}
\mathcal{R}^{t}: \Delta \mathbf{s}^{t} \leftarrow \mathbf{W}^t \big(\Phi^{t}(\mathbf{I},\mathbf{s}^{t-1})\big) + \mathbf{b}^t,$$ where $\Phi^{t}$ is the SIFT operator that extract SIFT feature at each facial point, and $\mathbf{W}^t$ is the *averaged* descent direction on the training set.
Actually, SDM bears some similarities to AAMs trained in a discriminative manner with linear regression [@cootes2001active], but differs from them in three aspects: (1) SDM is non-parametric in both shape and appearance; (2) SDM uses the part-based representation; (3) SDM learns different regressors $\mathcal{R}^t$ at different stages, while the original AAM [@cootes2001active] learns a constant regressor $\mathcal{R}$ for all stages.
Due to its concise formulation and state-of-the-art performance, SDM has been extensively investigated and extended. Xiong *et al.* [@xiong2015global] pointed out that SDM is a local algorithm that is likely to average conflicting gradient directions, and proposed an extension of SDM called Global SDM (GSDM) that divides the search space into regions of similar gradient directions. Yan *et al.* [@yan2013learn] proposed to generate multiple hypotheses, and then learn to rank or combine these hypotheses to get the final result. Asthana *et al.* proposed an incremental formulation for the cascaded linear regression framework [@xiong2013supervised], and presented multiple ways for incrementally updating a cascade of regression functions in an efficient manner. Zhu *et al.* [@zhu2015face] designed a cascaded regression framework that begins with a coarse search over a shape space that contains diverse shapes, and employs the coarse solution to constrain subsequent finer search of shape, which improves the robustness of cascaded linear regression in coping with large pose variations.
### Discussion
Arguably, cascaded regression is playing a prominent role among the state-of-the-art methods for face alignment *in-the-wild*. This is primarily because it has some distinct characteristics. (1) The training sample of cascaded regression is a triple defined by the face image, ground truth shape and the initial shape, which allows for convenient data augmentation by generating multiple initial shapes for one image. (2) It is capable of effectively leveraging large bodies of training data. (3) The shape constraints are encoded into regressors adaptively, which is more flexible than the parametric shape model that heuristically determines the model flexibility (e.g.,PCA dimension). (4) The cascaded regression framework is simple and generalizable, which allows different choices for the stage regressor $\mathcal{R}^t$ and convenient incorporation of feature learning techniques.
Although cascaded regression has achieved great success in face alignment, it is still not easy to perform regression from texture features to the whole shape update for some challenging faces with extreme expression or pose variation. This limitation can be partially confirmed by the fact that for some more flexible part localization task such as human pose estimation, the part detector-based methods still play a dominant role at present [@yang2013articulated; @liu2015articulated], rather than cascaded regression.
Deep neural networks {#subsec:deep_neural_networks}
--------------------
Deep neural networks, especially the deep convolutional network that can extract high-level image features, have been successfully utilized in many computer vision tasks, such as face verification [@taigman2014deepface; @sun2014deep], image classification [@krizhevsky2012imagenet; @simonyan2014very; @szegedy2015going], and object detection [@girshick2014rich]. Naturally, they are also an effective choice to model the nonlinear relationship between the facial appearance and the face shape (or shape update).
However, applying deep network directly to face alignment is nontrivial due to the follwoing reasons: (1) While fine-tuning an existing CNN architecture (e.g., AlexNet [@krizhevsky2012imagenet], GoogLeNet [@szegedy2015going]) to make it well adapted to the task at hand is very popular in computer vision [@girshick2014rich; @zhang2014part], such a strategy can hardly be applied for face alignment because the off-the-shelf large networks are typically trained for image classification while face alignment is a structural prediction problem. (2) Constructing a deep network-based system from scratch for face alignment should take into account the issue of over-fitting, and hence the network structures at each stage need to be carefully designed according to the task of this stage and the complexity involved.
{width="45.00000%"}
Focusing on above issues, Sun *et al.* [@sun2013deep] were pioneers in this area with their work called Deep Convolutional Network Cascade. They handled the face alignment task with three-level carefully designed convolutional networks, and fuse the outputs of multiple networks at each level for robust prediction (Fig. \[fig:DCNN\] illustrates one of the first-level CNN structures). The first level network takes the whole face image as input to predict the initial estimates of the holistic face shape, during which the shape constraints are implicitly encoded. Then, the following two level networks refine the position of each point to achieve higher accuracy. Several network structures critical for face alignment are investigated in [@sun2013deep], providing some important principles on the choice of convolutional network structures. For example, convolutional networks at the first level should be deeper than the following networks, since predicting facial points from large input regions is a high-level task.
Ever since the work of [@zhang2014facial], deep CNNs have been widely exploited for face alignment. Similar to [@zhang2014facial], Zhou *et al.* [@zhou2013extensive] designed a four-level convolutional network cascade to tackle the face alignment problem in a coarse-to-fine manner, where each network level is trained to locally refine a subset of facial points generated by previous network levels. Zhang *et al.* [@zhang2014facial] extended the work of [@sun2013deep] by jointly learning auxiliary attributes along with face alignment. Their work confirms that some heterogeneous but subtly correlated tasks, e.g., head pose estimation and facial attribute inference can aid the face alignment task through multi-task learning. Lai *et al.* [@lai2015deep] proposed an end-to-end CNN architecture to learn highly discriminative shape-indexed features, by encoding the image into high-level feature maps in the same size of the image, and then extracting deep features from these high level descriptors through a novel “Shape-Indexed Pooling" method. Despite of the great popularity and success, as mentioned before, we should take into account the tradeoff between the model complexity and training data size, since some deep models have been reported to be pre-trained with enormous quantity of external data sources [@sun2013deep; @zhang2014facial].
Summary and discussion
----------------------
We have reviewed discriminative methods for face alignment in six groups, i.e., *CLMs*, *constrained local regression*, *DPMs*, *ensemble regression-voting*, *cascaded regression* and *deep neural networks*. Among them, CLMs, constrained local regression and DPMs follow the “divide and conquer" principle to simplify the face alignment task by constructing individual local appearance model for each facial point. However, due to their small patch support and large appearance variation in training, these local appearance models are typically plagued by the problem of ambiguity. Furthermore, since further inference (or global shape optimization) is based on the detection responses of these local appearance models, the problem of ambiguity may create the most serious performance bottleneck for face alignment *in-the-wild*.
To break this bottleneck, another main stream in face alignment is to jointly estimate the whole face shape from image, implicitly exploiting the spatial constraints among facial points. In this line, we have first reviewed the *ensemble regression-voting* and *cascaded regression* methods, which learn a vectorial regression function to infer the whole face shape in an ensemble or cascaded manner. In particular, cascaded regression has emerged as one of the most popular and state-of-the-art methods, due to its speed, accuracy and robustness. Then, we briefly reviewed the deep learning-based approach for face alignment, which have the advantage of learning highly discriminative task-specific features, but should take into account the issue of over-fitting.
It is worth noting that some methods involve techniques motivated by different principles, which clearly overlap our category boundaries. For example, we classify the regression voting-based shape model matching method [@cootes2012robust] as CLM, since they fit a parametric shape model to a new image based on the response map for each facial point. However, since the response maps in [@cootes2012robust] are generated by random forest regression-voting, it can also be considered as an ensemble regression-voting method. Furthermore, some deep learning-based methods can also be classified as cascaded regression due to their cascaded structure [@zhang2014coarse; @lai2015deep].
Towards the development of a robust face alignment system {#sec_development}
=========================================================
Face alignment *in-the-wild* is very challenging due to many kinds of undesirable appearance variations, and hence it is often the case that no single modality is enough. In this section, we will focus on the practical aspects of constructing a robust face alignment system, which is mostly ignored in previous studies. Specifically, we first present a global system architecture for face alignment, and then have a close look at possible strategies to improve the robustness of face alignment under this architecture.
The global system architecture for face alignment
-------------------------------------------------
{width="95.00000%"}
Inspired by [@song2013literature; @fasel2003automatic], we give a global system architecture for face alignment, where a complicated system is divided into several substages. As shown in Fig. \[fig\_deve\_architecture\], the architecture can be roughly divided into three parts: face preprocessing, shape initialization, and the iterative process of feature extraction and shape prediction. We note that this architecture is only to illustrate a general pipeline for face alignment, while in practical not all components are mandatory. For example, the consensus of exemplar method [@belhumeur2011localizing] do not involve the shape initialization step.
While the feature extraction and shape prediction process have drawn a great deal of attention in literature, the face preprocessing and shape initialization steps are often ignored. Meanwhile, problems such as training data augmentation, and the accuracy and efficiency tradeoff are also essential for any practical face alignment system. In the following, we will have a closer look at these issues.
Training data augmentation
--------------------------
Due to the difficulty and cost of manual annotation, the number of training samples we *actually* have is often much smaller than that we *supposedly* have. In such a case, artificial data augmentation, which is usually done by label-preserving transforms, is the easiest and most common method to reduce over-fitting.
In general, there are four distinct forms of data augmentation to enlarge the training set: (1) generating image rotations from a small interval (e.g., \[-15 degrees, +15 degrees\] used in [@belhumeur2013localizing]); (2) synthesizing images by left-right flip to double the training set; (3) disturbing the bounding boxes by randomly scaling and translating the bounding box for each image, which also increases the robustness of face alignment algorithms to the bounding boxes; (4) sampling multiple initialization for each training image, which is typically used by cascaded regression methods.
Face preprocessing {#subsec_development_face_preprocessing}
------------------
For the task of face alignment, it is useful to remove the scaling variations of the detected faces, and enlarge the face region to ensure that all predefined facial points are enclosed.
### Handling scaling variations
Typically, for a face analysis system, the training and test faces are required to be roughly the same scale, by rescaling the bounding box produced by the face detector. We note that to help preserve more detailed texture information, the size of the normalized bounding box for high-resolution face databases is typically chosen to be larger than that for low-resolution face databases. For example, Belhumeur *et al.* [@belhumeur2013localizing] rescale the high-resolution images from the LFPW database so that the faces have an inter-ocular distance of roughly 55 pixels, while Dantone *et al.* [@dantone2012real] choose to rescale the bounding box of the low-resolution faces from the LFW database [@huang2007labeled] to 100$\times$100, which is slightly smaller than the size chosen by Belhumeur *et al.* [@belhumeur2013localizing].
### Enlarging face areas
The output of a face detector is a rough face region that might miss some facial points (e.g., the chin). This has little impact on cascaded regression, for which the bounding box only serves to rescale the face and compute the initial shape. However, for those methods based on exhaustive search or feature voting, it is necessary to enlarge the face bounding box to enclose all the facial points, or define the sampling region of image patches to cast votes. For this, Dantone *et al.* [@dantone2012real] suggest to enlarge the face bounding box by 30%, and we believe that this strategy may satisfy the requirements of all face alignment algorithms.
Shape initialization
--------------------
Most face alignment methods start from a rough initialization, and then refine the shape iteratively until convergence. The initialization step typically has great influence on the final result, and an initial shape far from the ground truth might lead to very bad alignment results.
The most common choice is to use the *mean shape* for initialization [@xiong2013supervised; @kazemi2014one; @ren2014face]. However, sometimes, the mean shape is likely to be far from the target shape, and leads to bad result. As an alternative, Cao *et al.* [@cao2012face] propose to run the algorithm several times using different initialisations *randomly* sampled from the training shapes, and take the median result as the final estimation to improve robustness. Burgos-Artizzu *et al.* [@burgos2013robust] proposed a smart restart method to further improve the multiple initialization strategy in [@cao2012face] by checking the variance between the predictions using different initializations.
Recently, some authors proposed to estimate an initial shape that is tailored to the input face. Zhang *et al.* [@zhang2014facial] showed that the five major facial points localized by their deep model can serve as anchor points to apply similarity transform to randomly sampled training shapes. Through this, very accurate initial shapes can be generated for other algorithms (e.g., [@burgos2013robust]) and lead to promising performance improvement. Zhang *et al.* [@zhang2014coarse] and Sun *et al.* [@sun2013deep] proposed to directly estimate a rough initial shape from the global image, which in general produces good initial shape that aids following alignment.
Accuracy and efficiency tradeoffs
---------------------------------
Face alignment in real time is crucial to many practical applications. The efficiency mainly depends on the feature extraction and shape prediction steps. In general, strong hand-designed feature (e.g., SIFT [@lowe2004distinctive]) captures detailed texture information that may aid detection, but have higher computational cost compared to simpler features (e.g., BRIEF [@calonder2010brief]). Zhu *et al.* [@zhu2015face] identified this phenomenon under the cascaded regression framework, and proposed to exploit different types of features at different stages to achieve a good trade-off between accuracy and efficiency, i.e., employ less accurate but computationally efficient BRIEF feature at the early stages, and use more accurate but relatively slow SIFT feature at later stages. Besides this hybrid strategy, a better choice is to learn highly efficient and discriminative features [@cao2012face; @ren2014face; @kazemi2014one]. In particular, Ren *et al.* [@ren2014face] propose to learn a set of highly discriminative local binary features for each facial point independently. Because extracting and regressing local binary features is computationally very cheap, [@ren2014face] achieves over 3,000 FPS while obtaining accurate alignment result.
In term of shape prediction, the regression-based methods in general are very efficient, while the exhaustive search based methods typically suffer from high computational cost [@belhumeur2011localizing; @zhou2013exemplar]. Dibeklio[ğ]{}lu *et al.* [@dibekliouglu2012statistical] propose to mitigate this issue through a coarse-to-fine search strategy. In [@dibekliouglu2012statistical], a three-level image pyramid from the cropped high-resolution face images is designed to reduce the search region, where the coarse-level images have lower resolution but much smaller size.
System Evaluation {#sec_evaluation}
=================
In this section, we first review the major wild face databases and evaluation metric in the literature, then summarize and discuss some of reported performance of current state-of-the-art, on the several popular wild face databases using the same evaluation metric for reference.
Databases and metric
--------------------
### Databases
[p[0.098]{}p[0.03]{}<p[0.065]{}<p[0.08]{}<p[0.045]{}<p[0.045]{}<p[0.485]{}]{} **[Databases]{} &**[Year]{} &**[\#Images]{} &**[\#Training]{} &**[\#Test]{} & **[\#Point]{} & **[Links]{}\
LFW [@huang2007labeled] & 2007 & 13,233 & 1,100 & 300 & 10 & <http://www.dantone.me/datasets/facial-features-lfw/>\
LFPW [@belhumeur2011localizing] & 2011 & 1,432 & - & - & 35 & <http://homes.cs.washington.edu/~neeraj/databases/lfpw/>\
AFLW [@kostinger2011annotated] & 2011 & 25,993 & - & - & 21 & <http://lrs.icg.tugraz.at/research/aflw>\
AFW [@zhu2012face] &2012 & 205 & - & - & 6 & <http://www.ics.uci.edu/~xzhu/face/>\
HELEN [@le2012interactive] &2012 & 2,330 & 2,000 & 300 & 194 & <http://www.ifp.illinois.edu/~vuongle2/helen/>\
300-W [@sagonas2013semi] & 2013 &3,837 & 3,148 & 689 & 68 &<http://ibug.doc.ic.ac.uk/resources/300-W/>\
COFW [@burgos2013robust] & 2013 &1,007 & - & - & 29 &<http://www.vision.caltech.edu/xpburgos/ICCV13/>\
MTFL [@zhang2014facial] & 2014 & 12,995 & - & - & 5 &<http://mmlab.ie.cuhk.edu.hk/projects/TCDCN.html>\
MAFL [@zhang2016learning] & 2016 &20,000 & - & - & 5 & <http://mmlab.ie.cuhk.edu.hk/projects/TCDCN.html>\
**************
LFPW is shared by web URLs, but some URLs are no longer valid.
Each face image in LFPW is annotated with 35 points, but only 29 points defined in [@belhumeur2011localizing] are used for the face alignment.
![Illustration of the example face images from eight wide face databases with original annotation.[]{data-label="fig_evaluation_annotaion"}](images/evaluation/all_annotation_examples.pdf){width="60.00000%"}
There have been many face databases developed for face alignment, with the ground truth facial points labelled manually by employing workers or through the tools such as Amazon mechanical turk (MTurk). Among them, some databases are collected under controlled laboratory conditions with normal lighting, neutral expression and high image quality, including the Extended M2VTS database (XM2VTS) [@messer1999xm2vtsdb], BioID face database [@jesorsky2001robust], PUT [@kasinski2008put], Multi-Pie [@gross2010multi], etc.
However, the goal of this paper is to investigate the problem of face alignment *in-the-wild*, so we are more concerned with the *uncontrolled* databases that exhibit large facial variations due to pose, expressions, lighting, occlusion and image quality. These uncontrolled databases are typically collected from social network such as google.com, flickr.com, facebook.com, which are more realistic and challenging for face alignment. In Tab. \[tab\_evaluation\_database\], we list the basic information of 9 wild face databases, including LFW [@huang2007labeled], LFPW [@belhumeur2011localizing], AFLW [@kostinger2011annotated], AFW [@zhu2012face], HELEN [@le2012interactive], 300-W [@sagonas2013semi], COFW [@burgos2013robust], MTFL [@zhang2014facial], and MAFL [@zhang2016learning], and also provide links to download them. The example face images from these databases with original annotation are illustrated in Fig. \[fig\_evaluation\_annotaion\]. It is worth noting that the LFPW, AFW and HELEN databases are re-annotated by Sagonas *et al.* [@sagonas2013semi] with 68 points.
### Evaluation metric
There have been several evaluation metrics for the alignment accuracy in the literature. For example, many authors reported the inter-pupil distance normalized facial point error averaged over all facial points and images for each database [@burgos2013robust; @ren2014face; @kazemi2014one; @zhu2015face; @lee2015face]. Specifically, the inter-ocular distance normalized error for facial point $i$ is defined as: $$\label{eq_evaluation_metric}
e_i = \frac{||\mathbf{x}_i-\mathbf{x}_i^*||_2}{d_{IOD}},$$ where $\mathbf{x}_i$ is the automatically localized facial point location, $\mathbf{x}_i^*$ is the manually annotated location, and $d_{IOD}$ is the inter-ocular distance. The normalization term $d_{IOD}$ in this formulation can eliminate unreasonable measurement variations caused by variations of face scales.
The cumulative errors distribution (CED) curve is also often chosen to illustrate the comparative performance, showing the proportion of the test images or facial points with the increase of the normalized error [@saragih2011deformable; @belhumeur2011localizing; @tzimiropoulos2014gauss; @tzimiropoulos2015project; @zhu2015face]. Some other evaluation metric can also been found in literature, such as the facial point error normalized by face size [@yu2013pose], the percentage of the test images or facial points less than given relative error level [@dibekliouglu2012statistical; @yu2013pose], and the percentage of accuracy improvement over other algorithm [@cao2012face].
Besides the accuracy, the efficiency is another important performance indicator of face alignment algorithms, which is typically measured by frames per second (FPS).
Evaluation and discussion
-------------------------
[p[0.4]{}p[0.05]{}<p[0.05]{}<p[0.4]{}]{} **[Methods]{} & **[Year]{} & **[$\#$Points]{} & **[Links]{}\
Boosted Regression with Markov Networks (BoRMaN) [@valstar2010facial] & 2010 & 22 & <http://ibug.doc.ic.ac.uk/resources/facial-point-detector-2010/>\
Constrained Local Model (CLM) [@saragih2011deformable] & 2011 & 66 & <https://github.com/kylemcdonald/FaceTracker>\
Tree Structured Part Model (TSPM) [@zhu2012face] & 2012 & 68 & <http://www.ics.uci.edu/~xzhu/face/>\
Conditional Random Forests (CRF) [@dantone2012real] & 2012 & 10 & <http://www.dantone.me/projects-2/facial-feature-detection/>\
Structured Output SVM [@uvrivcavr2012detector] & 2012 & 7 & <http://cmp.felk.cvut.cz/~uricamic/flandmark/>\
Cascaded CNN [@sun2013deep] & 2013 & 5 & <http://mmlab.ie.cuhk.edu.hk/archive/CNN_FacePoint.htm>\
Discriminative Response Map Fitting (DRMF) [@asthana2013robust] & 2013 & 66 & <https://sites.google.com/site/akshayasthana/clm-wild-code?>\
Supervised Descent Method (SDM) [@xiong2013supervised] & 2013 & 49 & [www.humansensing.cs.cmu.edu/intraface](www.humansensing.cs.cmu.edu/intraface)\
Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 2013 & 29 & <http://www.vision.caltech.edu/xpburgos/ICCV13/>\
Optimized Part Mixtures (OPM) [@yu2013pose] & 2013 & 68 & <http://www.research.rutgers.edu/~xiangyu/face_align/face_align_iccv_1.1.zip>\
Continuous Conditional Neural Fields (CCNF) [@baltruvsaitis2014continuous] & 2014 & 68 & <https://github.com/TadasBaltrusaitis/CCNF>\
Coarse-to-fine Shape Searching (CFSS) [@zhu2015face] & 2015 & 68 & [mmlab.ie.cuhk.edu.hk/projects/CFSS.html](mmlab.ie.cuhk.edu.hk/projects/CFSS.html)\
Project-Out Cascaded Regression (PO-CR) [@tzimiropoulos2015project] & 2015 & 68 & <http://www.cs.nott.ac.uk/~yzt/>\
Active Pictorial Structures (APS) [@antonakos2015active] & 2015 & 68 & <https://github.com/menpo/menpo>\
Tasks-Constrained Deep Convolutional Network (TCDCN) [@zhang2016learning] & 2016 & 68 & <http://mmlab.ie.cuhk.edu.hk/projects/TCDCN.html>\
********
We choose four common wild databases, i.e., LFW, LFPW, HELEN, 300-W and IBUG (challenging subset of 300-W) databases, to show comparative performance statistics of the state of the art. Table \[tab\_evaluation\_software\] lists some softwares published online, and Table \[tab\_evaluation\] summarizes the reported performance on above databases. Fig. \[fig:evaluation\_alignment\_examples\] shows some challenging images from IBUG aligned by eight state-of-the-art methods respectively.
For performance evaluation, we are mainly concerned with two key performance indicators, i.e., accuracy and efficiency. The former is measured by the normalized facial point error (cf. Eq. \[eq\_evaluation\_metric\]) averaged over all facial points and images for each database, while the later is measured by frames per second (FPS).
{width="98.00000%"}
[p[0.06]{}p[0.15]{}p[0.06]{}p[0.06]{}p[0.35]{} p[0.08]{}p[0.1]{}]{} **[Databases]{}& **[Challenges]{} & **[$\#$Test]{} & **[$\#$Points]{} & **[Methods]{} & **[Error (%)]{} & **[FPS]{}\
& &**************
&
& Conditional Random Forests (CRF) [@dantone2012real] & 7.00 & 10 (c++)\
& & & & Explicit Shape Regression (ESR) [@cao2012face] & 5.90 & 11 (Matlab)\
& & & & Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 5.30 & 15 (Matlab)\
& & & & & &\
& & &
& Consensus of Exemplar (CoE) [@belhumeur2013localizing] & 5.18 & -\
& &
&
& Consensus of Exemplar (CoE) [@belhumeur2011localizing] & 3.99 & $\approx$ 1 (C++)\
& & & & Explicit Shape Regression (ESR) [@cao2012face] & 3.47 & 220 (C++)\
& & & & Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 3.50 & 12 (Matlab)\
& & & & Supervised Descent Method (SDM) [@xiong2013supervised] & 3.49 & 160 (C++)\
& & & & Exemplar-based Graph Matching (EGM) [@zhou2013exemplar] & 3.98 & $<$ 1\
& & & & Local Binary Feature (LBF) [@ren2014face] & 3.35 & 460 (C++)\
& & & & Fast Local Binary Feature (LBF fast) [@ren2014face] & 3.35 & 4600 (C++)\
& & & & & &\
& & &
& Tree Structured Part Model (TSPM) [@zhu2012face] & 8.29 & 0.04 (Matlab)\
& & & & Discriminative Response Map Fitting (DRMF) [@asthana2013robust] & 6.57 & 1 (Matlab)\
& & & & Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 6.56 & 12 (Matlab)\
& & & & Supervised Descent Method (SDM) [@xiong2013supervised] & 5.67 & 70 (C++)\
& & & & Gauss-Newton Deformable Part Model (GN-DPM) [@tzimiropoulos2014gauss] & 5.92 & 70\
& & & & Coarse-to-fine Auto-encoder Networks (CFAN) [@zhang2014coarse] & 5.44 & 20\
& & & & Coarse-to-fine Shape Searching (CFSS) [@zhu2015face] & 4.87 & -\
& & & & CFSS Practical [@zhu2015face] & 4.90 & -\
& & & & Deep Cascaded Regression (DCR) [@lai2015deep] & 4.57 & -\
& &
&
& Stacked Active Shape Model (STASM) [@milborrow2008locating] & 11.10 & -\
& & & & Component-based ASM (ComASM) [@le2012interactive] & 9.10 & -\
& & & & Explicit Shape Regression (ESR) [@cao2012face] & 5.70 & 70 (C++)\
& & & & Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 6.50 & 6 (Matlab)\
& & & & Supervised Descent Method (SDM) [@xiong2013supervised] & 5.85 & 21 (C++)\
& & & & Ensemble of Regression Trees (ERT) [@kazemi2014one] & 4.9 & 1000\
& & & & Local Binary Feature (LBF) [@ren2014face] & 5.41 & 200 (C++)\
& & & & Fast Local Binary Feature (LBF fast) [@ren2014face] & 5.80 & 1500 (C++)\
& & & & Coarse-to-Fine Shape Searching (CFSS) [@zhu2015face] & 4.74 & -\
& & & & CFSS Practical [@zhu2015face] & 4.84 & -\
& & & & cascade Gaussian Process Regression Trees (cGPRT) [@lee2015face] & 4.63 & -\
& & & & & &\
& & &
& Tree Structured Part Model (TSPM) [@zhu2012face] & 8.16 & 0.04 (Matlab)\
& & & & Discriminative Response Map Fitting (DRMF) [@asthana2013robust] & 6.70 & 1 (Matlab)\
& & & & Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 5.93 & 12 (Matlab)\
& & & & Supervised Descent Method (SDM) [@xiong2013supervised] & 5.67 & 70 (C++)\
& & & & Gauss-Newton Deformable Part Model (GN-DPM) [@tzimiropoulos2014gauss] & 5.69 & 70\
& & & & Coarse-to-fine Auto-encoder Networks (CFAN) [@zhang2014coarse] & 5.53 & 20\
& & & & Coarse-to-Fine Shape Searching (CFSS) [@zhu2015face] & 4.63 & -\
& & & & CFSS Practical [@zhu2015face] & 4.72 & -\
& & & & Deep Cascaded Regression [@lai2015deep] & 4.25 & -\
& &
&
& Tree Structured Part Model (TSPM) [@zhu2012face] & 12.20 & 0.04 (Matlab)\
& & & & Discriminative Response Map Fitting (DRMF) [@asthana2013robust] & 9.10 & 1 (Matlab)\
& & & & Explicit Shape Regression (ESR) [@cao2012face] & 5.28 & 120 (C++)\
& & & & Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 8.35 & -\
& & & & Supervised Descent Method (SDM) [@xiong2013supervised] & 7.50 & 70 (C++)\
& & & & Ensemble of Regression Trees (ERT) [@kazemi2014one] & 6.4 & 1000\
& & & & Local Binary Feature (LBF) [@ren2014face] & 6.32 & 320 (C++)\
& & & & Fast Local Binary Feature (LBF fast) [@ren2014face] & 7.37 & 3100 (C++)\
& & & & Coarse-to-Fine Shape Searching (CFSS) [@zhu2015face] & 5.76 & 25\
& & & & CFSS Practical [@zhu2015face] & 5.99 & 25\
& & & & cascade Gaussian Process Regression Trees (cGPRT) [@lee2015face] & 5.71 & 93\
& & & & fast cGPRT [@lee2015face] & 6.32 & 871\
& & & & Tasks-Constrained Deep Convolutional Network (TCDCN) [@zhang2016learning] & 5.54 & 59\
& & & & Deep Cascaded Regression (DCR) [@lai2015deep] & 5.02 & -\
& & & & Megvii-Face++ [@huang2015coarse] & 4.54 & -\
& &
&
& Tree Structured Part Model (TSPM) [@zhu2012face] & 18.33 & 0.04 (Matlab)\
& & & & Discriminative Response Map Fitting (DRMF) [@asthana2013robust] & 19.79 & 1 (Matlab)\
& & & & Explicit Shape Regression (ESR) [@cao2012face] & 17.00 & 120 (C++)\
& & & & Robust Cascaded Pose Regression (RCPR) [@burgos2013robust] & 17.26 & -\
& & & & Supervised Descent Method (SDM) [@xiong2013supervised] & 15.40 & 70 (C++)\
& & & & Local Binary Feature (LBF) [@ren2014face] & 11.98 & 320 (C++)\
& & & & Fast Local Binary Feature (LBF fast) [@ren2014face] & 15.50 & 3100 (C++)\
& & & & Robust Discriminative Hough Voting (RDHV) [@jin2016face] & 11.32 & $<$ 1 (Matlab)\
& & & & Coarse-to-Fine Shape Searching (CFSS) [@zhu2015face] & 9.98 & 25\
& & & & CFSS Practical [@zhu2015face] & 10.92 & 25\
& & & & Tasks-Constrained Deep Convolutional Network (TCDCN) [@zhang2016learning] & 8.60 & 59\
& & & & Deep Cascaded Regression (DCR) [@lai2015deep] & 8.42 & -\
& & & & Megvii-Face++ [@huang2015coarse] & 7.46 & -\
For LFW, the reported performance of [@dantone2012real; @cao2012face; @burgos2013robust] follows the the evaluation procedure proposed in [@dantone2012real], consisting of a ten-fold cross validation using each time 1,500 training images and the rest for testing. In [@belhumeur2013localizing], the model is trained on Columbia’s PubFig [@kumar2009attribute], and tested on all 13,233 images of LFW.
Although used by [@belhumeur2013localizing], the 55 point annotation of LFW is not shared.
LFPW is shared by web URLs, but some URLs are no longer valid. So both the training and test images downloaded by other authors are less than the original version (1,100 training images and 300 test images).
LFPW and HELEN are originally annotated with 29 and 194 points respectively, while later Sagonas *et al.* [@sagonas2013semi] re-annotate them with 68 points. Some authors reported their performance on the 68 points version of these databases.
### Accuracy
As shown in Table \[tab\_evaluation\], the localization error on all these databases has been reduced to less than 10% of the inter-ocular distance by current state-of-the-art. Except for the extremely challenging IBUG database, the best performance on other databases is about 5% of the inter-ocular distance. To have an intuitive feeling of the extent of localization error, we exemplify the error range of 10% and 5% of the inter-ocular distance respectively in Fig. \[fig:error\_tolerance\] (a) and (b). This implies that most of the localized facial points by the state-of-the-art may lie in the error range depicted by the white circles in Fig. \[fig:error\_tolerance\] (a), while on LFPW annotated with 29 points, the mean error range goes to the white circles in Fig. \[fig:error\_tolerance\] (b). Besides the statistics listed in Table \[tab\_evaluation\], some authors also compared their methods with human beings and reported close to human performance on LFPW [@belhumeur2011localizing; @burgos2013robust] and LFW [@dantone2012real].
From Table \[tab\_evaluation\], we can observe that although generative methods (e.g., the GN-DPM [@tzimiropoulos2014gauss]) can produce good performance for face alignment *in-the-wild*, discriminative methods, especially those based on cascaded regression [@cao2012face; @burgos2013robust; @xiong2013supervised; @ren2014face; @kazemi2014one; @zhu2015face; @lai2015deep; @huang2015coarse], have been playing a dominate role for this task, partially due to recent development of large unconstrained databases. Furthermore, the deep learning-based approach [@sun2013deep; @zhang2014facial; @huang2015coarse; @zhang2016learning] have recently emerged as a popular and state-of-the-art method due to their strong feature learning capability, achieving very accurate (even the best) performance on the challenging 300-W and IBUG databases [@sagonas2013semi].
Fig. \[fig:evaluation\_alignment\_examples\] shows some extremely challenging cases on IBUG aligned by eight state-of-the-art methods, from which we can observe that large head poses, extreme lighting, and partial occlusions may pose major challenges for many advanced face alignment algorithms, but good results can still be achieved by some state-of-the-art, for example, by the Tasks-Constrained Deep Convolutional Network (TCDCN) method [@zhang2016learning]. Furthermore, we find the Fig. \[fig:evaluation\_alignment\_examples\] that: (1) Compared to other facial points, the points around the outline of the face are much more difficult to localize, due to the lack of distinctive local texture. (2) As the points around the mouth are heavily dependent on facial expressions, they are more difficult to localize than those points insensitive to facial expressions, such as the points along the eyebrows, outer corners of the eyes, and the nose tips.
Finally, we have to highlight that the accuracy statistics listed in Table \[tab\_evaluation\] may not fully characterize the behavior of these algorithms, since several factors can complicate the assessment. First, even for the same algorithm, different experimental details and programming skills may results in different performance. Secondly, while the number and variety of training examples have a direct effect on the final performance, the training data of some released software is not clear. Thirdly, as pointed by [@yang2015empirical], the performance of many algorithms is sensitive to the face detection variation, but different systems may employ different face detectors. For example, SDM [@xiong2013supervised] employs the Viola Jones detector [@viola2004robust], while GN-DPM [@tzimiropoulos2014gauss] uses the in-house face detector of the IBUG group.
### Efficiency
Besides accuracy, efficiency is another key performance indicator of face alignment algorithms. In the last column of Table \[tab\_evaluation\], we report the efficiency of some algorithms, and highlight the implementation types of them (Matlab or C++). In general, the running time listed here is consistent with the algorithm’s complexity. For example, algorithms that involves an exhaustive search of local detectors typically have a high time cost [@belhumeur2011localizing; @zhu2012face; @zhou2013exemplar], while the cascaded regression methods are extremely fast since both the shape-index feature and the stage regression are very efficient to compute [@cao2012face; @xiong2013supervised; @burgos2013robust]. It is worth noting that impressive speed (more than 1,000 FPS for 194 points on HELEN) has been achieved by the Local Binary Feature (LBF) [@ren2014face] and Ensemble of Regression Trees (ERT) [@kazemi2014one], using learning-based features.
Conclusion and prospect {#sec_conclusion}
=======================
Face alignment is an important and essential intermediary step for many face analysis applications. Such a task is extremely challenging in unconstrained environments due to the complexity of facial appearance variations. However, extensive studies on this problem have resulted in a great amount of achievements, especially during the last few years.
In this paper, we have focused on the overall difficulties and challenges in unconstrained environments, and provide a comprehensive and critical survey of the current state of the art in dealing with these challenges. Furthermore, we hope that the practical aspects of face alignment we organized can provide further impetus for high-performance, real-time, real-life face alignment systems. Finally, it is worth mentioning that some closely related problems are deliberately ignored in this paper, such as facial feature tracking in videos [@kapoor2002real; @ahlberg2001using] and 3D face alignment [@vezzetti20123d], which are also very important in practice.
Despite of many efforts devoted to face alignment during the last two decades, we have to admit that this problem is far from being solved, and several general promising research directions could be suggested.
- *Challenging databases collection:* Besides new methodologies, another notable development in the field of face alignment has been the collection and annotation of large facial datasets captured *in-the-wild* (cf., Table \[tab\_evaluation\_database\]). But even so, we argue that the collection of challenging databases is still important and has the potential to boost the performance of existing methods. This argument can be partially supported by the fact that: the performance of most algorithms on IBUG is inferior to that on other databases such as LFPW and HELEN, as the training set of these algorithms is typically less challenging compared to IBUG.
- *Feature learning:* One of the holy grails of machine learning is to automate more and more of the feature engineering process [@domingos2012few], i.e., to learn task-specific features in a data-driven manner. In the field of face alignment, many approaches that employ feature learning techniques, including both shallow feature learning [@cao2012face; @burgos2013robust; @ren2014face] and deep learning [@sun2013deep; @huang2015coarse] methods, have achieved state-of-the-art performances. We believe that, with the assistance of abundant manually labeled images, automatic feature learning techniques can be a powerful weapon for triumphing over various challenges of face alignment in the wild, and deserve the efforts and smarts of researchers.
- *Multi-task learning:* Multi-task learning aims to improve the generalization performance of multiple related tasks by learning them jointly, which has proven effective in many computer vision problems [@yuan2012visual; @zhang2013robust]. For face alignment *in-the-wild*, on the one hand, many factors such as pose, expression and occlusion may pose great challenges; while on the other hand, these factors can be considered jointly with face alignment to expect an improvement of robustness. This has been confirmed by the work of [@zhang2014facial], which proposes to exploit the power of multi-task learning under the deep convolutional network architecture, leading to a better performance compared to single task-based deep model. Although some attempts have been proposed, we believe that multi-task learning remains a meaningful and promising direction for face alignment in future.
We believe that face alignment *in-the-wild* is a very exciting line of research due to its inherent complexity and wide practical applications, and will draw increasing attention from computer vision, pattern recognition and machine learning.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is partially supported by National Science Foundation of China (61373060), Qing Lan Project, and the Funding of Jiangsu Innovation Program for Graduate Education (KYLX\_0289).
[0.75]{}
[2]{}
[^1]: A disadvantage of our extended definition for CLMs is that, the classical Deformable Part Models (DPMs) with the tree structured shape model [@yang2013articulated; @zhu2012face] will also be covered by our definition of CLMs, which however are traditionally treated as an independent approach relative to others. In this paper, we will still treat DPMs as an independent group of methods in face alignment, and describe them in Section \[subsec:DPMs\]
|
---
abstract: |
In this work, we investigate the Hölder spectrum of typical measure (in the Baire category sense) in a general compact set and we compute the multifractal spectrum of a typical measures supported by a self-similar set. Such mesures verify the multifractal formalism.
*2000 Mathematics Subject Classification*: 28A80.
*Key words and phrases*: Borel measures, Hausdorff dimension, singularity spectrum, multifractal formalism, self-similar sets, Baire categories.
$^{\dag}$ École Supérieure des Sciences et de Technologie de Hammam Sousse (Tunisia).
author:
- 'Moez Ben Abid$^{\dag}$'
title: 'Multifractal formalism for typical probability measures on self-similar sets '
---
Introduction and the main result
================================
Let $K$ be a compact set of ${\mathbb{R}}^{d}$ endowed with the metric induced by any norm on ${\mathbb{R}}^{d}$.
The local Hölder exponent of a positive measure $\mu$ on $K$ at $x\in K$, $h_{\mu}(x)$, is defined by $$h_{\mu}(x)=\liminf_{r\to0}\frac{\log\mu(B(x,r))}{\log r}$$ where $B(x,r)$ is the ball of center $x$ and radius $r$. The purpose of the multifractal analysis of a measure $\mu$ is to investigate the singularity spectrum $d_{\mu}$ of $\mu$, that is the map $$d_{\mu}:h\geq0\mapsto {\mbox{dim}_{\mathcal{H}}}(E_{\mu}(h))$$ where $E_{\mu}(h)=\left\{x\in K:\; h_{\mu}(x)=h\right\}$ and ${\mbox{dim}_{\mathcal{H}}}$ is the Hausdorff dimension.
Generally it is very difficult to obtain the singularity spectrum directly from the definition of the Hausdorff dimension. To avoid this difficulty, the multifractal formalism provide a formula which link the singularity spectrum to the Legendre transform of mapping defined by averaged quantities of the measure, precisely to the Legendre transform of the $L^{q}$ spectrum defined as follows. If $j$ is an integer greater than $1$ let $\mathcal{G}_{j}$ be the partition of ${\mathbb{R}}^{d}$ into dyadic boxes: $\mathcal{G}_{j}$ is the set of all cubes $$I_{j,\mathbf{k}}=\prod_{i=1}^{d}\left[ k_{i}2^{-j},(k_{i}+1)2^{-j}\right[$$ where $\mathbf{k}=(k_{1},k_{2},\cdots,k_{d})\in {\mathbb{Z}}^{d}$.
The $L^{q}$ spectrum of a measure $\mu\in\mathcal{M}(K)$ is the mapping defined for any $q\in{\mathbb{R}}$ by $$\tau_{\mu}(q)=\liminf_{j\to+\infty}-\frac{\log\sum_{Q\in\mathcal{G}_{j},\mu(Q)\neq0 }\mu(Q)^{q}}{j\log2}.$$
A classical result (see for example [@Fal1]) assert that for all measure $\mu$ for all $h\geq 0$, $$\label{majSpectrumLegendre}
d_{\mu}(h)\leq \left(\tau_{\mu}\right)^{*}(h):=\inf_{q\in{\mathbb{R}}}(qh-\tau_{\mu}(q)).$$
A important issue in multifractal analysis is to establish when the upper bound (\[majSpectrumLegendre\]) turns out to be an equality , when this happens we say that the measure $\mu$ *satisfies the multifractal formalism* at $h$. A lot of work has been achieved for specific measures.
In the few last years, a particular interest was allocated to generic results (in the sense of Baire or prevalence) on the space of the probability measures endowed with the weak topology or in some space of functions, see for examples [@BucNag], [@BucSeu1], [@BucSeu2], [@FraJaf], [@FraJafKah], [@Ols1], [@Ols2].
We denote by $\mathcal{M}(K)$ the space of probability measures on $K$ endowed with the weak topology. Recall that the weak topology on $\mathcal{M}(K)$ is induced by the metric $\varrho$ on $\mathcal{M}(K)$ defined as follows. Let $\mbox{Lip}(K)$ denote the family of Lipschitz functions $f:K\rightarrow{\mathbb{R}}$ with $\left|f\right|:=\sup_{x\in K}\left|f(x)\right|\leq 1$ and $Lip(f)\leq 1$ where $Lip(f)$ denotes the Lipschitz constant of $f$. If $\mu$ and $\nu$ belong to $\mathcal{M}(K)$ we set
$$\varrho(\mu,\nu)=\sup\left\{\left|\int f d\mu-\int f d\nu\right|: f\in \mbox{Lip}(K)\right\}.$$ then the space $\mathcal{M}(K)$ is complete and separable.
In [@BucSeu1], the authors determined the multifractal spectrum for typical measures $\mu$ (in the Baire sens) in $[0,1]^{d}$ and they showed that such measures satisfy the multifractal formalism. They also made the following conjecture
For any compact set $K\subset{\mathbb{R}}^{d}$, there exists a constant $0<D<d$ such that typical measures $\mu$ (in the Baire sens) in $\mathcal{M}(K)$ satisfy: for any $h\in[0,D]$, $d_{\mu}(h)=h$, and if $h>D$, $E_{\mu}(h)=\emptyset$.
Whether $D$ should be the Hausdorff dimension of $K$ or the lower box dimension of $K$ (or another dimension).
In this paper, we give a positive answer to this conjecture in the special case where $K$ is a self-similar set satisfying the open set condition.
Let $X$ be a complete metric space. We say that a set $A$ of $X$ is a $G_{\delta}$ set if it can be written as a countably intersection of dense open sets. We say that a property is typical in $X$ if it holds on residual set, i.e. a set with complement of first Baire category. By the Baire theorem any $G_{\delta}$ set is dense.
Our main result is the following
\[maintheorem\] Let $K$ be a self-similar set satisfying the open set condition. Let $s$ be the Hausdorff dimension of $K$. Then, there exists a $G_{\delta}$ set $\Omega$ of $\mathcal{M}(K)$ such that for all $\mu\in \Omega$,
- for all $h>s$, $E_{\mu}(h)=\emptyset$
- for all $h\in[0,s]$, $d_{\mu}(h)=h$.
In particular, for every $q\in[0,1]$, $\tau_{\mu}(q)=s(q-1)$, and $\mu$ satisfies the multifractal formalism at every $h\in[0,s]$, i.e. $d_{\mu}(h)=\left(\tau_{\mu}\right)^{*}(h)$.
Our paper is organized as follows: in the second section we show, for any compact $K$, that for typical measure $\mu$ in $\mathcal{M}(K)$, for $h>s$, $E_{\mu}(h)=\emptyset$ where $s$ is the upper box counting dimension, this can be en particular applied to self-similar sets. In the third section we recall some properties of self-similar sets that will be useful for us. Then, using the same approach as [@BucSeu1] with suitable modifications we prove the Theorem \[maintheorem\].
Results valid on compact $K$
============================
Let $0\leq s<+\infty$, $\lambda$ a borelian measure on $K$ and $a\in K$. We define the lower $s-$densities of $K$ at $a$ with respect to $\lambda$ by $$\Theta^{s}_{*}(K,a,\lambda)=\liminf_{r\downarrow 0}(2r)^{-s}\lambda\left(K\cap B(a,r)\right).$$
In this section we will prove the following theorems.
Let $K$ be a closed set of ${\mathbb{R}}^{d}$.
1. Let $a\in K$. Then, there exists a $G_{\delta}$ set $\Omega(a)$ of $\mathcal{M}(K)$ such that for all $\mu\in \Omega(a)$, $h_{\mu}(a)=0$.
2. Let $s\in]0,d]$ and $A\subset K$. Assume that there exists $\lambda$ a finite Borel measure on $K$ such that for all $a\in A$ $$\Theta^{s}_{*}(K,a,\lambda)>0.$$
Then, there exists a $G_{\delta}$ set $\Omega$ of $\mathcal{M}(K)$ such that for all $\mu\in \Omega$, for all $x\in A$, $h_{\mu}(x)\leq s$. That is, for all $\mu\in \Omega$, for all $h>s$, $E_{\mu}(h)\cap A=\emptyset$.
\[th1\]
Let $a\in K$. For all $h>0$ the set $\Lambda_{h}(a)=\left\{\mu\in \mathcal{M}(K);\;h_{\mu}(a)=h\right\}$ is of empty interior. Indeed, if not then using the dense $G_{\delta}$ set $\Omega(a)$ we get $\Lambda_{h}(a)\cap \Omega(a)\neq \emptyset$ which is impossible.
Let $E$ a non-empty bounded subset or ${\mathbb{R}}^{d}$ let $N_{r}(E)$ be the largest number of disjoint balls of radius $r$ with centers in $E$. The upper box-counting dimension of $E$ is defined as $$\overline{\mbox{dim}_{B}}E=\limsup_{r\to0}\frac{\log N_{r}(E)}{-\log r}.$$
Already we can prove the following result.
Let $K$ be a closed set of ${\mathbb{R}}^{d}$ let $s=\overline{\mbox{dim}_{B}}K$. Then, there exists a $G_{\delta}$ set $\Omega$ of $\mathcal{M}(K)$ such that for all $\mu\in \Omega $, for all $x\in K$, $h_{\mu}(x)\leq s$. That is, for all $\mu\in \Omega$, for all $h>s$, $E_{\mu}(h)=\emptyset$. \[th2\]
Proof of Theorem \[th1\]
------------------------
In the sequel, we will always denote by $B(x,r)$ (resp. $\overline{B}(x,r)$) the open (resp. closed) ball of center $x\in X$ and radius $r$, where $X$ any metric space.
1\) Let $a\in K$ and $s>0$. Let $(\nu_{n})$ be a dense sequence in $\mathcal{M}(K)$. Let $(d_{n})_{n}$ be a decreasing sequence to $0$.
Let $\theta>\frac{2}{s}$. Put $\beta_{n}=\frac{1}{\log\left|\log d_{n}\right|}$. We consider the following sequences $\alpha_{n}=d_{n}^{\beta_{n}}$, $r_{n}=d_{n}^{\theta s}$, $c_{n}=d_{n}^{\frac{\theta}{2}s}$. Remark that all the sequences are decreasing to $0$.
Denote by $$\mu_{n}=\alpha_{n}\delta_{a}+(1-\alpha_{n})\nu_{n}$$
where $\delta_{a}$ is the Dirac mass at $a$. Since $\rho(\mu_{n},\nu_{n})\leq 2\alpha_{n}\underset{n\to+\infty}{\rightarrow}0$, the sequence $(\mu_{n})_{n}$ is dense in $\mathcal{M}(K)$.
Now put
$$\Omega_{N}(a)=\bigcup_{k\geq N}B(\mu_{k},r_{k})\quad\mbox{and}\quad\Omega(a)=\bigcap_{N=1}^{+\infty}\Omega_{N}.$$ $\Omega(a)$ is a $G_{\delta}$ set in $\mathcal{M}(K)$ since for all $N$, $\Omega_{N}(a)$ is a dense open set.
Let $\mu\in \Omega$. There exists an increasing sequence $(m_{n})$ of integers such that for all $n$, $$\rho(\mu,\mu_{m_{n}})\leq r_{m_{n}}.$$
Since $0<c_{n}\leq d_{n}$, we can construct a Lipschitz function $f_{n}$ which satisfies $0\leq f_{n}(y)\leq c_{n}$ for all $y$ and $f_{n}(y)=c_{n}$ for all $y\in B(a,\frac{d_{n}}{2})$ and $f_{n}(y)=0$ for all $y\notin B(a,d_{n})$ and such that $Lip(f)\leq1$. (For example we can consider the restriction to $K$ of the function $f:{\mathbb{R}}^{d}\rightarrow {\mathbb{R}}$, $f(y)=c_{n}$ if $\left\|y-a\right\|\leq \frac{d_{n}}{2}$; $f(y)=-2\frac{c_{n}}{d_{n}}\left\|y-a\right\|+2c_{n}$ if $\frac{d_{n}}{2}<\left\|y-a\right\|\leq d_{n}$; $f(y)=0$ if $\left\|y-a\right\|>d_{n}$).
We have, $$\int f_{m_{n}}d\mu\leq c_{m_{n}}\mu(B(a,d_{m_{n}})).
\label{eq1}$$
In other part, using the property of the function $f$ we get for all $n$
$$\begin{aligned}
\int f_{m_{n}}d\mu_{m_{n}}&\geq &\alpha_{m_{n}}\int f_{m_{n}}d\delta_{a} \nonumber\\
&\geq&\alpha_{m_{n}}\int_{B(a,\frac{d_{m_{n}}}{2})} f_{m_{n}}d\delta_{a} \nonumber \\
&=&\alpha_{m_{n}}c_{m_{n}}.\label{eq2}\end{aligned}$$
We have $\rho(\mu,\mu_{m_{n}})\leq r_{m_{n}}$, thus using (\[eq1\]) and (\[eq2\]), for all $n$
$$\begin{aligned}
c_{m_{n}}\mu(B(a,d_{m_{n}}))&\geq& \int f_{m_{n}}d\mu\nonumber\\
&\geq& \int f_{m_{n}}d\mu_{m_{n}}-\rho(\mu,\mu_{m_{n}})\nonumber\\
&\geq& \alpha_{m_{n}}c_{m_{n}}-r_{m_{n}}\label{eq3}\end{aligned}$$
then by (\[eq3\]) $$\begin{aligned}
\mu(B(a,d_{m_{n}}))&\geq& \alpha_{m_{n}}-\frac{r_{m_{n}}}{c_{m_{n}}}\\
&=&d_{m_{n}}^{\beta_{m_{n}}}-d_{m_{n}}^{\frac{\theta}{2}s}\\
&=&d_{m_{n}}^{\beta_{m_{n}}}(1-d_{m_{n}}^{\frac{\theta}{2}s-\beta_{m_{n}}})\end{aligned}$$
Then for $n$ sufficiently large we get $$\mu(B(a,d_{m_{n}}))\geq \frac{1}{2}d_{m_{n}}^{\beta_{m_{n}}}.$$
Finally
$$h_{\mu}(a)=\liminf_{r\to0}\frac{\log\mu(B(a,r))}{\log r}\leq \liminf_{n\to+\infty}\frac{\log\mu(B(a,d_{m_{n}}))}{\log d_{m_{n}}}\leq \lim_{n\to+\infty}\beta_{m_{n}}=0.$$ This finish the proof of the first point.
2\) Now we must construct a dense $G_{\delta}$ set of $\mathcal{M}(K)$ which is independent of $a\in A$.
Let $(\nu_{n})$ be a dense sequence in $\mathcal{M}(K)$. Let $\nu=\frac{1}{\gamma(K)}\gamma\llcorner_{K}$, (remark that since $\Theta^{s}_{*}(K,a,\lambda)>0$ for some point $a$ then $\lambda(K)>0$).
Let $(\alpha_{n})_{n}$ be a sequence decreasing to $0$. Denote by $$\mu_{n}=\alpha_{n}\nu+(1-\alpha_{n})\nu_{n}.$$ Since $\rho(\mu_{n},\nu_{n})\leq 2\alpha_{n}\underset{n\to+\infty}{\rightarrow}0$, the sequence $(\mu_{n})_{n}$ is dense in $\mathcal{M}(K)$.
Let $\theta>1+\frac{2}{s}$. We consider the following sequences $d_{n}=\exp\left(-\frac{1}{\alpha_{n}}\right)$, $r_{n}=d_{n}^{\theta s}$ and $c_{n}=d_{n}^{\frac{\theta-1}{2}s}$. All the defined sequences are decreasing to $0$. Remark that $\alpha_{n}=d_{n}^{\beta_{n}}$ with $\lim_{n\to\infty}\beta_{n}=0$.
Now we set $$\Omega_{N}=\bigcup_{k\geq N}B(\mu_{k},r_{k})\quad\mbox{and}\quad\Omega=\bigcap_{N=1}^{+\infty}\Omega_{N}.$$ $\Omega$ is a $G_{\delta}$ set in $\mathcal{M}(K)$ since for all $N$, $\Omega_{N}$ is a dense open set.
Let $\mu\in \Omega$. There exists an increasing sequence $(m_{n})$ of integers such that for all $n$, $$\rho(\mu,\mu_{m_{n}})\leq r_{m_{n}}.$$
Let $a\in A$. By our hypothesis, there exist $c>0$ and $v>0$, such that $$\label{eqPropertyP2}
\mbox{if } 0<r<v,\quad \nu(B(a,r))\geq cr^{s}.$$ Let $f_{n}$ be the Lipschitz function as constructed in the first point of the theorem associated to the newer sequences $(c_{n})$ and $(d_{n})$.
We get for all $n$, $$\int f_{m_{n}}d\mu\leq c_{m_{n}}\mu(B(a,d_{m_{n}})).
\label{eq1secondpoint}$$
In other part, using the property of the function $f$ we get for all $n$ such that $d_{n}\leq v$,
$$\begin{aligned}
\int f_{m_{n}}d\mu_{m_{n}}&\geq &\alpha_{m_{n}}\int f_{m_{n}}d\nu \nonumber\\
&\geq&\alpha_{m_{n}}\int_{B(a,\frac{d_{m_{n}}}{2})} f_{m_{n}}d\nu \nonumber \\
&=&\alpha_{m_{n}}c_{m_{n}}\nu\left(B(a,\frac{d_{m_{n}}}{2})\right)\nonumber\\
&\geq&c'\alpha_{m_{n}}c_{m_{n}}d_{m_{n}}^{s}.\label{eq2secondpoint}\end{aligned}$$
Then by (\[eq1secondpoint\]) and (\[eq2secondpoint\]) we get
$$\begin{aligned}
\mu(B(a,d_{m_{n}}))&\geq& c'\alpha_{m_{n}}d_{m_{n}}^{s}-\frac{r_{m_{n}}}{c_{m_{n}}}\\
&=&c'd_{m_{n}}^{\beta_{m_{n}}+s}-d_{m_{n}}^{\frac{\theta+1}{2}s}\\
&=&d_{m_{n}}^{\beta_{m_{n}}+s}(c'-d_{m_{n}}^{\frac{\theta-1}{2}s-\beta_{m_{n}}})\end{aligned}$$
Then for $n$ sufficiently large we get $$\mu(B(a,d_{m_{n}}))\geq \frac{c'}{2}d_{m_{n}}^{\beta_{m_{n}}+s}.$$
Finally
$$h_{\mu}(a)=\liminf_{r\to0}\frac{\log\mu(B(a,r))}{\log r}\leq \liminf_{n\to+\infty}\frac{\log\mu(B(a,d_{m_{n}}))}{\log d_{m_{n}}}\leq \lim_{n\to+\infty}s+\beta_{m_{n}}=s.$$ This finish the proof of the second point.
Proof of Theorem \[th2\]
------------------------
For all $n\in{\mathbb{N}}$, we put $L_{n}=N_{2^{-n}}(K)$ and let $a_{1,n},\ldots, a_{L_{n},n}$ be $L_{n}$ points of $K$ such that for $i\neq j$ $$B(a_{i,n},2^{-n})\cap B(a_{j,n},2^{-n})=\emptyset.$$ Let $\alpha_{n}=2^{-\sqrt{n}}$. Let $(\nu_{n})$ be a dense sequence in $\mathcal{M}(K)$. We consider the probability measures $$\Pi_{n}=L_{n}^{-1}\sum_{i=1}^{L_{n}}\delta_{a_{i,n}}$$ and $$\mu_{n}=\alpha_{n}\Pi_{n}+(1-\alpha_{n})\nu_{n}.$$ Since $\rho(\mu_{n},\nu_{n})\leq 2\alpha_{n}\underset{n\to+\infty}{\rightarrow}0$, the sequence $(\mu_{n})_{n}$ is dense in $\mathcal{M}(K)$.
Now put $r_{n}=2^{-(s+2)n}$. We set $$\Omega_{N}=\bigcup_{k\geq N}B(\mu_{k},r_{k})\quad\mbox{and}\quad\Omega=\bigcap_{N=1}^{+\infty}\Omega_{N}.$$ $\Omega$ is a $G_{\delta}$ set in $\mathcal{M}(K)$ since for all $N$, $\Omega_{N}$ is a dense open set.
Let $\mu\in \Omega$. There exists an increasing sequence $(m_{n})$ of integers such that for all $n$, $$\rho(\mu,\mu_{m_{n}})\leq r_{m_{n}}.$$
Let $x\in K$. Since $L_{m_{n}}$ is the largest number of disjoint balls of radius $2^{-m_{n}}$ with centers in $K$ then there exists $i\in\left\{1,\ldots,L_{m_{n}}\right\}$ such that
$$B(x,2^{-m_{n}})\cap B(a_{i,m_{n}},2^{-m_{n}})\neq\emptyset.$$ Thus $a_{i,m_{n}}\in B(x,22^{-m_{n}})$.
Let $f_{n}\in Lip(K)$ such that for all $y\in B(x,22^{-m_{n}})$, $f_{n}(y)=2^{-m_{n}}$, for all $y\notin B(x,42^{-m_{n}})$, $f_{n}(y)=0$ and $0\leq f_{n}\leq 2^{-m_{n}}$.
We get for all $n$, $$\int f_{n}d\mu\leq 2^{-m_{n}}\mu(B(x,42^{-m_{n}})).
\label{eq1th2}$$
In other part, using the property of the function $f_{n}$ we get for all $n$,
$$\begin{aligned}
\int f_{n}d\mu_{m_{n}}&\geq &\alpha_{m_{n}}\int f_{n}d\Pi_{m_{n}} \nonumber\\
&\geq&\alpha_{m_{n}}\int_{B(x,22^{-m_{n}})} f_{n}d\Pi_{m_{n}}\nonumber\\
&\geq&\alpha_{m_{n}}L_{m_{n}}^{-1}\int_{B(x,22^{-m_{n}})} f_{n}d\delta_{a_{i,m_{n}}} \nonumber \\
&=&\alpha_{m_{n}}2^{-m_{n}}L_{m_{n}}^{-1}.\nonumber\end{aligned}$$
Let $t$ such that $\overline{\mbox{dim}_{B}}K=s<t<s+1$. Then there exists $v>0$ such that for all $0<r<v$, $$N_{r}(K)<r^{-t}.$$ Thus, for $n$ sufficiently large such that $2^{-m_{n}}<v$ we get $L_{m_{n}}^{-1}>2^{-tm_{n}}$. Then $$\int f_{n}d\mu_{m_{n}}\geq \alpha_{m_{n}}2^{-m_{n}}2^{-tm_{n}}.\label{eq2th2}$$
We have $\rho(\mu,\mu_{m_{n}})\leq r_{m_{n}}$, thus using (\[eq1th2\]) and (\[eq2th2\]), we get for $n$ sufficiently large
$$\begin{aligned}
2^{-m_{n}}\mu(B(x,42^{-m_{n}}))&\geq& \int f_{n}d\mu\nonumber\\
&\geq& \int f_{n}d\mu_{m_{n}}-\rho(\mu,\mu_{m_{n}})\nonumber\\
&\geq& \alpha_{m_{n}}2^{-m_{n}}2^{-tm_{n}}-r_{m_{n}}\label{eq3th2}\end{aligned}$$
then by (\[eq3th2\]) $$\begin{aligned}
\mu(B(x,42^{-m_{n}}))&\geq& \alpha_{m_{n}}2^{-tm_{n}}-2^{m_{n}}r_{m_{n}}\\
&=&2^{-tm_{n}(1+\frac{1}{t\sqrt{m_{n}}})}-2^{-(s+1)m_{n}}\\
&=&2^{-tm_{n}(1+\frac{1}{t\sqrt{m_{n}}})}(1-2^{(t-(s+1))m_{n}+\sqrt{m_{n}}})\end{aligned}$$ $\lim_{n\to+\infty}(t-(s+1))m_{n}+\sqrt{m_{n}}=-\infty$, then for $n$ sufficiently large we get $$\mu(B(x,42^{-m_{n}}))\geq \frac{1}{2}2^{-tm_{n}(1+\frac{1}{t\sqrt{m_{n}}})}.$$
Finally $$\begin{aligned}
h_{\mu}(x)&=&\liminf_{r\to0}\frac{\log\mu(B(x,r))}{\log r}\leq \liminf_{n\to+\infty}\frac{\log\mu(B(x,42^{-m_{n}}))}{\log42^{-m_{n}} }\\
&=&\liminf_{n\to+\infty}\frac{\log\mu(B(x,42^{-m_{n}}))}{\log2^{-m_{n}} }\\
&\leq& \lim_{n\to+\infty}t+\frac{1}{\sqrt{m_{n}}}=t.\end{aligned}$$ Then, for all $t$ such that $s=\overline{\mbox{dim}_{B}}K<t<s+1$, we have $h_{\mu}(x)\leq t$. Thus $h_{\mu}(x)\leq s$.
We conclude that for all $\mu\in \Omega$, for all $x\in K$, $h_{\mu}(x)\leq s$.
The mulitifractal spectrum of typical measures on self-similar sets
====================================================================
In this section we focus on the special case where the compact $K$ is a self-similar set. We recall the definition of such set and some related metric facts that will be useful for our purpose.
Recalls on self-similar sets
----------------------------
We refer the reader to [@Hut], [@Kus], [@Mos] for more properties of self-similar sets.
By ${\mathbb{R}}^{d}$, $d\geq 1$, we denote the $d-$dimensional Euclidean space, by $\mathcal{B}(x,r)$ the balls $\left\{y:\;\left|x-y\right|<r\right\}$, $x\in{\mathbb{R}}^{d}$, $r>0$, $\left|\;\right|$ the canonical Euclidean norm. Let $\mathbf{S}=\left\{S_{1},\cdots,S_{p}\right\}$ be a given set of contractive similitudes, that is $$\left|S_{i}(x)-S_{i}(y)\right|=\alpha_{i}\left|x-y\right|$$
$i=1,\ldots,p$, where we assume that $0<\alpha_{1}\leq\cdots\leq \alpha_{p}<1$. We will call briefly $S_{i}$ an $\alpha_{i}-$similitude.
We use the classical following notations: For $n\in\mathbb{N}^{*}$ we note $\mathbb{A}_n=\{\mathbf{i}=i_{1}\cdots
i_{n}: \ \forall \, k\in\{1,...,n\}, \ i_{k}\in\{1,...,p\}\}$ the sets of words of length $n$ in the alphabet $\{1,...,p\}$. $\mathbb{A}^{*}=\bigcup_{n\geq 1} \mathbb{A}_n$. Finally $\mathbb{A}=\{\mathbf{i}=i_{1}\cdots
i_{k}\cdots: \;i_{k}\in\{1,...,p\}\}$ the set of infinite words.
Without confusion we note in bold characters the elements of $\mathbb{A}$ and $\mathbb{A}^{*}$. For $\mathbf{i}=i_{1}\cdots i_{n}\in\mathbb{A}_n$, we set $|\mathbf{i}|=n$ the length of $\mathbf{i}$. For $\mathbf{i} \in\mathbb{A}$ and $n\geq 1$, we note $\mathbf{i}[n]=i_{1}\cdots i_{n}$.
If $\mathbf{i}=i_{1}\cdots i_{n}\in\mathbb{A}_n$, then $S_{{\bf i}}:= S_{i_{1}}\circ\cdots\circ
S_{i_{n}}$ is a contraction with ratio $\alpha_{{\bf i}}=\alpha_{i_1}\cdot\cdot\cdot \alpha_{i_n}$. If $T$ is any subset of ${\mathbb{R}}^{d}$, then $T_{\mathbf{i}}=S_{\mathbf{i}}(T)$ with the convention $T_{\emptyset}=T$ and $S_{\emptyset}=$Id. Particularly, for $\mathbf{i}\in\mathbb{A}_n$ $K_{\mathbf{i}}$ is called an $n-$complex.
We say that the self similar $K$ satisfy the open set condition, if there exists an open set $U$ such that for all $i\in\left\{1,\cdots,p\right\}$,
$$S_{i}(U)\subset U \quad \mbox{and}\quad S_{i}(U)\cap S_{j}(U)=\emptyset\; \mbox{ if }i\neq j.$$
Let $R>0$. We set $$I(R)=\left\{\mathbf{i}=i_{1}\cdots i_{n}\in \mathbb{A}^{*}\;:\;\alpha_{\mathbf{i}}\leq R<\alpha_{\mathbf{i}[n-1]} \right\}.$$ Let $s$ the real defined by $$\sum_{k=1}^{p}\alpha_{k}^{s}=1.$$ Under the open set condition we have ${\mbox{dim}_{\mathcal{H}}}(K)=s$ and $0<\mathcal{H}^{s}(K)<+\infty$ where $\mathcal{H}^{s}$ is the $s-$Hausdorff measure (see for example [@Fal1]). We set $$\lambda=\frac{1}{\mathcal{H}(K)}\mathcal{H}^{s}\llcorner K.$$
In the sequel we denote by $\sharp A$ the cardinality of the set $A$.
We gather the useful properties for us in the following proposition (see [@Mos], [@Quef]. Some results are also in the proof of the Theorem 9.3 in [@Fal1]).
We assume that the open set condition is satisfied with the open set $U$. Let $s={\mbox{dim}_{\mathcal{H}}}(K)$.
1. There exist two constants $c_{1},c_{2}>0$, such that for all $R>0$, $$\label{cardinalIR}
c_{1}R^{-s}\leq \sharp I(R)\leq c_{2}R^{-s}.$$
2. For all $R>0$, $$\sum_{\mathbf{i}\in I(R)}\alpha_{\mathbf{i}}=1.$$
3. There exist two constants $c_{1},c_{2}>0$ such that for all $R>0$, for all $x\in K$ $$\label{propLambda1}
c_{1}R^{s}\leq \lambda\left(B(x,R)\right)\leq c_{2}R^{s}.$$
4. Let $R>0$.
1. $K=\bigcup_{\mathbf{i}\in I(R)}K_{\mathbf{i}}$.
2. For all $\mathbf{i}, \mathbf{j}\in I(R)$ such that $\mathbf{i}\neq \mathbf{j} $, we have $$\label{propLambda2}
U_{\mathbf{i}}\cap U_{\mathbf{j}}=\emptyset\quad\mbox{and}\quad \lambda\left(K_{\mathbf{i}}\cap K_{\mathbf{j}}\right)=0.$$
\[propositionSelfsimilarset\]
Proof of Theorem \[maintheorem\]
--------------------------------
Let $K$ be a self-similar set associated to the system $\mathbf{S}=\left\{S_{1},\cdots,S_{p}\right\}$ of $\alpha_{i}-$simiitudes satisfying the open set condition, where we assume that $0<\alpha_{1}\leq\cdots\leq \alpha_{p}<1$. We adopt the notations of the previous section. Denote by $s={\mbox{dim}_{\mathcal{H}}}K$.
Since $s=\overline{\mbox{dim}_{B}}K$, then by Theorem \[th2\], we know that there exists a $G_{\delta}$ set $\Omega'$ of $\mathcal{M}(K)$ such that for all $\mu\in\Lambda$, for all $x\in K$, $h_{\mu}(x)\leq s$.
To achieve the proof of the Theorem \[maintheorem\], we will prove that there exists a $G_{\delta}$ set $\Omega''$ of $\mathcal{M}(K)$ such that for all $\mu\in\Omega''$, for all $h\in]0,s]$, $d_{\mu}(h)=h$. Then to recover $h=0$, we fix any point $x_{0}\in K$ and we consider the $G_{\delta}$ set $\Omega(x_{0})$ associated to $x_{0}$ in Theorem \[th1\]. We consider finally $\Omega=\Omega'\cap \Omega''\cap \Omega(x_{0})$ which stills a $G_{\delta}$ set of $\mathcal{M}(K)$.
In our proofs many constants will appear with no importance. To relieve the work, we will sometimes denote the constants by the same letter between consecutive inequalities even if the constants are different.
We adopt the same approach of [@BucSeu1] with suitable modifications.
For any $\mathbf{i}\in I(2^{-J_{N}})$ we pick an $x_{\mathbf{i}}\in K_{\mathbf{i}}$. The family of point $\bigcup_{N}\left\{x_{\mathbf{i}}:\; \mathbf{i}\in I(2^{-J_{N}})\right\}$ will be fixed in the rest of the paper.
We define the following probability measure $$\lambda_{n}=\sum_{\mathbf{i}\in I(2^{-J_{n}}) }\alpha_{\mathbf{i}}^{s}\delta_{x_{\mathbf{i}}}$$ where $\delta_{x_{\mathbf{i}}}$ is the Dirac mass at the point $x_{\mathbf{i}}$. $\lambda_{n}$ is probability measure since $\sum_{\mathbf{i}\in I(2^{-J_{n}}) }\alpha_{\mathbf{i}}^{s}=1$ (see Proposition \[propositionSelfsimilarset\]), and is supported by $K$.
Let $\beta_{n}=\frac{J_{n}}{n}$ and we denote by $$\mu_{n}=\beta_{n}\lambda_{n}+(1-\beta_{n})\nu_{n}$$ the sequence $(\mu_{n})_{n}$ is dense sequence in $\mathcal{M}(K)$ since $\varrho(\mu_{n},\nu_{n})\leq 2\beta_{n}$.
Let $n\in{\mathbb{N}}^{*}$. We introduce $$\Omega_{n}=\bigcup_{k\geq n}B(\mu_{k},2^{-sJ_{k}^{2}}) \quad\mbox{and}\quad \Omega''=\bigcap_{n\geq1}\Omega_{n}$$
$\Omega''$ is a $G_{\delta}$ set of $\mathcal{M}(K)$.
Let $\mu\in\Omega''$ be fixed. There exists a sequence $(J_{N_{p}})_{p\geq1}$ such that for all $p$, $$\varrho(\mu,\mu_{N_{p}})<2^{-sJ_{N_{p}}^{2}}.$$
Let $\theta\geq1$. Let us introduce the set of points $$\Lambda_{\theta,p}=\bigcup_{\mathbf{i}\in I(2^{-J_{N_{p}}})}\overline{B}(x_{\mathbf{i}},2^{-\theta J_{N_{p}}})\cap K.$$ and then let us define $$\Lambda_{\theta}=\bigcap_{P\geq 1}\bigcup_{p\geq P}\Lambda_{\theta,p}.$$
Let $\epsilon>0$. There exist $p_{\epsilon}$ and $c>0$, such that for all $p\geq p_{\epsilon}$ and for all $x\in\Lambda_{\theta,p}$, $${\label{majholdmu}}
\mu(B(x,2 2^{-\theta J_{N_{p}}}))\geq c 2^{-s(1+\epsilon)J_{N_{p}}}.$$
Let $\epsilon>0$ and $x\in \Lambda_{\theta,p}$. Then there exists $\mathbf{i}\in I(2^{-J_{N_{p}}})$ such that $x_{\mathbf{i}}\in \overline{B}(x,2^{-\theta J_{N_{p}}})$. Thus $$\mu_{N_{p}}(\overline{B}(x,2^{-\theta J_{N_{p}}}))\geq \beta_{N_{p}}\alpha_{\mathbf{i}}^{s}\geq c_{1}\beta_{N_{p}}2^{-sJ_{N_{p}}}=c_{1}2^{-s(1+\frac{1}{sN_{p}})J_{N_{p}}}\geq c_{1}2^{-s(1+\epsilon)J_{N_{p}}}$$ for $p$ so large that $\frac{1}{sN_{p}}< \epsilon$.
Let $f_{\theta,p}$ be a lipschitz function on $K$ with $f\in \mbox{Lip}(K)$ such that for all $z\in\overline{B}(x,2^{-\theta J_{N_{p}}})$, $f_{\theta,p}(z)=2^{-\theta J_{N_{p}}}$, for all $z\notin \overline{B}(x,22^{-\theta J_{N_{p}}})$, $f_{\theta,p}(z)=0$, and $0\leq f_{\theta,p}\leq 2^{-\theta J_{N_{p}}}$. By construction, $$\int f_{\theta,p}d\mu\leq 2^{-\theta J_{N_{p}}}\mu(B(x,22^{-\theta J_{N_{p}}}))$$ and
$$\int f_{\theta,p}d\mu_{N_{p}}\geq c_{1}2^{-\theta J_{N_{p}}}2^{-s(1+\epsilon)J_{N_{p}}}$$ thus $$\begin{aligned}
2^{-\theta J_{N_{p}}}\mu(B(x,22^{-\theta J_{N_{p}}}))&\geq&\int f_{\theta,p}d\mu\\
&\geq&\int f_{\theta,p}d\mu_{N_{p}}-\varrho(\mu,\mu_{N_{p}})\\
&\geq& c_{1}2^{-\theta J_{N_{p}}}2^{-s(1+\epsilon)J_{N_{p}}}-2^{-sJ_{N_{p}}^{2}}\end{aligned}$$ when $p$ is sufficiently large $$2^{-sJ_{N_{p}}^{2}}\leq \frac{1}{2}c_{1}2^{-\theta J_{N_{p}}}2^{-s(1+\epsilon)J_{N_{p}}}$$ thus there exists $p_{\epsilon}$ such that for all $p\geq p_{\epsilon}$, $$\mu(B(x,22^{-\theta J_{N_{p}}}))\geq \frac{1}{2}c_{1}2^{-s(1+\epsilon)J_{N_{p}}}.$$
\[propDimHausdorffLambdaTheta\] Let $\theta\geq 1$ and $x\in \Lambda_{\theta}$. Then $h_{\mu}(x)\leq \frac{s}{\theta}$.
If $x\in \Lambda_{\theta}$, then (\[majholdmu\]) is satisfied for infinite number of integer $p$. Hence, for all $\epsilon>0$, there is a sequence of infinite real numbers $(r_{p})$ decreasing to $0$ such that for all $p$ $$\mu(B(x,2 r_{p}))\geq c r_{p}^{\frac{s}{\theta}(1+\epsilon)}$$ this implies that $h_{\mu}(x)\leq\frac{s}{\theta}(1+\epsilon) $ for all $\epsilon>0$, the result follows.
For all $\theta\geq 1$, ${\mbox{dim}_{\mathcal{H}}}\Lambda_{\theta}\leq\frac{s}{\theta}$.
The result is obvious when $\theta=1$, since $\Lambda\subset K$ and $\mbox{dim}_{H}(K)=s$.
Let $\theta>1$ and $t>\frac{s}{\theta}$. For all $P\geq 1$, $\Lambda_{\theta}$ is covered by $\bigcup_{p\geq P}\Lambda_{\theta,p}$. Hence, for any $\delta>0$ and using the fact that $\sharp I(R)\leq cR^{-s}$ (see [@Quef]) we obtain $$\begin{aligned}
\mathcal{H}_{\delta}^{t}(\Lambda_{\theta})&\leq& \mathcal{H}_{\delta}^{t}\left(\bigcup_{p\geq P}\Lambda_{\theta,p}\right) \\
&\leq& \sum_{p\geq P}\sum_{\mathbf{i\in I(2^{-J_{N_{p}}})}}\left|\overline{B}(x_{\mathbf{i}},2^{-\theta J_{N_{p}}})\right|^{t}\\
&\leq& c\sum_{p\geq P}2^{-t\theta J_{N_{p}}}\sharp I(2^{-J_{N_{p}}})\\
&\leq&c \sum_{p\geq P}2^{-t\theta J_{N_{p}}}2^{s J_{N_{p}}}\end{aligned}$$ since $t>\frac{s}{\theta}$, this series is convergent. Hence, $\mathcal{H}_{\delta}^{t}(\Lambda_{\theta})\leq c\lim_{P\to\infty}\sum_{p\geq P}2^{-t\theta J_{N_{p}}}2^{s J_{N_{p}}}=0$. Thus, $\mathcal{H}^{t}(\Lambda_{\theta})=\lim_{\delta\to0}\mathcal{H}_{\delta}^{t}(\Lambda_{\theta})=0$. This implies that $\mbox{dim}_{H}(\Lambda_{\theta})\leq t$ for all $t>\frac{s}{\theta}$, then we conclude.
Let $m$ be any Borel measure on $K$. The Hausdorff dimension of $m$ is defined by $${\mbox{dim}_{\mathcal{H}}}m=\inf\left\{{\mbox{dim}_{\mathcal{H}}}E\;:\; E\subset K,\;m(E)>0\right\}.$$ When $m(E)>0$ then ${\mbox{dim}_{\mathcal{H}}}E\geq {\mbox{dim}_{\mathcal{H}}}m$. In other words, $$\label{propertyDimMesure}
\mbox{if }{\mbox{dim}_{\mathcal{H}}}E< {\mbox{dim}_{\mathcal{H}}}m,\;\mbox{then }m(E)=0.$$
As in [@BucSeu1] we have the following result
\[thMtheta\] For every $\theta\geq1$, there is a measure $m_{\theta}$ supported in $\Lambda_{\theta}$, a constant $C>0$ and a positive sequence $(\eta_{p})$ decreasing to $0$ such that for every Borel set $B$, $$\mbox{if }\left|B\right|\leq \eta_{p},\quad m_{\theta}(B)\leq C\left|B\right|^{\frac{s}{\theta}-\frac{2}{p-1}}.$$ In particular, ${\mbox{dim}_{\mathcal{H}}}m_{\theta}\geq \frac{s}{\theta}$.
We will construct a suitable Cantor set $\mathcal{C}_{\theta}$ included in $\Lambda_{\theta}$ and a measure $m_{\theta}$ supported on $\mathcal{C}_{\theta}$ with monofractal behaviour.
We suppose that the sequence $(J_{N_{p}})$ is sufficiently rapidly decreasing. Precisely, assume that $$\label{eqDecreJnp}
J_{N_{p+1}}>\max((p+1)\theta J_{N_{p}},e^{J_{N_{p}}}).$$
The following lemma will be useful for us to control the cardinality of some sets of balls.
\[lemNombreIntersecBoule\] Let $R>0$, $\mathbf{i}\in I(R)$. For every $c>0$, there exists $M\in{\mathbb{N}}^{*}$ independent of $R$ such that $$\sharp \left\{\mathbf{j}\in I(R);\;\overline{B}(x_{\mathbf{i}},cR)\cap\overline{B}(x_{\mathbf{j}},cR)\neq\emptyset\right\}\leq M.$$
In the sequel we denote by $\lambda$ the measure associated to the self similar $K$, see Proposition \[propositionSelfsimilarset\] for its properties.
Denote by $$T_{\mathbf{i}}=\sharp \left\{\mathbf{j}\in I(R);\;\overline{B}(x_{\mathbf{i}},cR)\cap\overline{B}(x_{\mathbf{j}},cR)\neq\emptyset\right\}.$$
Since $\overline{B}(x_{\mathbf{i}},cR)\cap\overline{B}(x_{\mathbf{j}},cR)\neq\emptyset$ and $\left|K_{\mathbf{j}}\right|=\left|K\right| \alpha_{\mathbf{i}}\leq \left|K\right| R$ then there exists a constant $a$ independent of $R$ such that $$K_{\mathbf{j}}\subset B(x_{\mathbf{i}},a R).$$ Hence $$T_{\mathbf{i}}\leq \sharp \left\{\mathbf{j}\in I(R):\; K_{\mathbf{j}}\subset B(x_{\mathbf{i}},a R)\cap K \right\}.$$ It follows that $$\lambda\left(\bigcup_{\mathbf{j};\;K_{\mathbf{j}}\subset B(x_{\mathbf{i}},a R)}K_{\mathbf{j}}\right)\leq \lambda\left(B(x_{\mathbf{i}},a R)\cap K\right)\leq cR^{s}.$$
Since for $\mathbf{j}\neq\mathbf{j'}\in I(R)$ $\lambda\left(K_{\mathbf{j}}\cap K_{\mathbf{j'}}\right)=0$, we obtain $$\sum_{\mathbf{j}\in I(R):\; K_{\mathbf{j}}\subset B(x_{\mathbf{i}},a R)}\lambda(K_{\mathbf{j}})\leq c R^{s}.$$ We know that for all $\mathbf{j}\in I(R)$, $c'R^{s}\leq \lambda(K_{\mathbf{j}})$, which gives $$c'\sharp \left\{\mathbf{j}\in I(R):\; K_{\mathbf{j}}\subset B(x_{\mathbf{i}},a R)\cap K\right\} R^{s}\leq c R^{s}.$$ Then $$\sharp \left\{\mathbf{j}\in I(R):\; K_{\mathbf{j}}\subset B(x_{\mathbf{i}},a R)\cap K\right\}\leq \frac{c}{c'}.$$ Since $T_{\mathbf{i}}\leq \sharp \left\{\mathbf{j}\in I(R):\; K_{\mathbf{j}}\subset B(x_{\mathbf{i}},a R)\cap K \right\}$ we get the desired result.
Denote by $\widetilde{F}_{1}$ a set formed by the largest number of disjoint balls $B(x_{\mathbf{i}},2^{- J_{N_{1}}})$, $\mathbf{i}\in I(2^{-J_{N_{1}}})$. We denote by $$D_{1}=\left\{\mathbf{i}\in I(2^{-J_{N_{1}}}):\;x_{\mathbf{i}}\mbox{ is a center of ball in }\widetilde{F}_{1}\right\}.$$ Then, we set $$F_{1}=\left\{\overline{B}(x_{\mathbf{i}},2^{-\theta J_{N_{1}}})\cap K:\;\mathbf{i}\in D_{1}\right\}.$$
We denote by $\Delta_{1}=\sharp F_{1}$. Remark that $\sharp F_{1}=\sharp\widetilde{F}_{1}$. Using the Lemma \[lemNombreIntersecBoule\], we get $$\frac{\sharp I(2^{-J_{N_{1}}})}{M}\leq\Delta_{1}\leq \sharp I(2^{-J_{N_{1}}}).$$ Hence $$c\frac{2^{sJ_{N_{1}}}}{M}\leq\Delta_{1}\leq c'2^{sJ_{N_{1}}}.$$
We define a probability measure $m_{1}$ by giving the value $m_{1}(V)=\frac{1}{\Delta_{1}}$ for each element $V\in F_{1}$ and then we extend $m_{1}$ to a Borel probability measure on the algebra generated by $F_{1}$, i.e. on $\sigma(V:\;V\in F_{1})$.
Assume that we have constructed $F_{1},\ldots, F_{p}$, $p\geq 1$ and a measure $m_{p}$ on the algebra $\sigma(V:\;V\in F_{p})$. Let $V\in F_{p}$. There exists $\mathbf{i}\in I(2^{-J_{N_{p}}})$ such that $V=\overline{B}(x_{\mathbf{i}},2^{-\theta J_{N_{p}}})\cap K$.
Let $r_{p}=2^{-\theta J_{N_{p}}}-\left|K\right|2^{-J_{N_{p+1}}}$. We have $\frac{1}{2}2^{-\theta J_{N_{p}}}\leq r_{p}\leq 2^{-\theta J_{N_{p}}}$. Let us consider $$D_{p,\mathbf{i}}=\left\{\mathbf{j}\in I(2^{-J_{N_{p+1}}}):\; K_{\mathbf{j}}\cap B(x_{\mathbf{i}},r_{p})\neq\emptyset \right\}.$$
\[lemCardinalDpi\] There exist two constants $c_{1},c_{2}>0$ such that for all $p$ $$c_{1}2^{-s\theta J_{N_{p}}}2^{sJ_{N_{p+1}}}\leq \sharp D_{p,\mathbf{i}}\leq c_{2}2^{-s\theta J_{N_{p}}}2^{sJ_{N_{p+1}}}.$$
Recall that $K=\bigcup_{\mathbf{j}\in I(2^{-J_{N_{p+1}}})}K_{\mathbf{j}}$. Then, $$\begin{aligned}
B(x_{\mathbf{i}},r_{p})\cap K&=&\bigcup_{\mathbf{j}\in I(2^{-J_{N_{p+1}}})}B(x_{\mathbf{i}},r_{p})\cap K_{\mathbf{j}}\\
&=&\bigcup_{\mathbf{j}\in D_{p,\mathbf{i}}}B(x_{\mathbf{i}},r_{p})\cap K_{\mathbf{j}}\end{aligned}$$ thus $$\begin{aligned}
\lambda\left(B(x_{\mathbf{i}},r_{p})\cap K\right)&=&\lambda\left(\bigcup_{\mathbf{j}\in D_{p,\mathbf{i}}}B(x_{\mathbf{i}},r_{p})\cap K_{\mathbf{j}}\right)\\
&\leq& \sum_{\mathbf{j}\in D_{p,\mathbf{i}}}\lambda(K_{\mathbf{j}})\\
&\leq& c \sharp D_{p,\mathbf{i}} 2^{-sJ_{N_{p+1}}}\end{aligned}$$ But $\lambda\left(B(x_{\mathbf{i}},r_{p})\cap K\right)\geq c'r_{p}^{s}\geq c''2^{-s\theta J_{N_{p}}}$, hence there exists $c_{1}$ such that $$c_{1}2^{-s\theta J_{N_{p}}}2^{sJ_{N_{p+1}}}\leq \sharp D_{p,\mathbf{i}}.$$
In the other hand, for $\mathbf{j}\in D_{p,\mathbf{i}}$, $K_{\mathbf{j}}\cap B(x_{\mathbf{i}},r_{p})\neq\emptyset$, this implies that $K_{\mathbf{j}}\subset B(x_{\mathbf{i}},2^{-\theta J_{N_{p}}}) $ (since $\left|K_{\mathbf{j}}\right|\leq \left|K\right|2^{-J_{N_{p+1}}}$). Thus, $$\bigcup_{\mathbf{j}\in D_{p,\mathbf{i}}} K_{\mathbf{j}}\subset B(x_{\mathbf{i}},2^{-\theta J_{N_{p}}})\cap K$$ hence $$\begin{aligned}
\lambda\left(\bigcup_{\mathbf{j}\in D_{p,\mathbf{i}}} K_{\mathbf{j}}\right)&\leq& \lambda\left(B(x_{\mathbf{i}},2^{-\theta J_{N_{p}}})\cap K\right)\\
&\leq& c 2^{-s\theta J_{N_{p}}}\end{aligned}$$ But, $\lambda\left(\bigcup_{\mathbf{j}\in D_{p,\mathbf{i}}} K_{\mathbf{j}}\right)=\sum_{\mathbf{j}\in D_{p,\mathbf{i}}}\lambda\left(K_{\mathbf{j}}\right)$ (since $\lambda\left(K_{\mathbf{j}}\cap K_{\mathbf{j'}}\right)=0$ for $\mathbf{j}\neq\mathbf{j'}\in I(2^{-J_{N_{p+1}}})$). Thus
$$\sum_{\mathbf{j}\in D_{p,\mathbf{i}}}\lambda\left(K_{\mathbf{j}}\right) \leq c 2^{-s\theta J_{N_{p}}}$$ but $$\sum_{\mathbf{j}\in D_{p,\mathbf{i}}}\lambda\left(K_{\mathbf{j}}\right)\geq c' \sharp D_{p,\mathbf{i}} 2^{-sJ_{N_{p+1}}}$$ thus, there exists $c_{2}$ such that $$\sharp D_{p,\mathbf{i}}\leq c_{2}2^{-s\theta J_{N_{p}}}2^{sJ_{N_{p+1}}}.$$
Let us define the set $\widetilde{F}_{p+1}(V)$ formed by the largest number of balls $B(x_{\mathbf{j}},2^{-\theta J_{N_{p+1}}}):\; \mathbf{j}\in D_{p,\mathbf{i}}$ such that if $\mathbf{j}\neq\mathbf{j}'\in D_{p,\mathbf{i}}$ we have $$B(x_{\mathbf{j}},2^{-J_{N_{p+1}}})\cap B(x_{\mathbf{j'}},2^{-J_{N_{p+1}}})=\emptyset.$$ Then, we set $$F_{p+1}(V)=\left\{\overline{U}\cap K:\; U\in \widetilde{F}_{p+1}(V)\right\}.$$ Remark that for all $U\in F_{p+1}(V)$, $U\subset V=\overline{B}(x_{\mathbf{i}},2^{-\theta J_{N_{p}}})\cap K$.
We have the following lemma
\[lemCardinalFp\] There exist two constants $c'_{1},c'_{2}>0$ such that for all $p$ $$c'_{1}2^{-s\theta J_{N_{p}}}2^{sJ_{N_{p+1}}}\leq \sharp F_{p+1}(V)\leq c'_{2}2^{-s\theta J_{N_{p}}}2^{sJ_{N_{p+1}}}.$$
We have $\sharp F_{p+1}(V)=\sharp \widetilde{F}_{p+1}(V)$. Since $\sharp \widetilde{F}_{p+1}(V)\leq \sharp D_{p,\mathbf{i}}$, we get obviously the second inequality.
Using the lemma \[lemNombreIntersecBoule\] we can pick at least $\frac{c'}{M}\sharp D_{p,\mathbf{i}}$ element of $\widetilde{F}_{p+1}(V)$ such that the balls of radius $2^{-J_{N_{p}}}$ are disjoint. Since $\widetilde{F}_{p+1}(V)$ is of largest cardinality then we conclude that $$\frac{c'}{M}\sharp D_{p,\mathbf{i}}\leq \widetilde{F}_{p}(V)$$ and then we get the first inequality by using the lemme \[lemCardinalDpi\].
Now we define $F_{p+1}=\bigcup_{V\in F_{p}}F_{p}(V)$. We define a probability measure $m_{p+1}$ by giving the mass $m_{p+1}(U)=\frac{m_{p}(V)}{\sharp F_{p+1}(V)}$, where $V$ is the unique element of $F_{p}$ containing $U$. We extend then $m_{p+1}$ to $\sigma(U:\;U\in F_{p+1})$.
Finally we set $$\mathcal{C}_{\theta}=\bigcap_{p\geq1}\bigcup_{V\in F_{p}}V.$$
By the Kolmogorov extension theorem, $(m_{p})_{p\geq1}$ converges weakly to a Borel probability measure $m_{\theta}$ supported on $\mathcal{C}_{\theta}$ and such that for every $p\geq 1$, for every $V\in F_{p}$, $m_{\theta}(V)=m_{p}(V)$.
### Hausdorff dimension of $\mathcal{C}_{\theta}$ and $m_{\theta}$
As is [@BucSeu1] we first prove that $m_{\theta}$ has an almost monofractal behavior on set belonging to $\bigcup_{p}F_{p}$.
When $p$ is sufficiently large, for every $V\in F_{p}$ $$\label{eq1Mtheta}
2^{-sJ_{N_{p}}(1+\frac{1}{p})}\leq m_{\theta}(V)\leq 2^{-sJ_{N_{p}}(1-\frac{2}{p})}$$ and $$\label{eq2Mtheta}
\left|V\right|^{\frac{s}{\theta}+\frac{1}{\left|\log\left|V\right|\right|}}\leq m_{\theta}(V)\leq \left|V\right|^{\frac{s}{\theta}-\frac{1}{\left|\log\left|V\right|\right|}}.$$
Let $V\in F_{p}$. We denote $\Delta_{p+1}(V)=\sharp F_{p+1}(V)$ (recall that $F_{p+1}(V)$ is the set of element of $F_{p+1}$ included in $V$). For $k\leq p$ denote by $V_{k}$ the unique element in $F_{k}$ containing $V$. By construction of the measure $m_{\theta}$ we obtain $$m_{\theta}(V)=\left(\prod_{k=1}^{p}\Delta_{k}(V_{k-1})\right)^{-1}.$$ Using Lemma \[lemCardinalFp\], there exist two constants $c'_{1},c'_{2}>0$ such that $$c'_{1}2^{-s\theta J_{N_{k}}}2^{sJ_{N_{k-1}}}\leq \Delta_{k}(V_{k-1})\leq c'_{2}2^{-s\theta J_{N_{k-1}}}2^{sJ_{N_{k}}}$$
by (\[eqDecreJnp\]) and the fact that $2^{-s\theta J_{N_{k-1}}}\leq1$ we get $$c'_{1}2^{sJ_{N_{k}}(1-\frac{1}{k})}\leq\Delta_{k}(V_{k-1})\leq c'_{2}2^{sJ_{N_{k}}}.$$ Hence $$(c'_{2})^{-p}\left(\prod_{k=1}^{p}2^{sJ_{N_{k}}}\right)^{-1}\leq m_{\theta}(V)\leq (c'_{1})^{-p}\left(\prod_{k=1}^{p}2^{sJ_{N_{k}}(1-\frac{1}{k})}\right)^{-1}.$$ Recalling that by (\[eqDecreJnp\]), $J_{N_{k}}>e^{J_{N_{k-1}}}$ for every $k$, thus $$\lim_{p\to+\infty}\frac{p}{J_{N_{p}}}\left(p\frac{\log c'_{2}}{s\log2}+\sum_{k=1}^{p-1}J_{N_{k}}\right)=0$$ (since $\sum_{k=1}^{p-1}J_{N_{k}}\leq (p-1)J_{N_{p-1}}\leq (J_{N_{p-1}})^{2}$, we get $\frac{p\sum_{k=1}^{p-1}J_{N_{k}}}{J_{N_{p}}}\leq p(J_{N_{p-1}})^{2}e^{-J_{N_{p-1}}}\leq (J_{N_{p-1}})^{3}e^{-J_{N_{p-1}}}\underset{p\to+\infty}{\rightarrow}0$). Thus there exists $p_{1}$ such that for all $p\geq p_{1}$, $$(c'_{2})^{-p}\left(\prod_{k=1}^{p-1}2^{sJ_{N_{k}}(1-\frac{1}{k})}\right)^{-1}\geq 2^{-\frac{s}{p}J_{N_{p}}}$$ hence, for all $p\geq p_{1}$, $$2^{-sJ_{N_{p}}(1+\frac{1}{p})}\leq m_{\theta}(V).$$
As previously, $\lim_{p\to+\infty}\frac{p}{J_{N_{p}}}\left(p\frac{\log c'_{1}}{s\log2}+\sum_{k=1}^{p-1}J_{N_{k}}(1-\frac{1}{k})\right)=0$. Thus there exists $p_{2}$ such that for all $p\geq p_{2}$ $$(c'_{1})^{-p}\left(\prod_{k=1}^{p-1}2^{sJ_{N_{k}}(1-\frac{1}{k})}\right)^{-1}\leq 2^{\frac{s}{p}J_{N_{p}}}$$ hence for all $p\geq p_{2}$ $$m_{\theta}(V)\leq 2^{-sJ_{N_{p}}(1-\frac{2}{p})}.$$
To prove (\[eq2Mtheta\]), remark that for all $V\in F_{p}$, $\left|V\right|\approx 2^{-\theta J_{N_{p}}}$ (where $\approx$ means that the ratio of the two quantities is bounded from below and above by two positives constants independents of $p$ ). Then, $p=o(\left|\log\left|V\right|\right|)$. Thus (\[eq1Mtheta\]) yields (\[eq2Mtheta\]).
Now we extend $(\ref{eq2Mtheta})$ to all Borel subsets of $K$.
There is two positive sequences $(\eta_{p})_{p}$, decreasing to $0$ and a constant $C>0$ such that for any Borel set $B\subset K$ with $\left|B\right|\leq \eta_{p}$ we have $$\label{eqMthetaBorel}
m_{\theta}(B)\leq \left|B\right|^{\frac{s}{\theta}-\frac{2}{p-1}}.$$
We follow the same ideas of [@BucSeu1]. Let $\eta_{p}=2^{-J_{N_{p}}}$. Let $B$ be a Borel set such that $B\subset K$ with $\left|B\right|<\eta_{p}$. Let $q\geq p+1$ the unique integer such that $$2^{-J_{N_{q}}}\leq \left|B\right|<2^{-J_{N_{q-1}}}.$$
Since for all $V,V'\in F_{q-1}$, $V=\overline{B}(x_{\mathbf{j}},2^{-\theta J_{N_{q-1}}})\cap K$, $V'=\overline{B}(x_{\mathbf{j'}},2^{-\theta J_{N_{q-1}}})\cap K$, we have $\overline{B}(x_{\mathbf{j}},2^{-J_{N_{q-1}}})\cap\overline{B}(x_{\mathbf{j'}},2^{-J_{N_{q-1}}})=\emptyset$ then $B$ intersect at most $C$ elements of $F_{q-1}$, where $C$ is a constant independent of $p$.
Let us distinguish two cases
$\bullet$ $2^{-\theta J_{N_{q-1}}}\leq \left|B\right|<2^{-J_{N_{q-1}}}$: if $B$ dont intersect no one of $F_{q-1}$ then $m_{\theta}(B)=0$. Otherwise, denoting by $V$ any one of $F_{q-1}$ intersecting $B$. Using (\[eq1Mtheta\]) we have $$\begin{aligned}
m_{\theta}(B)&\leq& C m_{\theta}(V)\leq C2^{-sJ_{N_{q-1}}(1-\frac{2}{q-1})}\\
&\leq&C\left|B\right|^{\frac{s}{\theta}(1-\frac{2}{q-1})}\leq \left|B\right|^{\frac{s}{\theta}(1-\frac{2}{p-1})}.\end{aligned}$$
$\bullet$ $2^{-J_{N_{q}}}\leq \left|B\right|<2^{-\theta J_{N_{q-1}}}$: Let $V\in F_{q-1}$ that intersect $B$ (if there is no such one then $m_{\theta}(B)=0$). We have proved that for any $U\in F_{q}$, such that $U\subset V$,
$$m_{\theta}(U)=\frac{m_{\theta(V)}}{\Delta_{q}(V)}.$$
By Lemma \[lemCardinalFp\] we have $\Delta_{q}(V)\geq c'_{1}2^{-s\theta J_{N_{q-1}}}2^{sJ_{N_{q}}}$, then $$\label{eqMthetaUFp}
m_{\theta}(U)\leq \frac{1}{c'_{1}} m_{\theta}(V)2^{s\theta J_{N_{q-1}}-sJ_{N_{q}}}.$$ In the other hand, since $B$ is within a ball of side length $C\left|B\right|$, where $C\geq \max\{2,\left|K\right|\}$, the number of elements of $F_{q}$ that intersecting $B$ is less than $c\left|B\right|^{s}2^{sJ_{N_{q}}}$.
Indeed, $B\subset B'$ where $B'$ is a ball of side length $C\left|B\right|$. If $U=\overline{B}(x_{\mathbf{j}},2^{-\theta J_{N_{q}}})\cap K$ such that $U\cap B\neq\emptyset$, then $K_{\mathbf{j}}\subset B'$ (since $\left|K_{\mathbf{j}}\right|\leq \left|K\right|2^{-J_{N_{q}}}$ and then $K_{\mathbf{j}}\subset B(x_{\mathbf{j}},\left|K\right|2^{-J_{N_{q}}})\subset B'$). Denote by $L$ the set of $U\in F_{p}$ such that $U\cap B=\emptyset$ and $S$ the set of $\mathbf{j}\in I(2^{-J_{N_{q}}})$ such that $K_{\mathbf{j}}\subset B'$, for all such $\mathbf{j}$ we have $K_{\mathbf{j}}\subset B'\cap K$. We have $\sharp L\leq \sharp S$. But $$\lambda\left(\bigcup_{\mathbf{j}\in S}K_{\alpha}\right)\leq \lambda\left(B'\cap K\right)\leq c\left|B'\right|^{s}\leq c'\left|B\right|^{s}$$ since $$\lambda\left(\bigcup_{\mathbf{j}\in S}K_{\alpha}\right)=\sum_{\mathbf{j}\in S}\lambda\left(K_{\mathbf{i}}\right)\geq c''\sharp S2^{-sJ_{N_{q}}}$$ we get $$\sharp L\leq \sharp S\leq c\left|B\right|^{s}2^{sJ_{N_{q}}}.$$
Hence, gathering all the estimation above and the fact that $\left|B\right|^{-\frac{1}{\theta}}>2^{J_{N_{q-1}}}$ we get
$$\begin{aligned}
m_{\theta}(B)&\leq&\sum_{U\in L}m_{\theta}(U)\\
&\leq& c'\left|B\right|^{s}2^{sJ_{N_{q}}}m_{\theta}(V)2^{s\theta J_{N_{q-1}}+sJ_{N_{q}}}\leq C\left|B\right|^{s}2^{s\theta J_{N_{q-1}}}m_{\theta}(V)\\
&\leq&C\left|B\right|^{s}2^{s\theta J_{N_{q-1}}}2^{-sJ_{N_{q-1}}(1-\frac{2}{p-1})}\leq C\left|B\right|^{s}2^{sJ_{N_{q-1}}(\theta-1+\frac{2}{q-1})}\\
&\leq&C\left|B\right|^{s}\left|B\right|^{-\frac{1}{\theta}s(\theta-1+\frac{2}{q-1})}\leq C\left|B\right|^{\frac{s}{\theta}(1-\frac{2}{p-1})}.\end{aligned}$$
Already we have all the ingredients to finish the proof of Theorem \[maintheorem\] with the same way as in [@BucSeu1] by considering the sets $$\widetilde{E}_{\mu}(h)=\left\{x\in K:\;h_{\mu}(x)\leq h\right\}=\bigcup_{h'\leq h}E_{\mu}(h).$$ For safe completeness we recover their idea.
\[propSpectre\] For any $h\in]0,s]$, $d_{\mu}(h)=h$.
Let $h\in]0,s]$, and $\theta=\frac{s}{\theta}$.
A standard result claim that for all $h\geq 0$, $$\label{eqdimEmuhtilde}
{\mbox{dim}_{\mathcal{H}}}\widetilde{E}_{\mu}(h)\leq \min\left\{h,d\right\}.$$
Then, by Proposition \[propDimHausdorffLambdaTheta\], $\Lambda_{\theta}\subset \widetilde{E}_{\mu}(h)$. Let us write $$\Lambda_{\theta}=\left(\Lambda_{\theta}\cap E_{\mu}(h)\right)\bigcup \left(\bigcup_{n\geq1}\Lambda_{\theta}\cap\widetilde{E}_{\mu}(h-\frac{1}{n})\right).$$ Now, consider the measure $m_{\theta}$ provided by Theorem \[thMtheta\] which is supported by the Cantor set $\mathcal{C}_{\theta}\subset \Lambda_{\theta}$. We have then $$m_{\theta}(\Lambda_{\theta})\geq m_{\theta}(\Lambda_{\theta})>0.$$ By (\[eqdimEmuhtilde\]), for any $n\geq 1$, ${\mbox{dim}_{\mathcal{H}}}\left(\Lambda_{\theta}\cap\widetilde{E}_{\mu}(h-\frac{1}{n})\right)\leq h-\frac{1}{n}<h$. Since ${\mbox{dim}_{\mathcal{H}}}m_{\theta}\geq \frac{s}{\theta}=h$, then by property (\[propertyDimMesure\]) we deduce that $m_{\theta}\left(\Lambda_{\theta}\cap\widetilde{E}_{\mu}(h-\frac{1}{n})\right)=0$.
Then, we get $m_{\theta}(\Lambda_{\theta})=m_{\theta}\left(\Lambda_{\theta}\cap E_{\mu}(h)\right)>0$. Hence, again by property (\[propertyDimMesure\]) $${\mbox{dim}_{\mathcal{H}}}E_{\mu}(h)\geq {\mbox{dim}_{\mathcal{H}}}\Lambda_{\theta}\cap E_{\mu}(h)\geq \frac{s}{\theta}=h.$$ Finally, the upper bound results from the inclusion $E_{\mu}(h)\subset \widetilde{E}_{\mu}(h)$. This finish the proof of the Proposition \[propSpectre\].
It remains the case $h=0$.
Let $x_{0}\in K$ any fixed point. By Theorem \[th1\], there exists a $G_{\delta}$ set $\Omega(x_{0})$ of $\mathcal{M}(K)$ such that for all $\mu\in \Omega(x_{0})$, $h_{\mu}(x_{0})=0$. Thus, for all $\mu\in \Omega(x_{0})$, $x_{0}\in E_{\mu}(0)$. Hence, for all $\mu\in \Omega(x_{0})$, $E_{\mu}(0)\neq\emptyset$. Thus, for all $\mu\in \Omega(x_{0})$, ${\mbox{dim}_{\mathcal{H}}}E_{\mu}(0)\geq 0 $. As already we have the upper bound, then for all $\mu\in \Omega(x_{0})$, ${\mbox{dim}_{\mathcal{H}}}E_{\mu}(0)=0$.
Consider $\Omega=\Omega'\cap\Omega''\cap\Omega(x_{0})$. $\Omega$ is a $G_{\delta}$ set of $\mathcal{M}(K)$ and for all $\mu in\Omega$, $\mu$ satisfy all the points of Theorem \[maintheorem\].
It remains for us to show that any $\mu\in \Omega$ satisfies the multifractal formalism.
Let $\mu\in \mathcal{M}(K)$. We denote by $N_{j}(K)$, the number of cubes of $\mathcal{G}_{j}$ that intersect $K$. By the concavity of $t\mapsto t^{q}$, for $q\in[0,1]$, we have for any $q\in[0,1]$ $$\begin{aligned}
\sum_{Q\in\mathcal{G}_{j},\mu(Q)\neq 0}\mu(Q)^{q}&=&\sum_{Q\in\mathcal{G}_{j},K\cap Q\neq \emptyset}\mu(Q)^{q}\\
&\leq& N_{j}(K)\left(\frac{1}{N_{j}(K)}\sum_{Q\in\mathcal{G}_{j},K\cap Q\neq \emptyset}\mu(Q)\right)^{q}=N_{j}(K)^{1-q}.\end{aligned}$$ Thus, for all $q\in[0,1]$ $$\begin{aligned}
\tau_{\mu}(q)=\liminf_{j\to+\infty}-\frac{\log\sum_{Q\in\mathcal{G}_{j},\mu(Q)\neq0 }\mu(Q)^{q}}{j\log2}&\geq&(q-1)\limsup_{j\to\infty}\frac{\log N_{j}(K)}{j\log2}\\
&=&(q-1){\overline{\mbox{dim}_{\mathcal{B}}}}(K)=(q-1)s.\end{aligned}$$
Let $\mu\in\Omega$. From (\[majholdmu\]) it follows that for all $h\in[0,s]$, $$\label{eqEncadLegendreSpectre}
d_{\mu}(h)=h\leq\left(\tau_{\mu}\right)^{*}(h)\leq \inf_{q\in[0,1]}(qh-\tau_{\mu}(q)).$$ Hence, for all $h\in[0,s]$, for all $q\in[0,1]$, $\tau_{\mu}(q)\leq (q-1)h$. In particular, for $h=s$, for all $q\in[0,1]$, $\tau_{\mu}(q)\leq (q-1)s$. As we already have the lower bound, we conclude that for all $q\in[0,1]$, $\tau_{\mu}(q)=(q-1)s$. Then, for all $h\in[0,s]$, $$\inf_{q\in[0,1]}(qh-\tau_{\mu}(q))=h=d_{\mu}(h).$$ Thus, the inequalities of (\[eqEncadLegendreSpectre\]) turn to be equalities. Hence, for all $h\in[0,s]$, $$d_{\mu}(h)=h=\left(\tau_{\mu}\right)^{*}(h).$$
[00]{} Z. Buczolich and J.Nagy, Hölder spectrum of typical monotone continuous functions. Real Anal. Exchange [**[26]{}**]{} (2000/1), 133-156.
Z. Buczolich and S. Seuret, Typical Borel measure on $[0,1]^{d}$ satisfy a multifractal formalism. Nonlinearity [**[23]{}**]{}(11), 2010.
Z. Buczolich and S. Seuret, Multifractal spectrum and generic properties of functions monotone in several variables. J. Math. Anal. and Applications, **382**(1), 110-126, 2011.
K. Falconer, Fractal Geometry. John Wiley, Second Edition, 2003.
A. Fraysse and S. Jaffard, How smooth is almost every function in Sobolev space?. Revista Matematica Iberoamericana, **22**, N.2, pp 663-682, (2006).
A. Fraysse, S. Jaffard and J.-P. Kahane, Some generic properties in analysis. C.R.A.S **340** série 1, pp 645-651, (2005).
J. E. Hutchinson, Fractals and self similarity. Indiana. Univ. Math. **30** (1981), 713-747.
S. Kusuoka, Diffusion processes in nested fractals, in “Statitistical Mecanics and Fractals”. Lect. Notes in Math. **1567**. Springer V., 1993.
P. Mattila, Geometry of sets and measures in euclidean spaces (fractals and rectifiability). Cambridge University Press. U. Mosco, Variational fractals, Anna. della Scu. Norm. Super. di Pisa, Classe di Scienze $4^{e}$ série, tome 25, no. 3-4 (1997), p. 683-712.
L. Olsen, Typical $L^{q}-$dimensions of measures. Monatsh. Math. [**[146]{}**]{} (2005), no. 2, 143-157. L. Olsen, Typical $L^{q}-$dimensions of measures for $q\in[0,1]$. Bull. Sci. Math. [**[132]{}**]{} (2008), no. 7, 551-561.
H. Queffélec, Topologie - 2e ed.: Cours et exercices corrigés. Dunod 2002.
|
---
abstract: 'In the framework of the symplectic extension of the Interacting Vector Boson Model (IVBM) a good description of the first excited positive and negative parity bands of the nuclei in the rare earth and the actinide region is achieved. The bands investigated in the model are extended to very high angular momenta as a result of their consideration as “yrast” bands with respect to the symplectic classification of the basis states. The analysis of the eigenvalues of the model Hamiltonian reveals the presence of an interaction between these bands. Due to this iteraction the $\Delta L=1$ staggering effect between the energies of thestates of two bands is also reproduced including the “beat” patterns.'
author:
- |
H. Ganev, V. P. Garistov, A. I. Georgieva\
*Institute of Nuclear Research and Nuclear Energy,*\
* Bulgarian Academy of Sciences, Sofia, Bulgaria*
title: |
Description of the Ground and Octupole Bands\
in the Symplectic Extension of the Interacting Vector Boson Model
---
Introduction
============
The existence of nuclei with stable deformed shapes was realized early in the history of nuclear physics. The observation of large quadrupole moments led to the suggestion that some nuclei might have spheroidal shapes, which was confirmed by the observation of rotational band structures and measurements of their properties. For most deformed nuclei, a description as an axial- and reflection-symmetric spheroid is adequate to reproduce the band’s spectroscopy. Because such a shape is symmetric under space inversion, all members of the rotational band have the same parity. However, with the first observation of negative parity states near the ground state, the possibility arose that some nuclei might have an asymmetric shape under reflection.
On the other hand, whenever symmetry breaking appears new behavior of the many-body system is expected. Reflection symmetry breaking is associated with a static octupole deformation which is expected to determine new collective features for the nuclear system.
Extensive investigations into the structure of nuclei with low-lying negative parity states has led to the conclusion that, while reflection asymmetric shapes can play a role in the band structure, they are not as stable as the familiar quadrupole deformations. The rotational spectra of some even-even nuclei in the rare earth and light actinide region exhibit, next to the ground band, a negative parity band which consists of the states with $I^{\pi }=1^{-},3^{-},5^{-},...$ . These two bands are displaced from each other, which means that fluctuations back to space symmetric shapes must also be significant. Experimentally the presence of “octupole” bands for some isotopes from the light actinide and rare earth region [@revexp] is firmly established.
There is a large variety of models that try to describe this behaviour of the low-ling states of deformed nuclei. Particularly successful are algebraic models based on symmetry principles. The introduction of an additional octupole degrees of freedom is a common feature of most of those models .
The prescription for describing negative parity states by the addition of an $f$ boson to the usual $s$ and $\ d$ of the IBM was first mentioned by Iachello and Arima [@IBM]. It was suggested [@Iach2] that the inclusion of a $p$ boson to the $s,d$ and $f$ bosons may play an important role in the description of these collective states.
The coherent state method (CSM) was applied by Alonso et al. to the *spdf* $SU(3)$ Hamiltonian with quadrupole and octupole interaction [alonso]{}. Recently A. A. Raduta and D. Ionescu [@raduta] have used a generalization of the CSM . They suggested that both ground and octupole bands may be considered as being projected from a single deformed intrinsic state that exhibits both quadrupole and octupole deformations.
Another collective model based on point symmetry group considerations [NS1]{} has also been used very successfully for the description of the energy levels of the ground and octupole bands and reproduces odd-even staggering between these levels [@NS2]. In this model the octupole field is parametrized by irreducible representations of the octachedron point symmetry group.
The introduction of an octupole degrees of freedom in the presence of comparatively large number of free parameters in all of these models allows for the reproduction of the experimental data of the energies of the negative parity states, at least in the low spin region.
In the beginning of the 1980’s a phenomenological algebraic model called the Interacting Vector Boson Model (IVBM) was introduced [@IVBMb]. This model is a generalization of the phenomenological broken $SU(3)$ symmetry model [@RuRa], which provided a good description of the low-lying ground and $\gamma $ bands [@NS3] of well deformed even-even nuclei. Its advantages were incorporated into the rotational limit of the IVBM [IVBMrl]{}, with an a good description of all the positive parity bands of nuclei in the rare earth and the actinide region. Moreover, the $U(6)$ extension of the model contains such sequences of $SU(3)$ multiplets, some of which proves to be convenient for the description of the low-lying negative parity bands [@oepb] .
With the recent advance of the experimental technique the investigated bands were extended to very high angular momenta [@revexp]. This motivated a new approach within the framework of the model aimed at a description of the first positive and negative bands, up to very high spins. In this new application, we make use of the symplectic extension of the IVBM [@ggg]. This allows these bands to be considered as yrast bands in the sense, that we take into account the states with a given $L,$ which minimize the energy values with respect to $N$. $N$ is the eigenvalue of the total number of bosons that build the basis states of the model. Its eigenvalue changes as $\Delta N=2$ in the infinite spaces of the boson representation of $Sp(12,R)$. When considering the dynamical symmetry of the symplectic extension of the model through the maximal compact subgroup $U(6)\supset Sp(12,R)$, we obtain the exactly solvable rotational limit with a Hamiltonian, diagonal in a basis defined by the irreducible representations of the corresponding chain of subgroups. The measured energies of the ground and octupole bands in even-even nuclei from the rare earth and actinide regions are reproduced in the model with rather good accuracy. The analysis of the obtained results shows that this is due to the appearance of a vibrational type term that influences the yrast energies. This term also plays the role of an interaction between the two considered bands, and is the reason for the correct reproduction of the odd-even staggering of their energies.
Algebraic Basis of the IVBM.
============================
We start with a brief review of the model’s assumptions and definitions. The IVBM is based on the introduction of two kinds of vector bosons (called $p$ and $n$ bosons), that built up the collective excitations in the nuclear system. The creation operators of these bosons are assumed to be $SO(3)$ vectors and they transform according to two independent fundamental representations (1,0) of the group $SU(3)$ . These bosons form a pseudospin doublet of the $U(2)$ group and differ in their pseudospin projection $\alpha =\pm \frac{1}{2}.$ The introduction of this additional degree of freedom leads to the extension of the $SU(3)$ symmetry to $U(6)$. Then the operators
$$u_{m}^{+}(\alpha =\frac{1}{2})=p_{m}^{+},\text{ \ \ \ \ \ }u_{m}^{+}(\alpha
=-\frac{1}{2})=n_{m}^{+} \label{bosons}$$
transform according to the fundamental representation $[1]_{6}$ of the group $U(6)$. The annihilation operators are obtained by the conjugation $%
(u_{m}^{+}(\alpha ))^{\dagger }=u_{m}(\alpha )$ and transform according to the conjugate $SU(3)$ representations (0,1)$.$ The bilinear products of the creation and annihilation operators of the two vector bosons generate the noncompact symplectic group $Sp(12,R)$ [@IVBMb]: $$F_{M}^{L}(\alpha ,\beta )=_{k,m}^{\sum }C_{1k1m}^{LM}u_{k}^{+}(\alpha
)u_{m}^{+}(\beta ),$$
$$G_{M}^{L}(\alpha ,\beta )=_{k,m}^{\sum }C_{1k1m}^{LM}u_{k}(\alpha
)u_{m}(\beta ),$$
$$A_{M}^{L}(\alpha ,\beta )=_{k,m}^{\sum }C_{1k1m}^{LM}u_{k}^{+}(\alpha
)u_{m}(\beta ), \label{generators}$$
where $C_{1k1m}^{LM}$ are the usual Clebsh-Gordon coefficients and $L$ and $%
M $ define the transformational properties of (\[generators\]) under rotations.
We consider $Sp(12,R)$ to be the group of the dynamical symmetry of the model [@IVBMb]. Hence the most general one and two-body Hamiltonian can be expressed in terms of its generators . Using commutation relations between the $F_{M}^{L}(\alpha ,\beta )$ and $G_{M}^{L}(\alpha ,\beta )$, the full range of number of bosons preserving Hamiltonian can be expressed in terms of operators $A_{M}^{L}(\alpha ,\beta )$: $$H=\sum_{\alpha ,\beta }h_{0}(\alpha ,\beta )A^{0}(\alpha ,\beta
)+\sum_{M,L}\sum_{\alpha \beta \gamma \delta }(-1)^{M}V^{L}(\alpha \beta
;\gamma \delta )A_{M}^{L}(\alpha ,\gamma )A_{-M}^{L}(\beta ,\delta ),
\label{general hamiltonian}$$ where $h_{0}(\alpha ,\beta )$ and $V^{L}(\alpha \beta ;\gamma \delta )$ are phenomenological constants.
Being a noncompact group, the representations of $Sp(12,R)$ are of infinite dimension, which makes it impossible to diagonalize the most general Hamiltonian. The operators $A_{M}^{L}(\alpha ,\beta )$ generate the maximal compact subgroup of $Sp(12,R)$, namely the group $U(6)$: $$Sp(12,R)\supset U(6)$$ So the even and odd unitary irreducible representations (UIR) of $Sp(12,R)$ split into an infinite but countable number of symmetric UIR of $U(6)$ of the type $[N,0,0,0,0,0]=[N]_{6}$, where $N=0,2,4,...$ for the even set (see Table 1) and $N=1,3,5,...$ for the odd set [@sp4nR]. These subspaces are of finite dimension, which simplifies the problem of diagonalization. Therefore the *complete* spectrum of the system can be calculated only trough the diagonalization of the Hamiltonian in the subspaces of *all* the UIR of $U(6)$, belonging to a given UIR of $Sp(12,R)$.
The rotational limit [@IVBMrl] of the model is further defined by the chain:
$$U(6)\supset SU(3)\otimes U(2)\supset SO(3)\otimes U(1) \label{chain}$$
$$\lbrack N]\ \ \ \ \ \ (\lambda ,\mu )\ \ \ \ \ (N,T)\ \ K\ \ \ \ \ L\ \ \ \
\ \ \ \ \ \ T_{0} \label{qnum}$$
where the labels below the subgroups are the quantum numbers (\[qnum\])corresponding to their irreducible representations. Their values are obtained by means of standard reduction rules and are given in [@IVBMrl]. In this limit the operators of the physical observables are the angular momentum operator $$L_{M}=-\sqrt{2}\sum_{\alpha }\ A_{M}^{1}(\alpha ,\alpha )$$ and the truncated (“Elliott”) quadrupole operator $$Q_{M}=\sqrt{6}\sum_{\alpha }A_{M}^{2}(\alpha ,\alpha ),$$ which define the algebra of $SU(3)$.
The operators of the pseudospincomponents and the number of bosons $N$ : $$\begin{aligned}
T_{+1} &=&\sqrt{\frac{3}{2}}A^{0}(p,n);\ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ T_{-1}=-\sqrt{\frac{3}{2}}A^{0}(n,p); \\
T_{0} &=&-\sqrt{\frac{3}{2}}[A^{0}(p,p)-A^{0}(n,n)];\ \ \text{\ \ \ \ \ \ \ }%
\ N=-\sqrt{3}[A^{0}(p,p)+A^{0}(n,n)],\end{aligned}$$ define the algebra of $U(2)$.
Since the reduction from $U(6)$ to $SO(3)$ is carried out by the mutually complementary groups $SU(3)$ and $U(2)$, their quantum numbers are related in the following way: $$T=\frac{\lambda }{2},\text{ \ \ \ \ \ \ }N=2\mu +\lambda \label{NTcon}$$ Making use of the latter we can write the basis as $$\mid \lbrack N]_{6};(\lambda ,\mu );K,L,M;T_{0}\rangle =\mid
(N,T);K,L,M;T_{0}\rangle \label{bast}$$ The ground state of the system is: $$\mid 0\text{\ \ }\rangle =\mid (N=0,T=0);K=0,L=0,M=0;T_{0}=0\text{ }\rangle
\label{GS}$$ which is the vacuum state for the $Sp(12,R)$ group.
The symplectic extension of IVBM
--------------------------------
The basis states associated with the even irreducible representation of the $%
Sp(12,R)$ can be constructed by the application of powers of raising generators $F_{M}^{L}(\alpha ,\beta )$ of the same group. Each raising operator will increase the number of bosons $N$ by two. As a result we get a realization of the reduction scheme [@sp4nR]:
$$Sp(12,R)\text{ \ }\underrightarrow{{\small N}}\text{\ }U(6)\text{\ \ }%
\underrightarrow{{\small T}^{2}}\text{\ \ \ }SU(2)\times SU(3)\text{\ \ }%
\underrightarrow{{\small T}_{0}}\text{\ \ \ \ }SU(3)$$
The $Sp(12,R)$ classification scheme for the $SU(3)$ boson representations for even value of the number of bosons $N$ is shown on Table 1. Each row (fixed $N$) of the table corresponds to a given irreducible representation of the $U(6)$. Then the possible values for the pseudospin are $T=\frac{N}{2}%
,\frac{N}{2}-1,...$ $0$ and are given in the column next to the respective value of $N$. Thus when $N$ and $T$ are fixed, $2T+1$ equivalent representations of the group $SU(3)$ arise. Each of them is labelled by the eigenvalues of the operator $T_{0}:-T,-T+1,...,T,$ defining the columns of Table 1. The same $SU(3)$ representations $(\lambda ,\mu )$ arise for the positive and negative eigenvalues of $T_{0}$.
Hence, in the framework of the discussed boson representation of the $%
Sp(12,R)$ algebra all possible irreducible representations of the group $%
SU(3)$ are determined uniquely through all possible sets of the eigenvalues of the Hermitian operators $N$ and $T^{2}.$ The equivalent use of the $%
(\lambda ,\mu )$ labels facilitates the final reduction to the $SO(3)$ representations, which define the angular momentum $L$ and its projection $%
M. $ The multiplicity index $K$ appearing in this reduction is related to the projection of $L$ in the body fixed frame and is used with the parity ($%
\pi $) to label the different bands ($K^{\pi }$) in the energy spectra of the nuclei. We define the parity of the states as $\pi =(-1)^{T}$. This allows us to describe both positive and negative bands.
The energy spectrum
-------------------
The Hamiltonian, corresponding to the considered, rotational limit of IVBM,is expressed in terms of the first and second order invariant operators of the different subgroups in the chain (\[chain\]): $$H=aN+\alpha _{6}K_{6}+\alpha _{3}K_{3}+\alpha _{1}K_{1}+\beta _{3}\pi _{3},
\label{Hl}$$ where $K_{n}$ are the quadratic invariant operators of the $U(n)$ - groups, $%
\pi _{3}$ is of the $SO(3)$ second order Casimir operator. As a result of the connections (\[NTcon\]) the Casimir operator $K_{3}$ with eigenvalue $%
(\lambda ^{2}+\mu ^{2}+\lambda \mu +3\lambda +3\mu ),$ is expressed in terms of the operators $N$ and $T$: $$K_{3}=2Q^{2}+\frac{3}{4}L^{2}=\frac{1}{2}N^{2}+N+T^{2}$$
Making use of the above relation, Hamiltonian (\[Hl\]) takes the form $$H=aN+bN^{2}+\alpha _{3}T^{2}+\beta _{3}\pi _{3}+\alpha _{1}T_{0}^{2},$$ and is obviously diagonal in the basis (\[bast\]) labelled by the quantum numbers of the subgroups of group-subgroup chain (\[chain\]). Its eigenvalues are the energies of the basis states of the boson representations of $Sp(12,R)$: $$E((N,T);KLM;T_{0})=aN+bN^{2}+\alpha _{3}T(T+1)+\beta _{3}L(L+1)+\alpha
_{1}T_{0}^{2} \label{Erot}$$
The energy of the ground state (\[GS\]) of the system is obviously $0$.
Application of IVBM for the description of the ground state and octupole bands energies
=======================================================================================
In this paper we modify the earlier application of the IVBM [@oepb] for the description of the first excited even and odd parity bands in order to reach much higher angular momentum states in both band types. We will apply the model to even- even deformed nuclei, which exhibit a low-lying negative parity band next to the ground band traditionally considered to be an octupole band [@revexp]. In order to do this we first have to identify these experimentally observed bands with the sequences of basis states for the even representation of $Sp(12,R)$ given in Table 1. We choose the $SU(3)$ multiplet $(0,\mu )$ for a description of the ground band, whereas for the octupole band the $SU(3)$ multiplet $(2,\mu -1)$ is used. In terms of $(N,T)$ this choice corresponds to $(N=2\mu ,T=0)$ for the positive $(K^{\pi
}=0^{+}) $ and $(N=2\mu +2,T=1)$ for the negative $(K^{\pi }=0^{-})$ parity band, respectively.
Yrast bands
------------
In this way, in the framework of the symplectic extension of boson representations of number preserving $U(6)$ symmetry we are able to consider all even eigenvalues of the number of vector bosons $N$ with the corresponding set of pseudospins $T.$
This approach is based on the fact that the energies (\[Erot\]) increase with increasing of $N$. We define the energies of each state with given $\
L$ as yrast energy with respect to $N$ in the two considered$\ $bands$.$ Hence their minimum values are obtained at $N=2L$ for the ground band, and $N=2L+2$ for the octupole band, respectively. So for the description of the ground band our choice corresponds to the sequence of states with different numbers of bosons, $N=0,4,8,...$ . and pseudospin $T=0$ in the column labelled $T_{0}=0$ of Table 1. Respectively for a description of the negative parity band, we choose the set of states with quantum numbers $%
N=8,12,...$ and $T=1$ from the same column $T_{0}=0$ . Since these quantum numbers uniquely define the ** **$SU(3)$ multiplets, which reduce to the corresponding values of the angular momenta $L$, the ground band belongs to the $SU(3)$ multiplet $(0,\frac{N}{2})$ and the octupole band to $(2,\frac{N}{2}-1)$. In the so defined $SU(3)$ representations for each $%
N$ the maximal values of $L$ appear for the first time (see Table 1.).
According to the correspondence identified above between the basis states $%
T_{0}=0$ and the experimental data on the ground and octupole bands, the last term in the energy formula (\[Erot\]) vanishes. The phenomenological model parameters $a,b,$ $\alpha _{3},$ and $\beta _{3}$ are evaluated by a fit to the experimental data. Their values obtained for some even-even deformed nuclei belonging to light actinides and rare earth region are given in Table 2. The second column gives the numbers of the experimental states used in the fitting procedure.
The comparison between the experimental spectra and our calculations using the values of the model parameters given in Table 2. for the ground and octupole bands of the nuclei $Ra^{224},Th^{226}$, $Sm^{152}$ and $Yb^{168}$ is illustrated in Figure 1. All experimental data are taken from [experdata]{}.
The agreement between the theoretical values obtained with only four model parameters and the experimental data for all the nuclei under consideration is rather good.
Applying the yrast conditions relating $N$ and $\ L$ the energies (\[Erot\]) for two considered bands can be rewritten as:
$$E(L)=\beta L(L+1)+(\gamma +\eta )L+\xi . \label{Evr}$$
The new free parameters $\beta ,\gamma ,\eta ,$ and $\xi $ are related to the previous ones as follows:
$$\beta =4b+\beta _{3},\text{ \ }\gamma =2a-4b,\text{ \ \ }\eta =8b,\text{ \ \
}\xi =2a+4b+2\alpha _{3}. \label{Lpar}$$
The values of $\beta $ and $\gamma $ can be determined only from a fit to the positive band energies, while $\eta $ and $\xi $ are estimated from the negative ones, respectively. The values of the parameters (\[Lpar\])determine the behavior of the energies of the two bands and their position with respect to each other. In some cases ($%
^{232}Th,^{234}U,^{236}U,^{238}U$) the two bands are almost parallel. The shift between them depends on the parameter $\xi .$ When they are very close they interact through the $L$ -dependent interaction with a strength $\gamma
+\eta .$
As a result of our theoretical assumptions we obtained a simple formula for the energy levels. From (\[Evr\]) we can see that eigenstates of the first positive and negative bands consists of rotational $L(L+1)$ and vibrational $L$ modes. The rotational interaction is with equal strength $\beta $ in both of the bands. The obtained values of the parameter $\eta $ are always negative, which means that the negative parity band is less vibrational than the positive one.
The staggering
--------------
In the collective rotational spectra of deformed even-even nuclei in this mass region some fine structure effects as back-banding and staggering behavior are observed . Odd-even staggering patterns between ground and octupole bands have been investigated recently [@NS2]. In order to test further our model we applied on the energies the staggering function defined as [@stag]:
$$Stg(L)=6\Delta E(L)-4\Delta E(L-1)-4\Delta E(L+1)+\Delta E(L+2)+\Delta
E(L-2),$$
where $\Delta E(L)=E(L)-E(L-1).$ This function is a finite difference of fourth order in respect to $\Delta E(L)$ or of fifth order in respect to energy $E(L)$ and is characteristic for the deviation of the rotational behavior from that of the rigid rotor. The calculated and experimental staggering patterns are illustrated in Figure 2. One can see a good agreement with experiment, as well as the reproduction of the beat patterns of the staggering behavior. They occur in the region where the interaction between the two considered bands is most strong or they cross. The correct reproduction of the experimental staggering patterns is due to the interaction term $\eta L$ in (\[Evr\]) between the positive and negative parity bands, which is a result of the introduced notion of yrast energies in the framework of the symplectic extension of the IVBM.
Conclusions
===========
We have applied the Interacting Vector Boson Model for the description of the ground and octupole bands in some even-even rare earth and actinide nuclei up to very high spins. In spite of the simplicity of the model without introducing additional degrees of freedom we are able to describe both positive and negative parity bands. This is due to the specific definition of the states parity depending on the pseudospin quantum number $T$.
The successful reproduction of the experimental energies and of their odd-even staggering was achieved as a result of their consideration as yrast energies in respect to the number of phonon excitation $N$ that build the collective states. The introduction of this notion was possible, as we extended the IVBM to its symplectic dynamical symmetry $Sp(12,R)$, which allows the change of the number of bosons that are the building blocks of the model Hamiltonian. Nevertheless the Hamiltonian remains with only few phenomenological parameters and is still exactly solvable. Through the algebraic properties of the dynamical symmetry chain relations between $%
SU(3) $ and $U(2)$ quantum numbers are established. Combining these relations with the notion of yrast energies the physical meaning of each term of the Hamiltonian is clarified. In the rotational limit of the model in addition to the rotational character of the considered bands an purely vibrational mode is appearing, which introduces also some interaction between them. This is the reason for the reproduction also of the fine effect of the structure of these bands. The obtained physically meaningful results are also simple and easy for use and they permit the application of the model to larger class of nuclei than the purely rotational ones.
The symplectic extension of the Interacting Vector Boson Model permits a richer classification of the states than its unitary version and gives the possibility for a further consideration of other collective bands. In general the model proves appropriate for the description of diverse nuclear structure problems.
Acknowledgements
================
The authors are grateful for fruitful discussions and help on the subject of this paper to professors J. P. Draayer, J. Cseh and D. Bonatsos. This work was partially supported by Bulgarian Science Committee under contract number $\Phi -905$.
[99]{} P. A. Butler and W. Nazarewicz, Rev. of Mod. Phys. *** *****68*** *(1996) 349.
F. Iachello, A. Arima, *”The Interacting Boson Model”,* Cambridge University Press, Cambridge, 1987.
J. Engel and F. Iachello, Phys. Rev. Lett. **54** (1985), 1126; J. Engel and F. Iachello, Nucl. Phys. **A472** (1987) 61; V. Zamfir and D. Kusnezov Phys. Rev. **C 63** (2001) 054306.
Alonso et al., Nucl. Phys. **A586** (1995) 100.
A. A. Raduta and D. Ionescu, Phys. Rev. **C 67** (2003) 044312.
N. Minkov, S. Drenska, P. Raychev, R. Roussev, and D. Bonatsos, Phys. Rev. **C 63** (2001) 044305.
N. Minkov, S. Drenska, Prog. Theor. Phys. Suppl. **146** (2002) 597.
A. Georgieva, P. Raychev, and R. Roussev,* [J. Phys. G: Nucl. Phys.]{} ***8*** *(1982) 1377.
P. Raychev, and R. Roussev, *J. Phys. G: Nucl. Phys.* ** 7*** *(1981) 1227.
N. Minkov, S. Drenska, P. Raychev, R. Roussev, and Dennis Bonatsos, Phys. Rev. **C 55** (1997) 2345.
A. Georgieva, P. Raychev, and R. Roussev, *J. Phys. G: Nucl. Phys.* ** 9*** *(1983) 521.
A. Georgieva, P. Raychev, and R. Roussev,* Interacting Vector Boson Model and Negative Parity Bands in Actinides, Bulg. J. Phys.*** 12*** *(1985) 2 147.
H.Ganev, V. Garistov, A. Georgieva, Proceedings of the XXII International Workshop on Nuclear Theory, Rila Mountains, Bulgaria, June 15-21 2003.
A. Georgieva, M. Ivanov, P. Raychev, and R. Roussev, * Int.J. Theor. Phys.* **28 **(1989) 769.
Mitsuo Sacai Atomic Data and Nuclear Data Tables 31, 399-432 (1984); **Level Retrieval Parameters **http://iaeand.iaea.or.at/nudat/levform.html
D. Bonatsos, C. Daskaloyannis, S. Drenska, N. Kaoussos, N. Minkov, P. Raychev, and R. Roussev, Phys. Rev. **C62** (2000) 024301.
$$\begin{tabular}{lllllll}
\textbf{Table 1.} & & & & & & \\ \hline\hline
\multicolumn{1}{||l}{$N$ (T)$\backslash $ $T_{_{0}}$} & \multicolumn{1}{||l}{%
$...$} & \multicolumn{1}{||l}{$\pm 4$} & \multicolumn{1}{||l}{$\pm 3$} &
\multicolumn{1}{||l}{$\pm 2$} & \multicolumn{1}{||l|}{$\ \pm 1$} &
\multicolumn{1}{||l||}{$\ \ 0$} \\ \hline\hline
\multicolumn{1}{||l}{$0%
\begin{tabular}{l}
$0$%
\end{tabular}%
\ \ $} & \multicolumn{1}{||l}{} & & & & & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(0,0)$%
\end{tabular}%
} \\ \cline{1-1}\cline{6-7}
\multicolumn{1}{||l}{$2%
\begin{tabular}{l}
$1$ \\
$0$%
\end{tabular}%
\ \ $} & \multicolumn{1}{||l}{} & & & & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(2,0)$%
\end{tabular}%
} & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(2,0)$ \\
$(0,1)$%
\end{tabular}%
} \\ \cline{1-1}\cline{5-7}
\multicolumn{1}{||l}{$4%
\begin{tabular}{l}
$2$ \\
$1$ \\
$0$%
\end{tabular}%
\ \ $} & \multicolumn{1}{||l}{} & & & \multicolumn{1}{|l}{%
\begin{tabular}{l}
$(4,0)$%
\end{tabular}%
} & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(4,0)$ \\
$(2,1)$%
\end{tabular}%
} & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(4,0)$ \\
$(2,1)$ \\
$(0,2)$%
\end{tabular}%
} \\ \cline{1-1}\cline{4-6}\cline{6-7}
\multicolumn{1}{||l}{$6%
\begin{tabular}{l}
$3$ \\
$2$ \\
$1$ \\
$0$%
\end{tabular}%
\ \ $} & \multicolumn{1}{||l}{} & & \multicolumn{1}{|l}{$%
\begin{tabular}{l}
$(6,0)$%
\end{tabular}%
\ \ $} & \multicolumn{1}{|l}{%
\begin{tabular}{l}
$(6,0)$ \\
$(4,1)$%
\end{tabular}%
} & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(6,0)$ \\
$(4,1)$ \\
$(2,2)$%
\end{tabular}%
} & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(6,0)$ \\
$(4,1)$ \\
$(2,2)$ \\
$(0,3)$%
\end{tabular}%
} \\ \cline{1-1}\cline{3-7}
\multicolumn{1}{||l}{$8%
\begin{tabular}{l}
$4$ \\
$3$ \\
$2$ \\
$1$ \\
$0$%
\end{tabular}%
\ \ $} & \multicolumn{1}{||l}{} & \multicolumn{1}{|l}{%
\begin{tabular}{l}
$(8,0)$%
\end{tabular}%
} & \multicolumn{1}{|l}{%
\begin{tabular}{l}
$(8,0)$ \\
$(6,1)$%
\end{tabular}%
} & \multicolumn{1}{|l}{$%
\begin{tabular}{l}
$(8,0)$ \\
$(6,1)$ \\
$(4,2)$%
\end{tabular}%
\ \ $} & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(8,0)$ \\
$(6,1)$ \\
$(4,2)$ \\
$(2,3)$%
\end{tabular}%
} & \multicolumn{1}{|l|}{%
\begin{tabular}{l}
$(8,0)$ \\
$(6,1)$ \\
$(4,2)$ \\
$(2,3)$ \\
$(0,4)$%
\end{tabular}%
} \\ \cline{1-1}\cline{3-7}
\multicolumn{1}{||l}{$...$} & \multicolumn{1}{||l}{$...$} &
\multicolumn{1}{|l}{$...$} & \multicolumn{1}{|l}{$...$} &
\multicolumn{1}{|l}{...} & \multicolumn{1}{|l|}{...} & \multicolumn{1}{|l|}{
...} \\ \cline{3-3}
\end{tabular}%
\ \$$
$$\begin{tabular}{llllll}
\textbf{Table 2.} & & & & & \\ \hline\hline
\multicolumn{1}{||l}{Nucleus} & \multicolumn{1}{||l}{$n_{s}$} &
\multicolumn{1}{||l|}{$a$} & \multicolumn{1}{||l|}{$b$} &
\multicolumn{1}{||l|}{$\alpha _{3}$} & \multicolumn{1}{||l||}{$\beta _{3}$}
\\ \hline\hline
\multicolumn{1}{||l}{$Ra^{224}$} & \multicolumn{1}{||l}{\small 13} &
\multicolumn{1}{|l}{\small 0.0119} & \multicolumn{1}{|l}{\small -0.0022} &
\multicolumn{1}{|l}{\small 0.0789} & \multicolumn{1}{|l||}{\small 0.0155} \\
\hline
\multicolumn{1}{||l}{$Ra^{226}$} & \multicolumn{1}{||l}{\small 18} &
\multicolumn{1}{|l}{\small 0.0269} & \multicolumn{1}{|l}{\small -0.0005} &
\multicolumn{1}{|l}{\small 0.0226} & \multicolumn{1}{|l||}{\small 0.0060} \\
\hline
\multicolumn{1}{||l}{$Th^{222}$} & \multicolumn{1}{||l}{\small 26} &
\multicolumn{1}{|l}{\small 0.0558} & \multicolumn{1}{|l}{\small 0.0000} &
\multicolumn{1}{|l}{\small -0.0557} & \multicolumn{1}{|l||}{\small 0.0030}
\\ \hline
\multicolumn{1}{||l}{$Th^{224}$} & \multicolumn{1}{||l}{\small 18} &
\multicolumn{1}{|l}{\small 0.0242} & \multicolumn{1}{|l}{\small -0.0011} &
\multicolumn{1}{|l}{\small 0.0362} & \multicolumn{1}{|l||}{\small 0.0100} \\
\hline
\multicolumn{1}{||l}{$Th^{226}$} & \multicolumn{1}{||l}{\small 20} &
\multicolumn{1}{|l}{\small 0.0194} & \multicolumn{1}{|l}{\small -0.0009} &
\multicolumn{1}{|l}{\small 0.0522} & \multicolumn{1}{|l||}{\small 0.0094} \\
\hline
\multicolumn{1}{||l}{$Th^{228}$} & \multicolumn{1}{||l}{\small 18} &
\multicolumn{1}{|l}{\small 0.0092} & \multicolumn{1}{|l}{\small -0.0020} &
\multicolumn{1}{|l}{\small 0.1470} & \multicolumn{1}{|l||}{\small 0.0138} \\
\hline
\multicolumn{1}{||l}{$Th^{232}$} & \multicolumn{1}{||l}{\small 29} &
\multicolumn{1}{|l}{\small 0.0155} & \multicolumn{1}{|l}{\small -0.0021} &
\multicolumn{1}{|l}{\small 0.3244} & \multicolumn{1}{|l||}{\small 0.0128} \\
\hline
\multicolumn{1}{||l}{$U^{234}$} & \multicolumn{1}{||l}{\small 19} &
\multicolumn{1}{|l}{\small 0.0124} & \multicolumn{1}{|l}{\small -0.0010} &
\multicolumn{1}{|l}{\small 0.3608} & \multicolumn{1}{|l||}{\small 0.0085} \\
\hline
\multicolumn{1}{||l}{$U^{236}$} & \multicolumn{1}{||l}{\small 25} &
\multicolumn{1}{|l}{\small 0.0154} & \multicolumn{1}{|l}{\small -0.0010} &
\multicolumn{1}{|l}{\small 0.2846} & \multicolumn{1}{|l||}{\small 0.0086} \\
\hline
\multicolumn{1}{||l}{$U^{238}$} & \multicolumn{1}{||l}{\small 27} &
\multicolumn{1}{|l}{\small 0.0142} & \multicolumn{1}{|l}{\small -0.0016} &
\multicolumn{1}{|l}{\small 0.2851} & \multicolumn{1}{|l||}{\small 0.0110} \\
\hline
\multicolumn{1}{||l}{$Yb^{168}$} & \multicolumn{1}{||l}{\small 41} &
\multicolumn{1}{|l}{\small 0.0235} & \multicolumn{1}{|l}{\small -0.0056} &
\multicolumn{1}{|l}{\small 0.6512} & \multicolumn{1}{|l||}{\small 0.0295} \\
\hline
\multicolumn{1}{||l}{$Sm^{152}$} & \multicolumn{1}{||l}{\small 15} &
\multicolumn{1}{|l}{\small 0.0194} & \multicolumn{1}{|l}{\small -0.0045} &
\multicolumn{1}{|l}{\small 0.4290} & \multicolumn{1}{|l||}{\small 0.0274} \\
\hline\hline
\end{tabular}%
\ \$$
|
---
abstract: 'We show that there is “no stable free field of index $\alpha\in (1,2)$”, in the following sense. It was proved in [@BPR18] that subject to a *fourth moment assumption*, any random generalised function on a domain $D$ of the plane, satisfying conformal invariance and a natural domain Markov property, must be a constant multiple of the Gaussian free field. In this article we show that the existence of $(1+\ve)$-moments is sufficient for the same conclusion. A key idea is a new way of exploring the field, where (instead of looking at the more standard circle averages) we start from the boundary and discover averages of the field with respect to a certain “hitting density” of Itô excursions.'
author:
- 'Nathanaël Berestycki[^1]'
- Ellen Powell
- 'Gourab Ray[^2]'
bibliography:
- 'EP\_bibliography.bib'
title: '$(1+\eps)$ moments suffice to characterise the GFF '
---
Introduction
============
The **Gaussian free field** (GFF) is a universal object believed (and in many cases proved) to govern the fluctuation statistics of many natural random surface models [@GOS; @NS; @MillerGL; @Kenyon_GFF; @dubedat_torsion; @BLRdimers; @BLRtorus; @DubedatGheissari; @Li] (see, e.g., [@LQGnotes; @LNWP] for an introduction and survey of some recent developments). Although the GFF can be defined in any dimension, this article is concerned with the planar continuum version, which satisfies two special properties; namely, **conformal invariance** and a **domain Markov property**. The former roughly entails that applying a conformal map to a GFF in any domain produces a GFF in the image domain. The latter says, informally, that for any $D' \subset D \subset \C$, the conditional law of the GFF on $D$ restricted to $D'$, given its behaviour outside of $D'$, is that of the harmonic extension of the GFF from $\partial D'$ to $D'$ plus an independent GFF in $D'$. However, one major technical issue with defining the GFF is that it cannot be made sense of as a random function. It is instead defined as a random generalised function, which in this article we view as a stochastic process indexed by smooth, compactly supported test functions. As a result, some preparation is required in order to rigorously formulate the above properties.
We will now formally state our assumptions, which are essentially the same as in [@BPR18] except for the moment condition and the Dirichlet boundary condition (we will comment after the theorem on the necessity of this adaptation).
Assume that for every simply connected domain $D\subset \C$, a stochastic process $h^D = (h^D_\phi)_{\phi \in C_c^\infty(D) }$ indexed by test functions is given. Assume further that each $h^D$ is linear in $\phi$: that is, for any $\lambda, \mu \in \R$ and $\phi, \phi'\in C_c^\infty(D)$, $$h^D_{\lambda \phi + \mu \phi'} = \lambda h^D_\phi + \mu h^D_{\phi'} \text{ almost surely. }$$ We then write, with an abuse of notation, $$(
h^D, \phi) := h^D_\phi \text{ for } \phi \in C_c^\infty(D).$$ We denote by $\Gamma^D $ the law of the stochastic process $h^D$. Thus $\Gamma^D$ is a probability distribution on $\R^{C_c^\infty(D)}$ equipped with the product topology. By Kolmogorov’s extension theorem $\Gamma^D$ is characterised by its consistent finite-dimensional distributions: i.e., by the joint law of $(h^D, \phi_1), \ldots, (h^D, \phi_k)$ for any $k \ge 1$ and any $\phi_1, \ldots, \phi_k \in C_c^\infty(D)$.
We finally recall that the $H^{-1}(D)$ norm of a function $f\in C_c^\infty(D)$ is given by $$\label{eqn:hminus1}
(f,f)_{-1}:=((-\Delta)^{-1/2}f,(-\Delta)^{-1/2}f)=(f,(-\Delta^{-1})f)= \iint_{D\times D} G_D(x,y) f(x)f(y) \, dx dy$$ where $G_D$ is the Green function with Dirichlet boundary conditions in $D$.
Let $D \subset \C$ be a proper simply connected open domain, and let $h^D$ be a sample from $\Gamma^D$.
We make the following assumptions. \[ass:ci\_dmp\]
(i) **(Moments)** For every $\phi\in C_c^\infty(D)$ and some $\xi>1$: $$\E[(h^D,\phi)]=0 \;\; \text{and} \;\; \E[|(h^D,\phi)|^\xi]<\infty.$$
(ii) **(Continuity and Dirichlet boundary conditions)** If $\phi_n\to \phi$ in $C_c^\infty(D)$, then $(h^D,\phi_n)\to (h^D,\phi)$ in probability as $n\to \infty$. Moreover, suppose that $(\phi_n)_{n \ge 1}$ is a sequence of non-negative test functions in $C_c^\infty(D)$, such that $d_n:=\sup\{d(z,\partial D)\, : \, z\in\text{Support}(\phi_n)\}\to 0$ as $n\to \infty$, and $\phi_n \to 0$ in $H^{-1}(D)$. Then we have that $(h^D,\phi_n) \to 0$ in probability and in $L^1$ as $n\to \infty$.
(iii) **(Conformal invariance.)** Let $f: D \to D'$ be a bijective conformal map. Then $
\Gamma^{D} = \Gamma^{D'}\circ f,
$ where $\Gamma^{D'} \circ f$ is the law of the stochastic process $(h^{D'}, |(f^{-1})'|^2 (\phi \circ f^{-1}))_{\phi \in C_c^\infty(D)}$.
(iv) **(Domain Markov property)**. Suppose $D' \subset D$ is a simply connected Jordan domain. Then we can decompose $
h^D= h^{D'}_D+\ph_D^{D'},
$ where:
- $ h^{D'}_D$ is independent of $\ph_D^{D'}$;
- $(\ph_D^{D'},\phi)_{\phi\in C_c^\infty(D)}$ is a stochastic process indexed by $C_c^\infty(D)$ that is a.s. linear in $\phi$ and such that when we restrict to $C_c^\infty(D')$, $$(\ph_D^{D'},\phi)_{\phi\in C_c^\infty(D')}$$ a.s. corresponds to integrating against a harmonic function in $D'$.
- $((h^{D'}_D,\phi))_{\phi\in C_c^\infty(D)}$ is a stochastic process indexed by $C_c^\infty(D)$, such that $(h^{D'}_D,\phi)_{\phi\in C_c^\infty(D')}$ has law $\Gamma^{D'}$ and $(h^{D'}_D,\phi)=0$ a.s. for any $\phi$ with $\operatorname{Support}(\phi)\subset D\setminus D'$.
Observe that in light of *(iii)*, the Dirichlet boundary condition *(ii)* holds in one simply connected domain $D$ if and only if it holds in all simply connected domains. Indeed, suppose that it holds in $D$ and let $f:D\to D'$ be a conformal map. Then if $(\phi_n)_{n}\to 0\in H^{-1}(D')$, we have by conformal invariance of the Green function that $\tilde{\phi}_n:=|f|^2 (\phi_n\circ f)$ converges to $0$ in $H^{-1}(D)$, and since $(h^{D'},\phi_n)$ is equal in law to $(h^D,\tilde{\phi}_n)$, that $(h^{D'},\phi_n)\to 0$ in probability and in $L^1$ as $n\to \infty$.
We now comment on the main changes with respect to the assumptions in [@BPR18]. As already mentioned, the main change is the fact that we have replaced a moment of order four in (i) with a moment of order $\xi$ where $\xi>1$. Beyond this, we have slightly adapted the Dirichlet boundary condition (assumption (ii)). Indeed, it may not even be apparent to the reader at first sight why we call (ii) a Dirichlet boundary condition. Suppose $\phi_n$ is a sequence of functions in $C_c^\infty(D)$, whose support converges to a subset of the boundary $\partial D$, in the sense that $d_n \to 0 $ (where $d_n$ is defined in (ii)). If $h$ is a Gaussian free field in $D$ (with Dirichlet boundary conditions), we may be tempted to believe that $(h, \phi_n) \to 0$. Unfortunately, without any additional assumption this is not necessarily the case, even if $\|\phi_n\|_1 $ is bounded (to see why, consider the uniform distribution in a ball of radius $\eps$ at distance $\eps$ from the boundary). Instead, in order for $(h, \phi_n)$ to converge to zero we need an extra condition which guarantees that the mass of $f_n$ is sufficiently “spread out”. In [@BPR18] we assumed that for $D= \D$, $(h, \phi_n)\to 0 $ for sequences $\phi_n $ which are bounded in $L^1$ and *rotationally symmetric*. However, in the present article, we will need $\phi_n$ to be asymptotically supported on a *proper* subset of the boundary (see the definition of $p_u$ in ) and so rotational invariance of the support of $\phi_n$ is not sufficient. Instead we need to quantify what “sufficiently spread out means”; this is exactly what convergence to 0 in $H^{-1}(D)$ ensures.
Before stating our results, we recall the definition of a Gaussian free field (with Dirichlet boundary conditions) on a domain $D \subset \C$.
\[def::gff\] A mean zero Gaussian free field $h_{\operatorname{GFF}} =h^D_{\operatorname{GFF}} $ with zero boundary conditions is a stochastic process indexed by test functions $(h_{\operatorname{GFF}}, \phi)_{\phi \in C_c^\infty(D)}$ such that:
- $h_{\operatorname{GFF}}$ is a centered Gaussian field; for any $n\ge 1$ and any set of test functions $\phi_1,\cdots, \phi_n \in C_c^\infty(D)$, $((h_{\operatorname{GFF}},\phi_1),\cdots, (h_{\operatorname{GFF}},\phi_n))$ is a Gaussian random vector with mean ${\mathbf{0}}$;
- for any two test functions $\phi_1,\phi_2 \in C_c^\infty(D)$, $$\E[(h_{\operatorname{GFF}},\phi_1) , (h_{\operatorname{GFF}},\phi_2)] = \int_{D} G^D(z,w) \phi_1(z)\phi_2(w)dzdw$$ where $G^D$ is the Green’s function with Dirichlet boundary conditions on $D$.
The main technical content of this paper is summarised by the following proposition, whose most important aspect states that moments of order $\xi$ as in Assumptions \[ass:ci\_dmp\], together with domain Markov property and conformal invariance, imply a moment of order 4.
\[prop:fourth\_moment\] Assume that $(\Gamma^D)_D$ satisfies Assumptions \[ass:ci\_dmp\]. Then in fact:
(1) $\E[(h^D,\phi)^4]<\infty $ for every $\phi\in C_c^\infty(D)$;
(2) the bilinear form $K_2^D$ on $C_c^\infty(D)\times C_c^\infty(D)$ defined by $$\E[(h^D,\phi)(h^D,\phi')]=K_2^D(\phi,\phi'), \quad \quad \phi, \phi' \in C_c^\infty(D)$$ is continuous; and
(3) the convergence in (ii) of Assumptions \[ass:ci\_dmp\] also holds in $L^2$.
As a direct consequence we obtain the following theorem, which is the main result of this paper.
\[thm::characterisation\_gff\] Suppose the collection of laws $\{\Gamma^D\}_{D\subset \mathbb{C}}$ satisfy Assumptions \[ass:ci\_dmp\] and let $h^D$ be a sample from $\Gamma^D$. Then there exists $\sigma\ge 0$ such that $h^D = \sigma h_{\operatorname{GFF}}^D$ in law, as stochastic processes.
This is a direct consequence of Proposition \[prop:fourth\_moment\] and [@BPR18 Theorem 1.6].
#### Proof idea:
In order to explain the new ideas required for Theorem \[thm::characterisation\_gff\], it is helpful to first recall the main steps in the proof of [@BPR18 Theorem 1.6].
*Sketch of proof of [@BPR18 Theorem 1.6].* The proof of Theorem 1.6 in [@BPR18] can be broken into two distinct parts: (1) showing that the field is Gaussian (i.e., that $h^D$ is a Gaussian process for each $D$) and (2) showing that it has the correct covariance structure. In fact, once Gaussianity is known, proving (2) is rather straightforward. It boils down to the fact that the Greens’ function is characterised by harmonicity away from the diagonal and logarithmic blow-up along the diagonal – see [@BPR18].
Proving (1) is rather more challenging. The key step in [@BPR18] is to show that “circle averages" around points are jointly Gaussian. That is, for any finite set of points, the joint law of the circle averages is Gaussian. The circle average process of a Gaussian free field $h^D$ around a point $z\in D$ is, roughly speaking, the process $(h,
\phi_t)_{t\ge 0}$, where $\phi_t$ is uniform measure on the circle of radius $\e^{-t}$ around $z$. More precision is required for a rigorous definition, since the $\phi_t$ are not smooth test functions, but this can be dealt with by approximating the $\phi_t$ appropriately. Once it is known that circle averages are jointly Gaussian, it is easy to deduce (1), because the field can be approximated by circle averages with small radii, and limits of Gaussians are Gaussian.
To address the question of showing Gaussianity of circle averages, let us consider the case where $D=\D$ is the unit disc, and we take averages around a single point: the origin. It is well known and easy to see that for a GFF in $\D$, the circle average process around $z = 0$ is a constant multiple of Brownian motion. For our given process $h^\D$, the domain Markov property together with scale invariance shows that the circle average process has independent and stationary increments. However, one cannot immediately deduce that it is Brownian motion, which would of course yield Gaussianity. More work is required to eliminate processes with jumps (e.g. compound Poisson processes, symmetric stable processes etc.) In [@BPR18], a fourth moment assumption on the field was used to apply Kolmogorov’s criterion, and thereby prove that the circle average process possesses an almost surely continuous modification. This modification must then be Brownian motion and, in particular, Gaussian. In fact, we can generalise this argument to show that arbitrary linear combinations of circle averages around multiple points must also be Gaussian, which completes the key step of the proof.
*Sketch of proof of Proposition \[prop:fourth\_moment\].* The major challenge in this article is to reach the same conclusion *without* the fourth moment assumption. In contrast to the above approach, we will simply aim to prove Gaussianity of single circle averages, rather than linear combinations of averages around multiple points. Note that this does not immediately imply *joint* Gaussianity of circle averages (for which significantly more work would be needed). However, it is enough (with a little extra work) to prove existence of fourth moments (\[prop:fourth\_moment\]) and given the result of [@BPR18], this concludes the proof of \[thm::characterisation\_gff\].
To summarise: the main step of the proof *in this article* is to show existence of an a.s. continuous modification of the circle average process around $z=0$ for $h^\D$ (the given field in the disk $\D$) assuming only $\xi$th moments of the field for some $\xi>1$. See \[cor:circ\_avg\_cont\] and \[prop:circ\_av\_bm\]. Achieving this is not merely a technical upgrade of the idea used in [@BPR18]; a new input is required.
Namely, in we introduce a certain **sine-average process** for the field $h^\H$, on semi-circles in the upper half plane. Its value at a given semi-circle can be viewed as the average of $h^\H$ with respect to a hitting measure for half-plane **Itô excursions** from $0$. As a result, one can easily construct a parametrisation (with respect to the semi-circle radius), under which the resulting process satisfies:
- (one-dimensional) Brownian scaling; and crucially
- a certain **“harness”** property, as introduced by Hammersley in [@harness] (see also [@Williams_harness]).
The increments of this process are easily checked to be independent; however, there is no reason *a priori* why they should be stationary. Nonetheless, we are able to formulate a (new) characterisation of Brownian motion in terms of this harness property and use this to show that the sine-average process must be a Brownian motion. This characterisation is given in Proposition \[prop:char\_BM\], and is an extension of a result proved in [@Wes93]. Crucially, our extension does not require as many moments as [@Wes93]; in fact moments of *any* order $\xi>0$ suffice.
From this point, we use rotational invariance and the domain Markov property to “average out” the semi-circle sine-averages of $h^\H$ and relate them to circle averages of $h^\D$. The consequence is existence of a continuous modification of the circle-average process around $0$ for $h^\D$. For this last step, one needs to precisely control the behaviour of the harmonic part in a domain Markov decomposition of $h^\D$, which forms the main technical part of the argument. This is where the assumption $\xi >1$ is used. Having done this, the proof of Proposition \[prop:fourth\_moment\] is concluded.
Consider a family of fields $(h^D)_D$ in simply connected domains $D$, that assign values $(h^D, \phi)$ to smooth test functions $\phi$. Theorem \[thm::characterisation\_gff\] shows that conformal invariance and the domain Markov property (in the sense of Assumptions \[ass:ci\_dmp\]) are incompatible with these $(h^D,\phi)$s having $\alpha$-stable (rather than Gaussian) distributions, for any value of the index $\alpha \in (1,2)$. Comparing to the better understood one-dimensional situation, a (1d) $\alpha$-stable process has different scaling properties to those of (1d) Brownian motion. Since scaling is a special type of conformal mapping, this suggests that “natural $\alpha$-stable analogues” of the GFF cannot enjoy conformal invariance. Our Theorem can be viewed as a rigourous justification of this informal heuristic when $\alpha \in (1,2)$.
We mention here that some variants of higher dimensional stable fields have been defined and studied before, see [@kumar1972stable] and also [@cipriani2016divisible] for a limiting construction. It will be interesting to find a suitable characterisation theorem for such fields.
In view of the above remark, it is natural to wonder whether *any* moments assumptions are needed to characterize the GFF.
\[Q:xi\] What are the minimal moment assumption necessary for \[thm::characterisation\_gff\] to hold? Do moments of order $\xi$ for any $\xi>0$ suffice?
#### Acknowledgements
We thank Scott Sheffield and Juhan Aru for some inspiring discussions. Part of this work was carried while all three authors visited Banff on the occasion of the programme “Dimers, Ising Model, and their Interactions”. We would like to thank the organisers as well as the team in BIRS for this opportunity and their hospitality.
Preliminaries
=============
Independent random variables
----------------------------
\[lem:indep\_moments\] Suppose that $(X,Y)$ are real-valued random variables defined on the same probability space, and that $X$ and $Y$ are independent. Then for any $\xi>0$, $$\mathbb{E}[|X+Y|^\xi]<\infty \Rightarrow \mathbb{E}[|X|^\xi]<\infty \text{ and } \mathbb{E}[|Y|^\xi]<\infty.$$
Fix some $M$ such that $\mathbb{P}(|Y|\le M)\ge 1/2$ and note that $|X/(X+Y)|\mathbf{1}_{\{|Y|\le M, |X|\ge 2M\}}\le 2$ (it is less than 1 if $X$ and $Y$ have the same sign, and less than $2$ otherwise). Then $\mathbb{E}[|X|^\xi \mathbf{1}_{\{|X|\le 2M\}}]\le (2M)^\xi$ and $$\mathbb{E}\left[|X|^\xi \mathbf{1}_{\{|X|\ge 2M\}}\right]\le 2 \mathbb{E}\left[\left|\frac{X}{X+Y}\right|^\xi \, \left|X+Y\right|^\xi \mathbf{1}_{\{|Y|\le M, |X|\ge 2M\}}\right] \le 4 \mathbb{E}\left[\left|X+Y\right|^\xi \right]<\infty.$$
Symmetrically, $\mathbb{E}[|Y|^\xi]<\infty$.
\[lem:vbe\] Let $r \in [1,2]$.
(i) Suppose that $X,Y$ are random variables with $\mathbb{E}[|X|^r]<\infty, \mathbb{E}[|Y|^r]<\infty, \mathbb{E}[Y|X]=0$ a.s. Then $\mathbb{E}[|X+Y|^r]\ge \mathbb{E}[|X|^r]$.
(ii) Suppose that $r\le 2$ and that $(X_1,\cdots, X_n)$ are independent, centred random variables with $\mathbb{E}[|X_j|^r]<\infty$ for $1\le j \le n$. Then $\mathbb{E}[|\sum_{j=1}^n X_j |^r]\le 2 \sum_{j=1}^n \mathbb{E}[|X_j|^r]$.
Immediate consequences of the domain Markov property
----------------------------------------------------
\[lem:unicity\_decomposition\] The assumption of zero boundary conditions implies that the domain Markov decomposition from (iv) is unique.
This is very similar to the proof of [@BPR18 Lemma 1.4], but we include it since some arguments are slightly different.
Suppose that we have two such decompositions: $$\label{eqn::DMP_uniqueness}
h^D = h^{D'}_D+\ph_D^{D'} = \tilde{h}^{D'}_D+\tilde{\ph}_D^{D'}.$$ Pick any $z\in D'$ and let $f:D'\to \D$ be a conformal map that sends $z$ to $0$. Further, let $(\phi_n)_{n\ge 1}$ be a sequence of nonnegative radially symmetric, mass one functions in $C_c^\infty(\D)$, that are eventually supported outside any $K\Subset \D$. It is easy to check that $\phi_n\to 0$ in $H^{-1}(\D)$ as $n\to \infty$, and if we set $\tilde{\phi}_n := |f'|^2 (\phi_n \circ f)$ for each $n$, then (as discussed below \[ass:ci\_dmp\]) $\tilde{\phi}_n$ converges to $0$ in $H^{-1}(D')$ as well. Hence, the assumption of Dirichlet boundary condition implies that $(h^{D'}_D-\tilde{h}^{D'}_D,\tilde{\phi}_n )\to 0$ in probability as $n\to \infty$. In turn, by (\[eqn::DMP\_uniqueness\]), this means that $(\ph_D^{D'}-\tilde{\ph}_D^{D'},\tilde{\phi}_n) \to 0$ in probability.
However, since $(\ph_D^{D'} - \tilde{\ph}_D^{D'})$ restricted to $D'$ is a.s. equal to a harmonic function, and since the $\phi_n$’s are radially symmetric with mass one, we have that $$(\ph_D^{D'}-\tilde{\ph}_D^{D'},\tilde{\phi}_n) =((\ph_D^{D'}-\tilde{\ph}_D^{D'})\circ f^{-1}, \phi_n)=(\ph_D^{D'}-\tilde{\ph}_D^{D'})\circ f^{-1}(0)=\ph_D^{D'}(z)-\tilde{\ph}_D^{D'}(z)$$ for every $n$. This implies that for each fixed $z\in D'$, $\ph_D^{D'}(z)=\tilde{\ph}_D^{D'}(z)$ a.s. Applying this to a countable dense subset of $z\in D'$, together with the fact that $(h^D,\phi)=(\ph_D^{D'},\phi)=(\tilde{\ph}_D^{D'},\phi)$ a.s. for any $\phi$ supported outside of $D'$, then implies that $\ph_D^{D'}$ and $\tilde \ph_D^{D'}$ are a.s. equal as stochastic processes indexed by $C_c^\infty(D)$.
Now, suppose that $D''\subset D'\subset D$ and $h^D$ is a sample from $\Gamma^D$. Applying the domain Markov property to $h^D$ in $D'$ and $D''$ respectively, we can write $h^D=h_D^{D'}+\varphi_D^{D'} \text{ and } h^D=h_D^{D''}+\varphi_{D}^{D''}.$ We can further decompose $h_D^{D'}=h_{D'}^{D''}+\varphi_{D'}^{D''}$ by applying the domain Markov property to $h_D^{D'}$ in $D''$.
\[lem:nested\_dmp\] As stochastic processes indexed by $C_c^\infty(D)$, we have that $h_D^{D''}=h_{D'}^{D''}$ and $\varphi_D^{D''}=\varphi_D^{D'}+\varphi_{D'}^{D''}$ a.s.
This follows by writing $h^D=h_D^{D''}+\varphi_D^{D''}$ and $h^D=h_D^{D'}+\varphi_{D}^{D'}=h_{D'}^{D''}+\varphi_{D'}^{D''}+\varphi_{D}^{D'}$ and applying Lemma \[lem:unicity\_decomposition\].
\[lem:harm\_ci\] Suppose $D$ is simply connected and that $D'\subset D$ is a simply connected Jordan domain. Then if $h^D=h^{D'}_D+\varphi_D^{D'}$ is the domain Markov decomposition of $h^D$ in $D'$ and $f:D\to f(D)$ is conformal, with $f(D')\subset f(D)$ a Jordan domain and $h^{f(D)}=h_{f(D)}^{f(D')}+\varphi_{f(D)}^{f(D')}$, we have that $$\varphi_D^{D'}=\varphi^{f(D')}_{f(D)}\circ f \text{ in law}$$ as harmonic functions in $D'$.
For $\phi\in C_c^\infty(D')$ let us denote $\phi^f(z) = |(f^{-1})'|^2 \phi \circ f^{-1}(z) $, so that $\phi^f\in C_c^\infty(f(D'))$. Then by conformal invariance (\[ass:ci\_dmp\](iii)) it follows that $$(h^{D} , \phi ) \overset{(d)}{=} (h^{f(D)}, \phi^f) \text{ and } (h^{D'} , \phi ) \overset{(d)}{=} (h^{f(D')}, \phi^f).$$ By uniqueness of the domain Markov decomposition (\[lem:unicity\_decomposition\]), it then follows that $$(\varphi_{D}^{D'} , \phi) \overset{(d)}{=} (\varphi_{f(D)}^{f(D')}, \phi^f)$$ and since $\varphi$ is harmonic, this is exactly the statement that $$\int_{D'} \varphi_{D}^{D'} (z)\phi(z)dz \overset{(d)}{=} \int_{f(D')} \varphi_{f(D)}^{f(D')}(z)\phi^f(z)dz = \int_{D'} \varphi_{f(D)}^{f(D')} (f(w))\phi(w)dw,$$ where the last equality is just the change of variables formula. Since this holds for all $\phi \in C_c^\infty(D')$, this completes the proof.
A priori moment bounds
----------------------
In the following, when $z$ lies in an open set $U\subset \C$, we write $d(z,\partial U):=\inf_{y\in \partial U} |y-z|$.
We are going to give some bounds on the moments of harmonic functions arising from the domain Markov property. Note that if $z\in D'\subset D$ and $\varphi_D^{D'}$ is such a function, then by harmonicity we can write $\varphi_D^{D'}(z)=(\varphi_D^{D'},\phi)=(h^D,\phi)-(h_D^{D'},\phi)$ for some properly chosen $\phi\in C_c^\infty(D')\subset C_c^\infty(D)$ (e.g., take $\phi$ to be a spherically symmetric bump function which integrates to 1). Therefore $$\mathbb{E}[|\varphi_D^{D'}(z)|^p]<\infty$$ for all $0\le p\le \xi$. Moreover, if $D''\subset D'$, then by \[lem:nested\_dmp\] and \[lem:vbe\](i), we have $$\label{eq:mono_moments}
\mathbb{E}[|\varphi_D^{D'}(z)|^p] \le \mathbb{E}[|\varphi_D^{D''}(z)|^p]$$ for all $p\in [1,\xi\vee 2]$.
\[lem:moment\_bound\] Suppose that $D'\subset D$ and that $z\in D'$. Then there exists a universal constant $C$ such that for all $p\in [0,\xi\vee 2]$ $$\mathbb{E}[|\varphi_D^{D'}(z)|^p]\le C \left(\log\left(\frac{d(z,\partial D)}{d(z,\partial D')}\right)\vee 1\right)$$
Let $r:=d(z,\partial D')/2$ and $R:=d(z,\partial D)/2$. By Jensen’s inequality we need only consider the case $p=\xi$. In this case, since $\xi>1$ and $B_z(r)\subset D'$, we may further assume by that $D'=B_z(r)$.
Now we iteratively apply \[lem:nested\_dmp\]. Let $B_k=B_z(2^k r)$ for $k\in \mathbb{N}_0$, and let $N:=\sup_{k\in \mathbb{N}_0} B_k\subset D$ so that $ N\le \log(R/r)/\log(2)$. Then we may write $$\varphi_{D}^{D'}(z)=\varphi_D^{B_N}(z)+\sum_{k=0}^{N-1} \varphi_k(z)$$ where the $\varphi_k(z)$ are independent and each distributed as $\varphi_{\D}^{\D/2}(0)$. Therefore by \[lem:vbe\](ii), it follows that $$\mathbb{E}[|\varphi_D^{D'}(z)|^\xi]\le \mathbb{E}[|\varphi_D^{B_N}(z)|^\xi] + N \mathbb{E}[|\varphi_{\D}^{\D/2}(0)|^\xi].$$ Now $\mathbb{E}[|\varphi_{\D}^{\D/2}(0)|^\xi]$ is bounded by some absolute constant, and so is $\mathbb{E}[|\varphi_D^{B_N}(z)|^\xi]$ (since by conformal invariance, the Koebe quarter theorem and \[lem:vbe\](i), it is less than or equal to $\mathbb{E}[|\varphi_{\D}^{(1/16)\D}(0)|^\xi]$). This completes the proof.
Sine-averages and harmonic functions {#sec:sine_avgs}
====================================
In the following we will denote the unit disc $\{z: |z|<1\}$ of $\C$ by $\D$, and the upper unit semi disc $\D\cap \H$ by $\D^+$. For $r>0$, we denote by $r\D^+$ the scaled disc $\{z\in \C: |z|<r\}$.
For $u>0$, we define $p_u$ to be the measure that integrates against $\phi\in C_c(\C)$ as $$\label{eq:sine}
(\phi,p_u)= p_u(\phi):=\sqrt{u} \int_{0}^{\pi} \sin(\theta) \phi\left(\frac{\e^{i\theta}}{\sqrt{u}}\right) \, d\theta.$$
Note that $p_u$ is supported on the circle of radius $r_u = 1/ \sqrt{u}$ and that its total mass is $2/r_u = 2 \sqrt{u}$.
The motivation for defining these measures comes from the fact that $f(re^{i\theta}) = \frac1r\sin(\theta)$ is harmonic in the upper half plane with zero boundary conditions (except at the origin). In fact, $f$ can be interpreted as the hitting density on a circle of radius $r$, for an Itô excursion in the upper-half plane starting from zero. While we our proofs can be written without referring to this interpretation, it may be useful for the intuition nonetheless, so we will now explain how to state this more precisely.
We start by recalling some background about such excursions (see Chapter 5.2 in [@Lawler-book] for further details). Let $\P_{i \eps}$ denote the law of Brownian motion starting from $i\eps$, killed when it leaves the upper-half plane $\H$. By definition, the **Itô excursion measure** from zero is the (infinite) measure $\N$ obtained as the vague limit $$\N : = \lim_{\eps \to 0} \frac1{\eps} \P_{i \eps}$$ which is supported on continuous trajectories $\omega $ starting from zero, such that $\omega (t) \in \H$ for $t \in (0, \zeta)$ where $\zeta = \zeta(\omega)$ is the lifetime of the excursion, and such that $\omega(t) = \omega(\zeta) \in \R$ for any $t \ge \zeta$. A “sample” from $\N$ will later be called a half-plane excursion. More generally, the corresponding excursion measure can be defined on any simply connected domain $D$ from a nice boundary point $z \in \partial D$, and we then denote it by $\N_{z, D}$.
Note that even though $\N$ has infinite mass we can easily make sense of conditional laws $\N(\cdot | E)$ when $\N(E)\, \in(0,\infty)$, thus resulting in probability measures. We record the following lemma.
\[L:exc\] The total mass of half-plane excursions reaching $\partial (r \D) \cap \H$ is $4/(\pi r)$. In fact, the mass of excursions leaving $r\D \cap \H$ through the arc $(re^{ia}, re^{ib})$ is precisely $$\frac2{\pi r} \int_a^b \sin (\theta) d\theta$$ for any $0 \le a \le b \le \pi$.
Note that $\N_{z;D}$ is conformally covariant: applying a conformal map $f: D \to D'$ such that $f$ is sufficiently nice near $z$, the image of $\N_{z, D}$ under $f$ is given by $|f'(z)| \N_{f(z); D' }$. Note also that when $D = \H$ and $z = \infty$, the measure $\N_{\infty, \H} ( X(\zeta_\H) \in [a, b]) = b-a$ on $\R$, is nothing but Lebesgue measure (here $\zeta_D$ denotes the first time that the excursion $X$ leaves the domain $D$, i.e., its lifetime). This is easy to check, as starting from a point $ir$ (with $r>0$) the hitting distribution of $\R$ by a Brownian motion has the Cauchy distribution scaled by $r$, which tends to $\pi^{-1}$ times Lebesgue measure on $\R$ as $r \to \infty$.
For $r >0$, consider the conformal maps $$f(z) = z + \frac{r^2}{z} = r( \frac{r}{z} + \frac{z}{ r}),$$ that map $\H \setminus (r\D)$ to $\H$ and satisfy $f(\infty) = \infty$ with $|f'(\infty) | =1$. Note that $f(r e^{i \theta}) = 2 r \cos (\theta)$. In particular $f$ sends the semicircle of radius $r$ to the interval $[-2r, 2r]$, of length $4r$. Hence if $\tau_r$ is the first hitting time of this circle, we have $$\N_{\infty, \H} ( \tau_{r} < \zeta) = 4r/\pi.$$ The first claim of the lemma follows from this after applying the inversion map $z \mapsto -1/z$ (which sends $\infty$ to 0, leaves $\H$ invariant, and transforms $r\D$ into $(1/r) \D$). The second claim follows easily after noting that the derivative in $\theta$ of $f(re^{i \theta})$ is $-2 r\sin (\theta)$.
\[R:exc\] For later reference, it may be useful to note that half-plane excursions enjoy the following Markov property: conditionally upon hitting the circle of radius $r$, the law of an excursion after this time is simply that of Brownian motion killed upon leaving $\H$.
Combined with the domain Markov property and scale invariance of our fields, the result is that when we “integrate $h^\H$ against $f$ on the semi-circle of radius $1/\sqrt{u}$ around $0$” - equivalently “test $h^\H$ against $p_u$” - and view this as a process in $u$, it will satisfy both Brownian scaling and a certain Markovian property (note that $u = 0$ corresponds to testing $h$ near the point at $\infty$). As a consequence, we may deduce that the process is Brownian motion – see \[sec:char\_BM\]. However, the reader may recall from the introduction that we really want *circle averages*, say for $h^\D$, to be Brownian motions. Since these processes are easily shown to have independent and stationary increments, this would be immediate if we knew that *they* satisfied Brownian scaling. Unfortunately, this seems very hard to deduce directly from \[ass:ci\_dmp\]. So, we introduce the measures $p_u$ (and associated sine-averages for $h^\H$, see below) instead, and will later relate them to circle averages in \[sec:circ\_avg\_gaussian\]. We remark that alternative measures to $p_u$, for example correctly defined variants in cones, could play the same role. The current set-up has been chosen as it seems to be the neatest.
Now, in order to make sense of “testing $h^\H$ against $p_u$” we need to first approximate $p_u$ by smooth test functions. For $\delta\in (0,\pi/2)$ we let $p_u^\delta$ be defined in the same way as $p_u$, but replacing $\sin(\theta)$ in the integral above with $\sin(\theta) \chi^\delta(\theta)$, where $\chi^\delta:[0,\pi]\to [0,1]$ is smooth, equal to $1$ in $[\delta, \pi-\delta]$, and equal to $0$ in $[0,\delta/2]\cup [\pi-\delta/2,\pi]$. Finally, for $\eta:[0,1]\to [0,1]$ a smooth bump function with $\int_{0}^{1}\eta(y) \, dy=1$, we define $\eta^{\delta}(\cdot):=\frac{1}{\delta}\eta(\frac{\cdot}{\delta})$ and denote by $p_u^{\delta,in},p_u^{\delta,out}$ the measures that integrate against $\phi\in C_c(\C)$ as $$(\phi,p_u^{\delta,in\,}):= \int_0^\delta (\phi,p^\delta_{u(1+x)})\, \eta^{\delta}(x) \, dx \;\; ; \;\; (p_u^{\delta,out},\phi):=\int_0^\delta (\phi,p^\delta_{u(1-x)})\, \eta^{\delta}(x) \, dx.$$ Thus $p_u^{\delta,in},p_u^{\delta,out}$ are smooth “fattenings” of the measure $p_u$ to the inside and outside of the arc $\partial (\frac1{\sqrt{u} } \D^+)$ respectively, that are also “cut off” away from the real line (so as to have compact support in $\H$). The reason for these definitions is the following:
\[rmk:fattening\_smooth\] We have that $$(p_u^{\delta,in/out},\phi)=\int_{\C} p_u^{\delta,in/out}(z)\phi(z) \, dz$$ for some $p_u^{\delta,in/out}\in C_c^\infty(\C)$ (note the abuse of notation $p_u^{\delta, in/out}$ for both measure and density here). We remark that it is possible to write down an explicit expression for $p_u^{\delta,in/out}(z)$, but we do not need it.
The upshot is that we can define $$(h^D,p_u^{\delta,in/out})$$ for any $D$ such that $\mathrm{Support}(p_u^{\delta,in/out})\Subset D$ (e.g., $D=\D^+$ or $D=\H$).
From here on in, we use the notation $$\mathsf D_u:=\frac{1}{\sqrt{u}}\D^+; \;\;\; u>0.$$
\[lem:harmonic\_sines\]
(a) Suppose that $u>0$ and $\varphi$ is a harmonic function in $\sou$, that can be extended continuously to a function on $\H\cup(-\frac{1}{\sqrt{u}}, \frac{1}{\sqrt{u}})$ that is equal to zero on $(-\frac{1}{\sqrt{u}}, \frac{1}{\sqrt{u}})$. Then $(\varphi,p_r)_{r\in (u,\infty)}$ is constant.
(b) Suppose that $u>0$ and $\varphi$ is a harmonic function in $\H\setminus \overline{\sou}$ that can be extended continuously to $0$ on $(-\infty,- \frac{1}{\sqrt{u}})\cup (\frac{1}{\sqrt{u}},\infty)$. Then $(\varphi,p_s)_{s\in (0,u)}$ is a linear function of $s$.
(c) Suppose that $0<s<r<\infty$ and $\varphi$ is a harmonic function in $\sos\setminus \overline{\sor}$ that can be extended continuously to 0 on $(-\frac{1}{\sqrt{s}},-\frac{1}{\sqrt{r}})\cup (\frac{1}{\sqrt{r}},\frac{1}{\sqrt{s}})$. Then $(\varphi,p_u)_{u\in (r,s)}$ is a linear function of $u$.
We observe that (a) is easily seen from the perspective of Itô excursions. By \[L:exc\], we can represent $(\varphi,p_r)$ for any $r>u$ by $\frac{\pi}{2}\N_{0,\H}(\varphi(X_{\tau_{(1/\sqrt{r})}\wedge \zeta}))$ where $\tau_{(1/\sqrt{r})}$ is the first hitting time of the semicircle of radius $(1/\sqrt{r})$ centred at $0$. For $s\ge r$, since $\varphi$ is assumed to be 0 on $(-1/\sqrt{u},1/\sqrt{u})$, we can apply the Markov property, \[R:exc\], of the excursion $X$ at $\tau_{(1/\sqrt{s})}\wedge \zeta$. This gives $(\varphi,p_r)=\sqrt{s} \int_0^\pi \sin(\theta) \E_{e^{i\theta/\sqrt{s}}}[\varphi(B_{\tau_{\sor}})] \, d\theta$ for $B$ a complex Brownian motion. By harmonicity of $\varphi$, this quantity is equal to $(\varphi,p_s)$ as required.
Actually, it can be seen from the argument above that the constant value of $(\varphi,p_r)$ for $r>u$, is equal to $\pi/2$ times the normal derivative, directed *into* $\H$, of $\varphi$ at the origin. Indeed, we saw that for any such $r$, $$(\varphi,p_r)=\frac{\pi}{2}\N_{0,\H}(\varphi(X_{\tau_{(1/\sqrt{r})}\wedge \zeta}))=\frac{\pi}{2}\lim_{\eps\to 0}\eps^{-1}\mathbb{E}_{i\eps}(\varphi(B_{\tau_{(1/\sqrt{r})}\wedge \zeta}))=\frac{\pi}{2}\lim_{\eps\to 0} \eps^{-1}\varphi(i\eps).$$ where the second equality is by definition of $\N_{0,\H}$ (with $B$ a Brownian motion) and the third is by harmonicity of $\varphi$.
[Since it is simpler for (b) and (c), the full proof of \[lem:harmonic\_sines\] below is of a more deterministic nature.]{}
Write $\varphi(r\e^{i\theta})=\varphi(r,\theta)$ and $f(u)=(\varphi,p_u)=\sqrt{u} \int_0^\pi \sin(\theta)\varphi(1/\sqrt{u},\theta) \, d\theta$. We will show that $f''\equiv 0$ on $(s,t)$, which implies (c).
Take any $u\in (s,r)$. Let us first remark, in order to justify differentiation under the integral and integration by parts in what follows, that $\varphi$ is in fact very regular in open neighbourhoods of $\pm (1/\sqrt{u})$ inside $\sos\setminus \overline{(\sor)}$. Indeed since $\varphi$ extends continuously to $0$ on neighbourhoods of $\pm (1/\sqrt{u})$ in $\R$, it can be extended by Schwarz reflection to a harmonic function in open balls $B_\eps(\pm 1/\sqrt{u})\subset \C$ for some $\eps$. See, for example, [@krantz §7.5.2]. In particular $\frac{\partial \varphi }{\partial \theta} $ remains bounded in neighbourhoods of $\pm 1/\sqrt{u}$. Now we compute $$\begin{aligned}
\frac{d^2}{du^2}(\sqrt{u} \varphi(1/\sqrt{u},\theta)) & = & \frac{1}{4u^{5/2}}\left( \frac{\partial^2}{\partial r^2}\varphi(1/\sqrt{u},\theta)+ \sqrt{u}\frac{\partial}{\partial r}\varphi(1/\sqrt{u},\theta)-u\varphi(1/\sqrt{u},\theta) \right) \\ & = & -\frac{1}{4u^{3/2}} \left(\frac{\partial^2}{\partial \theta^2}\varphi(1/\sqrt{u},\theta)+\varphi(1/\sqrt{u},\theta)\right)
,\end{aligned}$$ using harmonicity of $\varphi$ for the final identity. Differentiating under the integral in the expression for $f(u)$, and apply integration by parts twice with respect to $\theta$, we see that $f''(u)=0$.
\[prop:def\_sine\_avg\] Let $h^\H$ be a sample from $\Gamma^\H$. Then for any $u\in (0,\infty)$ the limits $$\label{eq:hpu_def} \lim_{\delta\downarrow 0}(h^\H,p_u^{\delta,in}) \text{ and }
\lim_{\delta\downarrow 0} (h^\H,p_u^{\delta,out})$$ exist in probability and in $L^1$, and are equal a.s. We define this limiting quantity to be the $(1/\sqrt{u})$-**sine average** of $h^\H$, and denote it (with a slight abuse of notation) by $(h^\H, p_u)$. Recall the notation $h^\H=h_\H^D+\varphi_\H^D$ for the domain Markov decomposition of $h^\H$ in $D\subset \H$. We also have that with probability one: $$\label{eq:hpu_alt_def} (h^\H,p_u)=(\varphi_{\H}^{\sou},p_r) \text{ for all } r> u \, \text{ \bfseries{and} } \, (h^\H,p_u)=\frac{u}{s}(\varphi_{\H}^{\H\setminus \overline{\sou}},p_s) \text{ for all } s< u.$$
\[rmk:hpu\_process\] This directly implies that for any finite collection $u_1,\cdots, u_n\in (0,\infty)$, the limits in hold jointly in probability, and holds jointly almost surely. In particular, this defines a consistent family of finite dimensional marginals, from which we may define the stochastic process $$(h^\H,p_u)_{u\in (0,\infty)}.$$
Before we begin the proof of \[prop:def\_sine\_avg\], we need the following lemma. It says (albeit in a more specific setting) that if we apply the domain Markov property to our field in a subdomain that shares a section of boundary with the original domain, then the harmonic function can be extended continuously to $0$ on the common section of boundary. This should seem very intuitive, but the proof is a little trickier than one might guess (see for example Fatou’s theorem for the kind of conditions that guarantee existence of non-tangential limits for harmonic functions at the boundary).
\[lem:harm\_0\_boundary\] Suppose that $h^\H=h_\H^{\D^+}+\varphi_{\H}^{\D^+}$ is the domain Markov decomposition of $h^\H$ in $\D^+$. Then $\varphi_{\H}^{\D^+}$ can almost surely be extended continuously to $0$ on $(-1,1)$.
We first show that for any $y\in (-1,1)$: $$\label{eq:bc_harm}
\varphi_{\H}^{\D^+}(y+i\delta)\to 0 \text { in distribution (so also in probability) as } \delta\to 0.$$
Without loss of generality, the other cases being very similar, let us assume that $y=0$. Observe that by \[lem:harm\_ci\] and harmonicity we have that $$\varphi_{\H}^{\D^+}(i\delta)\overset{(d)}{=}\varphi_{\H}^{(1/\delta)\D^+}(i)=(\varphi_{\H}^{(1/\delta)\D^+},\psi),$$ where $\psi\in C_c^\infty(\C)$ is non-negative with $\int_{\C} \psi=1$, supported in $B(i,1/2)$ and rotationally symmetric about $i$. Moreover, by definition of the domain Markov decomposition, we have that $$(h^\H, \psi_i)\overset{(d)}{=} (h^{(1/\delta)\D^+},\psi)+(\varphi_{\H}^{(1/\delta)\D^+},\psi) \text{ with } h^{(1/\delta)\D^+},\, \varphi_{\H}^{(1/\delta)\D^+} \text{ independent.}$$ On the other hand, it is easy to see by conformal invariance of $h$ that $(h^{(1/\delta)\D^+},\psi_i)$ converges in distribution to $(h^\H, \psi_i)$ as $\delta\to 0$. This implies (for example, by considering characteristic functions) that $$(\varphi_{\H}^{(1/\delta)\D^+},\psi)\to 0$$ in distribution and probability as $\delta\to 0$.
This completes the proof of . We immediately observe that the sequence in is uniformly integrable by \[lem:moment\_bound\], and so can be strengthened to say that $$\label{eq:bc_harm_e}
\mathbb{E}[|\ph(y+i\delta)|]\to 0 \text { as } \delta\to 0$$ With in hand, let us now take $I=[a,b]\subset (-1,1)$ arbitrary: we will show that $\varphi_{\H}^{\D^+}$ can almost surely be continuously extended to $0$ on $I$. We denote $\ph=\varphi_{\H}^{\D^+}$ from now on, and fix $J$ such that $I\subset J\subsetneq [-1,1]$.
First, observe that by dominated convergence and \[lem:moment\_bound\], implies that $\mathbb{E}[\int_J |\ph(y+i\delta)| \, dy]\to 0$ as $\delta\to 0$ and hence that for some sequence $\delta_k\to 0$, $a_k:=\int_J |\ph(y+i\delta_k)| \, dy$ converges to $0$ almost surely. We also have by \[lem:moment\_bound\] that if $S_J$ is the semicircle centered on $J$, then $M:=\int_{S_J} |\varphi(z)| \, dz$ is almost surely finite. Finally, by harmonicity we know that there exists some constant $C$ (deterministic, depending on $I,J$) such for any $z\in \D^+$ that is sufficiently close to $I$, $|\varphi(z)| \le M P(z) +C \Im(z)^{-1} a_k$ for all $k$ large enough, where $P(z)$ is the probability that a Brownian motion started from $z$ hits $S_J$ before $J$. Taking $k\to 0$ gives that $|\varphi(z)|\le MP(z)$ a.s. for all such $z$, and so $\ph$ can almost surely be continuously extended to $0$ on $I$.
Now we can use \[lem:harmonic\_sines\] to prove \[prop:def\_sine\_avg\].
Observe that for any $u >0$, $\varphi_\H^{\sou}$ can a.s. be extended continuously to $0$ on $(-1/\sqrt{u},1/\sqrt{u})$ by scaling and \[lem:harm\_0\_boundary\]. Hence by Lemma \[lem:harmonic\_sines\], on an event of probability one, $$\label{eq:vp_const}
(\varphi_\H^{\sou},p_r)=:c$$ is constant for all $r>u$. This implies (since $\eta^\delta$ has mass one and by definition of $p_u^{\delta, in}$) that with probability one, $$(\varphi_\H^{\sou},p_u^{\delta,in})-c= \int_0^\delta \left(\varphi_\H^{\sou},p^\delta_{u(1+x)})-(\varphi_\H^{\sou},p_{u(1+x)})\right)\, \eta^{\delta}(x) \, dx$$ for all $\delta$ small enough. Noting by \[lem:moment\_bound\] that the right-hand side goes to $0$ in $L^1$ as $\delta\to 0$, we can deduce that $$(\varphi_\H^{\sou},p_u^{\delta,in})\to c \text{ in probability and in } L^1$$ as $\delta\to 0$.
Therefore, to show that the first limit in exists in probability and in $L^1$, and is equal to $c$ almost surely, we need only show that $$\lim_{\delta\downarrow 0}(h^\H-\varphi_\H^{\sou},p_u^{\delta,in})= \lim_{\delta\downarrow 0}(h_\H^{\sou},p_u^{\delta,in})= 0$$ in probability and in $L^1$. However, this follows by applying the zero boundary condition assumption to the field $h_\H^{\sou}$.
An almost identical line of reasoning using part (b) of Lemma 3.2 implies that the second limit in exists a.s. and is equal to the constant value of the second expression in . Observe that $$(\varphi_{\H}^{\H\setminus \overline{\sou}},p_s)\to 0$$ in probability and in $L^1$ as $s\to 0$ (for example, by bounding its first moment using \[lem:moment\_bound\]).
Thus all that remains is to show that the two limits in (or equivalently in ) coincide a.s. For this, we will prove that $$\label{eqn:letters} c\overset{(a)}{=}\lim_{\delta \downarrow 0} (\varphi_{\H}^{\mathsf D_{u-\delta}},p_u)\overset{(b)}{=}\lim_{\delta\downarrow 0} (\varphi_{\H}^{\mathsf D_{u-\delta}},p_u^{\sqrt{\frac{u}{u-\delta}}-1, out})\overset{(c)}{=}\lim_{\delta\downarrow 0} (h^\H,p_u^{\sqrt{\frac{u}{u-\delta}}-1, out}),$$ where all limits are in probability. From this we may conclude, since we already showed that the first limit in was a.s. equal to $c$, and the right hand side above is equal to the second limit in (which we also know exists in probability.)
We will now prove the equalities (a), (b) and (c) from \[eqn:letters\] in turn. For (a), note that by \[lem:harmonic\_sines\] and scale invariance, $$\label{eqn:a}(\varphi_{\H}^{\mathsf D_{u-\delta}},p_u^{\delta,in})-(\varphi_{\H}^{\mathsf D_{u-\delta}},p_u) \overset{(d)}{=} (\varphi_{\H}^{\D^+}, f_\delta ),$$ where $f_\delta$ are a sequence of uniformly bounded smooth functions supported in vanishing neighbourhoods of $\{ \pm 1\}$. The difference therefore converges to $0$ in probability as $\delta\to 0$. Moreover, by Lemma \[lem:nested\_dmp\], we have
$$(\varphi_{\H}^{\mathsf D_{u-\delta}},p_u^{\delta,in})-(\varphi_{\H}^{\mathsf D_{u}},p_u^{\delta,in}) \overset{a.s.}{=} (\varphi_{\mathsf D_{u-\delta}}^{\mathsf D_{u}},p_u^{\delta,in}) \overset{(d)}{=} (h^{\mathsf D_{u-\delta}},p_u^{\delta, in} )- (h_{\mathsf D_{u-\delta}}^{\mathsf D_{u}}, p_u^{\delta, in}).$$ Both terms on the right-hand side also converge to $0$ in probability as $\delta\to 0$ by scaling again, and the Dirichlet boundary condition assumption. Putting these facts together gives (a).
Equality (b) follows by a very similar distributional equality to , again using Lemma \[lem:harmonic\_sines\]. Finally (c) holds, since $$(\varphi_{\H}^{\mathsf D_{u-\delta}},p_u^{\sqrt{\frac{u}{u-\delta}}-1, out})- (h^\H,p_u^{\sqrt{\frac{u}{u-\delta}}-1, out})=-(h_{\H}^{\mathsf D_{u-\delta}}, p_u^{\sqrt{\frac{u}{u-\delta}}-1, out})$$ almost surely and the right hand side (again by scaling) can be seen to converge to 0 in probability as $\delta \downarrow 0$.
A characterisation of Brownian motion {#sec:char_BM}
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\[prop:char\_BM\] Suppose that $(Y(u))_{u\in (0,\infty)}$ is a centred stochastic process. Write $\mathcal{F}_u^+:=\sigma(Y_s: s\ge u)$, $\mathcal{F}_u^-:=\sigma(Y_s: s\le u)$, and for $s<r$ let $\mathcal{F}_{s,r}$ be the $\sigma$-algebra generated by $\mathcal{F}_s^-$ and $\mathcal{F}_r^+$. Suppose that:
(i) $(Y(u))_{u\in (0,\infty)}$ is stochastically continuous, i.e., for any $u_0\in (0,\infty)$, $Y_u\to Y_{u_0}$ in probability as $u \to u_0$;
(ii) for some $\xi>0$, $\mathbb{E}[|Y(u)|^\xi]<\infty$ for all $u\in (0,\infty)$;
(iii) $Y$ satisfies Brownian scaling, that is, $(Y(cu))_{u> 0}$ has the same law as $(\sqrt{c}Y(u))_{u> 0}$ for any $c>0$;
(iv) for any $u>0$, $(Y(s)-Y(u))_{s\ge u}$ is independent of $\mathcal{F}_u^-$;
(v) for any $u>0$, $(Y(s)-\frac{s}{u}Y(u))_{s\le u}$ is independent of $\mathcal{F}_u^+$;
(vi) for any $s<r$ $(Y(u)-(\frac{u-s}{r-s}Y(r)+\frac{r-u}{r-s}Y(s)))_{u\in (s,r)}$ is independent of $\mathcal{F}_{s,r}$.
Then there exists a modification of $Y$ that is equal to $\sigma B$ in law for some $\sigma\ge 0$, where $B$ is a standard one-dimensional Brownian motion.
Observe that for this characterisation we only require $\xi>0$, we will comment later on why we need existence of $1+\eps$ moments for the main result of this paper. Also observe that by scaling, for any process $Y$ as in the statement of the proposition, $Y(\delta)$ is equal in distribution to $\sqrt{\delta} Y(1)$ for every $\delta$, and so tends to $0$ in probability as $\delta \to 0$.
This proposition is very close to the main result of [@Wes93], which is essentially the same but requires square-integrability of the process $Y$. Indeed, we will prove the proposition by showing square-integrability and then appealing to [@Wes93].
We also remark that there is a similar characterisation of Brownian motion in [@BPR18 Theorem 1.9]; the major difference being item $(vi)$. In [@BPR18] we assumed that the process in $(vi)$ has the law of a scaled version of the original process. This is stronger than the statement here, which assumes nothing about the law. On the other hand, only finiteness of logarithmic moments was assumed in [@BPR18], which is (slightly) weaker than the moment assumption $(ii)$ above.
For some motivation, let us first see the important corollary of this characterisation for the purposes of the present article. The proof of \[prop:char\_BM\] will follow immediately after.
\[cor:sine\_avg\_gaussian\] Let $h^\H$ be a sample from $\Gamma^\H$, and define the process $Y$ via $$Y(u):=(h^\H, p_u) \text{ for } u\ge 0,$$ where the right hand side is as defined in \[prop:def\_sine\_avg\] and \[rmk:hpu\_process\]. Then $Y$ satisfies the conditions of \[prop:char\_BM\], and hence has a modification with the law of $\sigma$ times a Brownian motion for some $\sigma\ge 0$.
We note that this result actually holds even if we only have $\xi>0$ in Assumption \[ass:ci\_dmp\], (i). This suggests that the answer to Question \[Q:xi\] is positive.
Since $Y(u)$ is the $L^1$ limit of $(h^\H,p_u^{\delta,in})$ as $\delta\to 0$, and $(h^\H,p_u^{\delta,in})$ is centred for every $\delta$ and $u$, it follows that $Y$ is a centred process. So, it suffices to prove the conditions (i)-(vi) of \[prop:def\_sine\_avg\].
(i) Equality (a) from in the proof of \[prop:def\_sine\_avg\], plus Lemma \[lem:harmonic\_sines\], tells us that $$(h^\H,p_1)-(h^\H,p_{1-\delta})\to 0$$ in probability as $\delta\to 0$. Moreover by scale invariance (see (iii) below) we have that $|(h^\H,p_s)-(h^\H,p_t)|$ is equal in distribution to $\sqrt{s\vee t}\, |(h^\H,p_1)-(h^\H,p_{(s\wedge t)/(s \vee t)})|$. This gives the stochastic continuity.
(ii) This holds with $\xi=1$ since $Y(u)$ is defined as a limit in $L^1$ for all $u$.
(iii) (Scale invariance) We assume without loss of generality that $c>1$. First, we claim that $$\label{eq:SI_harm}
(z\mapsto \varphi^{\mathsf D_{cu}}_\H(z), z \in \mathsf D_{cu})_{u\ge 0} \text{ and } (z\mapsto \varphi^{\sou}_\H(\sqrt{c}z) , z \in \mathsf D_{cu})_{u\ge 0}$$ have the same law as processes (of harmonic functions) in $u$, in the sense that the finite dimensional marginals of both sides have the same laws.
The statement for one dimensional marginals is a special case of \[lem:harm\_ci\]. For the higher dimensional marginals, since the argument with $n$ points is very similar, we will just show equality in law for the joint distribution at two points $u < u'$. For this, we use uniqueness of the domain Markov decomposition to write $$( \varphi_{\H}^{\mathsf D_{cu}} , \varphi_{\H}^{\mathsf D_{cu'}} ) \overset{(d)}{=} (\varphi_{\H}^{\mathsf D_{cu}} , \varphi_{\H}^{\mathsf D_{cu}} + \varphi_{\mathsf D_{cu}}^{\mathsf D_{cu'}} ) \; \text{ and } \; ( \varphi_{\H}^{\mathsf D_{u}} , \varphi_{\H}^{\mathsf D_{u'}} ) \overset{(d)}{=} (\varphi_{\H}^{\mathsf D_{u}} , \varphi_{\H}^{\mathsf D_{u}} + \varphi_{\mathsf D_{u}}^{\mathsf D_{u'}} )$$ where $\varphi_{\mathsf D_{cu}}^{\mathsf D_{cu'}}$ is independent of $ \varphi_{\H}^{\mathsf D_{cu}}$ and $\varphi_{\mathsf D_{u}}^{\mathsf D_{u'}}$ is independent of $ \varphi_{\H}^{\mathsf D_{u}}$ . Using this independence, and Lemma \[lem:harm\_ci\]/the one dimensional marginal case again, we obtain .
Now we complete the proof of scale invariance as follows. Fix $u>0$. By definition of the measures $p_u$, $$\begin{aligned}
\left((h^\H, p_{cu})\right) & \overset{\eqref{eq:hpu_alt_def}}{=} &
\left((\varphi_{\H}^{\frac{1}{\sqrt{cu}}\D^+},\; p_{2cu})\right)\\ & = & \left(\sqrt{2cu} \int_0^\pi \sin(\theta) \varphi^{\frac{1}{\sqrt{cu}}\D^+}_\H (\frac{e^{i\theta}}{\sqrt{2cu}}) \, d\theta\right) \\ &= &\left(\sqrt{c}\sqrt{2u} \int_0^\pi \sin(\theta) \varphi^{\frac{1}{\sqrt{u}}\D^+}_\H (\sqrt{c}\frac{e^{i\theta}}{\sqrt{2cu}}) \, d\theta\right) \\ & = & \left(\sqrt{c} (\varphi_{\H}^{\sou},\;p_{2u})\right) \\ & \overset{\eqref{eq:hpu_alt_def}}{=} & \left(\sqrt{c} (h^\H, p_u) \right)
\end{aligned}$$ where we used \[eq:SI\_harm\] in the third equality. Applying the same string of equalities for finite dimensional marginals, we get the result.
(iv) Fix $u\ge 0$ and observe that since $Y(s)=\lim_{\delta\downarrow 0}(h^\H,p_s^{\delta,out})=\lim_{\delta \downarrow 0} (\varphi_{\H}^{\sou},p_s^{\delta,out})$ for $s\le u$, $\mathcal{F}_u^-$ is independent of $h_\H^{\sou}$. This means that when we write (see \[lem:nested\_dmp\]) $$\varphi_{\H}^{\sor}=\varphi_{\H}^{\sou}+\varphi_{\sou}^{\sor}\; ; \;\; r\ge u,$$ we have that $\varphi_{\sou}^{\sor}$ is independent of $\mathcal{F}_u^-$. Then since $$Y(r) \overset{\eqref{eq:hpu_alt_def}}{=}(\varphi_{\H}^{\sor},p_{2r})=(\varphi_{\H}^{\sou},p_{2r})+(\varphi_{\sou}^{\sor},p_{2r})\overset{\eqref{eq:hpu_alt_def}}{=}Y(u)+(\varphi_{\sou}^{\sor},p_{2r}),$$ we reach the desired conclusion.
(v) Very similar to (iv).
(vi) Let us write $A_{r,s}:= \sos\setminus\overline{\sor}$. Reasoning as in the proof of (iv), we see that in the decomposition $$h^\H=h_\H^{A_{r,s}}+\varphi_\H^{A_{r,s}},$$ $h_\H^{A_{r,s}}$ is independent of $\mathcal{F}_{s,r}$. Hence, we must argue that $$\label{eq:harness_cond}(\varphi_\H^{A_{r,s}},p_u)=\frac{u-s}{r-s}Y(r)+\frac{r-u}{r-s}Y(s) \text{ for all } u\in (s,r).$$ Now, by \[lem:harmonic\_sines\] we know that the left hand side of is a.s.a linear function of $u\in (s,r)$, so we just need to prove that its limit as $u\downarrow s$ is equal to $Y(s)$, and as $u\uparrow r$ is equal to $Y(r)$.
Let us prove the first limit, the second one being very similar. For this, write $$\lim_{u\downarrow s} (\varphi_\H^{A_{r,s}},p_u)=\lim_{u\downarrow s} (\varphi_\H^{\sos},p_u)+\lim_{u\downarrow s}(\varphi_{\sos}^{A_{r,s}}, p_u)=Y(s)+\lim_{u\downarrow s}(\varphi_{\sos}^{A_{r,s}}, p_u)$$ and observe that by \[ass:ci\_dmp\] (iv), $$\varphi_{\sos}^{A_{r,s}} \text{ is harmonic in } A_{r,s} \text{ and } \to 0 \text{ on } \partial(\sos)\cup(-\frac{1}{\sqrt{s}},-\frac{1}{\sqrt{r}})\cup(\frac{1}{\sqrt{r}},\frac{1}{\sqrt{s}}).$$ This implies that $|\varphi_{\sos}^{A_{r,s}}|$ is uniformly bounded in a neighbourhood of $\partial(\sos)$ in $\sos$, and hence by dominated convergence that $\lim_{u\downarrow s}(\varphi_{\sos}^{A_{r,s}}, p_u)=0.$
This almost follows from [@Wes93 Theorem 1], except for the square integrability condition. So first, we will prove that $$\label{eqn:second_moment}
\mathbb{E}[|Y(u)|^2]<\infty \;\; \forall u\in [0,\infty).$$ To do this, pick some $n$ such that $2^{-n}\le \xi$, so that by assumption $\mathbb{E}[|Y(u)|^{2^{-n}}]<\infty$ for all $u$. We will prove that for any $m\ge 0$, $$\label{eq:induction}
\mathbb{E}[|Y(u)|^{2^{-m}}]<\infty \;\; \forall u\in [0,\infty) \; \Rightarrow \mathbb{E}[|Y(u)|^{2^{-m+1}}]<\infty \;\; \forall u\in [0,\infty),$$ from which the result follows by induction, starting with $m=n$.
So, let us take some $m\ge 0$ and assume that the left hand side of holds. Denote $\eta:=2^{-m}$ and first observe that $\mathbb{E}[|Y(2)-Y(1)|^\eta]<\infty$, since $|x+y|^\eta\le |x|^\eta + |y|^\eta$. By independence of $(Y(2)-Y(1))$ and $Y(1)$ (condition (iv) of \[prop:char\_BM\]), this implies that $\mathbb{E}[|Y(1)(Y(2)-Y(1))|^\eta]<\infty$. Now we apply condition (v) of \[prop:char\_BM\]. This tells us that we can write $Y(1)=Y(2)/2+Z$, where $Z$ is independent of $Y(2)$. Hence $$Y(1)(Y(2)-Y(1))=(\frac{Y(2)}{2}+Z)(\frac{Y(2)}{2}-Z)=\frac{Y(2)^2}{4} - Z^2$$ has a finite moment of order $\eta$. Applying \[lem:indep\_moments\], we obtain that $|Y(2)|^2$ has a finite moment of order $\eta$, and hence by scale invariance (condition (iii) of \[prop:char\_BM\]), that $\mathbb{E}[|Y(u)|^{2\eta}]<\infty$ for all $u\in [0,\infty)$. This completes the proof of the induction step, , and therefore of .
From here, we can appeal to the characterisation in [@Wes93 Theorem 1] of stochastic processes with linear conditional expectation and quadratic conditional variance. This says that if $Y$ is a process as in \[prop:char\_BM\], that in addition
- [is defined and stochastically continuous on $[0,\infty)$ with $Y(0)=0$,]{}
- has $Y(u)$ square integrable for every $u$,
- has $\mathbb{E}[Y(u)Y(s)]=\mathbb{E}[Y(u\wedge s)^2]=\sigma (u\wedge s)$ for some $\sigma\ge 0$ and all $u,s\in [0,\infty)$
then $Y$ must be $\sigma$ times a standard Brownian motion. [Note that by the discussion immediately after the statement of \[prop:char\_BM\], we can extend $Y$ to a stochastically continuous process on $[0,\infty)$ with $Y(0)=0$.]{} We also get the third point above by the assumption of Brownian scaling, plus the fact that the process is centred with independent increments. Hence [@Wes93 Theorem 1] provides the result.
Gaussianity of circle averages {#sec:circ_avg_gaussian}
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In this section we work with a sample $h^\D$ from $\Gamma^\D$. For any $\eps>0$ we can define the circle average $h_\eps(0)$ at radius $\eps$ around $0$ via $$h^\D_\eps(0):= \varphi_\D^{B_0(\eps)}(0)$$ as in [@BPR18]. Our next goal is to relate these circle averages to the sine averages from Section \[sec:sine\_avgs\]. This will allow us to show (using \[cor:sine\_avg\_gaussian\]) that the circle average process possesses a modification that is continuous in $\eps$, and will in turn imply that $(h^\D_{\e^{-t}}(0))_{t\ge 0}$ (which has independent and stationary increments by conformal invariance and the domain Markov property) is a Brownian motion. From this it will follow that $h^\D_\eps(0)$ is Gaussian for any $\eps>0$.
To begin, we will explain how the sine averages from \[sec:sine\_avgs\] can make sense for $h^D$ with some specific domains $D\ne \H$. Essentially, this is due to the domain Markov property, which allows us to relate $h^D$ with $h^\H$ in such a way that the sine average of one is the sine average of the other plus the sine average of a harmonic function.
For example, let us start with $D=\D^+$. By the domain Markov property, we can decompose $h^\H$ in the upper unit semi disc $\D^+$ as the independent sum $$h^\H=h_\H^{\D^+}+\varphi_\H^{\D^+},$$ and we already know that:
- for any $u\ge 1$, $(h^{\H},p_u^{\delta, in})\to (h^\H,p_u)$ in probability and in $L^1$ as $\delta\to 0$;
- for any $u>1$, $(\varphi_{\H}^{\D^+},p_u^{\delta,in})\to (\varphi_{\H}^{\D^+},p_u) \text{ a.s.\ and in } L^1 \text{ as } \delta\to 0$, where $(\varphi_\H^{\D^+},p_u)$ is a.s. constant in $u>1$;
- $(\varphi_{\H}^{\D^+},p_1^{\delta,in})$ converges to this constant value in probability and in $L^1$ as $\delta\to 0$ (using and the argument explained just after).
For the first bullet point we have used \[prop:def\_sine\_avg\], and for the second, \[lem:harmonic\_sines\] plus the fact that $\varphi_{\H}^{\D^+}$ is almost surely harmonic in $\D_+$ and can be extended continuously to $0$ on $(-1,1)$ (\[lem:harm\_0\_boundary\]).
This implies that for each $u\ge 1$, $$\lim_{\delta\to 0}(h_\H^{\D^+},p_u^{\delta, in})=:(h^{\D^+},p_u)$$ exists in probability and in $L^1$. Similarly, the joint limit can be defined for $(u_1,\cdots, u_n)$ with each $u_i\in [1,\infty)$, and the resulting process is equal in law to $(h^\H,p_u)_{u\ge 1}$ plus a (random) constant. On the other hand, we have that $(h_{\H}^{\D^+},p_1^{\delta,in})\to 0$ in $L^1$ and in probability as $\delta\downarrow 0$ (by the Dirichlet boundary condition assumption), so that $(h^{\D^+},p_1)=0$.
Putting all this together with \[cor:sine\_avg\_gaussian\], we obtain the following:
\[lem:bmdplus\] Let $h^{\D^+}$ be a sample from $\Gamma^{\D^+}$. Then for any $(u_1,\cdots, u_n)$ with $u_i\in [1,\infty)$ for $1\le i \le n$, the limit $$\lim_{\delta\downarrow 0}\left((h^{\D^+},p_{u_1}^{\delta,in}),\ldots, (h^{\D^+},p_{u_n}^{\delta,in})\right)=\left((h^{\D^+},p_{u_1}),\ldots,(h^{\D^+},p_{u_n}) \right)$$ exists in probability. Moreover, $(h^{\D^+},p_{1+t})_{t\ge 0}$ has the same finite dimensional distributions as some multiple (which is the same as that in \[cor:sine\_avg\_gaussian\]) of Brownian motion.
Next, we make sense of sine averages for $h^{\D}$. Again we can use the domain Markov property, and decompose $$\label{eqn:hDdecomp}
h^\D=h_\D^{\D^+}+\varphi_{\D}^{\D^+}.$$ However, deducing something from this is not quite so simple, since $\varphi_{\D}^{\D^+}$ is does *not* extend continuously to $0$ on $(-1,1)$. For example, since $(\varphi_{\D}^{\D^+},p_u)$ should correspond to integrating $\varphi_{\D}^{\D^+}$ on a contour that *does* touch the real line, it is not immediately obvious that this integral is well defined. We can manage this using that (a) $\varphi_{\D}^{\D^+}$ is not too badly behaved, and (b) the density $\sin(\theta)$ converges to $0$ as $\theta\to \{0,\pi\}$. For this some quantitative estimates are required, and we summarise them in the following lemma:
\[lem:phi\_der\] There exists a universal constant $C\in (0,\infty)$, such that for all $\eps>0$, $$\label{eqn:bound_sup_phi} \mathbb{E}[\sup_{w\in \D^+;\, \Im(w)>\eps}|\varphi_{\D}^{\D^+}(w)|]\le C \eps^{-1/\xi} \log(1/\eps)^{1/\xi}; \text{ and }$$
$$\label{eqn:bound_der_phi} \mathbb{E}[\sup_{r\in [0,1],\theta\in [0,\pi]; \, \Im(r\e^{i\theta})>\eps}|\frac{\partial}{\partial r}\varphi_{\D}^{\D^+}(r\e^{i\theta})|]\le C \eps^{-1-1/\xi}\log(1/\eps)^{1/\xi},$$
where $\xi>1$ is such that $\mathbb{E}[|(h^D,\phi)|^\xi]<\infty$ for all $D$ and $\phi\in C_c^\infty(D)$ (\[ass:ci\_dmp\](i)).
It is a standard fact (see e.g., [@Eva98 §2.2, Theorem 7]) that for a universal $C'>0$, for any function $\varphi$ that is harmonic in $B_z(r)\subset \C$ and for any $\mathbf{v}$ with modulus $1$, $|\partial_{\mathbf{v}}\varphi(z)|\le (C'/r) \sup_{y\in B_z(r)}|\varphi(y)|$. Hence follows from .
To prove , let $w\in \D^+$ with $\Im(w)>\eps$ be arbitrary, and denote by $D_\eps$ the domain $\D^+\cap\{z: \Im(z)>\eps/2\}$. Then by harmonicity and \[lem:harm\_0\_boundary\], if $f_w(y)$ is the density at $y+i\eps/2$, of the exit position from $D_\eps$ for a Brownian motion started from $w$, we have that $$\varphi_{\D}^{\D^+}(w)= \int_{-1}^{1} f_w(y)\varphi_{\D}^{\D^+}(y+i \eps/2) \, dy$$ and so $$|\varphi_{\D}^{\D^+}(w)|\le \left(\int_{-1}^1 f_w(y)dy\right)^{1/\xi^*} \left(\int_{-1}^{1} f_w(y)|\varphi_{\D}^{\D^+}(y+i \eps/2)|^\xi \, dy\right)^{1/\xi}$$ where $\xi^*$ is such that $1/\xi+1/\xi^*=1$. Moreover, by domination with respect to a Cauchy density, there exists a constant $M$ not depending on $\eps>0$, such that $0\le f_w(y)\le M/\eps$ for all $y\in [-1,1]$ and $w$ with $\Im(w)>\eps$. Putting this together, along with the fact that $\int_{-1}^1 f_w(y) \, dy \le 1$, we obtain that $$\sup_{w\in \D^+;\, \Im(w)>\eps} |\varphi_{\D}^{\D^+}(w)|^\xi \le \frac{M}{\eps} \int_{-1}^{1} |\varphi_{\D}^{\D^+}(y+i \eps/2)|^\xi \, dy.$$ To conclude, we observe that by \[lem:moment\_bound\] $$\mathbb{E}[|\varphi_{\D}^{\D^+}(y+i \eps/2)|^\xi]\le C'' \log(1/\eps) \;\; \forall y\in [-1,1],$$ with constant $C''$ not depending on $\eps>0$, so that $$\mathbb{E}[\sup_{w\in \D^+;\, \Im(w)>\eps} |\varphi_{\D}^{\D^+}(w)|]\le\mathbb{E}[\sup_{w\in \D^+;\, \Im(w)>\eps} |\varphi_{\D}^{\D^+}(w)|^\xi]^{1/\xi}\le C \eps^{-1/\xi} \log(1/\eps)^{1/\xi}$$ for some universal constant $C$, as required.
This allows us to deduce the following:
\[lem:cont\_sin\_avg\] Let $h^\D$ be a sample from $\Gamma^\D$ and recall the decomposition . Then for each $(u_1,\cdots, u_n)$ with $u_i\in [1,\infty)$ for $1\le i \le n$ the limit $$\label{eqn:sa_hd_1} \lim_{\delta\downarrow 0}\left((h^{\D^+}_{\D},p_{u_1}^{\delta,in}),\ldots,(h^{\D^+}_{\D},p_{u_n}^{\delta,in})\right) =:\left((h^{\D^+}_\D,p_{u_1}),\ldots,(h^{\D^+}_\D,p_{u_n})\right)$$ exists in probability, and the resulting finite dimensional distributions are those of a multiple (which is the same as that in \[cor:sine\_avg\_gaussian\]) of Brownian motion. Furthermore, on an event of probability one, $$\label{eqn:sa_hd_2} \left((\varphi^{\D^+}_{\D},p_{u}^{\delta,in})\right)_{u\ge 1} \text{ has a pointwise (in u) limit } \left((\varphi^{\D^+}_\D,p_{u})\right)_{u\ge 1} \text{ as } \delta\to 0,$$ and this limit is a continuous function. Finally, for any $1<v<w<\infty$, there exists $M(v,w)$ such that, $$\label{eqn:sa_hd_3}\mathbb{E}[ \sup_{s,t\in [v,w]}\frac{|(\varphi_{\D}^{\D^+},p_s)-(\varphi_{\D}^{\D^+},p_t)|}{|s-t|}] \le M(v,w) .$$
In words, this tells us that the sine-average process of $h^\D$ (defined by joint limits of $(h^\D,p_u^{\delta, in})$ as $\delta\to 0$) makes sense and is a Brownian motion plus a nicely behaved continuous function whose derivative is bounded in expectation, . The role of this key lemma is to show that when we “average" the sine-average process over rotations (as will soon be made precise) we obtain a process with a continuous modification. The control given by is important here to ensure that we retain continuity after averaging, and it is for this that we need the existence of moments with order strictly greater than $1$. (We remark that we have also used it in several other places for simplicity).
This is really the crux of the proof, since the resulting “averaged” process will actually turn out to be the circle average process for $h^\D$ around $0$ (recall from the introduction that establishing continuity of circle averages is the main step in our argument.)
Since $h_{\D}^{\D^+}$ has the same law as $h^{\D^+}$, the statement concerning the limit follows from \[lem:bmdplus\]. To show that holds with probability one note that by Markov, for any $\xi^{-1}<a<1$, $$\mathbb{P}[\sup_{w\in \D^+;\, \Im(w)>\eps}|\varphi_{\D}^{\D^+}(w)| > \ve^{-a}]\le C \eps^{a-1/\xi} \log(1/\eps)^{1/\xi}$$ Thus applying Borel–Cantelli (to the sequence $\ve_n = 2^{-n}$) we conclude that a.s., for any $\xi^{-1}<a<1$, $$|\varphi_{\D}^{\D^+}(z)|\le \Im(z)^{-a}$$ for all $z$ with $\Im(z)$ sufficiently small. This implies (since $\sin(\arg(z))\Im(z)^{-a}\to 0$ as $\Im(z)\to 0$). Similarly, an application of Borel–Cantelli and allows us to deduce that, on an event of probability one, $F(u):=(\varphi_{\D}^{\D^+},p_u)$ is differentiable in $u$, and for some finite deterministic constants $\{M'(v,w)\}_{1<v<w<\infty}$, $$|F'(r)|\le M'(v,w) \int_0^\pi \sin(\theta) |\frac{\partial}{\partial r}\varphi_{\D}^{\D^+}(\e^{i\theta}/\sqrt{r})| \, d\theta \text{ for all } r\in [v,w]$$ From this and , follows in a straightforward manner.
Now we will relate these quantities to circle averages, by averaging over rotations. Let $h^\D$ be a sample from $\Gamma^\D$ and for $\alpha \in [0,2\pi)$, let $h^{\D,\alpha}$ be the image of $h^\D$ under an anti-clockwise rotation by angle $\alpha$. That is, $(h^{\D,\alpha},\phi)_{\phi\in C_c^\infty(\D)}=(h^{\D},\phi\circ f_\alpha)_{\phi\in C_c^\infty(\D)}$ where $f_\alpha$ denotes the isometry $z\mapsto \e^{-i\alpha}z$.
Then by conformal (specifically, rotation) invariance, $$\label{eq:law_ind_a} h^{\D,\alpha}\overset{(d)}{=}h^\D$$ for each fixed $\alpha$. Write $h^{\D^+}_{\D,\alpha}+\varphi^{\D^+}_{\D,\alpha}$ for the domain Markov domain decomposition of $h^{\D,\alpha}$ in $\D^+$.
Now let $A$ be uniformly distributed on the interval $[0,2\pi]$ (independently from $h^\D$). Then we have that:
- for each $(u_1,\cdots, u_n)$ with $u_i\in [1,\infty)$ for $1\le i \le n$ $$\lim_{\delta\downarrow 0}\left((h^{\D^+}_{\D,A},p_{u_1}^{\delta,in}),\ldots, (h^{\D^+}_{\D,A},p_{u_n}^{\delta,in})\right)=:\left((h^{\D^+}_{\D,A},p_{u_1}),\cdots, (h^{\D^+}_{\D,A},p_{u_n})\right)$$ exists a.s. and for any $s,t\ge 1$ $$\label{eqn:fourth_moment_circ} \mathbb{E}[|(h_{\D,A}^{\D^+},p_s)-(h_{\D,A}^{\D^+},p_t)|^4]\le c|s-t|^2$$ for some universal constant $c$ (because for each angle $\alpha$ the process $(h^{\D^+}_{\D,\alpha},p_s)_s$ is a fixed, i.e. not depending on $\alpha$, multiple of Brownian motion);
- $((\varphi^{\D^+}_{\D,A},p_{u}^{\delta,in}))_{u\ge 1}$ has a pointwise limit $((\varphi^{\D^+}_{\D,A},p_{u}))_{u\ge 1}$ with probability one as $\delta\to 0$, and for any $1<v<w<\infty$, there exists $M(v,w)$ such that, $$\label{eqn:circ_der_bound}\mathbb{E}[ \sup_{s,t\in [v,w]}\frac{|(\varphi_{\D,A}^{\D^+},p_s)-(\varphi_{\D,A}^{\D^+},p_t)|}{|s-t|}] \le M(v,w) .$$
This allows us to reach the following conclusion.
For every $u\in [1,\infty)$, the conditional expectation $$\mathbb{E}[(h^{\D,A},p_u) \, | \, h^\D ]:=\mathbb{E}[(h_{\D,A}^{\D^+},p_u)+(\varphi_{\D,A}^{\D^+},p_u) \, | \, h^\D ]$$ is well defined. This defines a stochastic process in $u$ which possesses an a.s. continuous modification.
Since $(h_{\D,A}^{\D^+},p_u)$ and $(\varphi_{\D,A}^{\D^+},p_u)$ are random variables in $L^1(\P\times dA)$ (as can be seen using , by first taking expectation over the field given $A$, and then over $A$) the conditional expectations $$\mathbb{E}[(h_{\D,A}^{\D^+},p_u) \, | \, h^\D ] \text{ and } \mathbb{E}[(\varphi_{\D,A}^{\D^+},p_u) \, | \, h^\D ]$$ are well defined for any fixed $u$. By , the fact that conditioning is a contraction in $L^4$, and Kolmogorov’s continuity criterion, the first of these two stochastic processes has an a.s. continuous modification. To deal with the second process, observe that by , for any $1<v<w<\infty$, we have $$\begin{aligned}
& \mathbb{E}\left[\sup_{s,t\in [v,w]} \frac{\left|\mathbb{E}[(\varphi_{\D,A}^{\D^+},p_t) \, | \, h^\D]-\mathbb{E}[ (\varphi_{\D,A}^{\D^+},p_s)\, | \, h^\D]\right|}{|s-t|}\right] \\ & \le \mathbb{E}\left[ \mathbb{E}[\sup_{s,t\in [v,w]}\frac{|(\varphi_{\D,A}^{\D^+},p_t) - (\varphi_{\D,A}^{\D^+},p_s)|}{|s-t|}\, | \, h^\D]\right] \le M(v,w).\end{aligned}$$ Hence the process $\mathbb{E}[(\varphi_{\D,A}^{\D^+},p_u) \, | \, h^\D ]$ in $u$ has a modification which is a.s. continuous.
The connection to circle averages is the following. Recall that $h_\eps^\D(0)$ denotes the radius $\eps$ circle average of $h^\D$ around $0$. Recall that this is defined to be equal to $\varphi_{\D}^{\eps\D}(0)$ if $h^\D$ has domain Markov decomposition $h^{\eps\D}_\D+\varphi_{\D}^{\eps\D}$ in $\eps\D$.
\[lem:cond\_sin\_av\_equals\_circ\_av\] For any $u\in [1,\infty)$, $\mathbb{E}[(h^{\D,A},p_u) \, | \, h^\D]=\sqrt{u}h^\D_{\frac{1}{\sqrt{u}}}(0)$ a.s.
Fix $u\in [1,\infty)$. Since $(h^{\D,A},p_u^{\delta,in})\to (h^{\D,A},p_u)$ in probability and in $L^1$ as $\delta\to 0$, we have that $$\mathbb{E}[(h^{\D,A},p_u) \, | \, h^\D]=\mathbb{E}[\lim_{\delta\downarrow 0}(h^{\D,A},p_u^{\delta,in}) \, | \, h^\D]=\lim_{\delta\downarrow 0} \mathbb{E}[(h^{\D,A},p_u^{\delta,in})\, | \, h^\D]$$ where the rightmost limit holds in probability and in $L^1$. By definition of $A$, the right hand side is equal to $$\lim_{\delta\downarrow 0}\frac{1}{2\pi}\int_0^{2\pi} (h^{\D,\alpha},p_u^{\delta,in}) \, d\alpha = \lim_{\delta\downarrow 0}\frac{1}{2\pi}\int_0^{2\pi} (h^{\D},p_u^{\delta,in}\circ f_\alpha) \, d\alpha$$ where $f_\alpha(z)=e^{-i\alpha}z$ is rotation by $\alpha$. By linearity of $h^\D$ this is equal to $$\lim_{\delta\downarrow 0}(h^\D, \frac{1}{2\pi} \int_0^{2\pi} p_u^{\delta,in}\circ f_\alpha \, d\alpha)=\lim_{\delta\downarrow 0}(\varphi_{\D}^{\frac{1}{\sqrt{u}}\D},\frac{1}{2\pi}\int_0^{2\pi} p_u^{\delta,in}\circ f_\alpha \, d\alpha)+\lim_{\delta\downarrow 0}(h_{\D}^{\frac{1}{\sqrt{u}}\D},\frac{1}{2\pi}\int_0^{2\pi} p_u^{\delta,in}\circ f_\alpha \, d\alpha),$$ where the second term above goes to $0$ in probability as $\delta\to 0$ by the Dirichlet boundary condition assumption. Moreover, the function $\frac{1}{2\pi}\int_0^{2\pi} p_u^{\delta,in}\circ f_\alpha \, d\alpha$ is radially symmetric with total mass tending to $\sqrt{u}$ as $\delta\to 0$. By harmonicity, it then follows that $$\lim_{\delta\downarrow 0}(\varphi_{\D}^{\frac{1}{\sqrt{u}}\D},\frac{1}{2\pi}\int_0^{2\pi} p_u^{\delta,in}\circ f_\alpha \, d\alpha)= \sqrt{u} \varphi_{\D}^{\frac{1}{\sqrt{u}}\D}(0)=\sqrt{u}h^\D_{\frac{1}{\sqrt{u}}}(0)$$ a.s., as required.
(We emphasise that the process in \[lem:cond\_sin\_av\_equals\_circ\_av\] above is not Brownian motion, but rather a time change of it). The corollary is the following:
\[cor:circ\_avg\_cont\] The process $(h^\D_\eps(0))_{\eps\in (0,1]}$ possesses a continuous modification.
\[prop:circ\_av\_bm\] The process $(h_{e^{-t}}^\D(0))_{t\ge 0}$ has a modification with the law of $(\sigma B_t)_{t\ge 0}$, where $\sigma\ge 0$ and $B$ is a standard one-dimensional Brownian motion.
By the assumptions of conformal invariance and the domain Markov property, this process has independent increments, and it is also centred. By \[cor:circ\_avg\_cont\], it possesses a continuous modification. Since any continuous centred Lévy process must be a multiple of Brownian motion, this implies the result.
\[cor:circ\_avg\_gaussian\] For any $D$ and $z\in D$, let $F_z^D$ be the conformal map from $D\to \D$ with $z\mapsto 0$ and $(F_z^D)'(z)\in \R_+$. Then the process $$\label{eqn:he} \hat{h}_{e^{-t}}^D(z)=\varphi_D^{(F_z^D)^{-1}(B_0(e^{-t}))}(z)$$ defined for $t\ge 0$, has a modification with the law of $\sigma$ times a Brownian motion.
This follows from conformal invariance, \[ass:ci\_dmp\](iii).
Conclusion of the proof
=======================
Without loss of generality we assume that $D=\D$. For $z\in \D$ and $\eps = \eps(z) <d(z,\partial \D)$. Let $$\label{eq:rz}
r_z(\eps):=\sup\{r\in [0,1]\, : (F_z^\D)^{-1}(B_0(r)) \subset B_z(\eps)\}.$$ Also set $h^\D_{\eps}(z)=\varphi_{\D}^{B_z(\eps)}(z)$ and define $\hat{h}^\D_{r_z(\eps)}(z)$ via and .
For $\delta>0$, define $\eta_\delta$ to be a smooth radially symmetric function that approximates uniform measure on the unit circle as $\delta\to 0$. For concreteness, $\eta_\delta$ can be taken to be a smooth radially symmetric function equal to 1 on the annulus $\{z: 1 - \delta \le |z| \le 1 -\delta/2 \}$ that is 0 outside a $\delta/10$ neighbourhood of this annulus. We assume that each $\eta_{\delta}$ is normalised to have total integral one. For $\eps\in (0,1)$, further define $$\eta^\eps_\delta(\cdot)=\frac{1}{\eps^2} \eta_{\delta}(\frac{\cdot}{\eps})$$
Take $\phi\in C_c^\infty(\D)$. Recall that for Proposition \[prop:fourth\_moment\](1) we need to show that $(h^\D, \phi)$ has finite fourth moment. The idea is to show that $$\label{eq:fourth_moment_1}
\int \hat{h}_{r_\eps(z)}^\D(z) \phi(z) \, dz \to (h^\D, \phi) \text{ in probability as } \eps\to 0$$ and that $$\label{eq:fourth_moment_two} \left(\int_{\D} \phi(z) \hat{h}^\D_{r_\eps(z)}(z)\, dz
\right)^4 \text{ is uniformly integrable in } \eps$$ This means that $(\int_{\D} \phi(z) \hat{h}^\D_{r_\eps(z)}(z))^4$ converges in $L^1$ to $(\phi,h^\D)^4$, and in particular, that $(\phi,h^\D)^4$ is integrable.
*Proof of .* We bound, for $\delta>0$:
$$\begin{aligned}
\label{eq:triangle}
&\left| \int \hat{h}_{r_\eps(z)}^\D(z) \phi(z)\, dz - (h^\D,\phi)\right| \nonumber \\
\le & \left| \int (\hat{h}_{r_\eps(z)}^\D(z)-h^\D_{\eps}(z))\phi(z) \, dz \right| + \left| \int h^\D_{\eps}(z)\phi(z)\, dz - (h^\D, \phi*\eta^\eps_\delta)\right| +
\left| (h^\D, \phi*\eta^\eps_\delta)-(h^\D,\phi)\right|
\end{aligned}$$
We start by showing that the first term in goes to 0 in probability as $\eps\to 0$. For this, one can check explicitly that for every $\delta<d(z,\partial \D)$ we must have $r_z(\delta)\ge \delta/(\delta+\textrm{CR}(z,D))$, and therefore (by another calculation) that $(F_z^\D)^{-1}(B_0(r_z(\delta)))$ contains the ball of radius $\delta(1-\frac{\delta}{\delta+\frac{1}{2}\textrm{CR}(z,D)})$ around $z$. Hence, by conformal invariance and \[lem:nested\_dmp\], $h_\delta^\D(z)-\hat h_{r_z(\delta)}^\D(z)$ is distributed as $\varphi_{\D}^{D_\delta^z}(0)$, where for some $f(\delta)$ tending to $0$ as $\delta\to 0$ and every $z$ in the support of $\phi$, $D_\delta^z\subset \D$ contains the ball of radius $1-f(\delta)$ around 0. By , it then follows that $$\mathbb{E}[|h_\delta^\D(z)-\tilde{h}_{r_z(\delta)}^\D(z)|]\le \mathbb{E}[|\varphi_{\D}^{B_0(1-f(\delta))}(0)|]= \mathbb{E}[|h_{(1-f(\delta))}^\D(0)|],$$ and this tends to $0$ as $\delta\to 0$ by \[prop:circ\_av\_bm\]. By boundedness of $\phi$, this proves that the first term of goes to 0 in probability as $\eps\to 0$.
We also have that the third term of goes to 0 in probability as $\eps\to 0$, for any fixed $\delta$. Indeed, $\phi*\eta_{\delta}^\eps \to \phi$ in $C_c^\infty(\D)$ as $\eps\to 0$ because $\eta_{\delta}$ is a smooth approximation to the identity for every $\delta$: see, eg. [@Eva98 §5.3]. Thus by \[ass:ci\_dmp\](i) (stochastic continuity), $(h^\D,\phi*\eta^\eps_\delta)\to (h^\D,\phi)$ in probability as $\eps\to 0$.
So to show we are left to prove that the middle term of goes to $0$ in probability as $\delta\to 0$, *uniformly* in $\eps$. (That is, for any $c>0$ the probability that this term is bigger than $c$ goes to $0$ as $\delta\to 0$, uniformly in $\eps$.) To do this, we note that $\phi*\eta_\delta^\eps(z)=\int \phi(w) \eta_{\delta}^\eps(w-z) \, dw$ and so by linearity of $h^\D$, $$(h^\D,\phi*\eta^\eps_\delta)=\int_w (h^\D, \eta_{\delta}^\eps(w-\cdot))\phi(w) \, dw.$$ Moreover, by the Dirichlet boundary condition assumption and scale invariance, for every $w$ in the support of $\phi$ $$(h^\D,\eta_{\delta}^\eps(w-\cdot))-h^\D_{\delta}(w)\to 0$$ in probability and in $L^1$ as $\delta\to 0$, uniformly in $\eps$. Combined with the boundedness of $\phi$, this completes the proof.
*Proof of .* For this, we will show that $\int_{\D} \phi(z) \hat{h}^\D_{r_\eps(z)}(z)\, dz$ is uniformly bounded in $L^6$.
For $(z_1,\cdots, z_6)$ in $\operatorname{Support}(\phi)^6$, write $R=R(z_1,\cdots, z_6)$ for the largest $r$ such that the balls $B_{z_i}(r)$ are all disjoint. Then for $\eps<R$, by the domain Markov property and \[lem:nested\_dmp\], we have that $$\mathbb{E}[\prod_{i=1}^6 \hat h_{r_\eps(z_i)}^\D(z_i)]=\mathbb{E}[\prod_{i=1}^6 \hat h_R^\D(z_i)].$$ By repeated application of Hölder’s inequality, the term on the right hand side above is less than $\prod_{i=1}^6(\mathbb{E}[(h^\D_R(z_i))^6])^{1/6}$, and since each $h_R^\D(z_i)$ is Gaussian with variance less than some universal constant times $\log(1/R)$, we obtain that
$$\mathbb{E}[\left(\int \hat{h}^\D_{r_\eps(z)}(z) \phi(z) \, dz \right)^6]= C(\phi) \left(1+ \iint_{D^6} |\log(R(z_1,\cdots, z_6))|^3 \, d\mathbf{z} \right) <\infty$$ where $C(\phi)$ is a finite constant depending on $\phi$ but not $\eps$. Since this bound is uniform in $\eps$, the proof is complete.
Suppose that $\phi_n$ is a sequence of functions in $C_c^\infty(\D)$ converging to $\phi\in C_c^\infty(\D)$. Then by the previous part of this proof, $$\mathbb{E}[(h^\D,\phi_n)^4]=\lim_{\eps\to 0}\mathbb{E}[(\int_{\D} \phi(z) \hat{h}^\D_{r_\eps(z)}(z)\, dz
)^4]$$ for each $n$, and this expectation is easily seen to be uniformly bounded in $n$ (using Hölder’s inequality and the fact that we know the marginal distributions of the $\hat{h}^\D$’s; as above). By the stochastic continuity assumption, we have that $(h^\D,\phi_n)\to (h^\D,\phi)$ in probability as $n\to \infty$. Putting this together with the uniform boundedness in $L^4$, we can deduce in particular that $(h^D,\phi_n)$ converges in $L^2$ to $(h^D,\phi)$ as $n\to \infty$. This implies the continuity of $K_2^D$ by Cauchy–Schwarz.
The same arguments can be used to show that $(h^\D,\phi_n)$ is uniformly bounded in $L^4$ when $\phi_n$ is as in \[ass:ci\_dmp\](ii). This implies that the convergence of this assumption also holds in $L^2$.
[^1]: Supported in part by EPSRC grant EP/L018896/1, the University of Vienna, and FWF grant “Scaling limits in random conformal geometry”.
[^2]: Supported in part by NSERC 50311-57400 and University of Victoria start-up 10000-27458
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---
author:
- |
A. Gemes[^1],$^a$ ,$^{bc}$ S. Frey,$^{c}$ T. An,$^{d}$ Z. Paragi,$^{e}$ and A. Moór$^{c}$\
Trinity College, University of Cambridge, Cambridge, United Kingdom\
MTA-ELTE Extragalactic Astrophysics Research Group, Budapest, Hungary\
Konkoly Observatory, MTA Research Centre for Astronomy and Earth Sciences, Budapest, Hungary\
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China\
Joint Institute for VLBI ERIC, Dwingeloo, The Netherlands\
E-mail: , , , , ,
title: 'High-resolution radio imaging of the gamma-ray blazar candidate J1331+2932'
---
Introduction
============
When looking for evidence for $\gamma$-ray flaring activity in the stellar binary system DG CVn, recently Loh et al. [@Loh17] found a transient source using *Fermi* Large Area Telescope (LAT) data from 2012 November. However, since simultaneous flaring of DG CVn was not reported at any other waveband that time, the background quasar J1331+2932 fell under suspicion as the possible source of the $\gamma$-rays. Among the $\gamma$-ray emitting extragalactic sources, blazars constitute the most populous group (Ackermann et al. 2015). Blazars are active galactic nuclei (AGN) with relativistic plasma jets directed at small inclination angles to the observer. Milliarcsecond (mas) resolution radio interferometric observations using the technique of very long baseline interferometry (VLBI) are the best suited for providing the ultimate evidence to discriminate between blazar and non-blazar radio-emitting AGN.
Observations and data reduction
===============================
We observed J1331+2932 with the European VLBI Network (EVN) at 5 GHz on 2017 Apr 11 (project code: RSG08, PI: K. É. Gabányi). Nine telescopes of the EVN were used in e-VLBI mode: Jodrell Bank Mk2 (United Kingdom), Westerbork (the Netherlands), Medicina, Noto (Italy), Toruń (Poland), Yebes (Spain), Hartebeesthoek (South Africa), Irbene (Latvia), and Tianma (China). The long intercontinental baselines from the European stations to Hartebeesthoek and Tianma provided high angular resolution ($\sim 1.5$ mas) in both north–south and east–west directions. The maximum data rate was 2048 Mbits$^{-1}$ but four antennas (Jodrell Bank, Westerbork, Toruń, and Tianma) operated at half of that value. The corresponding total bandwidth of 256 MHz was divided into 8 intermediate frequency (IF) channels in both right and left circular polarizations. Each IF was further divided into 64 spectral channels. Since the brightness and compactness of the source was previously unknown, the method of phase referencing was applied. This involved regular observations of the nearby ($1.4^\circ$ angular separation) compact calibrator J1334+3044 within the atmospheric coherence time, with a duty cycle of $\sim$6.5 min. The total time spent on the target source J1331+2932 was about 140 min.
The EVN data were calibrated in the NRAO Astronomical Image Processing System (AIPS) [@Greisen03] and hybrid mapping was performed in Difmap [@Shepherd97] according to standard procedures. More details will be published elsewhere (A. Gemes et al., in prep.). The phase-referenced image of J1331+2932 was used to determine its accurate astrometric position with respect to the calibrator source. Our EVN observation prove that the target source was sufficiently bright and compact for direct fringe-fitting in AIPS. Therefore we made our final image of J1331+2932 this way.
Results and discussion
======================
The coordinates of the blazar candidate could be determined more accurately than before: right ascension 13$^{\rm h}$ 31$^{\rm m}$ 01.83259$^{\rm s}$ and declination 29$^\circ$ 32$^\prime$ 16.5099$^{\prime\prime}$. The estimated error in the position is 0.5 mas. The 5-GHz EVN image of J1331+2932 using fringe-fitted data is displayed in Fig. \[image\]. The source shows a compact radio structure typical for blazars, a bright core and a weak jet component to the south-west.
![5-GHz EVN image of J1331+2932. The peak brightness is 14.08 mJy beam$^{-1}$. The first contours are at $\pm 0.165$ mJy beam$^{-1}$ ($\sim 3\sigma$ image noise), the positive contour levels increase by a factor of 2. The Gaussian restoring beam shown in the lower-left corner is 1.5 mas $\times$ 1.3 mas (FWHM) with a major axis position angle $-5.5^\circ$.[]{data-label="image"}](Gemes-Fig1.png){width="0.5\columnwidth"}
We fitted circular Gaussian brightness distribution model components to the self-calibrated visibility data in Difmap. For the core, the component shrank to a point regardless of whether a second Gaussian component was fitted to the jet or not. It implies that the compact core is practically unresolved in this experiment. We derived an upper limit of the size of the core (0.09 mas) by taking the minimum resolvable size [@Kovalev05] into account. This allowed us to estimate a lower limit to its brightness temperature [@Condon82]: $$T_{\rm B}=\frac{2\ln{2}}{\pi}\frac{c^{2}S}{k_{B}\nu^{2}\vartheta^{2}}(1+z),$$ where $c$ is the speed of light, $k_{B}$ the Boltzmann constant, $z=0.48$ the redshift of J1331+2932 [@Alam15], $S$ the core flux density (14.7 mJy), $\nu$ the observing frequency, and $\vartheta$ the full width at half maximum (FWHM) angular size of the source (an upper limit in our case). This yields that the minimum brightness temperature of J1331+2932 is $T_B\geq1.3\times10^{11}$ K. Assuming an intrinsic brightness temperature $T_{\rm B,int}= 3 \times 10^{10}$ K [@Homan06], the lower limit of the Doppler factor is $\delta = T_{\rm B} / T_{\rm B,int} \geq 4.3$. This indicates relativistic beaming which implies that the candidate is indeed a blazar. Using typical values of $5 \leq \Gamma \leq 15$ for the Lorentz factor, the jet inclination angle $\theta$ can be estimated using $$\delta=\frac{1}{\Gamma\left(1-\beta\cos{\theta}\right)},$$ where $\beta$ is the bulk speed of the jet in units of the speed of light. From this, we obtain $\theta \leq 14^{\circ}$ for the jet in J1331+2932.
Based on our EVN data, we confirm that J1331+2932 is a blazar and thus the most likely counterpart of the *Fermi* LAT source. We also investigated mid-infrared monitoring data taken by the *Wide-field Infrared Survey Explorer (WISE)* satellite [@Wright10]. These show significant flux density variations from daily to yearly time scales, strengthening the case for J1331+2932 being a blazar. Moreover, the *WISE* colours of the source are close to typical values observed for blazars [@Massaro11; @Massaro12].
[99]{} M. Ackermann, M. Ajello, W. B. Atwood, et al., *The Third Catalog of Active Galactic Nuclei Detected by the Fermi Large Area Telescope*, *ApJ* [**810**]{} (2015) 14 \[[arXiv:1501.06054]{}\].
S. Alam, F. D. Albareti, C. Allende Prieto, et al., *The Eleventh and Twelfth Data Releases of the Sloan Digital Sky Survey: Final Data from SDSS-III*, *ApJS* [**219**]{} (2015) 12 \[[arXiv:1501.00963]{}\].
J. J. Condon, M. A. Condon, G. Gisler, J. J. Puschell, *Strong radio sources in bright spiral galaxies. II. Rapid star formation and galaxy-galaxy interactions*, *ApJ* [**252**]{} (1982) 102.
E. W. Greisen, *AIPS, the VLA, and the VLBA*, *Information Handling in Astronomy - Historical Vistas, Astrophysics and Space Science Library* [**285**]{} (2003), Kluwer, Dordrecht, 109.
D. C. Homan, Y. Y. Kovalev, M. L. Lister, et al., *Intrinsic Brightness Temperatures of AGN Jets*, *ApJ* [**642**]{} (2006) L115 \[[arXiv:astro-ph/0603837]{}\].
Y. Y. Kovalev, K. I. Kellermann, M. L. Lister, et al., *Sub-Milliarcsecond Imaging of Quasars and Active Galactic Nuclei. IV. Fine-Scale Structure*, *AJ* [**130**]{} (2005) 2473 \[[arXiv:astro-ph/0505536]{}\].
A. Loh, S. Corbel, G. Dubus, *Fermi/LAT detection of a transient gamma-ray flare in the vicinity of the binary star DG CVn*, *MNRAS* [**467**]{} (2017) 4462 \[[arXiv:1702.03754]{}\].
F. Massaro, R. D’Abrusco, M. Ajello, J. E. Grindlay, H. E. Smith, *Identification of the Infrared Non-thermal Emission in Blazars*, *ApJ* [**740**]{} (2011) L48 \[[arXiv:1203.0304]{}\].
F. Massaro, R. D’Abrusco, G. Tosti, et al., *The WISE Gamma-Ray Strip Parameterization: The Nature of the Gamma-Ray Active Galactic Nuclei of Uncertain Type*, *ApJ* [**750**]{} (2012) 138 \[[arXiv:1203.1330]{}\].
M. C. Shepherd, *Difmap: an Interactive Program for Synthesis Imaging*, *Astronomical Data Analysis Software and Systems VI, ASP Conference Series* [**125**]{} (1997), Astron. Soc. Pacific, San Francisco, 77.
E. L. Wright, P. R. M. Eisenhardt, A. K. Mainzer, et al., *The Wide-field Infrared Survey Explorer (WISE): Mission Description and Initial On-orbit Performance*, *AJ* [**140**]{} (2010) 1868 \[[arXiv:1008.0031]{}\].
[^1]: The EVN is a joint facility of independent European, African, Asian, and North American radio astronomy institutes. Scientific results from data presented in this publication are derived from the following EVN project code: RSG08. This publication makes use of data products from the *WISE*, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 730562 (RadioNet). We thank the Hungarian National Research, Development and Innovation Office (OTKA NN110333 and 2018-2.1.1-UK\_GYAK) for support. KÉG was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
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---
author:
- |
$^a$, Raquel de los Reyes$^{b}$[^1], Dalibor Nosek$^{a}$ for the CTA Consortium[^2]\
Charles University, Faculty of Mathematics and Physics\
V Holesovickach 2, 180 00 Prague, Czech Republic\
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany\
E-mail:
title: Atmospheric monitoring and array calibration in CTA using the Cherenkov Transparency Coefficient
---
Introduction
============
The Cherenkov Telescope Array (CTA) is a ground-based very high energy (VHE) gamma-ray observatory in the pre-construction phase [@OngICRC2017]. It will consist of two arrays on both Earth’s hemispheres including not only different optical systems and detector hardware, but also different sizes of telescopes. A complete strategy on the calibration of the full array [@CettinaICRC2017] as well as the atmosphere [@EbrICRC2017] is currently under development in CTA. The Cherenkov Transparency Coefficient (CTC) is included within the calibration strategy of both:
- Removing the system dependency of the stereo trigger rates, the CTC will depend only on the atmospheric extinction of the Cherenkov light emitted by the air showers.
- Utilizing the optical throughput dependency of the CTC, normally neutralized by the throughput estimated with the muons, will allow the CTC to monitor the variations of the detection efficiency of the telescopes, alternatively to their assessment through the muon analysis.
In the next sections, we will describe the steps to apply the results of the H.E.S.S. collaboration [@Hahn] and extend them for their use in more complex systems, like the CTA. This study makes use of Monte Carlo (MC) simulations of protons observed by a candidate array of telescopes located at the northern CTA site (CTA-N) at La Palma [@MC]. The array consists of 4 large-sized (LST) and 15 medium-sized (MST) telescopes with the mirror dish diameters of 23 m and 12 m, respectively [@HassanICRC2017].
In Sec. \[Sec:Stereo\], we will give a brief introduction about the concepts involved in the CTC, including the discussion of the effects of the array layout geometry (Sec. \[Sec:Geometry\]), the influence of the Earth’s magnetic field (Sec. \[Sec:Bfield\]) and the array hardware (Sec. \[Sec:Hardware\]). In Sec. \[Sec:Intercalibration\], we will describe the role of the CTC in the inter-calibration of the CTA telescopes of the same size. Sec. \[Sec:Atmosphere\] deals with the use of the CTC for the monitoring of the atmospheric transparency. In Sec. \[Sec:Conclusions\], a summary of the status of this feasibility study will be given together with future steps to be undertaken.
Trigger rates of Cherenkov telescopes {#Sec:Stereo}
=====================================
During the scientific observations of the CTA, the telescopes will record events seen by at least two telescopes (stereo trigger events). In contrast with the single telescope triggers, the stereo requirement will partially eliminate random fluctuations due to the night sky background and accidental triggers. In a first approximation, the rate of triggered events is mainly determined by the lowest detectable energy of cosmic rays (energy threshold). This energy threshold depends inversely on the detection efficiency of the stereo partners (their effective area). Variations of the effective area depend on the transparency of the atmosphere $T = e^{-\mathrm{AOD}}$, where AOD is the aerosol optical depth.
Based on these assumptions, the H.E.S.S. experiment defined the transparency coefficient [@Hahn] as $CTC = (N\cdot k_{N})^{-1}\cdot\sum_{i}{ R_{i}^{\frac{1}{1.7}}\cdot( \mu_{i} \cdot g_{i} } )^{-1}$, where the sum runs over each of the $N$ active telescopes, $R_{i}$ is the stereo trigger rate at zenith of all events triggering the $i$-th telescope together with at least one other telescope, $\mu_{i}$ is the muon-estimated normalised optical throughput, $g_{i}$ is the average pixel gain and $k_{N}$ accounts for layout-related changes.
The normalization factor $k_{N}$ also includes a dependency of the system rate on the distance between telescopes ($D$), their orientation with respect to the shower ($\beta$) and the effects of the Earth’s magnetic field ($\vec{B}$). For arrays with tens of telescopes, like CTA, this results in a vast number of possible realizations which cannot be straightforwardly included in the H.E.S.S. formula. In order to account for all possible dependencies in $k_{N}$, the stereo trigger rate $R$ used in the definition of the CTC must be modified (by a function $F$) to ensure the independence of the transparency estimate $\tau$ for CTA from hardware and observation-related quantities: $$\label{Eq:transparency}
\tau (\mathrm{AOD}) = R \left( \mathrm{AOD}, D, \theta, \beta, \vec{B}, \varepsilon \right) \cdot F^{-1}\left( D, \theta, \beta, \vec{B}, \varepsilon \right).$$ where $\theta$ is the zenith angle (see Fig. \[Fig:coordinates\]) and $\varepsilon$ describes the hardware dependency studied in Sec. \[Sec:Hardware\].
Geometrical configuration {#Sec:Geometry}
-------------------------
{width="0.78\columnwidth"} \[Fig:coordinates\]
{width="\columnwidth"} \[Fig:zenith\]
In this section, we examine the relationship between the trigger rates, the distance $D$ between telescopes and their relative position $\beta$ with respect to the shower direction given by $\theta$ and $\phi$ (see Fig. \[Fig:coordinates\]). The study utilizes 2-telescope trigger rates obtained from a set of MC simulations [@MC] of proton showers for the CTA-N site. The shower directions were given by $\theta \in [0^{\circ},60^{\circ}]$ with $\phi = 0^{\circ}$ or $180^{\circ}$. The positions of telescopes were chosen such that the different pairs were aligned with respect to the showers at angles $\beta =0^{\circ}$, $30^{\circ}$, $60^{\circ}$ or $90^{\circ}$. In order to cover a larger range of telescope distances, MSTs were simulated at 9 different positions for each orientation $\beta$.
Since the telescope trigger efficiency decreases with the distance from the shower impact point on the ground [@KonradB], the stereo rate is reduced for larger separations between the detectors ($D$). This is shown in Fig. \[Fig:zenith\], where for $\theta < 10^{\circ}$, the rate for telescopes at 381 m distance (blue) is $\sim 17\%$ of the rate at 125 m (red). The effective area increases with $\theta$ roughly as $\propto 1 / \mathrm{cos{\theta}}$ and the more advantageous configurations are those with larger separations of telescopes. These effects are coupled and their combined impact on the stereo rate is a matter of the separation $D$ and orientation $\beta$ of telescopes with respect to the shower direction ($\theta$, $\phi$), illustrated in the left plot in Fig. \[Fig:ZA-AZ\_noMF\].
![ Left: trigger rate of two MSTs vs the telescope separation ($D$) and their relative orientation ($\beta$, different markers) with respect to the shower direction given by the zenith angle ($\theta$, different colours). Right: solid and dashed lines show the fit results of Eq.(\[Eq:rate\_fit\]) to the data vs $d_{\mathrm{SP}}$ for $\theta = 0^{\circ}$ (red) and $60^{\circ}$ (blue), respectively. $1\sigma$ statistical uncertainties (contours) of all rates from the left plot with the same $\theta$ are shown vs $d_{\mathrm{SP}}$. []{data-label="Fig:ZA-AZ_noMF"}](fig3_1.png "fig:"){width="0.49\columnwidth"} ![ Left: trigger rate of two MSTs vs the telescope separation ($D$) and their relative orientation ($\beta$, different markers) with respect to the shower direction given by the zenith angle ($\theta$, different colours). Right: solid and dashed lines show the fit results of Eq.(\[Eq:rate\_fit\]) to the data vs $d_{\mathrm{SP}}$ for $\theta = 0^{\circ}$ (red) and $60^{\circ}$ (blue), respectively. $1\sigma$ statistical uncertainties (contours) of all rates from the left plot with the same $\theta$ are shown vs $d_{\mathrm{SP}}$. []{data-label="Fig:ZA-AZ_noMF"}](fig3_2.png "fig:"){width="0.49\columnwidth"}
For showers incident from the zenith ($\theta=0^{\circ}$, red markers in the left plot in Fig. \[Fig:ZA-AZ\_noMF\]), the Cherenkov pool on ground is roughly circular and there is no dependence of rates on the telescope alignment relative to the shower direction ($\beta$). At higher zenith angles ($\theta=60^{\circ}$), the pool is an ellipse and telescope pairs along its major axis (CT2, CT3 in Fig. \[Fig:coordinates\]; red circles in Fig. \[Fig:ZA-AZ\_noMF\], left) trigger in coincidence more likely than the pairs which are oriented in an orthogonal direction (CT1, CT2; blue triangles in Fig. \[Fig:ZA-AZ\_noMF\], left), although their separations may be the same.
The dependence on $\beta$ and $D$ can be eliminated if the rates are examined in terms of telescope separations projected in the shower plane $d_{\mathrm{SP}}$, illustrated in the right plot in Fig. \[Fig:ZA-AZ\_noMF\] for all data with different $\beta$ ($0^{\circ}$ and $90^{\circ}$) combined into sets according to $\theta$. Unlike the fixed separation of telescopes $D$ in the ground plane, the distance in the shower plane is a function of the pointing of the telescopes: $d_{\mathrm{SP}} (\theta,\beta) = D . \sqrt{ 1 - \sin^{2}{\theta} . \cos^{2}{\beta} }$. It follows that $d_{\mathrm{SP}}=D$ when $\theta = 0^{\circ}$ (compare the red markers and contours in Fig. \[Fig:ZA-AZ\_noMF\]) or $\beta = 90^{\circ}$ (compare the triangles and contours).
The geometrical dependence of the rate $R_{\mathrm{Fit}}(d_{\mathrm{SP}})$ can be fit by the functions (lines in Fig. \[Fig:ZA-AZ\_noMF\]) $$\label{Eq:rate_fit}
R_{\mathrm{Fit}} (d_{\mathrm{SP}}) =
\begin{cases}
A_0 \cdot e^{A_1 \cdot (d_{\mathrm{SP}} - A_3)}, & \text{if}\ d_{\mathrm{SP}}<A_3 \\
A_0 \cdot e^{A_2 \cdot (d_{\mathrm{SP}} - A_3)}, & \text{if}\ d_{\mathrm{SP}}>A_3.
\end{cases}$$ While this effective description removes the dependence on the distance and relative alignment of telescopes, the coefficients $A_{i}$ still depend on the zenith angle. Functions $A_{i} (\cos{\theta})$ were found by fitting the values of $A_{i}$ obtained from fits of the trigger rate to the Eq.(\[Eq:rate\_fit\]) for six values of $\theta$ in the range $[0^{\circ},60^{\circ}]$. The fit results provide four sets of look-up parameters (Table \[Tab:fit\]) which together with Eq.(\[Eq:rate\_fit\]) allow to estimate the correction $F\left( D, \theta, \beta \right)$ of the stereo trigger rate for the geometrical configuration, leaving Eq.(\[Eq:transparency\]) as $\tau = \tau(\mathrm{AOD}, \vec{B}, \epsilon)$. Note that a similar study was performed with the LSTs providing equivalent results.
Earth’s magnetic field {#Sec:Bfield}
----------------------
The Earth’s magnetic field also affects the distribution of Cherenkov light on the ground [@KonradB]. Here, we investigate these effects using the MC set of Sec. \[Sec:Geometry\]. In addition to this we include another MC set in this study, in which the magnetic field intensity was changed from $|\vec{B}| \equiv B\simeq 0~\mu$T to $38.7~\mu$T, consistently with the CTA-N site[^3]. The effects of the magnetic field are compared for the configurations with $(\theta = 20^{\circ}, \phi = 180^{\circ})$ and $(\theta = 50^{\circ}, \phi = 0^{\circ})$, corresponding to an angle between the vector $\vec{B}$ and the shower direction $\Omega$ (see Fig.\[Fig:coordinates\]) of $5^{\circ}$ and $72^{\circ}$, respectively.
![ Residuals $(R_{\mathrm{Fit}}-R_{\mathrm{True}}) / R_{\mathrm{True}}$ between the simulated trigger rate ($R_{\mathrm{True}}$) and the fit value ($R_{\mathrm{Fit}}$) from Eq. (\[Eq:rate\_fit\]) which includes only the geometrical effects ($B\simeq0$). Red crosses are the residuals for data with $B\simeq0$. Circles and triangles correspond to data with $B\neq0$ for different telescope alignment $\beta$. Angle $\Omega$ (Fig. \[Fig:coordinates\]) is $5^{\circ}$ and $72^{\circ}$ for the left and right plot, respectively. []{data-label="Fig:residuals"}](fig4_1.png "fig:"){width="0.49\columnwidth"} ![ Residuals $(R_{\mathrm{Fit}}-R_{\mathrm{True}}) / R_{\mathrm{True}}$ between the simulated trigger rate ($R_{\mathrm{True}}$) and the fit value ($R_{\mathrm{Fit}}$) from Eq. (\[Eq:rate\_fit\]) which includes only the geometrical effects ($B\simeq0$). Red crosses are the residuals for data with $B\simeq0$. Circles and triangles correspond to data with $B\neq0$ for different telescope alignment $\beta$. Angle $\Omega$ (Fig. \[Fig:coordinates\]) is $5^{\circ}$ and $72^{\circ}$ for the left and right plot, respectively. []{data-label="Fig:residuals"}](fig4_2.png "fig:"){width="0.49\columnwidth"}
Using Eq. (\[Eq:rate\_fit\]) and Table \[Tab:fit\], we estimated the trigger rate $R_{\mathrm{Fit}}$ for data with $B\simeq 0$, accounting only for the geometrical effect. The residuals between $R_{\mathrm{Fit}}$ and the simulated rate $R_{\mathrm{True}}$ were calculated for the data with $B\simeq 0$ and $B\neq 0$, illustrated in Fig. \[Fig:residuals\] as a function of the distance $d_{\mathrm{SP}}$. For the sake of simplicity, the statistical error bars are omitted in Fig. \[Fig:residuals\]. These uncertainties rise with the distance from roughly $10\%$ to as much as $20\%$ and $40\%$ for $\Omega=5^{\circ}$ and $\Omega=72^{\circ}$, respectively.
For the shower directions nearly parallel with the magnetic field ($\Omega = 5^{\circ}$, left plot in Fig. \[Fig:residuals\]), the Lorentz force exerted on the particles is negligible and the trigger rates in the magnetic field (circles and triangles) are expected to be consistent with the case $B \simeq 0$ (crosses). The deviations of $R_{\mathrm{True}}$ from the geometrical fit $R_{\mathrm{Fit}}$ are randomly distributed and are independent of $d_{\mathrm{SP}}$ and $\beta$. The RMS of the residuals for $B \simeq 0$ is at the level of $2\%$ for any value of $\Omega$.
For a non-zero magnetic field in a non-parallel orientation relative to the shower direction ($\Omega = 72^{\circ}$, right plot), the mean deviation of $R_{\mathrm{Fit}}$ from $R_{\mathrm{True}}$ differs by $\sim 10\%$ (circles and triangles) compared to the case $B\simeq0$. The residuals also show a tendency with $d_{\mathrm{SP}}$ depending on the relative alignment $\beta$ of the telescopes with respect to the shower direction. In the studied configuration, the deflection of particles in the magnetic field causes the Cherenkov pool to broaden roughly orthogonally to the shower and magnetic field direction. A telescope alignment parallel to the shower direction (blue circles) is then less favourable than the perpendicular orientation (green triangles), as opposed to the instances with a small magnetic field effect (red crosses).
A correction of the trigger rates for the magnetic field effects requires to take into account both the magnitude and the direction of the Lorentz force for all possible pointings of telescopes and their distances ($d_{\mathrm{SP}}$). However, the residual distributions are rather flat for examined angles $\Omega$ up to $d_{\mathrm{SP}} \approx 200$ m (Fig. \[Fig:residuals\]). The $\vec{B}$–dependence in Eq.(\[Eq:transparency\]) may be neglected by imposing a cut on the maximum separation of telescope pairs which will be considered in the calibration. The systematic uncertainties due to the magnetic field are then expected to be within the residual RMS for $B\simeq0$. For a better correction we need more MC simulations at different $\Omega$.
Hardware dependence {#Sec:Hardware}
-------------------
![ Trigger rate of two MSTs with $d_{\mathrm{SP}} = 150$ m vs the mirror degraded efficiencies $\varepsilon_1$ and $\varepsilon_2$ for $\Omega = 5^{\circ}$ and $72^{\circ}$ (Fig. \[Fig:coordinates\]). Rates were corrected for the geometric and magnetic field effects and normalised by the value of $R_{\mathrm{Fit}}$ in Eq. \[Eq:rate\_fit\] for $\varepsilon_{1}=\varepsilon_{2}=1$. The line represents $R = \varepsilon_{1} \cdot \varepsilon_{2}$. []{data-label="Fig:Rate_Eff"}](fig5.png){width="\columnwidth"}
$P$ $p_0$ $p_1$ $p_2$
------- ---------------------------- -------------- ---------------
$A_0$ $934\pm14$ $15.9\pm9.9$ $1.13\pm0.09$
$A_2$ $(7.4\pm0.4)\text{e}^{-3}$ $-3.7\pm0.4$ $0.58\pm0.03$
$A_3$ $184\pm9$ $-6.5\pm2.2$ $0.48\pm0.04$
\[Tab:fit\]
![ Trigger rate of two MSTs with $d_{\mathrm{SP}} = 150$ m vs the mirror degraded efficiencies $\varepsilon_1$ and $\varepsilon_2$ for $\Omega = 5^{\circ}$ and $72^{\circ}$ (Fig. \[Fig:coordinates\]). Rates were corrected for the geometric and magnetic field effects and normalised by the value of $R_{\mathrm{Fit}}$ in Eq. \[Eq:rate\_fit\] for $\varepsilon_{1}=\varepsilon_{2}=1$. The line represents $R = \varepsilon_{1} \cdot \varepsilon_{2}$. []{data-label="Fig:Rate_Eff"}](fig6.png "fig:"){width="\columnwidth"} \[Fig:intercalibration\]
As stated in Eq.(\[Eq:transparency\]), the rate depends also on the hardware state of the instrument quantified by the detection efficiency $\epsilon$ which changes due to the degradation or maintenance activities of various components. Variations in the reflectivity of optical elements and the responses of camera photodetectors have to be accounted for in dedicated calibration procedures [@CettinaICRC2017]. Other influence of the hardware is the plan to maintain trigger rates constant by adapting trigger thresholds to the measured individual pixel rates. This is foreseen for the LSTs, but it is not discussed in this work.
We examined hardware effects using the MC simulations for two telescopes with varying optical efficiencies. In Fig. \[Fig:Rate\_Eff\], the stereo rate exhibits a linear dependence on the product of telescope efficiencies $\varepsilon_{1} \cdot \varepsilon_{2}$ for two configurations $\Omega$ ($5^{\circ}$ and $72^{\circ}$) with fixed $d_{\mathrm{SP}}=150$ m. This behaviour may be coincidental since it follows from the energy spectrum of cosmic rays that $R \propto E^{-1.7}_{\mathrm{th}}$, where the energy threshold is approximately inversely proportional to the telescope efficiency, i.e $E_{\mathrm{th}} \propto \varepsilon^{-1}$. It further follows that $R \propto \left( \varepsilon_{1} \cdot \varepsilon_{2} \right)^{0.85} = \varepsilon^{1.7}$ for $\varepsilon_{1} = \varepsilon_{2} \equiv \varepsilon$. This proportionality may differ for instances when $\varepsilon_{1} \neq \varepsilon_{2}$ and it is only the final superposition of all possibilities that behaves linearly. Considering this linear dependence as an appropriate approximation, for the moment we will assume it as an effective description of the stereo trigger rate.
Inter-calibration of telescope responses {#Sec:Intercalibration}
========================================
In the following, we will describe the inter-calibration principle for responses of telescopes of the same type using the effective rate correction $F\left( D, \theta, \beta, \vec{B}, \varepsilon \right)$ in Eq.(\[Eq:transparency\]).
Only stereo rates retrieved for pairs of telescopes are assumed in the calibration. A cut on the maximum distance of 200 m between telescopes is applied to select such pairwise stereo rates for calibration. Using the estimates from Eq.(\[Eq:rate\_fit\]) with Table \[Tab:fit\] in Eq.(\[Eq:transparency\]), the rates are corrected for geometrical dependencies ($D, \theta, \beta$) with the systematic uncertainties, including the effect of the magnetic field, assumed to be within $2\%$ (Sec. \[Sec:Bfield\]). This way, pairwise transparency coefficients $\tau_{ij}$ are obtained for the hardware conditions $\varepsilon_{i} = \varepsilon_{j} = 1$, where $i$ and $j$ label the two telescopes in coincidence. On the assumption that the atmospheric conditions over the array are uniform, the transparency observed by all telescope pairs is expected to be the same (within the systematic uncertainties), i.e. $\tau_{ij} = \tau_{kl} = T$, where $T$ is the true atmospheric transparency.
Possible degradations of the instrument performance are not included in the definition of $\tau_{ij}$. As shown in Sec. \[Sec:Hardware\], trigger rates modified by hardware changes can be expressed as $R_{ij} (\varepsilon_{i}, \varepsilon_{j}) \approx \varepsilon_{i} \cdot \varepsilon_{j} \cdot R_{ij} (\varepsilon_{i} = \varepsilon_{j} = 1)$, implying $\tau_{ij} (\varepsilon_{i}, \varepsilon_{j}) \approx \varepsilon_{i} \cdot \varepsilon_{j} \cdot T$. Variations of telescope efficiencies from the nominal values are then quantified by the asymmetry in coefficients $\tau_{ij}$: $$\label{Eq:asymmetry}
a_{ij/kl} = \frac{\tau_{ij}-\tau_{kl}}{\tau_{ij}+\tau_{kl}}.$$
Efficiencies $\varepsilon_i$ are treated as free parameters allowing the relative inter-calibration of telescope responses by means of minimisation of the sum of squared residuals $$\label{Eq:chi2}
\chi^{2} = \sum_{\mathrm{pairs}}{ \left( a_{ij/kl} - \frac{\varepsilon_{i} \cdot \varepsilon_{j} - \varepsilon_{k} \cdot \varepsilon_{l}}{\varepsilon_{i} \cdot \varepsilon_{j} + \varepsilon_{k} \cdot \varepsilon_{l}} \right)^{2} \cdot \sigma^{-2}_{ij/kl}} ,$$ where $\sigma^{2}_{ij/kl}$ are the variances of asymmetries $a_{ij/kl}$ and the sum runs over all selected pairs of telescopes present during the data acquisition.
In this work, the described procedure is applied only to the telescopes of the same sub-system. In the simulated CTA-N layout 3AL4M15-5F, each of the 15 MSTs and 4 LSTs has at least two neighbours within 200 m, providing in total 24 and 6 pairs, respectively. As the inter-calibration is performed in a relative manner, it requires to fix the value of efficiency of one telescope. Neither the choice of the reference telescope nor the exact value of the fixed efficiency is relevant for the inter-calibration.
The outlined method was applied to the full CTA-N array [@HassanICRC2017]. The atmospheric transparency $T$ was constant in the simulations. All telescopes got randomly assigned optical efficiencies from the normal distribution $\mathcal{N} (0.7, 0.1)$. Sets of telescope efficiencies were reconstructed from the minimisation in Eq.(\[Eq:chi2\]) per each sub-system and then compared to the initial MC values (Fig. \[Fig:intercalibration\]). Both true ($\varepsilon_{\mathrm{True}}$) and recovered ($\varepsilon_{\mathrm{Reco}}$) sets of efficiencies were normalised by their respective mean values. Their RMS is $\sim3\%$ for the inter-calibration of both the MST and LST sub-systems.
Atmospheric calibration {#Sec:Atmosphere}
=======================
The calibration of the atmospheric transparency to Cherenkov light is achieved using the relative telescope efficiencies $\varepsilon_{\mathrm{Reco}}$ (see Sec. \[Sec:Intercalibration\]). As the absolute value of the normalisation of efficiencies is not specified in this way, it has to be fixed by another calibration procedure [@CettinaICRC2017]. For the purpose of online monitoring, the normalisation can be chosen from the previous observation run[^4], assuming these efficiencies do not change significantly within the same night ($<5\%$).
Using the data set from the previous section, we re-scaled each $\varepsilon_{i}$ obtained in the calibration by a factor $\overline{\varepsilon_{\mathrm{True}}} / \overline{\varepsilon_{\mathrm{Reco}}}$. Inverse values of the re-scaled efficiencies were used to correct transparency coefficients $\tau_{ij}$. The mean value of corrected $\tau_{ij}$ for all selected pairs of telescopes determines the estimate of the atmospheric transparency in the observation run $T (\mathrm{AOD}) = 1.02 \pm 0.04$, consistent with the fact that all simulations in this study assumed the same atmospheric conditions.
Conclusions {#Sec:Conclusions}
===========
The impact of atmospheric conditions on the trigger rates and reconstructed effective areas of IACTs has been previously addressed by different methods [@Dorner; @CAT; @Nolan]. In this work, we have presented another approach using the Cherenkov Transparency Coefficient, successfully implemented in the H.E.S.S. experiment [@Hahn], as an atmospheric sensitive quantity in CTA.
Necessitated by the more complex CTA layout, an effective correction of the trigger rate for the geometrical and hardware effects has been found in order to maintain the CTC independence from these issues. Neglecting the effects of the magnetic field, the systematic uncertainty of the trigger rate has been found to be $\sim2\%$ (for $d_{\mathrm{SP}}<200$ m). For the simulated configuration, the recovered CTC is consistent with the expectation. In addition, the comparison of transparency coefficients obtained per telescope pairs has been shown to be a viable inter-calibration procedure for relative telescope responses with the resolution of reconstructed efficiencies being $\sim3\%$ for the CTA-N. All results in this study have been obtained assuming fixed atmospheric conditions.
Since the CTC is calculated using the output of scientific observations, it provides a crosscheck for other calibration methods [@CettinaICRC2017] without the need for additional devices or interference with the regular data taking. Currently under investigation are the robustness of the CTC under different aerosol concentrations and air density profiles and its application for other telescope types anticipated in CTA (especially for CTA-S) as well as the cross-calibration of different sub-systems. Future study is also foreseen to investigate the feasibility of the method under varying trigger thresholds of the LSTs.
This work was conducted in the context of the CTA Central Calibration Facilities Work Package. We gratefully acknowledge financial support from the agencies and organizations listed here: http://www.cta-observatory.org/consortium\_acknowledgments
[99]{}
R. A. Ong et al., these proceedings, (2017) M. C. Maccarone et al., these proceedings, (2017) J. Ebr et al., these proceedings, (2017) J. Hahn et al., Astropart. Phys., [**54**]{} (2014) 25 \[[1310.1639](https://arxiv.org/abs/1310.1639)\] K. Bernlöhr, Astropart.Phys., [**30**]{} (2008) 149 \[[0808.2253](https://arxiv.org/abs/0808.2253)\] P. Cumani et al., these proceedings, (2017) K. Bernlöhr, Astropart. Phys., [**12**]{} (2000) 255 \[[astro-ph/9908093](https://arxiv.org/abs/astro-ph/9908093)\] S. Lombardi, Ph.D. thesis, Padua University (2010) D. Dorner, K. Nilsson and T. Bretz, A&A, [**493**]{} (2009) 721 \[[0808.0279](https://arxiv.org/abs/0808.0279)\] S. Le Bohec, et al., NIM A, 416:425 (1998) \[[astro-ph/9804133](https://arxiv.org/abs/astro-ph/9804133)\] S.J. Nolan et al., Astropart.Phys., [**34**]{} (2010) 304 \[[1009.0517](https://arxiv.org/abs/1009.0517)\] A.M.W. Mitchell et al., Astropart. Phys., [**75**]{} (2016) 1 \[[1510.06526](https://arxiv.org/abs/1510.06526)\]
[^1]: Now with the German Aerospace Center (DLR), Earth Observation Center (EOC), D-82234, Wessling, Germany
[^2]: http://cta-observatory.org
[^3]: <https://ngdc.noaa.gov/geomag-web/#igrfwmm>
[^4]: Observation run refers to the observation time unit applied in current IACT systems ($\approx 20 - 30$ min).
|
---
abstract: 'We use the quantum threshold laws combined with a classical capture model to provide an analytical estimate of the chemical quenching cross sections and rate coefficients of two colliding particles at ultralow temperatures. We apply this quantum threshold model (QT model) to indistinguishable fermionic polar molecules in an electric field. At ultracold temperatures and in weak electric fields, the cross sections and rate coefficients depend only weakly on the electric dipole moment $d$ induced by the electric field. In stronger electric fields, the quenching processes scale as $d^{4(L+\frac{1}{2})}$ where $L>0$ is the orbital angular momentum quantum number between the two colliding particles. For $p-$wave collisions ($L=1$) of indistinguishable fermionic polar molecules at ultracold temperatures, the quenching rate thus scales as $d^6$. We also apply this model to pure two dimensional collisions and find that chemical rates vanish as $d^{-4}$ for ultracold indistinguishable fermions. This model provides a quick and intuitive way to estimate chemical rate coefficients of reactions occuring with high probability.'
author:
- 'Goulven Qu[é]{}m[é]{}ner and John L. Bohn'
title: Strong Dependence of Ultracold Chemical Rates on Electric Dipole Moments
---
=cmr7
Introduction
============
Ultracold samples of bi-alkali polar molecules have been created very recently in their ground electronic $^1\Sigma$, vibrational $v=0$, and rotational $N=0$ states [@Sage05; @Ni08; @Deiglmayr08]. This is a promising step before achieving Bose-Einstein condensates or degenerate Fermi gases of polar molecules, provided that further evaporative cooling is efficient. For this purpose, elastic collision rates must be much faster than inelastic quenching rates. This issue is somewhat problematic for the bi-alkali molecules recently created, since they are subject to quenching via chemical reactions. If a reaction should occur, the products are no longer trapped.
For alkali dimers that possess electric dipole moments, elastic scattering appears to be quite favorable, since elastic scattering rates are expected to scale with the fourth power of the dipole moment [@Hensler03; @Bohn09]. Inelastic collisions of polar species can originate from two distinct sources. The long-range dipole-dipole interaction itself is anisotropic and can cause dipole orientations to be lost. This kind of loss generally leads to high inelastic rates, and is regarded as the reason why electrostatic trapping of polar molecules is likely not feasible [@Avdeenkov02]. Moreover, these collisions also scale as the fourth power of dipole moment in the ultracold limit [@Hensler03], meaning that the ratio of elastic to inelastic rates does not in general improve at higher electric fields. This sort of loss can be prevented by trapping the molecules in optical dipole traps.
More serious is the possibility that collisions are quenched by chemical reactions. Chemical reaction rates are known to be potentially quite high even at ultracold temperatures [@Bala01; @Soldan02; @Quemener04; @Quemener05; @Cvitas05a; @Cvitas05b; @Lara06; @Quemener07; @Hutson07; @QuemenerCHAPTER; @Quemener09]. Indeed, for collision energies above the Bethe–Wigner threshold regime, it appears that many quenching rates, chemical or otherwise, of barrierless systems are well described by applying Langevin’s classical model [@Langevin05]. In this model the molecules must surmount a centrifugal barrier to pass close enough to react, but are assumed to react with unit probability when they do so. This model has adequately described several cold molecule quantum dynamics calculations [@Quemener05; @Cvitas05a; @Lara06; @QuemenerCHAPTER; @Quemener09].
Within the Bethe–Wigner limit, scattering can be described by an elegant Quantum Defect Theory (QDT) approach [@Julienne89; @Burke98; @Mies00; @Julienne09]. This approach makes explicit the dominant role of long-range forces in controlling how likely the molecules are to approach close to one another. Consequently, quenching rate constants can often be written in an analytic form that contains a small number of parameters that characterize short-range physics such as chemical reaction probability. For processes in which the quenching probability is close to unity, the QDT theory provides remarkably accurate quenching rates [@Orzel99; @Hudson08]. For dipoles, however, the full QDT theory remains to be formulated.
In this article we combine two powerful ideas – suppression of collisions due to long-range physics, and high-probability quenching inelastic collisions for those that are not suppressed – to derive simple estimates for inelastic/reactive scattering rates for ultracold fermionic dipoles. The theory arrives at remarkably simple expressions of collision rates, without the need for the full machinery of close-coupling calculations. Strikingly, the model shows that quenching collisions scale as the [*sixth*]{} power of the dipole moment for ultracold $p-$wave collisions. On the one hand, this implies a tremendous degree of control over chemical reactions by simply varying an electric field, complementing alternative proposals for electric field control of molecule-molecule [@Hudson06] or atom-molecule [@Tscherbul08] chemistry. On the other hand, it also implies that evaporative cooling of polar molecules may become more difficult as the field is increased. In section II, we formulate the theoretical model for three dimensional collisions. In section III, we apply this model to pure two dimensional collisions and conclude in section IV. In the following, quantities are expressed in S.I. units, unless explicitly stated otherwise. Atomic units (a.u.) are obtained by setting $\hbar = 4 \pi \varepsilon_0 = 1$.
Collisions in three dimensions
==============================
Cross sections and collision rates
----------------------------------
In quantum mechanics, the quenching cross section of a pair of colliding molecules (or any particles) of reduced mass $\mu$ for a given collision energy $E_c$ and a partial wave $L,M_L$ is given by $$\begin{aligned}
\sigma^{qu}_{L,M_L} &=& \frac{\hbar^2 \pi}{2 \mu E_c} \ |T^{qu}_{L,M_L}|^2 \times \Delta % \nonumber \\
\label{cross}\end{aligned}$$ where $T^{qu}$ is the transition matrix element of the quenching process, $|T^{qu}_{L,M_L}|^2$ represents the quenching probability, and the factor $\Delta$ represents symmetrization requirements for identical particles [@Burke99]. If the two colliding molecules are in different internal quantum states (distinguishable molecules), $\Delta=1$ and if the two colliding molecules are in the same internal quantum state (indistinguishable molecules), $\Delta=2$. The total quenching cross section of a pair of molecules is $\sigma^{qu} = \sum_{L,M_L} \sigma^{qu}_{L,M_L} $. The quenching rate coefficient of a pair of molecules for a given temperature $T$ (collisional event rate) is given by $$\begin{aligned}
K^{qu}_{L,M_L} = < \sigma^{qu}_{L,M_L} \times v >
%\nonumber \\
= \int_0^\infty \sigma^{qu}_{L,M_L} \, v \, f(v) \, dv
\label{rate}\end{aligned}$$ where $$\begin{aligned}
f(v) = 4 \pi \, \left( \frac{\mu}{2 \pi k_B T} \right)^{3/2} \, v^2 \, \exp{[-(\mu v^2)/(2 \, k_B \, T)]} \end{aligned}$$ is the Maxwell–Boltzmann distribution for the relative velocities for a given temperature and $k_B$ is the Maxwell–Boltzmann constant. The total quenching rate coefficient of a pair of molecules is $K^{qu} = \sum_{L,M_L} K^{qu}_{L,M_L} $. To avoid confusion, we will also write the corresponding rate equation for collisions between distinguishable and indistinguishable molecules. First, we consider collisions between two distinguishable molecules in quantum states $a$ and $b$ ($\Delta=1$ in Eq. ). During a time $dt = \tau$, where $\tau$ is the time of a quenching collisional event, the number of molecules $N_a$ lost in each collision is one and the number of molecules $N_b$ lost in each collision is one. Then $dN_a/dt=-1/\tau$ and $dN_b/dt=-1/\tau$. The volume per colliding pairs of molecules is $V/(N_a \, N_b)$, where $V$ stands for the volume of the gas. During the time $\tau$, the quenching collisional event is associated with a volume $<\sigma^{qu} \times v> \times \tau = K^{qu} \times \tau $. By definition of $\tau$, this volume should be equal to the one occupied by just one colliding pair of molecules. Then we get $K^{qu} \times \tau = V/(N_a \, N_b)$. The rate equation for the number of molecule $N_a$ or $N_b$ is then given by $$\begin{aligned}
\frac{dN_{a,b}}{dt} = - K^{qu} \times \frac{N_a \, N_b}{V} .\end{aligned}$$ If $n_a = N_a / V$ and $n_b = N_b / V$ are the densities of molecule $a$ and $b$ in the gas, then $$\begin{aligned}
\frac{dn_{a,b}}{dt} = - K^{qu} \times n_a \, n_b.\end{aligned}$$ We consider now the case of collisions between two indistinguishable molecules ($\Delta=2$ in Eq. ). During the time $dt = \tau$ the number of molecules $N$ lost in each collision is two. Then we get $dN/dt=-2/\tau$. The volume per colliding pairs of molecules is $V/(N(N-1)/2)$ where we have taken into account the indistinguishability of the molecules. For the same reason explained above, the volume associated with the collisional event during the time $\tau$ should be equal to the volume occupied by just one colliding pair of molecules. And then $K^{qu} \times \tau = V/(N(N-1)/2)$. The rate equation for the number of molecule $N$ is then given by $$\begin{aligned}
\frac{dN}{dt} = - 2 \, K^{qu} \times \frac{N(N-1)/2}{V} .\end{aligned}$$ If $n = N / V$ and $N(N-1) \approx N^2$, then $$\begin{aligned}
\frac{dn}{dt} = - K^{qu} \times n^2.\end{aligned}$$
Quantum threshold model
-----------------------
\[t\]
![ (Color online) Effective potential barrier $V(R)$ as a function of the intermolecular separation $R$. $V_b$ and $R_b$ denote the height and the position of the centrifugal barrier. \[spag-FIG\] ](figure1.eps){width="8cm"}
We consider the case of two identical ultracold fermionic polar molecules, as has been achieved very recently for KRb dimers [@Ni08; @Ospelkaus09] in their ro-vibronic ($^1\Sigma,v=0,N=0$) ground state. Under these circumstances, because of Fermi exchange symmetry, the relative orbital angular momentum quantum number $L$ between the two molecules must take odd values $L = 1,3,5,7 ...$. These molecules are polar molecules and can be controlled by an electric field $\cal E$. In the usual basis set of partial waves $|L M_L \rangle$, the long-range behavior of two colliding polar molecules in a presence of an electric field is governed by an interaction potential matrix whose elements are $$\begin{gathered}
\langle L M_L | V(R) | L' M_L' \rangle = \\
\left\{ \frac{\hbar^2 L(L+1)}{2 \mu R^2} - \frac{C_6}{R^6} \right\} \delta_{L,L'} \, \delta_{M_L,M_L'} \\
- \frac{{C_3}(L,L';M_L)}{R^3} \delta_{M_L,M_L'}
\label{barrierpot-efield}\end{gathered}$$ where $R$ denotes the distance between the two molecules. The diagonal elements represent effective potentials for the colliding molecules and the non-diagonal elements represent couplings between them. The coefficient $C_6$ is the van der Waals coefficient, assumed to be isotropic in the present treatment. The $- {C_3} / R^3$ is the term corresponding to the electric dipole-dipole interaction expressed in the partial wave basis set $\langle L M_L | V_{dd}(R,\theta,\varphi) | L' M_L' \rangle$ between two polarized molecules in the electric field direction, with $V_{dd}(R,\theta,\varphi)= d^2 (1 - 3 \cos^2{\theta}) / (4 \pi \varepsilon_0 \, R^3)$, where $d=d({\cal E})$ is the induced electric dipole moment, and $\theta,\varphi$ represent the relative orientation between the molecules. In the basis set of partial waves, $C_3$ takes the form $$\begin{gathered}
%\begin{eqnarray}
{C_3}(L,L';M_L)
= \alpha(L,L';M_L) \, \frac{d^2}{4 \pi \varepsilon_0} \\
= 2 \ (-1)^{M_L} \ \sqrt{2L+1} \ \sqrt{2L'+1} \\
\ \left( \begin{array}{ccc} L & 2 & L' \\ 0 & 0 & 0 \end{array} \right)
\ \left( \begin{array}{ccc} L & 2 & L' \\ -M_L & 0 & M_L' \end{array} \right) \ \frac{d^2}{4 \pi \varepsilon_0} .
\label{C3coef}
%\end{eqnarray}\end{gathered}$$ The large bracket symbols denote the usual 3$-j$ coefficients. The coefficient $\alpha$ is introduced to simplify further notations. The combination between repulsive and attractive terms in the effective potentials (diagonal terms) of Eq. generate a potential barrier of height $V_b$ which is plotted schematically in Fig. \[spag-FIG\]. The height of this barrier plays a crucial role as it can prevent the molecule from accessing the short range region where reactive chemistry occurs.
The quantum threshold (QT) model consists of two conditions. First, for $E_c < V_b$, we use the Bethe–Wigner threshold laws [@Bethe35; @Wigner48] for ultracold scattering. Second, we use the classical capture model (Langevin model) [@Langevin05] to estimate the probability of quenching for $E_c \ge V_b$. A classical capture model is indiferent to collision energies $E_c < V_b$ since the barrier prevents the molecules from coming close together. In real-life quantum scattering, collisions do occur at these energies due to quantum tunneling, and they are the ones relevant to ultracold collisions. Moreover, collisions in this energy regime are dictated by the the Bethe–Wigner quantum threshold laws. For quenching collisions, the threshold laws [@Bethe35; @Wigner48; @Sadeghpour00] state that $|T^{qu}_{L,M_L}|^2 \propto E_c^{L+\frac{1}{2}}$. For $E_c \ge V_b$, a classical capture model will guarantee to deliver the molecule pair to small values of $R$, where chemical reactions are likely to occur with unit probability (see Fig. \[spag-FIG\]). Following this classical argument, we will assume that when $E_{c} \ge V_b $, the quenching probability reaches unitarity $|T^{qu}|^2=1$. Using this assumption together with the quantum threshold laws, the QT quenching tunneling probability below the barrier can be written as $$\begin{aligned}
|T^{qu}_{L,M_L}|^2 = \left( \frac{E_c}{V_b} \right)^{L+1/2} .
\label{Tqusq}\end{aligned}$$ Consequently, the quenching cross section and rate coefficient are approximated by $$\begin{aligned}
\sigma^{qu}_{L,M_L}
& = & \frac{\hbar^2 \pi}{2 \mu V^{L+\frac{1}{2}}_b} \ E_c^{L-\frac{1}{2}} \times \Delta \nonumber \\
K^{qu}_{L,M_L} &=&
\frac{\hbar^2 \pi}{\sqrt{2 \mu^3} V_b^{L+\frac{1}{2}}} \ < E_c^{L} > \times \Delta
\label{BetheWigner-n6}\end{aligned}$$ for $E_c < V_b$. The QT model has the simple and intuitive advantage of showing how the cross sections and rate coefficients scale with the height of the entrance centrifugal barrier. For $E_c \ge V_b$, it is easy to find the corresponding expression of the cross section in Eq. by setting $|T^{qu}|^2=1$. The cross section $\sigma^{qu}_{L,M_L}$ will reach the unitarity limit at $E_c \ge V_b$. It is also easy to find the corresponding expression of the rate coefficient in Eq. . The QT model is general for any collision between two particles provided that there is a barrier in the entrance collision channel and that chemical reactions occur with near unit probability at short range. The only information on short range chemistry is that chemical reactions occur at full and unit probability and the only information on long range physics is provided by the height of the entrance barrier $V_b$. The QT model describes the background scattering process, it does not take into account scattering resonances. Note that the model will not be appropriate in the present form for barrierless ($s-$wave) collisions since $V_b=0$. For this particular type of collisions that do not possess a centrifugal barrier, the QDT theory can be usefully applied [@Ospelkaus-chemistry-09; @Julienne09; @Idziaszek09]. The present form of the QT model does not take into account the anisotropy of the intermolecular potential at intermediate range and/or the electronic and nuclear spin structure of the molecular complex but remains suitable as far as the entrance centrifugal barrier takes place at long range. The QT model will have to be modified if longer range interactions takes place. For example, collisions between $N=0$ and $N=1$ polar molecules can have long range interactions between hyperfine states due to dipolar and hyperfine couplings [@Ospelkaus09; @Aldegunde08]. However, for collisions between rotationless $N=0$ polar molecules, the hyperfine couplings are weak and the QT model can be applied without further modifications.
Rates in zero electric field
----------------------------
\[h\]
![ (Color online) Quenching cross section of $^{39}$K + $^{39}$K$_2$ as a function of the collision energy for the partial wave $L=1-5$: (i) calculated with a full quantum calculation (solid lines), reproduced from Ref. [@Quemener05] (ii) using the QT model (dashed lines) (iii) fitting the QT model (dotted line), using $p=0.34$ in Eq. . \[crossK3-FIG\] ](figure2.eps){width="8cm"}
\[h\]
![ (Color online) Quenching probability of $^{39}$K + $^{39}$K$_2$ as a function of the collision energy for the partial wave $L=1$: comparison between the full quantum calculation (solid line) and the QT model (dashed line). The fitted QT model appears as a dotted line. The height of the barrier $V_b$ and the corrected height $\gamma \times V_b$ ($\gamma=2.06$) appear as vertical lines. \[probaK3-FIG\] ](figure3.eps){width="8cm"}
In the absence of an electric field in Eq. , the long range potential reduces to a diagonal term in the basis set of partial waves. The position and height of the barrier are given by $$\begin{aligned}
R_b &=& \left( \frac{6 \, \mu \, C_6}{\hbar^2 L(L+1)} \right)^{1/4} \nonumber \\
V_b &=& \left( \frac{ \left(\hbar^2 L (L+1)\right)^3}{54 \, \mu^3 \, C_6} \right)^{1/2} .
\label{Vn6}\end{aligned}$$ We can insert Eq. in Eq. to get analytical forms of the quenching cross section or rate coefficient. For two indistinguishable fermionic polar molecules at ultracold temperatures when $L=1$, we get $$\begin{aligned}
K^{qu}_{L=1,M_L} & = &
\frac{\pi}{8} \,
\left( \frac{3^{13} \, \mu^3 \, C_6^3}{\hbar^{10}} \right)^{1/4}
\ k_B T \times \Delta .
\label{ratequn6L1}\end{aligned}$$ In Eq. , we used the fact that $<E_c> = 3 k_B T / 2$ in three dimensions. Note that to get the overall contribution for a given $L$, we have to multiply Eq. by the degeneracy factor $(2L+1)$ corresponding to all values of $M_L$. We can get similar expressions for any partial wave $L$.
To test the validity of the model, we compare in Fig. \[crossK3-FIG\] the quenching cross sections of $^{39}$K$(^2S)$ + $^{39}$K$_2(^3\Sigma_u^+,v=1,N=0)$ collisions as a function of the collision energy for the partial waves $L=1-5$: (i) calculated in Ref. [@Quemener05] with a full quantum time-independent close-coupling calculation based on hyperspherical democratic coordinates [@Launay89] and the full potential energy surface of K$_3$ (solid lines) (ii) using the simple QT model (dashed lines) with a value of $C_6=9050$ a.u. given in Ref. [@Quemener05] (1 a.u. = 1 $E_h \, a_0^6$ where $E_h$ is the Hartree energy and $a_0$ is the Bohr radius). In this example, the QT model provides an upper limit to the cross sections. This is due to the fact that the quenching cross section does not reach a maximum value at the height of the barrier $V_b$, but rather at somewhat higher energy, say $\gamma \times V_b$, with $\gamma > 1$ (see Ref. [@Quemener05]). For all partial waves, there is a worse agreement for collision energies in the vicinity of the height of the barrier where the passage from the ultralow regime to the unitarity limit is smoother than for the QT model. This smoother passage is visible in Fig. \[probaK3-FIG\] for the full quantum $L=1$ quenching probability compared to the QT model, which has a sharp corner in the vicinity of $V_b$.
To account for more flexibility in the QT model, one can replace $V_b$ in Eq. by $\gamma \times V_b$ ($\gamma > 1$), and use the coefficient $\gamma$ as a fitting parameter to reproduce either full quantum calculations or experimental observed data. Alternatively, we can correct the QT quenching tunneling probability with an overall factor $p$, $$\begin{aligned}
{|T^{qu}_{L,M_L}|}^2_{\text{fit}} & = & p \times \left( \frac{E_c}{V_b} \right)^{L+1/2}
\label{Tqusqfit}\end{aligned}$$ with $p=\gamma^{-(L+1/2)}$. $p<1$ can be interpreted as the quenching probability reached at the height of the barrier $V_b$ in the QT model, rather than the rough full unit probability ($p=1$). As an example, we find that $\gamma \approx 2.06$ reproduces the quantum $L=1$ partial wave cross section for $^{39}$K + $^{39}$K$_2$ (dotted line in Fig. \[crossK3-FIG\] and Fig. \[probaK3-FIG\]). This yields a maximum quenching probability of $p = \gamma^{-3/2} \approx 0.34$ instead of 1. In other words, the QT model is only a factor of $p^{-1} \approx 2.96$ higher than the full quantum calculation for $^{39}$K + $^{39}$K$_2$ collisions at ultralow energies.
Given the fact that full quantum calculations are computationally demanding [@Soldan02; @Quemener04; @Quemener05; @Cvitas05a; @Cvitas05b; @Lara06; @Quemener07] and impossible at the present time for alkali molecule-molecule collisions, the accuracy of the QT model is satisfactory and can be a quick and powerful alternative way to estimate orders of magnitude for the scattering observables. Besides, agreement between the QT model with experimental data or full quantum calculations is expected to be satisfactory for collisions involving alkali species, because it is likely that short range quenching couplings will dominate and lead to high quenching probability in the region where the two particles are close together [@QuemenerCHAPTER]. Very recently, Eq. of the QT model has been applied for the evaluation of ultracold chemical quenching rate of collisions of two $^{40}$K$^{87}$Rb molecules in the same internal quantum state, and provided good agreement with the experimental data [@Ospelkaus-chemistry-09].
Rates in non-zero electric field
--------------------------------
### Numerical expressions
\[h\]
![ (Color online) Diabatic (dashed lines) and adiabatic (solid lines) barrier heights $V_b^{d,a}$ as a function of the induced dipole moment $d$ for the partial waves $L=1, M_L=0$ (red curves) and $L=1, |M_L|=1$ (blue curves). \[barriers-FIG\] ](figure4.eps){width="8cm"}
\[h\]
![ (Color online) Quenching rate coefficients of two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules as a function of the induced electric dipole moment for $L=1$ and for a temperature of $T=350$ nK (black curves). The rates have been calculated using the barrier heights of Fig. \[barriers-FIG\]. The red lines represent the $L=1, M_L=0$ partial wave contribution. The blue lines represent the sum of $L=1, M_L=1$ and $L=1,M_L=-1$ partial wave contributions. The dashed lines represent the rates calculated with the diabatic barriers while the solid lines with the adiabatic barriers (see text for detail). The total, $M_L=0$ and $|M_L|=1$ curves have been indicated in the left hand side. \[rate-num-FIG\] ](figure5.eps){width="8cm"}
In the presence of an electric field in Eq. , the long-range interaction potential matrix is no more diagonal and couplings between different values of $L$ occur. $M_L$ is still a good quantum number. A first approximation (diabatic approximation) consists of neglecting these couplings and using only the diagonal elements of the diabatic matrix directly. Then one can find numerically for which $R$ the centrifugal barriers are maximum and evaluate the height of the diabatic barriers $V_b^d$. This is repeated for all values of the induced dipole moment $d$. A second approximation (adiabatic approximation) is to diagonalize this matrix (including the non-diagonal coupling terms) for each $R$ and again find the maximum of the centrifugal barriers to get the height of the adiabatic barriers $V_b^a$. As an example, we compute these barrier heights for $^{40}$K$^{87}$Rb$-$$^{40}$K$^{87}$Rb collisions, using a value of $C_6 = 16130$ a.u. [@Kotochigova09]. We plot in Fig. \[barriers-FIG\] the heights of the diabatic (dashed lines) and adiabatic (solid lines) barriers for the quantum numbers $M_L=0$ (red curves) and $|M_L|=1$ (blue curves). The adiabatic barriers have been calculated using five partial waves $L=1-9$ in Eq. . The effect of the couplings can be clearly seen in this figure by comparing diabatic and adiabatic barriers. Especially for the $|M_L|=1$ case for $d \approx 0.16$ D (1 D = 1 Debye = $3.336 \, 10^{-30}$ C.m), couplings with higher partial waves make the adiabatic barrier decrease as the dipole increases while the diabatic barrier continues to increase.
Using these heights of the barriers, we use Eq. to plot in Fig. \[rate-num-FIG\] the total quenching rate coefficients (black curves) as a function of $d$ for two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules in the same quantum state for $L=1$ and at a typical experimental temperature of $T=350$ nK [@Ospelkaus-chemistry-09]. For $T=350$ nK, the mean collision energy $<E_c> = 3 k_B T /2 = 525$ nK, and the maximum dipole moment for which $V_b < 525$ nK (that is for which Eq. does not apply anymore) is around $d \approx 0.24$ D (see Fig. \[barriers-FIG\]). The dashed curves correspond to rates calculated with the diabatic approximation while the solid curves correspond to rates calculated with the adiabatic approximation. The $M_L=0$ contribution is plotted in red and the contribution of $M_L=+1$ and $M_L=-1$ is plotted in blue. The rates highly reflect the behavior of the centrifugal barriers in the entrance collision channel. When the barrier increases with the dipole, it prevents the molecules from getting close together and the quenching rates decreases. When the barrier decreases, the tunneling probability is increased allowing the molecules to get close together, and the quenching rates increases.
### Analytical expressions
\[t\]
![ (Color online) Same as Fig. \[rate-num-FIG\] but we use the analytical expressions for the rates (see text for detail). The total, $M_L=0$ and $|M_L|=1$ curves have been indicated in the left hand side. The individual analytical curves have been indicated in the right hand side by roman numbers. \[rate-anal-FIG\] ](figure6.eps){width="8cm"}
In order to have an intuitive sense of how the chemical quenching rate scales with the induced dipole moment (and the electric field), we evaluate analytical expressions of the barriers and the rates as it has been done in the previous section for a zero electric field. The analytical expression of the height of the diabatic barrier $V_b^d$ is complicated by the occurrence of two distinct long-range potentials in the diagonal matrix term of Eq. . We circumvent this difficulty by looking in the two limits where one dominates over the other. For small electric fields, we use the zero electric field limit discussed in the preceding section by setting $C_3 = 0$. For larger electric fields we ignore the $C_6$ coefficient in Eq. if the electric dipole-dipole interaction is attractive (positive $C_3$). We ignore the centrifugal term in Eq. if the electric dipole-dipole interaction is repulsive (negative $C_3$). These two cases are discussed below. In between, to accommodate the transition between the low-field and high-field limit, we will simply add the rate coefficients derived in the two limiting cases.
For positive $C_3$ coefficients in Eq. , $-C_3/R^3$ is attractive in Eq. . For $L=1$ partial waves for example, this occurs when $M_L=0$, which favors an attractive orientation of dipoles. We consider $$\begin{aligned}
\left|\frac{C_3}{R^3}\right| \gg \left|\frac{C_6}{R^6}\right|
\label{c3ggc6}\end{aligned}$$ in Eq. . In this case, the position and height of the barrier are given by $$\begin{aligned}
R_b &=& \frac{3 \, \mu \, C_3}{\hbar^2 L(L+1)} \nonumber \\
V_b &=& \frac{\left(\hbar^2 L(L+1)\right)^{3}}{54 \, \mu^3 \, C_3^2} \propto d^{-4} .
\label{Vn3}\end{aligned}$$ The position of the barrier in Eq. has to be in the region where Eq. is satisfied. This happens for suitably large dipole moments, $d > d_a$ where $$\begin{aligned}
d_a = \left( \frac{ \left( \hbar^2 L(L+1)\right)^3 \, C_6 \, (4 \pi \varepsilon_0)^{4} }{27 \, \mu^3 \, \alpha^4}\right)^{1/8} .\end{aligned}$$ The subscript [*a*]{} stands for the attractive interaction. For two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules, and for $L=1$ and $M_L=0$, $\alpha(1,1;0)=4/5$ and we get $d_a = 0.103$ D. The threshold laws for quenching collisions in an electric field are the same as in the zero-field limit. Consequently, the quenching cross sections and rate coefficients behaves as in Eq. except that $V_b$ is given now by Eq. and varies with $d$. We can insert Eq. in Eq. to get the corresponding analytical expressions. For a partial wave $L>0$, the quenching rate scales as $d^{4(L+\frac{1}{2})}$. For two indistinguishable fermionic polar molecules at ultracold temperatures when $L=1$ and $M_L=0$, we get $$\begin{gathered}
K^{qu}_{L=1,M_L=0} = \\
\frac{3 \pi}{8} \,
\left( \frac{6^9 \, \mu^6} {5^6 \, \hbar^{14}} \right)^{1/2}
\ \left(\frac{d^{2}}{4 \pi \varepsilon_0}\right)^{3}
\ k_B T \times \Delta .
\label{ratequn3L1}\end{gathered}$$ Thus the $L=1, M_L=0$ quenching rate increases as $d^{6}$. This is a more rapid dependence on dipole moment than for purely long-range dipolar relaxation in dipolar gases [@Hensler03].
For negative $C_3$ coefficients in Eq. , $-C_3/R^3$ is repulsive in Eq. . For $L=1$ partial waves for example, this occurs when $M_L=\pm 1$, which favors a repulsive orientation of dipoles. We consider $$\begin{aligned}
\left|\frac{C_3}{R^3}\right| \gg \left|\frac{\hbar^2 L(L+1)}{2 \mu R^2}\right|
\label{c2ggc3}\end{aligned}$$ in Eq. . The long-range potential again experiences a barrier, but now it is generated by the balance between the repulsive dipole potential at large $R$, and the attractive van der Waals potential at somewhat smaller $R$. In this case, the position and height of this barrier are given by $$\begin{aligned}
R_b &=& \left( \frac{2 \, C_6}{|C_3|} \right)^{1/3} \nonumber \\
V_b &=& \frac{|C_3|^2} {4 \, C_6} \propto d^4 .
\label{Vn3rep}\end{aligned}$$ For this approximation to hold, the position of the barrier in Eq. has to be in the region where Eq. is satisfied. This requires that $d > d_r$ where $$\begin{aligned}
d_r = \left( \frac{ \left(\hbar^2 L(L+1)\right)^3 \, C_6 \, (4 \pi \varepsilon_0)^{4} } {4 \, \mu^3 \, \alpha^4} \right)^{1/8} .\end{aligned}$$ The subscript [*r*]{} stands for the repulsive interaction. For two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules, and for $L=1$ and $M_L=1$ or $M_L=-1$, $\alpha(1,1;\pm1)=2/5$ and we get $d_r = 0.186$ D. We can replace Eq. in Eq. to get the corresponding analytical expressions. For a partial wave $L>0$, the quenching processes scale as $d^{-4(L+\frac{1}{2})}$. For two indistinguishable fermionic polar molecules at ultracold temperatures when $L=1$ and $|M_L|=1$, we get $$\begin{gathered}
K^{qu}_{L=1,|M_L|=1} = \\
\frac{3 \pi}{8}
\left( \frac{50 \, \hbar^{\frac{4}{3}} \, C_6}{\mu} \right)^{3/2}
\ \left(\frac{d^{2}}{4 \pi \varepsilon_0}\right)^{-3}
\ k_B T \times \Delta.
\label{ratequn3L1rep}\end{gathered}$$ The $L=1, |M_L|=1$ quenching rate decreases as $d^{-6}$ as the electric field grows. These analytical expressions use the diabatic barriers. If we consider that at large $d$, the total rate is mostly given by the $M_L=0$ contribution (we neglect the $|M_L|=1$ contributions at large $d$), one can have an analytical expression using the adiabatic barrier. If we take into account the couplings between $L=1, M_L=0$ and $L=3, M_L=0$, we can diagonalize analytically the $2\times2$ matrix in Eq. . It can be shown that for each dipole moment $d$, the coupling with $L=3,M_L=0$ lower the diabatic barrier of $L=1,M_L=0$ by a factor of 0.76 at the position of the barrier, to give rise to the adiabatic barrier. Inserting this correction of the barrier in Eq. , this yields a correction of $0.76^{-3/2} \approx 1.51$ for $K^{qu}_{L=1,M_L=0}$. The difference between diabatic and adiabatic calculations can be already seen in Fig. \[barriers-FIG\] and Fig. \[rate-num-FIG\] for the numerical barriers at large dipole moment.
In Fig. \[rate-anal-FIG\] the black curve corresponds to the total quenching rate coefficient as a function of $d$ for two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules in the same quantum state for $L=1$ and at a temperature of $T=350$ nK. The analytical expressions I, II, III, IV, V (green thin lines) correspond respectively to Eq. , 2 $\times$ Eq. , Eq. , 2 $\times$ Eq. , 1.51 $\times$ Eq. . The curves III and IV are for the diabatic barriers, while curve V is to account for the adiabatic barrier. The red dashed line (I+III) represents the $L=1, M_L=0$ partial wave contribution for the diabatic barriers while the blue dashed line (II+IV) represents the sum of $L=1, M_L=1$ and $L=1,M_L=-1$ partial wave contributions for the diabatic barriers. The analytical sum I+II+III+IV is represented as a black dashed line. To account for the adiabatic barriers we assume that the correction for the total rate comes only from the $L=1, M_L=0$ partial wave, and we replace I+III by I+V (red solid line). The analytical sum I+II+V+IV is represented as a black solid line. Neglecting the $d^{-6}$ contribution at larger $d$, the analytical $p-$wave quenching rate (taking into account the adiabatic barriers) is given by the simple expression $$\begin{gathered}
K^{qu}_{L=1} =
\frac{\pi}{8} \,
%\left\{
\bigg\{
p_1
\left( \frac{3^{17} \, \mu^3 \, C_6^3}{\hbar^{10}} \right)^{1/4} \\
+
1.51 \, p_2 \left( \frac{2^9 \, 3^{11} \, \mu^6} {5^6 \, \hbar^{14}} \right)^{1/2}
\, \frac{d^{6}}{(4 \pi \varepsilon_0)^{3}}
%\right\} \\
\bigg\}
\, k_B T \times \Delta.
\label{rateanal}\end{gathered}$$ $p_1$ ($p_2$) is the quenching probability reached at the height of the barrier in the QT model for the zero (non-zero) electric field regime. The QT model assumes that $p_1=p_2=1$ but become fitting parameters ($p_1,p_2 < 1$) when compared with full quantum calculations or experimental data. The limiting value $d_a=0.103$ D ($d_r=0.186$ D), where the $d^6$ ($d^{-6}$) behavior begins, has also been indicated with an arrow. It turns out that the total rates for $L=1$ calculated analytically (for both the use of diabatic and adiabatic barriers) are very similar to the numerical ones of Fig. \[rate-num-FIG\] (10 % difference at most, around $d_a$). However, the sub-components $L=1, M_L=0$ and $L=1, |M_L|=1$ have different behaviors. For example the numerical $L=1, M_L=0$ ($L=1, |M_L|=1$) component starts to increase (decrease) at earlier dipole moment (typically at 0.02 D) than their analytical analogs (typically after 0.06 D). The use of the simple analytical expressions (using the diabatic or adiabatic barriers) can be useful to estimate the total rate coefficients, while the numerical ones are prefered to estimate the $L=1, M_L=0$ and $L=1, |M_L|=1$ individual rates.
Prospects for collisions in two dimensions
==========================================
\[t\]
![ (Color online) Barrier heights as a function of the induced dipole moment $d$ for the partial waves $|M|=1$ in two dimensions. The green thin curves represent the analytical Eq. (constant) and Eq. ($d^4$). The dashed black curve is the sum of them. The solid black curve is the height of the barrier in Eq. . \[barriers2D-FIG\] ](figure7.eps){width="8cm"}
\[h\]
![ (Color online) Two dimensional quenching rate coefficient (black curves) of two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules as a function of the induced electric dipole moment for the $M=1$ and $M=-1$ components at a temperature of $T=350$ nK. The dashed lines represent the rate using analytical expressions while the solid line represents the rate using the numerical expression (see text for detail). The individual analytical curves have been indicated in the right hand side by roman numbers. \[2D-FIG\] ](figure8.eps){width="8cm"}
In three dimensional collisions, the quenching loss is largely due to incident partial waves with angular momentum projection $M_L=0$, emphasizing head-to-tail orientations of pairs of dipoles. These are the kind of collisions that are largely suppressed in traps with a pancake geometry, with the dipole polarization axis orthogonal to the plane of the pancake [@Micheli07]. If these collisions can be removed, then it is likely that increasing the electric field will suppress quenching collisions, making evaporative cooling possible. If we assume an ideal pancake trap that confines the particles to move strictly on a plane, one can apply the present model to estimate the behavior of the quenching processes. We assume that the molecules are polarized along the electric field axis, perpendicular to the two dimensional plane. In this case, the long range potential is given by $$\begin{aligned}
V(\rho) = \frac{\hbar^2 (|M|^2 - 1/4)}{2 \mu \rho^2} - \frac{C_6}{\rho^6}
+ \frac{d^2}{4 \pi \varepsilon_0 \, \rho^3}
\label{barrierpot-efield-2D}\end{aligned}$$ where $\rho$ stands for the distance between 2 particles in a two dimensional plane, $M$ stands for the angular momentum projection on the electric field axis. The last term comes from the repulsive dipole-dipole interaction when the dipoles are pointing along the electric field and approach each other side by side. The height of this barrier has been plotted as a function of $d$ in Fig. \[barriers2D-FIG\] (black solid line). At ultralow energy and large molecular separation, the Bethe–Wigner laws for quenching processes depend only on the long-range repulsive centrifugal term $1/R^2$ [@Sadeghpour00; @Rau84]. The repulsive centrifugal terms are different in Eq. and Eq. . As the repulsive centrifugal term in Eq. leads to the threshold laws in Eq. , the replacement $L(L+1) \to |M|^2-1/4$ (that is $L \to |M|-1/2$) in Eq. leads to $$\begin{aligned}
|T^{qu}_{M}|^2 = \left( \frac{E_c}{V_b} \right)^{|M|}
\label{Tqusq-2D}\end{aligned}$$ where $V_b$ denotes the height of the centrifugal barrier in two dimensions. This result requires that the centrifugal potential is repulsive, i.e., that $|M|>0$. For $M=0$ the threshold law exhibits instead a logarithmic divergence [@Sadeghpour00].
In two dimensions, quenching cross sections and rate coefficients have respectivelly units of length and length squared per unit of time, and are given by [@Naidon06] $$\begin{aligned}
\sigma^{qu}_{M} &=& \frac{\hbar}{\sqrt{2 \mu E_c}} \ |T^{qu}_{M}|^2 \times \Delta \nonumber \\
K^{qu}_{M}
&=& \frac{\hbar}{\mu} \ |T^{qu}_{M}|^2 \times \Delta.
\label{cross-2D}\end{aligned}$$ Within this model, it follows that the quenching cross section and rate coefficient for $|M| > 0$ are given by $$\begin{aligned}
\sigma^{qu}_{M}
& = & \frac{\hbar}{\sqrt{2 \mu} \, V_{b}^{|M|}} \ E_c^{|M|-1/2} \times \Delta \nonumber \\
K^{qu}_{M} &=&
\frac{\hbar}{\mu \, V_{b}^{|M|}} \ < E_c^{|M|} > \times \Delta .
\label{crossrate2D}\end{aligned}$$ The energy dependence is in agreement with the one found in Ref. [@Li09]. In Eq. , $$\begin{aligned}
V_{b} = \left(\frac{(\hbar^2 (|M|^2 - 1/4))^3}{54 \, \mu^3 \, C_6}\right)^{1/2}
\label{barrier2D-const}\end{aligned}$$ for the zero-electric field regime and $$\begin{aligned}
V_{b} = \frac{1}{4 \, C_6} \left(\frac{d^2}{4 \pi \varepsilon_0}\right)^2
\label{barrier2D-d4}\end{aligned}$$ for the non-zero electric field regime. The height of these barriers has been reported in Fig. \[barriers2D-FIG\] (green thin lines). These results imply that for $|M|=1$ the quenching processes within this model will be independent of the dipole moment in the zero electric field regime, where $$\begin{aligned}
K^{qu}_{|M|=1} & = &
\left( \frac{2^{7} \, \mu \, C_6 }{\hbar^4} \right)^{1/2}
\ k_B T \times \Delta
\label{rate2D-c6}\end{aligned}$$ and will scale as $d^{-4}$ in the non-zero electric field regime, where $$\begin{aligned}
K^{qu}_{|M|=1} & = &
\frac{4 \, \hbar \, C_6}{\mu}
\ \left(\frac{d^{2}}{4 \pi \varepsilon_0}\right)^{-2}
\ k_B T \times \Delta.
\label{rate2D-c3}\end{aligned}$$ We use the fact that $<E_c> = k_B T$ in two dimensions. The non-zero electric field regime is reached when $d > d_{2D}$ where $$\begin{aligned}
d_{2D} = \left( \frac{ \left(\hbar^2 (|M|^2 - 1/4) \right)^3 \, C_6 \, (4 \pi \varepsilon_0)^{4}} {4 \, \mu^3} \right)^{1/8}\end{aligned}$$ For two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules, and for $|M|=1$, we get $d_{2D} =0.081$ D. The behavior of the quenching rate (black lines) is shown in Fig. \[2D-FIG\] for two indistinguishable fermionic polar $^{40}$K$^{87}$Rb molecules as a function of the induced electric dipole moment for $M=1$ and $M=-1$ components at a temperature of $T=350$ nK. The dashed line represents the analytical rate which is the sum of the analytical expression VI corresponding to $2 \times $ Eq. and analytical expression VII corresponding to $2 \times $ Eq. . The solid line represents the rate using the general expression Eq. and the numerical height of the barrier calculated in Eq. . The limiting value $d_{2D} = 0.081$ D, where the $d^{-4}$ behavior for the quenching rate begins, has also been indicated with an arrow. The difference between the numerical calculation and the analytical expression reflects the difference in the calculation of the height of the barrier, already seen in Fig. \[barriers2D-FIG\]. The numerical calculation is more exact, while the other is analytical. However, at large $d$, the numerical rate tends to the analytical $d^{-4}$ behavior. The quenching rate decreases rapidly as the dipole moment increases and this may be promising for efficient evaporative cooling of polar molecules since the elastic rate is expected to grow with increasing dipole moment [@Ticknor09].
Conclusion
==========
We have proposed a simple model which combines quantum threshold laws and a classical capture model to determine analytical expressions of the chemical quenching cross section and rate coefficient as a function of the collision energy or the temperature. We also provide an estimate as a function of the induced electric dipole moment $d$ in the presence of an electric field. We found that the quenching rates of two ultracold indistinguishable fermionic polar molecules grows as the sixth power of $d$. For weaker electric field, quenching processes are independent of the induced electric dipole moment. Prospects for two dimensional collisions have been discussed using this model and we predict that the quenching rate will decrease as the inverse of the fourth power of $d$. This fact may be useful for efficient evaporative cooling of polar molecules. This model provides a general and comprehensive picture of ultracold collisions in electric fields. Preliminary data suggest that this model gives good agreement with experimental chemical rates for three dimensional collisions in an electric field [@Ni-dipolar-09].
Acknowledgements
================
We acknowledge the financial support of NIST, the NSF, and an AFOSR MURI grant. We thank K.-K. Ni, S. Ospelkaus, D. Wang, M. H. G. de Miranda, B. Neyenhuis, P. S. Julienne, J. Ye, and D. S. Jin for helpful discussions.
[14]{}
J. M. Sage, S. Sainis, T. Bergeman, and D. DeMille, Phys. Rev. Lett. [**94**]{}, 203001 (2005).
K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science [**322**]{}, 231 (2008).
J. Deiglmayr, A. Grochola, M. Repp, K. Mörtlbauer, C. Glück, J. Lange, O. Dulieu, R. Wester, and M. Weidemüller, Phys. Rev. Lett. [**101**]{}, 133004 (2008).
S. Hensler et al., Appl. Phys. B [**77**]{}, 765 (2003).
J. L. Bohn, M. Cavagnero, and C. Ticknor, New J. Phys. [**11**]{}, 055039 (2009).
A. V. Avdeenkov and J. L. Bohn, Phys. Rev. A [**66**]{}, 052718 (2002).
N. Balakrishnan and A. Dalgarno, Chem. Phys. Lett. [**341**]{}, 652 (2001).
P. Soldán, M. T. Cvitaš, J. M. Hutson, P. Honvault, and J.-M. Launay, Phys. Rev. Lett. [**89**]{}, 153201 (2002).
G. Quéméner, P. Honvault, and J.-M. Launay, Eur. Phys. J. D [**30**]{}, 201 (2004).
G. Quéméner, P. Honvault, J.-M. Launay, P. Soldán, D. E. Potter, and J. M. Hutson, Phys. Rev. A [**71**]{}, 032722 (2005).
M. T. Cvitaš, P. Soldán, J. M. Hutson, P. Honvault, and J.-M. Launay, Phys. Rev. Lett. [**94**]{}, 033201 (2005).
M. T. Cvitaš, P. Soldán, J. M. Hutson, P. Honvault, and J.-M. Launay, Phys. Rev. Lett. [**94**]{}, 200402 (2005).
M. Lara, J. L. Bohn, D. Potter, P. Sold[á]{}n and J. M. Hutson, Phys. Rev. Lett. [**97**]{}, 183201 (2006).
G. Quéméner, J.-M. Launay, and P. Honvault, Phys. Rev. A [**75**]{}, 050701(R) (2007).
J. M. Hutson and P. Soldán, Int. Rev. Phys. chem. [**26**]{}, 1 (2007).
G. Qu[é]{}m[é]{}ner, N. Balakrishnan, and A. Dalgarno, in [*Cold Molecules: Theory, Experiment, Applications*]{}, R. V. Krems, W. C. Stwalley, and B. Friedrich (Eds.), CRC Press (2009).
G. Quéméner, N. Balakrishnan, and B. K. Kendrick, Phys. Rev. A [**79**]{}, 022703 (2009).
P. Langevin, Ann. Chim. Phys. [**5**]{}, 245 (1905).
P. Julienne and F. H. Mies, J. Opt. Soc. Am. B [**6**]{}, 2257 (1989).
J. P. Burke, Jr., C. H. Greene, and J. L. Bohn, Phys. Rev. Lett. [**81**]{}, 3355 (1998).
F. H. Mies and M. Raoult, Phys. Rev. A [**62**]{}, 012708 (2000).
P. S. Julienne, Faraday Discuss. [**142**]{}, 361 (2009).
C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L. Rolston, Phys. Rev. A [**59**]{}, 1926 (1999).
E. R. Hudson, N. B. Gilfoy, S. Kotochigova, J. M. Sage, and D. DeMille, Phys. Rev. Lett. [**100**]{}, 203201 (2008).
E. R. Hudson et al., Phys. Rev. A [**73**]{}, 063404 (2006).
T. V. Tscherbul and R. V. Krems, J. Chem. Phys. [**129**]{}, 034112 (2008).
J. P. Burke, Jr., PhD Thesis, University of Colorado (1999), available online at: http://jilawww.colorado.edu/pubs/thesis/burke.
S. Ospelkaus, K.-K. Ni, G. Quéméner, B. Neyenhuis, D. Wang, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, Phys. Rev. Lett. [**104**]{}, 030402 (2010).
H. A. Bethe, Phys. Rev. [**47**]{}, 747 (1935).
E. P. Wigner, Phys. Rev. [**73**]{}, 1002 (1948).
H. R. Sadeghpour et al., J. Phys. B: At. Mol. Opt. Phys. [**33**]{}, R93 (2000).
S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis, G. Qu[é]{}m[é]{}ner, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye, arXiv:0912.3854, Science, in press (2010).
Z. Idziaszek and P. S. Julienne, arXiv:0912.0370, submitted (2009).
J. Aldegunde, B. A. Rivington, P. S. $\dot{Z}$uchowski, and J. M. Hutson, Phys. Rev. A [**78**]{}, 033434 (2008).
J.-M. Launay and M. Le Dourneuf, Chem. Phys. Lett. [**163**]{}, 178 (1989).
S. Kotochigova, private communication (2009).
A. Micheli, G. Pupillo, H. P. Büchler, and P. Zoller, Phys. Rev. A [**76**]{}, 043604 (2007).
A. R. P. Rau, Comments At. Mol. Phys. [**14**]{}, 285 (1984).
P. Naidon and P. S. Julienne, Phys. Rev. A [**74**]{}, 062713 (2006).
Z. Li, S. V. Alyabyshev, and R. V. Krems, Phys. Rev. Lett. [**100**]{}, 073202 (2008).
C. Ticknor, Phys. Rev. A [**80**]{}, 052702 (2009).
K.-K. Ni, S. Ospelkaus, D. Wang, G. Qu[é]{}m[é]{}ner, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, arXiv:1001.2809, submitted (2010).
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---
abstract: 'We investigate parameterizing hard combinatorial problems by the size of the solution set compared to all solution candidates. Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in expected time $O^*(\varepsilon^{-{0.617}})$ where $\varepsilon$ is the fraction of assignments that are satisfying. This improves significantly over the trivial sampling bound of expected $\Theta^*(\varepsilon^{-1})$, and on all previous algorithms whenever $\varepsilon = \Omega(0.708^n)$. We also consider algorithms for 3-SAT with an $\varepsilon$ fraction of satisfying assignments, and prove that it can be solved in $O^*(\varepsilon^{-2.27})$ deterministic time, and in $O^*(\varepsilon^{-0.936})$ randomized time. Finally, to further demonstrate the applicability of this framework, we also explore how similar techniques can be used for vertex cover problems.'
author:
- 'Jean Cardinal$^1$, Jerri Nummenpalo$^2$, and Emo Welzl$^3$'
bibliography:
- 'references.bib'
date:
title: 'Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems[^1]'
---
Introduction {#sec:introduction}
============
In order to cope with the computational complexity of combinatorial optimization and satisfiability problems without sacrificing correctness guarantees, one can consider a family of instances for which a certain parameter is bounded, and analyze the complexity of algorithms as a function of this parameter. While it is now commonplace in combinatorial optimization to define the parameter as the [*size*]{} of a solution, we here consider computationally hard problems parameterized by the [*number*]{} of solutions. More precisely, we will consider satisfiability problems in which we are promised that a fraction at least $\varepsilon$ of all possible assignments are satisfying, and graph covering problems in which a fraction at least $\varepsilon$ of all vertex subsets of a certain size are solutions.
Counting and sampling solutions to CNF formulas and more generally to CSP formulas has important practical applications. For example, in verification and artificial intelligence [@naveh2007constraint]; and Bayesian inference [@sang2005performing]. Recent algorithmic developments have made possible practical algorithms that can tackle industrial scale problems [@meel2016constrained].
In contrast to that line of work we focus on the exact complexity of sampling, in particular to sampling solutions for 2-CNF formulas, and show that we can significantly improve on the *trivial sampling algorithm* that repeatedly samples uniformly in the search space and terminates after $\varepsilon^{-1}$ steps on average. A few previous works have also considered satisfiability problems under the promise that there are many solutions, most notably from Hirsch [@hirsch1998fast], and more recently from Kane and Watanabe [@kane2013short]. Their focus has been on deterministic algorithms and we extend their work while also adding the consideration of randomized algorithms for $k$-SAT.
Before detailing our contributions more precisely, we briefly summarize the current state of knowledge regarding this family of questions.
Background and previous work on satisfiability
----------------------------------------------
Hirsch [@hirsch1998fast] developed a deterministic algorithm that finds a satisfying assignment for a $k$-CNF formula $F$ with an $\varepsilon$ fraction of satisfying assignments in time $O^*(\varepsilon^{-\delta_k})$ where $(\delta_k)_{k =2}^{\infty}$ is a positive increasing sequence defined by the roots of the characteristic polynomials of certain recurrence relations. The constant obtained for $k=3$ is $\delta_3 \approx 7.27$. The main idea in his algorithm is that such formulas $F$ have *short implicants* which are satisfying assignments that need to fix only few variables — in this case only $O(\log \varepsilon^{-1})$ many — and such assignments can be found relatively fast with a branching algorithm. Trevisan [@trevisan2004note] proposed a similar algorithm to that of Hirsch but with an explicit running time of $O^*(\varepsilon^{-(\ln 4)k2^k})$. Although his algorithm is slightly simpler, the performance guarantees, at least for small $k$, are worse.
Kane and Watanabe [@kane2013short] looked at general CNF formulas in a similar setting. They assume that $\varepsilon \geq 2^{-n^{\delta}}$, that the number of clauses is bounded by $n^{1+\delta'}$ and that $\delta + \delta' < 1$. Under these conditions they show that the formula has a short implicant that only fixes a linear fraction of the variables and they provide a $O^*(2^{n^\beta})$ time algorithm for finding a solution with $\beta < 1$.
Classical derandomization tools naturally apply in this context. For arbitrary CNF formulas on $n$ variables with $\varepsilon 2^n$ satisfying assignments, one can obtain a deterministic algorithm by using a pseudorandom generator that $\varepsilon$-fools depth-2 circuits. A result by De et al. [@DeETT10] provides such pseudorandom generators with seed length $O\left(\log n + \log^2 \frac{m}{\varepsilon}\log\log \frac{m}{\varepsilon}\right)$. By enumerating over all seeds, we obtain a running time of $O^*\left(\left( \frac{n}{\varepsilon}\right)^{c\cdot \log {\frac{n}{\varepsilon}}}\right)$ for some constant $c$ (assuming there are $\mathrm{poly} (n)$ clauses). A recent result of Servedio and Tan improves this running time to $n^{\tilde{O}(\log\log n)^2}$ for any $\varepsilon \geq 1/\mathrm{poly}\log(n)$ [@ST16].
We let *Sample-2-SAT* denote the problem of sampling exactly and uniformly a satisfying assignment. Due to self-reducibility of satisfiability, any algorithm for the counting problem \#2-SAT can be used to solve Sample-2-SAT with only a multiplicative polynomial loss in runtime. In fact, so far the best algorithm for Sample-2-SAT is Wahlstr[ö]{}m’s \#2-SAT algorithm [@wahlstrom2008tighter] that runs in time $O(1.238^n)$. In contrast to the exponential time algorithms, 2-SAT can be solved in linear time with the classical algorithm of Aspvall et al. [@aspvall1979linear]. We note that while Sample-2-SAT is between 2-SAT and \#2-SAT in complexity, under the assumption $RP \not= NP$ it is not possible to uniformly or even almost uniformly sample satisfying assignments in polynomial time. We can use a simple threefold reduction to prove this:
- The constraints for an independent set in a graph can be modeled as a 2-SAT formula. Therefore a polynomial time algorithm for Sample-2-SAT would give a polynomial time algorithm for *Sample-IS*. (sampling uniformly among independent sets of any size). The same holds for approximate versions of the problems.
- Such sampling algorithms would yield a fully polynomial randomized approximation scheme (FPRAS) for \#IS. See for example the article of Jerrum et al. [@jerrum1986random].
- Lastly, such an FPRAS exists only if $RP = NP$. For details see for example the book by Jerrum [@jerrum2003counting Chapter 7, Proposition 7.7].
Even when relaxing Sample-2-SAT to almost uniform sampling, the best algorithm is still the one based on Wahlstr[ö]{}m’s counting algorithm. This is in contrast to $k$-CNF formulas with $k \geq 3$ which have an exponential gap between exact and almost uniform sampling. More precisely, the gap is between exact and approximate counting. See Schmitt and Wanka [@schmitt2013exploiting] for a table of the best algorithms.
Our results
-----------
In Section \[sec:extending\_hirsch\] we recall Hirsch’s [@hirsch1998fast] algorithm for finding a satisfying assignment for a $k$-CNF $F$ with a fraction $\varepsilon$ of satisfying assignments. We slightly generalize his analysis to also cover improved branching rules for $k$-SAT. The resulting deterministic algorithms have running times of $O^*(\varepsilon^{-\lambda_k})$ for some positive increasing sequence $(\lambda_k)_{k=2}^{\infty}$, where for instance $\lambda_3 \leq 2.27$. We demonstrate how similar techniques can be used for finding vertex covers and we give a deterministic algorithm running in time sublinear in $\varepsilon^{-1}$ for instances of $k$-vertex cover with at least $\varepsilon \binom{n}{k}$ solutions and $k$ bounded by some fraction of $n$.
In Section \[sec:rand\_alg\] we prove our main result, Theorem \[thm:Sample-2-SAT\_epsilon\], which describes an algorithm for Sample-2-SAT that runs in expected time $O^*(\varepsilon^{-{0.617}})$. It therefore improves on the algorithm based on Wahlstr[ö]{}m’s algorithm [@wahlstrom2008tighter] when $\varepsilon = \Omega(0.708^n)$, or equivalently when $F$ has $\Omega(1.415^n)$ satisfying assignments. We leave it as an open problem to decide whether sampling solutions to 3-CNF formulas can be done in time $O^*(\varepsilon^{-\delta})$ with $\delta < 1$ and discuss why the 2-CNF case does not generalize. In Proposition \[prop:simple\_randomized\_3-SAT\] we show how to solve 3-SAT in time $O(\varepsilon^{-0.936}(m+n))$ using similar ideas.
Notation
--------
For a Boolean variable $x$ we denote its *negation* by $\bar{x}$ and for a set $V$ of Boolean variables let $\overline{V}$ be the set of negated variables. A *literal* is either a Boolean variable or its negation and in the former case we call the literal *positive* and in the latter we call it *negative*. We think of a *CNF formula*, or simply a *formula*, $F$ over a variable set $V$ as a set $F = \{C_1,C_2,\ldots,C_m\}$ of *clauses* where each clause $C_i \subset V\cup \overline{V}$ is a set of literals without both $x$ and $\bar{x}$ in the same clause for any variable $x \in V$. By a $k$-CNF formula and by a $(\leq k)$-CNF we denote CNF formulas in which every clause has cardinality exactly $k$ or at most $k$, respectively. We let ${\textnormal{vbl}}(F) \subseteq V$ denote the set of variables that appear in $F$ either as a positive or negative literal. The *empty formula* is denoted by $\{\}$ and the *empty clause* by $\square$. An *assignment* to the variables in the formula $F$ is a function $\alpha : V \rightarrow \{0,1\}$ and it is said to *satisfy* $F$ if every clause $C \in F$ is satisfied, namely, if the clause contains a literal whose value is set to 1 under the assignment. A satisfying assignment is also called a *solution*. The empty formula is satisfied by any assignment to the variables and the empty clause by none. The set of all satisfying assignments of a formula $F$ over $V$ is denoted ${\textnormal{sat}}_V(F)$, and we omit the subscript $V$ when it is clear from the context. A *partial assignment* to $F$ is a function $\beta : W \rightarrow \{0,1\}$ with $W \subseteq V$ and we let $F^{[\beta]}$ be the formula over the variables $V\setminus W$ which is attained from $F$ by removing each clause of $F$ that is satisfied under $\beta$ and then removing all literals assigned to $0$ from the remaining clauses. If $u \in V\cup \overline{V}$ is a literal and $i \in \{0,1\}$ we let $F^{[u \mapsto i]}$ denote $F^{[\beta]}$ where $\beta$ is the partial assignment that maps only $u$ to $i$. By *unit clause reduction* we refer to the process of repeatedly setting variables to satisfy the unit clauses until finishing the process by exhausting the unit clauses or finding the empty clause.
All the logarithms are in base 2 unless noted otherwise.
Deterministic algorithms and Hirsch’s method {#sec:extending_hirsch}
============================================
In this section we consider Hirsch’s method [@hirsch1998fast] for finding a satisfying assignment to a $k$-CNF formula, and extend the analysis to accommodate any branching rule.
We first briefly recall basic definitions on branching algorithms. A *complexity measure* $\mu$ is a function that assigns a nonnegative value $\mu(F)$ to every instance $F$ of some particular problem. Given a problem and a complexity measure $\mu$ for it, we say that an algorithm correctly solving the problem is a *branching algorithm* (with respect to $\mu$) if for every instance $F$ the algorithm computes a list $(F_1,\ldots,F_t)$ of instances of the same problem, recursively solves the $F_i$’s, and finally combines the results to solve $F$. Finding the list $(F_1,\ldots,F_t)$ and recursively solving each of them is called a *branching*. Letting $b_i = \mu(F) - \mu(F_i)$ we call the vector $(b_1,\ldots,b_t)$ the *branching vector* associated to the branching. Lastly, the *branching number* $\tau(b_1,\ldots,b_t)$ is defined as the smallest positive solution of the equation $\sum_{i=1}^t x^{-b_i} = 1$. If $\lambda$ is the largest branching number of any possible branching in the algorithm and $T(F)$ is the time used to find the branching and to combine the results after the recursive calls, then the running time of the algorithm can be bounded by $O(T(F)\lambda^{\mu(F)})$.
Following Hirsch [@hirsch1998fast], we consider a [*breadth-first*]{} version of such a branching algorithm, taking a $k$-CNF Boolean formula $F$ as input. We use the number of variables as a measure, and branch on partial assignments $\beta_i$, each fixing exactly $b_i$ variables. The set $\Phi_{\ell}$ in the algorithm below eventually contains the formulas constructed from input $F$ after fixing exactly $\ell$ variables.
1. set $\ell \leftarrow 0$, $\Phi_0 \leftarrow \{F\}$, and $\Phi_{\ell}\leftarrow\emptyset$ for all $\ell >0$.
2. \[bfalg:rec\] if $\{\}\in \Phi_{\ell}$, then stop and return the so far fixed variables
3. for each $F\in \Phi_{\ell}$ such that $\square \not\in F$:
1. find a collection of $t$ partial assignments of the form $\beta_i : W_i\to \{0,1\}$, where $W_i\subseteq {\textnormal{vbl}}(F)$
2. for each $i\in [t]$:
1. $\Phi_{\ell + b_i}\leftarrow\Phi_{\ell + b_i} \cup \{ F^{[\beta_i]} \}$
4. $\ell\leftarrow \ell +1$; if $\ell\leq n$ then go to step \[bfalg:rec\]
For this algorithm to be correct, the partial assignments in 3a have to of course be chosen according to a correct branching rule. The complete collection $\Phi_{\ell}$ can be seen as a collection of nodes of the search tree of the recursive algorithm, and is referred to as the $\ell$th [*floor*]{} of the tree. The following lemma holds [@hirsch1998fast].
\[lem:floor\] $|\Phi_{\ell} |\leq \lambda^{\ell}$ where $\lambda$ is the maximum branching number of the recursion tree.
The following result was proved by Hirsch in the special case of the simple Monien-Speckenmeyer algorithm [@monien1985solving], in which the branching vector was $(1,2,\ldots ,k)$. We generalize it to arbitrary branching vectors.
\[thm:branching\_algorithm\] Consider a $k$-CNF formula $F$ with $n$ variables and $m$ clauses, and suppose it has at least $\varepsilon 2^n$ satisfying assignments. Then any breadth-first branching algorithm for $k$-SAT with maximum branching number $\lambda_k < 2$ runs in time $O^*(\varepsilon^{-B})$ on this instance, where $B := 1/(\log_{\lambda_k}2 -1)$.
After the $(\ell-1)$th step, we created all nodes of the tree in the $\ell$th floor, and all nodes that have a parent in the $(\ell -1)$th floor or above. There are at most $2^{n-\ell -j}$ assignments for each node of the $(\ell +j)$th floor. Using Lemma \[lem:floor\], we have that the total number of assignments for the remaining nodes is at most $$\sum_{j=0}^n \lambda^{\ell +j} 2^{n-\ell -j}
= \lambda^{\ell} 2^{n-\ell} \sum_{j=0}^n \lambda^j 2^{-j}
\leq c_{\lambda} \lambda^{\ell} 2^{n-\ell} ,$$ for the constant $c_{\lambda}=2/(2-\lambda )$. Note that we use $\lambda < 2$ which holds for any nontrivial branching rule. Because the algorithm has not terminated yet, all the $\varepsilon 2^n$ assignments are still to be found and therefore we can bound $\ell$ by $$\begin{aligned}
c_{\lambda} \lambda^{\ell}2^{n-\ell} & \geq & \varepsilon 2^n\\
(\lambda / 2)^{\ell} & \geq & \varepsilon / c_{\lambda} \\
\ell & \leq & \log_{\lambda / 2} (\varepsilon / c_{\lambda}). \\\end{aligned}$$ We then have $$\begin{aligned}
\lambda^{\ell} & = & \lambda^{\log_{\lambda / 2} (\varepsilon / c_{\lambda})} \\
& = & \lambda^{\log_{\lambda} (\varepsilon / c_{\lambda}) / \log_{\lambda} (\lambda / 2)} \\
& = & (\varepsilon / c_{\lambda})^{1 / \log_{\lambda} (\lambda / 2)} \\
& = & (c_{\lambda} / \varepsilon)^B ,\end{aligned}$$ and the total number of nodes is at most $\sum_{i=0}^{\ell} \lambda^i = O(\lambda^{\ell})$.
To get concrete bounds from Theorem \[thm:branching\_algorithm\] it remains to find good branching rules for $k$-SAT. The improved algorithm by Monien and Speckenmeyer [@monien1985solving] for $k$-SAT uses the notion of autarkies and the branching vectors appearing in the algorithm are $(1)$ and $(1,2,\ldots,k-1)$ of which the latter has the worse branching number. This directly yields the following result for $k=3$.
\[thm:branching\_algorithm\_3\_sat\] Given a $3$-CNF formula $F$ on $n$ variables and an $\varepsilon > 0$ with the guarantee that $|{\textnormal{sat}}(F)| \geq \varepsilon 2^n$, one can find a satisfying assignment for $F$ in deterministic time $O^*\left(\varepsilon^{-2.27}\right)$.
Vertex cover {#sec:VC}
------------
The technique we have seen is not unique to satisfiability but extend easily to known graph problems. As an example, we now consider the *vertex cover problem*: given a graph $G$ and an integer $k$, does there exist a subset $S\in \binom{V(G)}{k}$ such that $\forall e\in E(G), e\cap S\not=\emptyset$? The optimization version consists of finding a smallest subset $S$ satisfying the condition. We consider exact algorithms, hence the problem is equivalent to the maximum independent set problem (consider $V(G)\setminus S$). This is naturally related to the previous results on 2-SAT: the vertex cover problem can be cast as finding a minimum-weight satisfying assignment for a monotone 2-CNF formula.
We first briefly recall a standard algorithm for finding a minimum vertex cover in a graph $G$ on $n$ vertices, if one exists, in time $O^*(1.3803^n)$. First note that if the maximum degree of the graph is 2, then the problem can be solved in polynomial time. Otherwise, pick a vertex $v$ of degree at least 3, and return the minimum of $1+VC(G-v)$ and $VC(G-v-N(v))$, where $VC$ are recursive calls, and $N(v)$ is the set of neighbors of $v$ in $G$. The running time $T(n)$ obeys the recurrence $T(n)=T(n-1)+T(n-4)$, solving to the claimed bound. We can also analyze it with respect to the size $k$ of the sought cover, yielding $T(k)=T(k-1)+T(k-3)$, solving to $1.4656^k$. In the latter, we do not count the total number of vertices that are processed, but only those that are part of the solution. Hence we can distinguish the branching number $\lambda$ related to the number of vertices processed and the branching number $\rho$ related to the number of vertices included in the vertex cover (equivalently, the weight of the current partial assignment). In our case, we have $\rho < 1.4656$.
We now consider instances of the vertex cover problem in which we are promised that there are at least $\varepsilon \binom{n}{k}$ vertex covers. Given a branching algorithm, we can parse its search tree in breadth-first order, by associating with each node the number of vertices included in $S$ so far (that is, the weight of the partial assignment). We define $\Phi_{\ell}$ as the set of nodes with such value $\ell$, and call it the $\ell$th floor. The following lemma is similar to Lemma \[lem:floor\].
$|\Phi_{\ell}|\leq \rho^{\ell}$.
After generating the $\ell$th floor $\Phi_{\ell}$, there are at most $\rho^{\ell} \binom{n - \ell}{k -\ell}$ remaining covers to check. If this is less than the total number of solutions of size $k$, we are done. The following statement gives an upper bound on the number of levels of the tree we need to parse.
\[lem:height\] Let $\ell^* := \ln (\frac 1{\varepsilon}) / \ln (\frac{n}{\rho k})$. Then for $k,n>> \ell^*$ and $k\leq n/\rho$, we have $$\rho^{\ell} \binom{n - \ell}{k -\ell} \geq \varepsilon \binom{n}{k} \Rightarrow \ell \leq \ell^*.$$
$$\begin{aligned}
\rho^{\ell} \binom{n - \ell}{k -\ell} & \geq & \varepsilon \binom{n }{k}\\
\rho^{\ell} \frac{(n - \ell)!}{(k -\ell)! (n-k)!} & \geq & \varepsilon \frac{n!}{k! (n-k)!} \\
\rho^{\ell} \frac{(n - \ell)!}{(k -\ell)!} & \geq & \varepsilon \frac{n!}{k!} \\
\rho^{\ell} \frac{k!}{(k - \ell)!} & \geq & \varepsilon \frac{n!}{(n-\ell)!}\end{aligned}$$
For $k, n$ sufficiently large, this holds whenever $$\begin{aligned}
\rho^{\ell} k^{\ell} & \geq & \varepsilon n^{\ell} \\
(\rho k /n)^{\ell} & \geq & \varepsilon \\
\ell \ln (\rho k / n) & \geq & \ln \varepsilon \\
\ell & \leq & \ell^* ,
\end{aligned}$$ where the last line uses the assumption that $k<n/\rho$.
For $n$ large enough, Lemma \[lem:height\] implies that if $\ell > \ell^*$ then the number of remaining solutions is smaller than the promised number $\varepsilon \binom{n}{k}$, and either we have found one already, or greedily completing any partial solution leads to a solution. Hence the running time is within a linear factor of $\rho^{\ell^*}$, which simplifies as follows.
Given a Vertex Cover instance composed of a graph $G$ on $n$ vertices, a number $k<n/\rho$, and an $\varepsilon > 0$ with the guarantee that $G$ has at least $\varepsilon \binom{n}{k}$ vertex covers of size $k$, one can find such a vertex cover in deterministic time $$O^*\left( \varepsilon^{-\frac{\log \rho}{\log (\frac n {\rho k})}}\right),$$ where $\rho$ is the branching number of an exact branching algorithm for $k$-vertex cover. In particular, this holds for $\rho = 1.4656$.
Note that the running time remains sublinear in $1/\varepsilon$ for all values of $k$ such that $\frac{\log \rho}{\log (\frac n {\rho k})} < 1 \Leftrightarrow k < n / \rho^2$. Hence for those values of $k$, and in particular when $k=o(n)$, we have a deterministic algorithm for $k$-vertex cover whose complexity improves on the trivial sampling algorithm.
Randomized algorithms for Sample-2-SAT and for 3-SAT {#sec:rand_alg}
====================================================
In this section we present our algorithm for Sample-2-SAT with an expected running time of $O\left(\varepsilon^{-{0.617}}(m+n)\right)$ on $2$-CNF formulas with more than $\varepsilon$ fraction of satisfying assignments. The parameter $\varepsilon$ does not need to be a constant and the algorithms can be easily modified so that they do not need to know $\varepsilon$ in advance. Before stating and proving our main result we consider a warm-up algorithm that gives a weaker bound but already highlights some of the main ideas. In the end we discuss the complications of generalizing our method to Sample-3-SAT and see how to solve 3-SAT in expected time $O\left(\varepsilon^{-0.940}(m+n)\right)$ using similar techniques as for Sample-2-SAT.
Schmitt and Wanka [@schmitt2013exploiting] have used analogous ideas to approximately count the number of solutions in $k$-CNF formulas.
A warm-up algorithm for Sample-2-SAT {#sec:warm-up}
------------------------------------
We will start with a warm-up algorithm that we then improve. Let $F$ be a 2-CNF formula over the variable set $V$ with $n:=|V|$ and with $m$ clauses. Let $S \subseteq F$ be a greedily chosen maximal set of variable disjoint clauses. We make the following remarks.
- Any satisfying full assignment for $F$ must in particular satisfy $S$ and is therefore an extension of one of the $3^{|S|}$ partial assignments to ${\textnormal{vbl}}(S)$ that satisfy all clauses in $S$.
- Because of maximality any partial assignment of the form $\alpha : {\textnormal{vbl}}(S) \rightarrow \{0,1\}$ has the property that $F^{[\alpha]}$ is a ($\leq 1$)-CNF.
- Counting and sampling of solutions of a ($\leq 1$)-CNF is easily done in linear time.
The set $S$ allows us on one hand to do improved rejection sampling and on the other hand to device a branching based sampling. More concretely, consider the following two algorithms that use $S$.
1. Sample uniformly among all full assignments for $F$ that satisfy all the clauses in $S$ until finding one that satisfies $F$.
2. Go through all $3^{|S|}$ partial assignments $\alpha : {\textnormal{vbl}}(S) \rightarrow \{0,1\}$ that satisfy $S$ and for each $\alpha$ compute $A_{\alpha} := |{\textnormal{sat}}_{V \setminus {\textnormal{vbl}}(S)}(F^{[\alpha]})|$, i.e., the number of satisfying assignments in $F^{[\alpha]}$. Then $A := \sum_{\alpha} A_{\alpha}$ is the number of satisfying assignments in $F$. Draw one partial assignment $\alpha^*$ at random so that Pr$(\alpha^* = \alpha) = A_{\alpha} / A$. For the remaining variables choose an assignment $\beta^* : V \setminus {\textnormal{vbl}}(S) \rightarrow \{0,1\}$ uniformly among all assignments satisfying $F^{[\alpha^*]}$. Output the full assignment which when restricted to ${\textnormal{vbl}}(S)$ is $\alpha^*$ and when restricted to $V\setminus{\textnormal{vbl}}(S)$ is $\beta^*$.
The correctness of the first algorithm is clear since any assignment satisfying $F$ must also satisfy $S$. One sample can also be drawn in linear time. Because the clauses of $S$ are variable disjoint, the pool of assignments we are sampling from has $(\frac{3}{4})^{|S|}2^n$ assignments and it contains all the at least $\varepsilon 2^n$ satisfying assignments. Therefore the probability of one sample being satisfying is at least $(\frac{4}{3})^{|S|}\varepsilon$, implying an expected runtime of $O\left(\varepsilon^{-1}(\frac{3}{4})^{|S|}(m+n)\right)$ for the first algorithm.
We need the second algorithm to balance the first one when $|S|$ is small. For the correctness we observe that the partial assignments $\alpha$ partition the solution space in the sense that $A = \sum_\alpha A_{\alpha} = |{\textnormal{sat}}_V(F)|$ and a simple calculation shows that the output distribution is uniform over ${\textnormal{sat}}_V(F)$. With the remarks made before the algorithm description we conclude that the runtime of the second algorithm is $O(3^{|S|}(m+n))$. If space is a concern, the sampling of $\alpha^*$ can be done in linear space without storing the numbers $A_{\alpha}$ as follows: Sample a uniform number $r$ from $\{1,\ldots,A\}$ and go through the partial assignments $\alpha$ again in the same order and output the first $\alpha$ for which the total number of assignments counted up to that point reaches at least $r$.
For any given $S$ we can choose the better of the two algorithms which gives an expected runtime guarantee of $$\begin{aligned}
O\left(\max_{|S|} \left\{3^{|S|}, \varepsilon^{-1}\left(\frac{3}{4}\right)^{|S|}\right\} \cdot (m+n)\right) = O\left(\varepsilon^{-\log_4 3} (m+n) \right)\end{aligned}$$ where $\log_4 3 < 0.793$. Note that we do not need to know $\varepsilon$ in advance to get the same runtime guarantee as we can simulate running both of the algorithms in parallel until one finishes.
A faster algorithm for Sample-2-SAT
-----------------------------------
In the warm-up algorithm we used the set $S$ on the one hand to reduce the size of the set of assignments we are sampling from and on the other hand we used it as a small size *hitting set* for the clauses in $F$: every clause in $F$ contained at least one variable from ${\textnormal{vbl}}(S)$. To improve we will do two things. Firstly, we will consider more complicated independent structures that improve on both aspects above, giving us both a smaller size sampling pool and a better hitting set. Secondly, we notice that it is not necessary to always use an exact hitting set in the counting procedure but an “almost hitting set” is enough. Namely, if some small set of variables hits almost all clauses we can count the number of solutions to the remaining relatively small $(\leq 2)$-SAT with a good exponential time algorithm for \#2-SAT.
We introduce first some notation. For $i \in \mathbb{N}$ we call a set of clauses $S$ an *$i$-star* if $|S| = i$ and if there exists a variable $x$ such that for any pair of distinct clauses $C,D \in S$ we have $\{x\} = {\textnormal{vbl}}(C)\cap {\textnormal{vbl}}(D)$. A *star* is an $i$-star for some $i$. For $i \geq 2$ we call the variable $x$ the *center* of the star and any other variable is called a *leaf*. For 1-stars we consider both of the variables as centers and neither of them as leaves. A star is called *monotone* if the center appears as the same literal in every clause of the star. We call a set $T$ of exactly three clauses a *triangle* if every 2-element subset of $T$ is a star and $T$ is not itself a star. Finally, we call a family ${\mathcal{M}}$ of CNF formulas *independent* if no two formulas in ${\mathcal{M}}$ share common variables.
\[thm:Sample-2-SAT\_epsilon\] Let $F$ be a 2-CNF formula on $n$ variables and $m$ clauses and let $\varepsilon > 0$ be such that $|{\textnormal{sat}}(F)| \geq \varepsilon 2^n$. A uniformly random satisfying assignment for $F$ can be found in expected time $O\left(\varepsilon^{-\delta}(m+n)\right)$ where $\delta < {0.617}$.
Let $V$ be the variable set of $F$ and let $k \geq 2 $ be a constant independent of $\varepsilon$ that we fix later. We start by constructing a sequence $({\mathcal{M}}_0,{\mathcal{M}}_1,\ldots,{\mathcal{M}}_k)$ of $k+1$ independent families of formulas where every family consists of subformulas of $F$.
Let ${\mathcal{M}}_0$ be any independent 1-maximal family of $1$-stars (clauses) in $F$. That is, in addition to maximality we require further that there is no clause in the family whose removal would allow the addition of two clauses in its place. We can find ${\mathcal{M}}_0$ with a greedy algorithm in linear time[^2].
To construct ${\mathcal{M}}_1$ from ${\mathcal{M}}_0$, we add clauses of $F$ to the 1-stars of ${\mathcal{M}}_0$ greedily to update them into non-monotone 2-stars or triangles while maintaining independence. As a result ${\mathcal{M}}_1$ is an independent family of subformulas of $F$ that consists of 1-stars, non-monotone 2-stars, and triangles and no 1-star can be turned into the other two types by adding clauses of $F$ to it without revoking independence.
For $i = 2,\ldots,k$ we construct ${\mathcal{M}}_i$ from ${\mathcal{M}}_{i-1}$ by greedily adding clauses of $F$ to the monotone $(i-1)$-stars to turn them into monotone $i$-stars while ensuring independence. Since $k$ is a constant, and all since greedily adding clauses can be done in linear time, the total time taken to construct the families is $O(m+n)$. An example of ${\mathcal{M}}_4$ can be seen in Figure \[fig:ind\_family\_construction\]. We describe the structural properties of the families later in the proof.
![A possible construction of ${\mathcal{M}}_4$ for a formula $F$ that is displayed as a graph with the variables as vertices and edges between variables appearing in the same clause. The subformulas of $F$ that make up ${\mathcal{M}}_4$ are given by the components defined by the black bold edges. The edges that form up ${\mathcal{M}}_0$ are the horizontal black bold edges. There is one non-monotone 2-star in ${\mathcal{M}}_4$ and it is denoted by the square center vertex.[]{data-label="fig:ind_family_construction"}](figures/sample_construction_ipe){width="\textwidth"}
Analogously to the warm-up algorithm in the previous section we describe two different algorithms that both make use of the independent families we have constructed and that complement each other in terms of their running times. The second algorithm describes in fact $k$ different algorithms, determined by the choice of a parameter $\ell \in \{1,\ldots,k\}$. For each $i = 1,\ldots,k$ we let $s_i$ denote the number of monotone $i$-stars in ${\mathcal{M}}_k$. By construction the parameter $r_i := \sum_{j=i}^k s_j$ then denotes the number of monotone $i$-stars in $M_i$. We further let $t$ be the number of triangles and $q$ be the number of non-monotone $2$-stars in ${\mathcal{M}}_k$, and therefore in every ${\mathcal{M}}_i$ with $i = 1,\ldots,k$. The two algorithms we consider are:
1. Sample uniformly among all full assignments for $F$ that satisfy all the clauses in ${\mathcal{M}}_k$ until finding one that satisfies $F$.
2. Fix $\ell \in \{1,\ldots k\}$. Define further the variable set $W := {\textnormal{vbl}}({\mathcal{M}}_{\ell})$ and let $W' \subseteq W$ be the set of variables of ${\mathcal{M}}_{\ell}$ that appear in a clause of $F$ that has exactly one variable of ${\mathcal{M}}_{\ell}$ in them. Go through all $2^{|W'|}$ partial assignments $\alpha : W' \rightarrow \{0,1\}$ and compute $A_\alpha := |{\textnormal{sat}}_{V\setminus W'}(F^{[\alpha]})|$ by using Wahlstr[ö]{}m’s \#2-SAT algorithm [@wahlstrom2008tighter]. Let $A := \sum_{\alpha} A_{\alpha}$ and choose one partial assignment $\alpha^*$ at random so that Pr$(\alpha^* = \alpha) = A_{\alpha} / A$. For the remaining variables choose an assignment $\beta^* : V \setminus W' \rightarrow \{0,1\}$ uniformly among all assignments satisfying $F^{[\alpha^*]}$. This can be done by branching on a variable, using Wahlstr[ö]{}m’s algorithm to count the number of assignments in the two branches, flipping a biased coin weighed by the counts to decide on the branch and repeating the same on the resulting formula until all variables have been set. Output the full assignment which when restricted to $W'$ is $\alpha^*$ and when restricted to $V\setminus W'$ is $\beta^*$.
The correctness analysis for both of these two algorithms is essentially the same as in our warm-up in Section \[sec:warm-up\] and it remains to discuss the running times.
Starting with the first algorithm we note that the stars and triangles in ${\mathcal{M}}_k$ have constant size so the sampling of an assignment can be done in linear time in each iteration. Out of the $2^{i+1}$ possible assignments to the variables in any monotone $i$-star it can be easily checked that $2^i + 1$ satisfy all the clauses in the star. Both for a triangle or for a non-monotone 2-star there are 8 possible assignments out of which at most 4 are satisfying. Therefore from the independence of ${\mathcal{M}}_k$ we know that there are at most $$\begin{aligned}
\label{eqn:number_of_assignments}
2^{-t-q}\prod_{i=1}^{k} \left(\frac{2^i+1}{2^{i+1}}\right)^{s_i}2^n\end{aligned}$$ full assignments to the variables in $F$ that satisfy everything in ${\mathcal{M}}_k$. Since $F$ has at least $\varepsilon 2^n$ satisfying assignments and the size of the universe we are sampling from is given by we conclude that the first algorithm takes expected time $$\begin{aligned}
\label{eqn:2-sat_more_complicated_sampling}
O\left(\varepsilon^{-1}2^{-t-q}\prod_{i=1}^{k} \left(\frac{2^{i}+1}{2^{i+1}}\right)^{s_i}(m+n)\right)\end{aligned}$$ until returning a uniform satisfying assignment.
Consider now the runtime of the second algorithm. This is the more intricate part of the analysis and we will make use of the structure of the families that we have set up. It may be helpful to consider Figure \[fig:ind\_family\_construction\]. Let $F' \in {\mathcal{M}}_{\ell}$ be one of the subformulas in the family ${\mathcal{M}}_{\ell}$. We claim that $|{\textnormal{vbl}}(F')\cap W'| \leq 1$ and that if ${\textnormal{vbl}}(F')\cap W' = \{x\}$, then $F'$ is either an $\ell$-star or a non-monotone 2-star and $x$ is the center of the star. Towards showing the claim let $\{u,v\}$ be a clause with ${\textnormal{vbl}}(u) \in W$ and ${\textnormal{vbl}}(v) \in V\setminus W$ so that $\{u,v\}$ is a witness for ${\textnormal{vbl}}(u) \in W'$. If ${\textnormal{vbl}}(u)$ was a leaf of a star of ${\mathcal{M}}_{\ell}$, then we could have made ${\mathcal{M}}_0$ larger which would contradict the 1-maximality when ${\textnormal{vbl}}(u)\in {\textnormal{vbl}}({\mathcal{M}}_0)$ or just maximality in the case of ${\textnormal{vbl}}(u)\not\in {\textnormal{vbl}}({\mathcal{M}}_0)$. For the same reasons the variable ${\textnormal{vbl}}(u)$ can not appear in any triangle. For any $j < \ell$ the variable ${\textnormal{vbl}}(u)$ can also not be the center of a $j$-star as otherwise we would have updated that star into a monotone ($j+1$)-star when constructing ${\mathcal{M}}_{j+1}$ or we would have created a non-monotone $2$-star already in the beginning while constructing ${\mathcal{M}}_1$. The options for ${\textnormal{vbl}}(u)$ that remain are the centers of $\ell$-stars and the centers of the non-monotone 2-stars. In the case of $\ell = 1$ we still have to argue that at most one center may appear in $W'$. If both of the centers appeared in $W'$, it would either violate the 1-maximality of ${\mathcal{M}}_0$ or we could have turned the 1-star into a triangle which proves the claim. Therefore we have the bound $|W'| \leq r_{\ell} + q$.
We can observe from the argumentation above that if $\alpha : W' \rightarrow \{0,1\}$ is a partial assignment for $F$, then doing unit clause reduction on the formula $F^{[\alpha]}$ results in a $2$-CNF formula over some variable set $W_{\alpha} \subseteq W \setminus W'$. Computing $A_{\alpha}$ with Wahlstr[ö]{}m’s algorithm takes time $O(c^{|W_{\alpha}|})$ [@wahlstrom2008tighter]. Therefore we want to bound $|W_{\alpha}|$ as tightly as possible. If the assignment $\alpha$ sets the center literal of a monotone $\ell$-star to 0, then the values of the $\ell$ remaining variables in the star are determined and will be set to their required values with unit clause reduction. For a non-monotone 2-star either assignment of the center will force the value of one of the leaves and one leaf stays undetermined. If $\alpha$ sets $i$ of the $r_{\ell}$ literals in the centers of the monotone $\ell$-stars to 0 we get the bound $$\begin{aligned}
\label{eqn:bound_on_remaining_variables}
|W_{\alpha}| \leq q + 3t + \ell(r_{\ell}-i)+\sum_{j = 1}^{\ell-1}(j+1)s_j.\end{aligned}$$ Among the assignments $\alpha$ that we consider there are $\binom{r_{\ell}}{i}2^{q}$ different ones that set $i$ of the central literals of the monotone $\ell$-stars to $0$. Using formula we conclude that the runtime cost of going over the assignments $\alpha$ and computing the numbers $A_{\alpha}$ is $$\begin{aligned}
\label{eqn:2-sat_more_complicated_listing_alg}
&O\left(\sum_{i=0}^{r_{\ell}} \binom{r_{\ell}}{i} 2^{q}\cdot c^{q+3t+\ell(r_{\ell}-i)+\sum_{j = 1}^{\ell-1}(j+1)s_j}\cdot(m+n)\right) \notag \\
= \: &O\left(c^{3t}(2c)^{q}\left(1+c^{\ell}\right)^{r_{\ell}}\left[\prod_{j=1}^{\ell-1}c^{(j+1)s_j}\right]\cdot(m+n)\right)\end{aligned}$$ where we used the binomial theorem. We can again use the same trick as in the warm-up algorithm to sample $\alpha^*$ without storing all the values of $A_{\alpha}$ to keep the space requirement linear. The running time of finding $\beta^*$ with the branching procedure takes time $O(c^{|W_{\alpha^*}|}|W_{\alpha^*}| + (m+n))$ which is subsumed by .
We have now one algorithm with running time given by and for any $\ell \in \{1,\ldots,k\}$ we have an algorithm with running time given by . Given the sequence $({\mathcal{M}}_1,\ldots,{\mathcal{M}}_k)$ we choose the algorithm with the best runtime. To find a worst case upper bound on the runtime we look for the runtime in the form $$\begin{aligned}
\label{eqn:worst_case_runtime}
O\left(\varepsilon^{-\delta}(m+n)\right)\end{aligned}$$ and compute the nonnegative parameters $s_1,\ldots,s_k; t$ and $q$ that maximize the minimum of the different runtimes. Write $\sigma_i := s_i / \log \frac{1}{\varepsilon}, \tau := t / \log \frac{1}{\varepsilon}, \rho := q / \log \frac{1}{\varepsilon}$. By taking logarithms of the runtimes , and we can write the problem of finding $\delta$ and the worst case parameters $\sigma_i, \tau, \rho$ as the linear program $$\begin{array}{rl}
\displaystyle \max_{\delta,\,\sigma_i,\tau,\rho} \quad\delta \hfill & \\
\text{s.t.} \hfill - \tau - \rho + \sum_{i = 1}^k \sigma_i \log \left(\frac{2^i+1}{2^{i+1}}\right) &\geq \:\, \delta - 1 \\
3\tau\log c + \rho\log(2c)+ \sum_{i = 1}^{\ell-1} \sigma_i \log \left(c^{i+1}\right) + \sum_{i = \ell}^k \sigma_i \log \left(1+c^{\ell}\right) &\geq \:\, \delta \quad \textnormal{for all } \ell = 1,\ldots,k \\
\delta,\sigma_i,\tau,\rho &\geq \:\, 0 \quad \textnormal{for all } i = 1,\ldots,k \enspace .
\end{array}$$ It turns out that we only need to consider $k=7$ due to the fact that $c^{j+1} > 1+c^j$ in the integers when $j \geq 7$ which implies that the running time for higher values of $k$ no longer improves. For $k=7$ the linear program has in the optimum $\delta < 0.61618$. The approximate values of the other variables in the optimum are $\sigma_1 \approx 0.131, \sigma_2 \approx 0.127,, \sigma_3 \approx 0.111, \sigma_4 \approx 0.084,\sigma_5 \approx 0.051, \sigma_6 \approx 0.022, \sigma_7 \approx 0.004$ and exact values of $\tau = 0$ and $\rho = 0$. This finishes the proof.
We attempted to improve the analysis by constructing families that do not consist only of stars and triangles but the runtimes we achieved were not better. In some sense stars seem particularly good for the efficient use of Wahlstr[ö]{}m’s \#2-SAT algorithm as a subroutine because the set $W'$ is not too big. We also note that while we could consider adding the option of choosing $\ell = 0$ in the second algorithm, it is easily verified that choosing $\ell = 1$ instead gives a better performance.
A randomized algorithm for 3-SAT
--------------------------------
One could say that our Sample-2-SAT algorithm works because counting and sampling solutions for a $(\leq 1)$-CNF is trivial. Direct generalizations of our method to Sample-3-SAT do not work because the same is not true for $(\leq 2)$-CNF formulas. Instead of solving Sample-3-SAT we apply our method for 3-SAT.
\[prop:simple\_randomized\_3-SAT\] Let $F$ be a $3$-CNF formula on $n$ variables and $m$ clauses and let $\varepsilon > 0$ be such that $|{\textnormal{sat}}(F)| \geq \varepsilon 2^n$. A satisfying assignment for $F$ can be found in expected time $O\left(\varepsilon^{-\log_{8} 7}(m+n)\right)$.
Let $S$ be a maximal set of variable disjoint clauses in $F$. Either sample among those assignments that satisfy $S$ until finding a satisfying assignment or go through all the $7^{|S|}$ partial assignments to ${\textnormal{vbl}}(S)$ and check the satisfiability of the resulting $(\leq 2)$-CNF.
Checking through the partial assignments takes time $O(7^{|S|}\cdot(m+n))$ because each of the $7^{|S|}$ instances of $(\leq 2)$-SAT can be solved in linear time [@aspvall1979linear]. The rejections sampling takes expected time $O\left(\varepsilon^{-1}\left(\frac{7}{8}\right)^{|S|}(m+n)\right)$ because we are sampling from a pool of $\left(\frac{7}{8}\right)^{|S|}2^n$ assignments that contain all the at least $\varepsilon 2^n$ many satisfying assignments. Choosing always the better of the two methods, depending on $|S|$, gives a worst case running time of $O\left(\varepsilon^{-\log_{8} 7}(m+n)\right)$.
Proposition \[prop:simple\_randomized\_3-SAT\] gives an algorithm that works for any $\varepsilon$, but there exist better algorithms for certain ranges of $\varepsilon$. The PPSZ algorithm for 3-SAT runs in expected time $O(1.308^n)$ [@hertli20143] which is faster in the case that $\varepsilon = O(0.750^n)$. It is also possible to analyze Sch[ö]{}ning’s algorithm [@schoning2002probabilistic] for 3-SAT to get a dependence on $\varepsilon$ by using an isoperimetric inequality for the hypercube by Frankl and F[ü]{}redi [@frankl1981short]. The computation can be found in Appendix \[appendix:Schoening\]. The runtime guarantee that results is $O\left(\left(\frac{4}{3}\cdot 2^{-H^{-1}(\delta)}\right)^n\right)$ in expectation where $\delta$ is the solution to $\varepsilon = 2^{(\delta-1)n}$ and where $H : (0,1/2] \rightarrow (0,1]$ is the bijective *binary entropy function* defined by $H(x) = -x\log_2(x) - (1-x)\log_2(1-x)$. The range where Sch[ö]{}ning’s algorithm is better than Proposition \[prop:simple\_randomized\_3-SAT\] is when $\varepsilon = O(0.929^n)$.
Conclusion {#sec:conclusion}
==========
An interesting open problem is whether Sample-3-SAT can be solved time $O^*\left(\varepsilon^{-\delta}\right)$ for some $\delta < 1$. Similarly, can we achieve such a running time for 3-SAT with a deterministic algorithm?
We also believe that parameterizing by the number of solutions should be a fruitful approach to other problems besides satisfiability or vertex cover.
##### Acknowledgments {#acknowledgments .unnumbered}
We would like to thank Noga Alon and József Solymosi for discussions on the problem. We also thank the reviewers of IPEC 2017 for valuable remarks that improved the exposition.
Analysis of Sch[ö]{}ning’s algorithm with many assignments {#appendix:Schoening}
==========================================================
Sch[ö]{}ning’s algorithm for $k$-SAT is as follows. Start by picking random assignment and as long as there are unsatisfied clauses pick one and flip the value of a random literal in the clause. Keep on flipping $3n$ times and if no satisfied assignment has been found, restart the process from a new random assignment. Sch[ö]{}ning[@schoning2002probabilistic] showed that if $\alpha^{*}$ is some satisfying assignment of $F$ and if we choose $\alpha$ as our initial random assignment, then the probability that we find a satisfying assignment within the $3n$ steps is at least $$\begin{aligned}
\label{eqn:schoening_success_probability}
(1/(k-1))^{ d_{H}(\alpha^{*},\alpha)}\end{aligned}$$ where $d_H(\alpha^{*},\alpha)$ is the Hamming distance of $\alpha^{*}$ and $\alpha$. As the distance of a random assignment to a fixed satisfying assignment is binomially distributed, the probability of finding a satisfying assignment in one iteration before restarting is at least $$\begin{aligned}
\frac{1}{2^n}\sum_{j=0}^{n}\binom{n}{i}\left(\frac{1}{k-1}\right)^j = \left( \frac{k}{2(k-1)} \right)^n \enspace .\end{aligned}$$ Therefore, with the restarts, we see that Sch[ö]{}ning’s algorithm runs in expected time $$\begin{aligned}
O^{*}\left(\left( \frac{2(k-1)}{k} \right)^n\right)\end{aligned}$$ when $F$ is satisfiable. When there are many satisfying assignments we need to be able to compute or approximate the distribution of the distance of the initial random assignment to its closest satisfying assignment. To do this we will use an isoperimetric inequality for the hypercube which allows us to reduce the analysis to the case where the satisfying assignments are arranged very regularly. We use the formulation from Frankl and F[ü]{}redi [@frankl1981short]. A *Hamming ball* in $\{0,1\}^n$ with center $\alpha \in \{0,1\}^n$ is a set $B \subseteq \{0,1\}^n$ such that for some $r$ we have that $$\begin{aligned}
\{\beta \in \{0,1\}^n \: | \: d_{H}(\beta,\alpha) \leq r\} \subseteq B \subseteq \{\beta \in \{0,1\}^n \: | \: d_{H}(\beta,\alpha) \leq r+1\}.\end{aligned}$$ Note that this possibly less common definition of a Hamming ball allows for Hamming balls of any cardinality. We call $r$ the *radius* of the Hamming ball. The cardinality of a Hamming ball of radius $r = \rho n$ for a constant $\rho \in [0,1]$ is $O^*(2^{H(\rho)n})$ where $H : [0,1] \rightarrow [0,1]$ is the binary entropy function defined for $x \in (0,1)$ by $H(x) := -x\log_2(x) -(1-x)\log_2(1-x)$ and $H(0) = H(1) = 0$. See Chapter 10 §11 of the book by MacWilliams and Sloane [@macwilliams1977theory] for a proof. We define the inverse $H^{-1} : [0,1] \rightarrow [0,0.5]$ by restricting the domain of $H$ into $[0,0.5]$ on which $H$ is injective.
\[thm:isoperimetric\] Let $A,B \subseteq \{0,1\}^n$ be sets and define $$\begin{aligned}
d_{H}(A,B) := \min\{d_H(\alpha,\beta) \: | \: \alpha \in A, \beta \in B\}.\end{aligned}$$ There are two Hamming balls $A_0$ with center $\textbf{0}$ and $B$ with center $\textbf{1}$ such that $|A| = |A_0|$ and $|B| = |B_0|$ and $d_H(A_0,B_0) \geq d_{H}(A,B)$.
We are now ready to prove the following theorem which is most likely known to experts but we find it noteworthy to write a proof.
\[thm:schoening\_analysis\] Given a $k$-SAT instance $F$ on $n$ variables and a $\delta \in [0,H(1/k)]$ with the guarantee that $|{\textnormal{sat}}(F)| \geq 2^{\delta n}$, one can find a satisfying assignment for $F$ in expected time $$\begin{aligned}
O^*\left(\left(\frac{2(k-1)}{k}\right)^{n} \cdot (k-1)^{-H^{-1}(\delta)n}\right).\end{aligned}$$
Let $\sigma = \sigma(\delta)$ be a constant that we decide later on, let $A$ be the set of at least $2^{\delta n}$ assignments that satisfy $F$, let $B = \{\beta \in \{0,1\}^n \:|\: d_H(\alpha,\beta) \geq \sigma n + 1 \: \forall \alpha \in A\}$, and define $A_0$ and $B_0$ with respect to $A$ and $B$ as in Theorem \[thm:isoperimetric\]. Because $|A_0| = |A| \geq 2^{\delta n}$, the Hamming ball $A_0$ has radius at least $H^{-1}(\delta) n + O(\log n)$. Define $\rho := H^{-1}(\delta)$. We want to analyze the probability that a u.a.r. assignment $\alpha \in \{0,1\}^n$ is at most at a distance of $\sigma n$ from some point in $A$. Using basic properties of distances between two Hamming balls, that $d_{H}(A_0,B_0) \geq d_H(A,B) = \sigma n + 1$ and that $A_0$ has radius $\rho n + O(\log n)$ we can compute: $$\begin{aligned}
&\text{Pr}(d_{H}(\alpha,A) \leq \sigma n) \nonumber\\
&= \text{Pr}(\alpha \not\in B) \nonumber \\
&= \text{Pr}(\alpha \not\in B_0) \nonumber \\
&\geq \text{Pr}(d_{H}(\alpha,A_0) \leq d_H(A_0,B_0) - 1) \nonumber \\
&\geq \text{Pr}(d_{H}(\alpha,A_0) \leq \sigma n) \nonumber \\
&= \text{Pr}(d_H(\textbf{0},\alpha) \leq (\sigma + \rho)n + O(\log n)) \nonumber \\
&= \Omega^*(2^{(H(\sigma + \rho)-1)n}). \label{eqn:probability_close}\end{aligned}$$
The probability that a random assignment is at most at a distance $\sigma n$ of $A$ is given by and the probability of finding a satisfying assignment when starting from such an assignment is by at least $(1/(k-1))^{\sigma n}$. The inverse of the product of these two probabilities is $$\begin{aligned}
O^{*}\left(2^{(1-H(\sigma + \rho))n}(k-1)^{\sigma n}\right)\end{aligned}$$ which is a bound on the expected number of times we need a restart in Sch[ö]{}ning’s algorithm before finding a satisfying assignment. We can still choose $\sigma$ and the best choice is to define $\sigma :=\max\{1/k - \rho,0\} = \max\{1/k - H^{-1}(\delta),0\}$ which makes the expected running time equal to $$\begin{aligned}
\label{eqn:runtime_our_alg}
\begin{cases}
O^*\left(2^{(1-H(1/k))n}(k-1)^{ (1/k- H^{-1}(\delta))n}\right) & \quad \text{if } \delta \leq H(1/k) \\
O^*\left(2^{(1-\delta)n}\right) & \quad \text{otherwise.}\\
\end{cases}\end{aligned}$$ It is easy to check that $2^{(1-H(1/k))n}(k-1)^{n/k} = \left(\frac{2(k-1)}{k}\right)^{n}$ which finishes the proof.
[^1]: This work started at the 2016 Gremo Workshop on Open Problems (GWOP), on June 6-10 at St. Niklausen, OW, Switzerland.
[^2]: This is equivalent to finding a 1-maximal matching in a graph: first find a maximal matching and then find a maximal set of independent augmenting paths of length 3 and augment them.
|
---
abstract: |
We report the analysis of the magnetic fluctuations in the superconducing $%
\mathrm{La_{2-x}Sr_xCuO_4}$ and the related lanthanum cuprates having the different symmetry of the low temperature structure. The NMR and ESR investigations revealed the dynamical coexistence of the superconductivity and the antiferromagnetic correlations in the large part of superconductivity region of the phase diagram. We show that for all compounds, independent on their low temperature symmetry and on their superconducting properties, the enhancement of the spin stiffness near 1/8 doping takes place.
address:
- |
E.K.Zavoiskii Institute for Technical Physics of the RAS,\
Sibirskii Trakt 10/7,\
Kazan 420029, RUSSIA
- |
NHMFL, 1800 E P.Dirac Dr.,\
Tallahassee FL 32310, USA
- |
E.K.Zavoiskii Institute for Technical Physics of the RAS,\
Sibirskii Trakt 10/7,\
Kazan 420029, RUSSIA
- |
NHMFL, 1800 E P.Dirac Dr.,\
Tallahassee FL 32310, USA
- |
E.K.Zavoiskii Institute for Technical Physics of the RAS,\
Sibirskii Trakt 10/7,\
Kazan 420029, RUSSIA
- |
NHMFL, 1800 E P.Dirac Dr.,\
Tallahassee FL 32310, USA
author:
- 'G. B. Teitel’baum,[^1] V. E. Kataev, E. L. Vavilova,'
- 'P. L. Kuhns, A. P. Reyes, and W. G. Moulton'
title: |
Interplay between the magnetic fluctuations and superconductivity\
in the lanthanum cuprates
---
The interest to the microscopic phase separation in the high-$T_{c}$ superconducting materials has received a strong impetus after the discovery of stripe correlations [@gb1]. They were observed only in the compounds specially doped with the rare earth ions whose role is to induce the low temperature tetragonal phase favorable for the pinning of the stripe fluctuations. Recent neutron scattering experiments [@gb2] in the low temperature orthorhombic phase of $\mathrm{%
La_{2-x}Sr_{x}CuO_{4}}$ with $x=0.12$ reveal the presence of modulated antiferromagnetic order very similar to that found in compound $%
\mathrm{{La_{1.6-x}Nd_{0.4}Sr_{x}CuO_{4}}}$. But on the larger time scale the magnetic fluctuations in $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ are dynamical especially for the superconducting state and their relevance to the stripe structure is a matter of debate. In particular, the dynamical character of the microscopic phase separation hinders the investigation of its properties by means of low frequency local methods such as conventional NMR [gb3,gb4]{}.
The main aim of the present work is to analyze the phase diagram and the properties of magnetic fluctuations for superconducting $\mathrm{%
La_{2-x}Sr_{x}CuO_{4}}$ and related compounds with a help of experiments whose characteristic frequency is shifted to larger values in comparison with the conventional NMR. We consider ($\nu \backsim 10$ $%
\mathrm{GHz}$) and high field ($\nu \backsim 0.1$ $\mathrm{GHz}
$) measurements which are focused on a comparative analysis of the magnetic fluctuations for the different metalloxides. With this purpose we discuss the ESR data obtained for such compounds as $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ [@gb5], $\mathrm{La_{2-x}Ba_{x}CuO_{4}}$ [@ad3], $\mathrm{La_{2-x-y}Eu_{y}Sr_{x}CuO_{4}}$ [@ad4] together with the conventional NMR data for $\mathrm{%
La_{2-x-y}Nd_{y}Sr_{x}CuO_{4}}$ [@gb8] and new high field NMR data for superconducting All the measurements were carried out on powder samples with various hole doping. For the doping level covers the entire superconducting region of the phase diagram, for we studied the doping region in the vicinity of the well known $T_{c\text{ }}$dip, whereas the and series correspond to the nonsuperconducting phase. The samples which were used for the measurements were doped with 1 at. % of , used as probe [@gb5]. Such tiny concentration of Gd ensured only the small suppression of $T_{c}$ via pair breaking.
We analyzed the temperature and concentration dependence of the width of the most intense component of multiline $\mathrm{Gd^{3+}}$ spectrum, corresponding to the fine splitting of the spin states $S=7/2$ in the crystalline electric field [@gb5]. The typical temperature dependence of the linewidth $\delta H$ is shown in Fig.\[Fig1\].
The temperature behaviour for $T>T_{c}$ is qualitatively very similar for all samples under study: a linear dependence of $\delta H$ on temperature which is followed by the rapid growth of the linewidth at low $T$. But after cooling below 40K the behaviour of superconducting and nonsuperconducting samples becomes different: the linewidth of superconducting exhibits the downturn starting at a temperature $T_{m}$ dependent on $x$ whereas for other samples which are not bulk superconductors the linewidth continues to grow upon further lowering temperature (See Fig.1).
This behaviour may be explained if to take into account that in addition to the important but temperature independent residual inhomogeneous broadening the linewidth is given by different homogeneous contributions linked to the magnetic properties of $\mathrm{CuO_{2}}$ planes:
i\) the interaction of $\mathrm{Gd^{3+}}$ spins with the charge carriers, i.e. the Korringa relaxation channel. The simplest Korringa term in the linewidth is $\delta H=a+bT$ with $b=4\pi {(}JN_{F}{)}^{2}P_{M}$ (Ref.), where $P_{M}=[S(S+1)-M(M+1)]$ - is the squared matrix element of the spin transitions between the $M$ and $M+1$ states, $\ N_{F}$ is the density of states at the Fermi level, $\ J$ is the coupling constant between the and charge carriers spins [gb5]{}. The factor $P_{M}$ describes the Barnes-Plefka enhancement [@gb6] of the relaxation with respect to the standard Korringa rate. Such an enhancement occurs in exchange-coupled crystal field split systems where the g-factors of localized and itinerant electrons are approximately equal but the relaxation of conduction electrons towards the ”lattice” is strong enough to inhibit bottleneck effects. For the system under study it was discussed in Ref.. Note, that the enhancement of the linear slope for compound relative to that for LSCO seen in Fig.1 is due to the influence of the depopulation of the first excited magnetic level [@ad4].
ii\) the interaction of Gd with copper spins, giving rise to homogeneous broadening of line (a close analogue of nuclear spin-lattice relaxation):
$$\delta H=\frac{1}{2}{(\gamma H)}^{2}P_{M}\left[ \left( \tau /3\right)
+\left( 2\tau /3\right) /(1+{(\omega \tau )}^{2})\right]
\label{eq}$$
where $\tau $ is the magnetic fluctuations life-time, $H$ is the internal magnetic field at site. Following Ref. we assume the activation law for the fluctuation lifetime temperature dependence $\tau
=\tau _{\infty }\exp (E_{a}/kT)$ with $\tau _{\infty }$ being the lifetime at the infinite temperature and $E_{a}$ - the activation energy, proportional to the spin stiffness $\rho _{s}$ $\left( E_{a}=2\pi \rho
_{s}\right) $.
The second contribution describes the standard Bloembergen-Purcell-Pound (BPP) behaviour: the broadening of the ESR line upon cooling with the downturn at certain freezing temperature $T_{m}$ corresponding to $\omega
\tau =1$. Here $\omega $ is the resonant frequency. This expression is written for the case when the fluctuating magnetic fields responsible for spin relaxation are induced by local moments. In the polycrystalline samples the averaging over the random orientation of the local moments with respect to the external magnetic field yields by a factor of 2 larger probability of their perpendicular orientation as compared to the collinear one.
We observed that depending on the Sr content the linewidth behaviour transforms from the -like (with the maximum at $T_{m}$) to the pure Korringa (linear) temperature dependence. Basing on the observation that the relative weight of the -contribution, compared with the Korringa one, decreases with increasing doping we conclude that at low $x$ the spin probes almost magnetically correlated state and at the high $x$ end - almost nonmagnetic metal. Such a transformation may be explained it terms of the microscopical phase separation to the metallic and AF correlated phases. It is worth to remind that very soon after the discovery of the high $T_{c}$ superconductivity in cuprates it was suggested [@gb10], that the microscopical phase coexistence is the inherent feature of these materials. Note that according to the phase diagram shown in Fig.\[Fig2\] the obtained $T_{m}$ values are lower than the respective $T_{c}$, although for certain hole doping they are lying close to each other. The relative amount of the AF phase falls abruptly in the vicinity of $x=0.20$ so that for $x=0.24$ any traces of it are absent. One cannot exclude that this boundary is connected with the existence of the widely discussed quantum critical point [@ad7] at this doping values.
The different temperature dependences of the linewidths for the superconducting and nonsuperconducting compounds may be consistently explained assuming that for the superconducting samples the linewidth below $%
T_{c}$ is governed by fluctuating fields which are transversal to the constant field responsible for the Zeeman splitting of the spin states (the second term in Eq.(\[eq\]) for $\delta H$). Since these fluctuations are induced by moments lying in the $_{%
\text{2}}$ planes, it means that ions are subjected to the constant magnetic field normal to these planes. This may indicate that the magnetic flux lines penetrating in the layered superconduting sample tend to orient normally to the basal planes where the circulating superconducting currents flow (it is also possible, that ions pin the magnetic fluctuations connected with the normal vortex cores). The important argument in favor of the magnetic fluctuations contribution to the Gd ESR linewidth is given by the fact that the BPP peak at $T=T_{m}(x=0.10)\approx 16$ K is in a reasonable agreement with that observed near 4 K in the $^{139}$La nuclear spin relaxation rate temperature dependence for LSCO compound with $x=0.10$ at the frequency of 140 MHz [@gb11].
In principle there might be also a second possibility of the different low temperature behaviour of the linewidth for superconducting samples in comparison with that for nonsuperconducting ones. The nonresonant field dependent microwave absorption in the superconducting state may distort the shape of the spectrum. But these distortions should be especially pronounced for the broad lines, typically for the samples with the small amount of holes, whereas the temperature $T_{m}$, characteristic for small $x$, is considerably lower than $T_{c}$. Thus the possible distortion of lineshape owing to the nonresonant microwave absorption as the main reason for the apparent narrowing of the line below $T_{c}$ seems to be improbable.
Since the measurements were carried out at nonzero external field it is very important to consider the flux lattice effects. At typical fields of approximately $0.3$ , oriented normally to the $_{\text{2}}$ layers, the period of lattice is 860 , whereas the vortex cores sizes for are approximately 20 . As the upper critical field amounts to 62 , it is clear that in the case of ESR the vortex cores occupy only 0.5% of the $_{\text{2}}$ planes. According to Ref. the spins in the vortex cores may be ordered. Therefore the phase diagram in Fig.2 indicates that not only the spins in the normal vortex cores are correlated, but the correlations are spread over the distances of the order of magnetic correlation length which at low doping reaches 600-700 [@ad6].
Numerical simulations of the linewidths for the compounds with the different Sr content enable us to estimate the values of the parameters in the expression for the linewidth. For example the maximal effective internal field $H$ in the rare earth positions is about 200 ; the life time $\tau _{\infty }$, which was found to be material dependent, for LSCO is equal to $\tau _{\infty }=0.3\cdot
10^{-12}$ sec, and the activation energies $E_{a}$ for all investigated compounds are shown in Figs.2, 3. Note that since the influence of the Nd magnetic moments for the LNSCO compound hinders the ESR measurements the activation energy for this compound was estimated from the measurements of the nuclear spin relaxation on Cu and La nuclei.
The enhancement of $E_{a}$ (that is of a spin stiffness $\rho _{s}$) near $%
x=0.12$ shown in Fig.3 gives evidence of the developed antiferromagnetic correlations for all investigated compounds and explains both the anomalously narrow peak in inelastic neutron scattering [@gb7] and the elastic incommensurate peak with a narrow q-width [@gb2] reported for the superconducting $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ for this Sr doping. This indicates the important role of the commensurability and gives evidence of the plane character of the inhomogeneous spin and charge distributions. The maximal activation energies are 80 K for , 144 for , 160 for and 143 for . Note that for and the signatures of the bulk superconductivity [@ad3; @ad4] become visible upon the suppression (in course of the or doping) of the activation energy down to 80-85 . Therefore it is plausible to assume that these values of the activation energy are probably the critical ones for the realization of the bulk superconducting state. The corresponding boundary is shown in Fig.\[Fig3\]. Fluctuations with the higher activation energies (spin stiffness) are effectively pinned and suppress the superconductivity.
To obtain the information about the ordered magnetic moments for the compounds with the enhanced spin stiffness the NMR measurements at 20-25 T were carried out in a high homogeneity resistive magnet of the NHMFL in Tallahassee FL. The temperature and doping dependencies of $%
^{63,65}$Cu and $^{139}$La NMR field sweep spectra of the oriented powders $%
\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ were studied. According to the previous La NQR results [@gb3; @gb11] the measurements of oriented powder samples in a magnetic field perpendicular to **c** axis revealed that for Sr content near 1/8 the central lines of the observed spectra both for Cu and La exhibit the broadening upon cooling below 40-50 K (Fig.\[Fig4\]).
Such a behaviour is connected with the slowing down of the magnetic fluctuations, which are gradually slowing down upon ordering. The broadening of the La NMR line allows us to estimate that the additional magnetic field at La nucleus is 0.015 T. If we consider that for the antiferromagnet $\mathrm{La_{2}CuO_{4}}$ the copper moment of 0.64$\mu _{B}$ induces at the La site the field of 0.1 [@gb12], then the effective magnetic moment in the present case is $\sim 0.09\mu _{B}$. Note that the manifestation of the magnetic order only in the vicinity of $%
x=1/8$, when the AF structure is commensurate with the lattice, indicates that the magnetic inhomogeneities are of a plane character.
In conclusion our investigation reveals that for all studied compounds independent on the symmetry type (LTO or LTT) in the neighbourhood of $1/8$ doping the enhancement of the spin stiffness takes place. The compounds with the spin stiffness larger than the certain critical value (See Fig.3) reveal no bulk superconductivity.
According to the phase diagram the inherent feature of the superconducting state in cuprates is the presence of frozen antiferromagnetic correlations. Such a coexistence seems to be a result of phase separation at the microscopic scale as it was discussed in pioneering paper of Gor’kov and Sokol [@gb10].
In the neighbourhood of $1/8$ doping this coexistence may be realized in a form of dynamic stripes, since the corresponding enhancement of the spin-stiffness reveals the plane character of the spin (and charge) inhomogeneities.
One of the authors (G.B.T.) is grateful to L.P.Gor’kov for the valuable discussions of the phase separation specifics for the cuprates. This work is partially supported through the RFFR Grant N 01-02-17533.
J.M.Tranquada B.J. Sternlieb B.J., J.D.Axe [*et al*]{}, Nature [**375,**]{} 561 (1995).
T.Suzuki, T.Goto., K.Chiba [*et al*]{}, Phys.Rev. B[** 57,**]{} R3229 (1998).
S.Oshugi, Y. Kitaoka, H.Yamanaka [*et al.*]{}, J. Phys. Soc.Jpn. [**63,** ]{}2057 (1994).
T.Goto, K.Chiba, M.Mori [*et al*]{}, J. Phys. Soc. Jpn. [**66,**]{} 2870 (1997).
V.Kataev, Yu. Greznev, E.Kukovitskii [*et al*]{}, JETP Lett. [**56,**]{} 385 (1992); V.Kataev, Yu.Greznev, G.Teitel’baum [*et al*]{}, Phys. Rev. B[** 48,**]{} 13042 (1993).
B.Z.Rameev, E.F.Kukovitskii, V.E.Kataev, G.B.Teitel’baum, Physica C, [**246**]{}, 309 (1995).
V.E.Kataev, B.Z.Rameev, B.Buechner, M.Huecker, R.Borowski, Phys.Rev. B[** 55**]{}, R3394 (1997); V.E.Kataev V.E., B.Z.Rameev, A.Validov [*et al*]{}, Phys.Rev. B[** 58**]{}, R11876 (1998).
G.B.Teitel’baum, I.M.Abu-Schiekah I.M., O.Bakharev [*et al*]{}, Phys.Rev. B[** 63,**]{} 020507(R) (2001).
S.E.Barnes, Adv. Phys. [**30,** ]{} 801 (1981).
L.P.Gor’kov, A.V.Sokol, JETP Lett. [**46,** ]{}420 (1987).
S.Sachdev, Rev. Mod. Phys., [** 75,**]{} 913 (2003).
M.-H.Julien, A.Campana, A.Rigamonti [*et al.*]{}, Phys.Rev.[** **]{}B[** 63,** ]{}144508 (2001).
D.P.Arovas, A.J.Berlinsky, C.Kallin, S.C.Zhang, Phys. Rev. Lett. [**79**]{}, 2871 (1997).
B.Lake, G.Aeppli, K.N.Clausen [*et al*]{}, Science, [**291**]{}, 832 (2001).
K.Yamada, C.H.Lee, K.Kurahashi [*et al*]{}, Phys.Rev. B[**57,**]{} 6165 (1998).
T.Tsuda, T.Shimizu, H.Yasuoka [*et al*]{}, J.Phys.Soc.Jpn. [**57,** ]{} 2908 (1988).
[^1]: E-mail: grteit@dionis.kfti.knc.ru
|
---
abstract: 'We compute the stresses in an elastic medium confined in a vertical column, when the material is at the Coulomb threshold everywhere at the walls. Simulations are performed in 2 dimensions using a spring lattice, and in 3 dimensions, using Finite Element Method. The results are compared to the Janssen model and to experimental results for a granular material. The necessity to consider elastic anisotropy to render qualitatively the experimental findings is discussed.'
author:
- 'G. Ovarlez$^{1,2}$[^1], E. Clément$^2$'
title: 'Elastic medium confined in a column versus the Janssen experiment.'
---
Introduction
============
The mechanical status of granular matter is presently one of the most open and debated issues [@PDM]. This state of matter exhibits many unusual mechanical and rheological properties such as stress induced organization at the microscopic [@Oda] or at the mesoscopic [@Radjai] level which may yield macroscopic effects such as stress induced anisotropy [@Geng03; @Attman04a]. This issue sets fundamental questions relevant to the understanding of many other systems exhibiting jamming such as dense colloids or more generally soft glassy materials [@Nagel; @Cates]. For practical applications, the quasistatic rheology of granular assemblies is described using a phenomenological approach, based on an elasto-plastic modelling of stress-strain relations [@Wood]. So far, there is no consensus on how to express correctly the macroscopic constitutive relations solely out of microscopic considerations and under various boundary conditions or loading histories. This very basic issue was illustrated in a recent debate on how to understand the stress distribution below a sand pile and especially how to account for the dependence on preparation protocols [@Vaneltroudutas]. A new mechanical approach was proposed based on the concept of ”fragile matter” [@Cates] and force chains propagation modelling [@OSL]. But recent experiments have dismissed this approach and evidenced results more consistent with the traditional framework of general elasticity [@Guiguir].
In this paper we focus on the predictions for stresses measurements at the bottom for an elastic material confined in a rigid cylinder. When the column is filled with granular material it corresponds to the classical Janssen’s problem [@Janssen]. Recently, this issue has received a lot of attention either experimentally [@Vaneljanssen; @OvarlezRheo01; @OvarlezRheo03; @OvarlezSurp03; @Kolb99; @bertho02; @Zenith03] or numerically [@landry04; @Radjai04], the confined material being either pushed or pulled down. Surprisingly, so far to our knowledge, there are very few systematic comparison or even direct relation with the outcome of standard elasticity in the same situation of confinement. Note that experimentally, it was found that essentially elastic materials like gels, may display a Janssen stress saturation effects that could well predict the onset of self-collapsing under gravity [@Allain]. In a recent paper [@Evesque], Evesque and de Gennes proposed a model for the slow filling of an elastic medium modelling a granular packing. As the material is poured in the column, displacements of the material at the bottom are induced by the weight of the new material added so that it mobilizes friction forces. Within the assumption of a minimal anchorage length, it leads to partial and inhomogeneous mobilization of friction at the walls: friction is fully mobilized at the bottom, and partially in the upper part of the column. A Janssen like pressure profile can be derived in the case where the saturation length $\lambda$ is high compared to the column radius $R$; this last condition is actually restrictive and inappropriate for usual experimental cases [@Vaneljanssen; @OvarlezSurp03].
We propose to study in detail the elastic predictions and to compare them to the Janssen model and to experimental results for a granular material in *the same situation* i.e. when the material is at the Coulomb threshold *everywhere* at the walls. The main comparison features with the experimental results have already been presented in [@OvarlezSurp03]; here, we detail much more the elastic predictions.
Two kinds of situations are considered. First, the mass at the bottom of the column is measured as a function of the material filling mass. Second, similar measurements are produced with an overweight on the top of the material. We recently performed the corresponding experiments for a granular material [@OvarlezSurp03] and it was shown that one could obtain quite reproducible data provided a good control of the packing fraction homogeneity and a polarization of all the friction forces at the walls in the upwards direction. Our measurements confirmed, for the first time, the general validity of Janssen’s saturation curve. They also evidenced an overshoot effect of spectacular amplitude induced by a top mass equal to the saturation mass. These experimental results are actually strong tests for any theory of granular matter.
Experimental results and Janssen model
======================================
We first summarize our recent experimental results [@OvarlezSurp03] and compare them to the Janssen model predictions. In [@OvarlezSurp03], a slightly polydisperse assembly of 1.5 mm glass beads was poured at controlled packing fraction in steel columns of various friction coefficient, and of diameter varying between 38 mm and 80 mm. The aim of the experimental procedure is to achieve the Coulomb threshold everywhere at the walls when the apparent mass $M_{a}$ is measured; see [@OvarlezSurp03] for details on the procedure. The apparent mass $M_{a}$ measured at the bottom of the column was plotted as a function of the filling mass $M_{fill}$ of the material. The typical results obtained when $M_{fill}$ is varied are shown on Fig. \[fig1\]. The apparent mass $M_{a}$ saturates exponentially with $M_{fill}$. When an overweight equal to the saturation mass $M_{sat}$ is added on top of the granular material, $M_{a}$ increases with $M_{fill}$, up to a maximum $M_{max}$, which is about $20\%$ higher than $M_{sat}$, then decreases slowly towards the saturation mass $M_{sat}$.
The simple model which captures the physics of this saturation phenomenon was provided in 1895 by Janssen [@Janssen]. This model is based on the equilibrium of a granular slice taken at the onset of sliding everywhere at the walls; we attempted to realize as best as possible this last condition in our experiment [@OvarlezSurp03]. In cylindrical coordinates with origin at the top surface and the cylinder axis being the $z$ axis, the relation, at the slipping onset, between the shear stress $\sigma_{rz}$ and the horizontal stress $\sigma_{rr}$ at the walls is $$\sigma_{rz}(r\!\!=\!\!R,z)=\mu_{s}\sigma_{rr}(r\!\!=\!\!R,z)$$ where $\mu_{s}$ is the Coulomb static friction coefficient between the grains and the walls. It results in a relation between the filling mass $M_{fill}$ and the apparent mass at the bottom $M_{a}$ of the form: $$M_{a}=M_{sat}(1-\exp(-\frac{M_{fill}}{M_{sat}})) \label{janssen}$$ with $$M_{sat}=\frac{\rho\pi R^{3}}{2K\mu_{s}}$$ where $\rho$ is the mass density of the granular material, and $K$ is the Janssen parameter rendering the average horizontal redirection of vertical stresses: $$\begin{aligned}
\sigma_{rr}=K\sigma_{zz} \label{redirection}\end{aligned}$$ From a mechanical point of view, a major simplification of this model comes from the assumption that the redirection parameter $K$ would stay constant along the vertical direction. But on the other hand, it provides a clear and simple physical explanation for the existence of an effective screening length $\lambda$ $=R/2K\mu_{s}$ above which the mass weighted at the bottom saturates.
In [@OvarlezSurp03], several saturation profiles were measured for various packing fractions, columns sizes and friction coefficients between the grains and the walls. When the apparent mass rescaled by the saturation mass is plotted as a function of the filling mass also rescaled by the saturation mass, we obtain a *universal rescaling* of all data on a curve which is precisely the one predicted by Janssen (Fig. \[fig1\]): $M_{a}/M_{sat}=f(M_{fill}/M_{sat})$, with $f(x)=1-\exp(-x)$. The rescaling with radius $R$ and friction coefficient $\mu_{s}$ was also checked, and good agreement with the Janssen model rescaling was found. The Janssen constant $K$ was found to depend on packing fraction $\bar\nu$ and an effective relation was derived: $\Delta
K/K\simeq5\Delta\bar\nu/\bar\nu$. On the other hand, when a top mass equal to the saturation mass is added on the top of the granular material, the apparent mass $M_{a}$ displays a maximum $M_{max}$ $20\%$ higher than $M_{sat}$, whereas the Janssen model predicts $M_{a}=M_{sat}$ whatever the filling mass $M_{fill}$ is. Therefore, this overshoot goes beyond the possibilities of Janssen’s model which seems adapted to a unique configuration.
In the next section, we study in detail the predictions of isotropic homogeneous elasticity.
Elasticity: theory and simulation methods
=========================================
Theory {#elastheory}
------
We first recall the general framework of homogeneous isotropic elasticity, and then predict the behavior of an elastic material confined in a column.
The elastic theory gives, in the limit of small deformations, a linear relation between the stress tensor components $\sigma_{ij}$ and the strain tensor components $\epsilon_{ij}$. For an isotropic elastic material, we get $$E\epsilon_{ij}=(1+\nu_{p})\sigma_{ij}-\nu_{p}\delta_{ij}\sigma_{kk}
\label{stressstrain}$$ where E is the Young modulus, and $\nu_{p}$ the Poisson ratio which takes its value between $-1$ and $1/2$ in 3D, and between -1 and 1 in 2D.
In a uniaxial homogeneous compression experiment (Fig. \[Fig2\]), where $\sigma_{zz}=-p$ is imposed everywhere, the other stress tensor components being null, we get $$\epsilon_{zz}=-p/E$$ everywhere and $$\epsilon_{xx}=\epsilon_{yy}=-\nu_{p} \epsilon_{zz}$$ which signifies that the material expands in the transverse direction.The Young modulus $E$ is thus characteristic of the material’s stiffness; a cylinder of length $l$ and section $S$ has stiffness $k=ES/l$ in the axial direction. The Poisson ratio is linked to the material compressibility: the volume variation is: $\delta V/V=-(1-2\nu_{p})p/E$. Therefore, an incompressible material has Poisson ratio $1/2$ in 3D (1 in 2D).
![Sketch of uniaxial compression of a free elastic material (top) and of a confined elastic material (bottom).[]{data-label="Fig2"}](fig2.eps){width="8cm"}
If we now confine an elastic medium of Young modulus $E$ and Poisson ratio $\nu_{p}$, in a rigid cylinder of radius $R$, no more radial displacement at the walls is allowed: $u_{r}(r\!=\!R)=0$. Stresses and displacements can actually be calculated in the *limit of high depths $z$* under the assumption that they then should be independent of $z$. The boundary conditions we impose are the Coulomb condition everywhere at the walls $$\sigma_{rz}(r\!\!=\!\!R)=\mu_{s}\sigma_{rr}(r\!\!=\!\!R)$$ and infinitely rigid walls i.e. $u_{r}(r\!\!=\!\!R)=0$. The stress tensor components are then $$\begin{aligned}
\sigma_{rz}(r,z) & =-\frac{1}{2}\rho g r\\
\sigma_{rr}=\sigma_{\theta\theta} & =\frac{\nu_{p}}{1-\nu_{p}}\;\sigma
_{zz}\\
\sigma_{zz}^{sat}(r,z) & =-\frac{(1-\nu_{p})\rho
gR}{2\nu_{p}\mu_{s}} \label{stressasympt}\end{aligned}$$ And the asymptotic displacements are $$\begin{aligned}
u_{z}(r,z) & =-\frac{1\!+\!\nu_{p}}{2E}\rho gr^{2}-\frac{1\!-\!\nu
_{p}\!-\!2\nu_{p}^{2}}{2\mu_{s}\nu_{p}E}\rho gRz+u_{0}\label{depasympt}\\
u_{r}(r,z) & =u_{\theta}(r,z)=0\end{aligned}$$ This can be checked by injecting this solution in the stress-strain relation (\[stressstrain\]) and internal equilibrium relation $$\partial_{i}\sigma_{ij}=-\rho g_{j}$$
Thus, we obtain a Janssen’s like redirection phenomenon due to a Poisson’s ratio effect with a local Janssen’s parameter $K_{el}=\sigma_{rr}(r,z)/\sigma_{zz}(r,z)$, being for large depths: $$K_{el}=\frac{\nu_{p}}{1-\nu_{p}}$$ At 2D, we obtain the same saturation and stress redirection phenomena with $K_{el}=\nu_{p}$.
For a free elastic medium, the Poisson ratio effect is a transverse dilatation; for a confined elastic material, the Poisson ratio effect is a transverse redirection of stresses (Fig. \[Fig2\]).
In the following, $K_{el}$ will design the elastic stress redirection constant, whereas $K$ is devoted to design the stress redirection constant in the Janssen framework.
In this section, we obtained the asymptotic values of stresses and displacements. For the vertical stress, the limit is similar to the Janssen asymptotic vertical stress if one identifies $K$ an $K_{el}$. But we cannot say anything from this calculation about the whole pressure profile. We thus need to perform numerical computation. In the next section, we present the numerical methods we employed to simulate Janssen experiments for an elastic column.
Numerical methods
-----------------
Two different methods were employed: we first computed the stresses in 2D with a spring lattice. We also computed the stresses in 3D, using Finite Element Method thanks to CAST3M [@Castem].
### 2D: spring lattice
The 2D computations are performed in order to provide elastic predictions for a direct comparison with 2D simulations of granular materials.
In order to avoid any confusion, let us recall that we are not trying here to give a microscopic model for a granular material, but we use a discrete system behaving like an effective elastic medium in the continuous limit in order to perform simple numerical simulations.
However, this system can describe the most simple of granular materials: a 2D hexagonal piling of frictionless disks (with non-hertzian contacts so that contact elasticity is linear). Note also that systems of springs have been recently studied by Goldenberg and Goldhirsch [@Elastochaines1]; they showed that large forces inhomogeneities (i.e. like forces chains) can be found at the discrete scale in these systems, which are elastic in the continuous limit.
![Sketch of the discrete 2D elastic medium.[]{data-label="fig3"}](fig3.eps){width="7cm"}
In order to simulate a 2D elastic medium, we put point masses $m$ on an hexagonal lattice of link size $a$ (Fig. \[fig3\]). Every particle is linked to her 6 neighbors with identical springs of stiffness $k$ and length $a$ at rest. Therefore, the potential interaction energy between particles placed at $0$ and $x_{i}$ (such that $x_{i}x_{i}=a^{2}$) at rest, submitted to infinitesimal displacements $u_{i}$ and $v_{i}$, is: $$E_{p}=\frac{1}{2}k
\Bigl(\sqrt{(x_{i}+v_{i}-u_{i})(x_{i}+v_{i}-u_{i})} -a\Bigr)^{2}$$
By varying the stiffness $k$, we can only vary the young modulus $E$ of the effective elastic medium. As we also need to vary the Poisson ratio $\nu$, an elastic torsion potential between neighbor springs separated by angle $\theta$ (Fig. \[fig4\]) is added: $$E_{p}=\frac{1}{2}k_{b}\cos^{2}(\theta-\frac{\pi}{3})$$ This potential tries to maintain an angle $\pi/3$ between 2 neighbors springs if $k_{b}<0$. Thus the limit $k_{b}\rightarrow\-infty$ corresponds to a contractive medium of Poisson ratio -1, whereas the limit $k_{b}\rightarrow\infty$ corresponds to a incompressible medium.
![Displacements of the points of an elementary cell of the 2D elastic medium.[]{data-label="fig4"}](fig4.eps){width="8cm"}
An elementary area $A=a^{2}\sqrt{3}/2$ can be associated to each particle. The surface energy $\omega$ then reads: $$\omega=\frac{2}{\sqrt{3}a^{2}}\biggl(\frac{1}{2}\sum E_{p}
(springs)+\sum E_{p}(angles)\biggr)$$
If we consider the continuum limit of this system, the displacement of a particle located at $x_{i}$ is: $$u_{i}=(\partial_{j}u_{i}) x_{j}$$ The surface potential energy can be easily computed at second order in $\epsilon_{ij} =(1/2)(\partial_{i}u_{j}+\partial_{j}u_{i})$, and one obtains: $$\omega=
\frac{\sqrt{3}}{2}\biggl(\frac{3}{4}(k+2k_{b})(\epsilon_{xx}^{2}+\epsilon_{yy}^{2})+\frac{1}{2}(k-6k_{b})\epsilon_{xx}\epsilon
_{yy}+(k+6k_{b})\epsilon_{xy}^{2}\biggr)$$
For an elastic medium, stress-strain linearity reads $$\sigma_{ij}=C_{ijkl}\epsilon_{kl}$$
There are *a priori* 9 independent parameters in $C_{ijkl}$ (but only 2 if it is an isotropic elastic medium). The surface energy then reads $$\omega=\frac{1}{2}\sigma_{ij} \epsilon_{ij}=\frac{1}{2}
\epsilon_{ij} C_{ijkl}\epsilon_{kl} \label{omega}$$
If we identify this energy to the one we obtained for the spring lattice, we see that our system, in the continuous limit, is an isotropic elastic medium of Young modulus E and Poisson ratio $\nu_{p}$: $$\begin{aligned}
\nu_{p} & =\frac{1}{3}\frac{k-6k_{b}}{k+2k_{b}}\\
E & =\frac{2\sqrt{3}}{3}k\frac{k+6k_{b}}{k+2k_{b}}\end{aligned}$$
In order to solve the equilibrium problem of the lattice, we need to express the internal forces. They can be deduced from the surface energy (\[omega\]): to the usual elastic forces due to compression (or decompression) of springs, add forces which tends to drive the triangles angles to their equilibrium value (if $k_{b}<0$) or out of the $\pi/3$ value (if $k_{b}>0$). The force on A, due to the out of equilibrium angles of triangle ABC (Fig. \[fig4\]) reads: $$F_{A_{i}}=-\frac{3}{2}k_{b} (u_{A_{i}}+R_{{-\pi/3}_{ij}}u_{B_{j}}+R_{{\pi
/3}_{ij}}u_{C_{j}})$$ where $R_{{\theta}_{ij}}$ is the rotation matrix of angle $\theta$.
For the numerical computation, we impose the balance of forces (gravity, elastic forces, torsion forces) everywhere. At the bottom, we impose a null vertical displacement (rigid bottom), and either a perfectly stick (i.e. nullity of horizontal displacement) or perfectly slip (i.e. nullity of horizontal projection of forces) bottom. At the walls, we impose a null horizontal displacement (rigid wall), and we impose the Coulomb condition: the forces projected vertically are proportional to the forces projected horizontally with proportionality factor $\mu_{s}$. On top of the material, the forces projected horizontally are null (perfectly slip overweight), and the overweight is simulated by imposing the same vertical displacement for each particle on top (stiff overweight), which is close to the experimental situation. We thus obtain a linear system on the point displacements. We can vary stiffness $k$ and $k_{b}$ in order to simulate elastic mediums of different Young modulus and Poisson ratio. Note that for varying $\nu_{p}$ between 0 and 1, we need to vary $k_{b}$ between $-k/6$ and $k/6$. We also vary the friction at the walls $\mu_{s}$.
### 3D: FEM
In order to get the whole stress saturation curve, finite element numerical simulations [@Castem] were performed. The column is modelled as an isotropic elastic medium. We vary the friction at the walls $\mu_{s}$, the Young modulus $E$ and the Poisson ratio $\nu_{p}$. We imposed a rigid (nullity of vertical displacements), either perfectly stick (nullity of horizontal displacements) or perfectly slip (nullity of horizontal stresses) bottom. We found no appreciable difference between these two previous cases. The condition $\sigma_{rz}=\mu_{s}\sigma_{rr}$ is imposed everywhere at the walls. The cylinder is modelled as a steel elastic medium. We verified that in all the simulations performed, there is no traction in the elastic medium, so that this can actually be a model for a granular material.
For the simulations performed without overweight, the top surface is set free (no stress); the overweight is modelled as a perfectly slip (no horizontal stress) brass elastic medium.
In order to impose the Coulomb condition at the walls, we first set $F_{z}=0$ at each point of the mesh at the walls; we then obtain in these conditions the value $F_{r}$ exerted by the elastic medium on the walls. We then iterate in order to obtain $F_{z}=\mu_{s} F_{r}$ where $\mu_{s}$ is the friction coefficient at the walls: the vertical force imposed at step (i+1) is: $$F_{z}(i+1)=(1-\epsilon)\times F_{z}(i) + \epsilon\times\mu_{s} F_{r}(i)$$ We choose $\epsilon=0.2$. This procedure ensures convergence towards the Coulomb condition: if at step (i) the Coulomb condition $F_{z}(i)=\mu_{s}
F_{r}(i)$ is fulfilled, then at step (i+1): $F_{z}(i+1)=(1-\epsilon) F_{z}(i)
+ \epsilon\mu_{s} F_{r}(i)=(1-\epsilon)\mu_{s} F_{r}(i) + \epsilon\mu_{s}
F_{r}(i)=\mu_{s} F_{r}(i)=F_{z}(i)$. This boundary condition is the same as the one at step (i): this yields $F_{r}(i+1)=F_{r}(i)$, and thus $F_{z}(i+1)=\mu_{s} F_{r}(i+1)$.
### Remarks
Note that in these simulations we imposed the Coulomb condition everywhere at the walls. This allows comparison with the Janssen model in the same situation. But, regarding the experimental results [@OvarlezSurp03], i) nothing really insures that our piling preparation is strictly isotropic and ii) in spite of the careful procedure, we are never absolutely sure that all the friction forces at the wall are actually mobilized upwards. Moreover, the modelling of the contacts may seem rudimentary, as elasticity of contact should be included.
In these simulations, imposing dynamical friction at the walls or the static Coulomb threshold is actually the same: the material obeys the same equilibrium equations (if we consider a steady sliding at the walls), and the same condition at the walls, with just a change in the name (and the experimental value) of the friction coefficient.
Simulation results
==================
In this section, we present the results obtained from numerical computations at 2D and 3D.
In the following, we vary mainly the friction coefficient, and the Poisson ratio. The 2D simulations are in arbitrary units. The 3D simulations were all performed, if no contrary mention, for an isotropic elastic medium of mass density $\rho=1.6\ \mbox{g\,cm}^{-3}$ (which corresponds to an assembly of glass beads of packing fraction $\bar\nu=64\%$), of Young modulus 100 MPa, in a steel cylinder of radius $R=4$ cm, Poisson ratio 0.3, Young modulus 210 GPa, and thickness 3 mm. These data, which correspond to the display used in [@OvarlezSurp03], will not be specified anymore in the following. Simulations will also be performed in order to study the influence of the variation of the cylinder radius and the Young modulus of the elastic medium.
Simulations without overweight {#Nooverweight}
------------------------------
We first perform simulations similar to the original Janssen experiment: we plot the weight at the bottom as a function of the weight of elastic material in the column (Fig. \[fig5\]).
![3D simulation of a Janssen experiment (squares) for an elastic material of Poisson ratio $\nu _{p}=0.45$; the friction at the walls is $\mu_{s}=0.5$. The data are fitted by a Janssen curve of coefficient $K=0.89$ (line). The Janssen curve for $K$ corresponding to the elastic stress redirection constant $K_{el}=\nu _{p}/(1-\nu_{p})=0.82$ is also displayed (dotted line).[]{data-label="fig5"}](fig5.eps){width="8cm"}
We see on Fig. \[fig5\], for friction coefficient $\mu_{s}=0.5$ and Poisson ratio $\nu_{p}=0.45$, that the apparent mass $M_{a}$ saturates exponentially with the filling mass. The data are perfectly fitted by the Janssen model for these parameters (Fig. \[fig5\]), but the Janssen coefficient $K$ extracted from the fit is $9\%$ higher than the elastic stress redirection constant $K_{el}=~\nu_{p}/(1-\nu_{p})=0.82$. This is *a priori* unexpected as the elastic saturation pressure and the Janssen one should be identical with $K=K_{el}$.
### Effect of the bottom
In order to understand this feature, we study the whole mean vertical pressure profile in the same column (Fig. \[fig6\]).
![Mean pressure profile in a simulated elastic material of height 31 cm and Poisson ratio $\nu _{p}=0.45$ (squares); the friction at the walls is $\mu_{s}=0.5$. Depth $z=0$ cm corresponds to the top of the column. Depth $z=31$ cm corresponds to the bottom. We display the integral of vertical stresses $F_{z}$ at height $z$. The data are compared to a Janssen curve of coefficient $K=K_{el}=0.82$ corresponding to the elastic stress redirection constant (dotted line).[]{data-label="fig6"}](fig6.eps){width="8cm"}
Regarding mean vertical pressure, we see that the asymptotic value is now the expected theoretical asymptotic value, and the Janssen curve with $K=K_{el}$ gives a good though not perfect fit of the profile. The pressure saturates at high depths but decreases suddenly near the bottom; this is actually the value on the bottom we measure in a Janssen experiment. This feature explains why the saturation mass is lower than the expected one on Fig. \[fig5\]. The reason for this change of pressure near the bottom is that the asymptotic vertical displacement is parabolic whereas the bottom is rigid and imposes a flat displacement.
It is thus important to note that the usual Janssen experiment, in which one measure is made for one given height, is not equivalent to measuring a pressure profile, and results in a lower saturation stress (i.e. higher Janssen’s constant $K$) than the pressure profile. For more clarity on this last point, we illustrate this difference on Fig. \[fig7\].
![Sketch of comparison between a pressure profile and measures at the bottom for an elastic medium.[]{data-label="fig7"}](fig7.eps){width="10cm"}
![Study of the rescaling with the Poisson ratio $\nu_{p}$ for 3D simulations of an elastic material. We plot $M_{a}\times\mu_{s} \nu_{p}/(1-\nu_{p})$ vs. $M_{fill}\times\mu_{s} \nu_{p}/(1\!-\!\nu_{p}) $ for $\nu_{p}=0.26$ (squares), $\nu_{p}=0.35$ (open circles), $\nu_{p}=0.4$ (triangles), $\nu_{p}=0.45$ (open down triangles) and $\nu_{p}=0.49$ (stars); the friction at the walls is $\mu_{s}=0.5$.[]{data-label="fig8"}](fig8.eps){width="8cm"}
![Janssen constant $K$ extracted form 2D (squares) and 3D (triangles) simulations for various Poisson ratios. In the 2D simulations, the friction at the walls is $\mu_{s}=1.0$; in the 3D simulations, the friction at the walls is $\mu_{s}=0.5$. $K$ is plotted vs. the elastic stress redirection constant $K_{el}=\nu_{p}$ in 2D, $K_{el}=\nu_{p}/(1-\nu_{p})$ in 3D.[]{data-label="fig9"}](fig9.eps){width="8cm"}
In the following, we study the rescaling law of simulated Janssen experiments with friction coefficient $\mu_{s}$ at the walls and the Poisson ratio $\nu_{p}$. The data are compared to the Janssen model predictions by plotting $M_{a}\times\mu_{s}\times\nu_{p}/(1-\nu_{p})=f(M_{fill}\times\mu_{s}\times
\nu_{p}/(1-\nu_{p}))$ for different $\mu_{s}$ and $\nu_{p}$ (since $K_{el}\!=\!\nu_{p}/(1-\nu_{p})$ and $M_{sat}\!\propto\! 1/(K_{el}\mu_{s})$).
### Effect of the Poisson ratio
On Fig. \[fig8\], we study the rescaling with the Poisson ratio $\nu_{p}$ for 3D simulations. The rescaling law is rather good, the differences may not be observable experimentally. On Fig. \[fig9\] we plot the Janssen coefficient $K$ extracted from the fit of data in 2D and 3D versus the elastic stress redirection constants $K_{el}$ ($\nu_{p}$ in 2D, $\nu_{p}/(1-\nu_{p})$ in 3D).
We observe that for a given friction coefficient, $K$ hardly depends on $\nu_{p}$ in 2D: $K$ variation is $1\%$ for $K_{el}$ varying from $0.33$ to $0.77$. In 3D, $K/K_{el}$ increases roughly linearly with $K_{el}$ for $K_{el}$ varying from $0.25$ to $0.96$; however, K variation is less than 10% in this range. We remark that $K>K_{el}$: we explained it in the preceding section as a consequence of the presence of the rigid bottom.
![Study of the rescaling with the friction coefficient $\mu_{s}$ at the walls for 2D simulations of an elastic material. We plot $M_{a}\times\mu_{s}$ vs. $M_{fill}\times\mu_{s}$ for $\mu_{s}=0.4$ (squares), $\mu_{s}=0.6$ (open circles), $\mu_{s}=0.8$ (triangles) and $\mu_{s}=1.0$ (open down triangles); the Poisson ratio is $\nu_{p}=0.77$.[]{data-label="fig10"}](fig10.eps){width="8cm"}
![Study of the rescaling with the friction coefficient $\mu_{s}$ at the walls for 3D simulations of an elastic material. We plot $M_{a}\times\mu_{s}\nu_{p}/(1-\nu_{p})$ vs. $M_{fill}\times\mu_{s}\nu_{p}/(1-\nu_{p})$ for $\mu_{s}=0.1$ (open down triangles) $\mu_{s}=0.25$ (triangles), $\mu_{s}=0.5$ (open circles) and $\mu_{s}=0.8$ (squares); the Poisson ratio is $\nu_{p}=0.45$.[]{data-label="fig11"}](fig11.eps){width="8cm"}
### Effect of friction at the walls {#Nooverweightvaryfriction}
On Fig. \[fig10\] and Fig. \[fig11\], we study the rescaling with the friction coefficient $\mu_{s}$ at the walls respectively for 2D and 3D simulations. We now see that the proposed rescaling law is not good: the saturation is more abrupt as the friction coefficient is higher.
![Janssen constant $K$ extracted form 2D (squares) and 3D (triangles) simulations for various friction coefficient $\mu_{s}$. In the 2D simulations, the Poisson ratio is $\nu_{p}=0.77$; in the 2D simulations, the Poisson ratio is $\nu_{p}=0.45$. $K$ is rescaled by the elastic stress redirection constant $K_{el}=0.77$ in 2D, $K_{el}=0.82$ in 3D.[]{data-label="fig12"}](fig12.eps){width="8cm"}
On Fig. \[fig12\] we plot the Janssen coefficient $K$ extracted from the saturation mass value in 2D and 3D versus $\mu_{s}$. We now observe that the Janssen constant (and thus the saturation mass) depends strongly on the friction coefficient at the walls: $K$ increase is $12\%$ in 2D for $\mu_{s}$ varying between $0.1$ and $1.0$, and $20\%$ in 3D for $\mu_{s}$ varying from $0.1$ to $0.8$. $K$ depends roughly quadratically on $\mu_{s}$. Moreover, the Janssen constant seems to tend towards the elastic stress redirection constant $K_{el}$ when $\mu_{s}$ goes to 0.
In order to understand these features, we plot on Fig. \[fig13\] the mean vertical pressure $F_{z}$ versus the depth, and its rescaling with $\mu_{s}$.
![Integral of vertical stresses $F_{z}$ vs. depth $z$, for elastic materials of Poisson ratio $\nu_{p}=0.45$, and friction at the walls $\mu_{s}=0.25$ (triangles), $\mu_{s}=0.5$ (open circles), et $\mu_{s}=0.8$ (squares). We plot $F_{z}/F_{sat}$ vs. $2K_{el}\mu_{s}\,z/R$ where $F_{sat}=\rho g\pi
R^{3}/(2K_{el}\mu_{s})$ is the theoretical saturation value for $F_{z}$. $z/R=0$ corresponds to the top of the material. We also plot the Janssen model prediction with $K=K_{el}=\nu_{p}/(1-\nu_{p})$ (line).[]{data-label="fig13"}](fig13.eps){width="8cm"}
We see that, regarding the pressure profile, the Janssen rescaling law is correct for the asymptotic value, but the profiles are slightly different: for low friction at the walls $\mu_{s}\approx0.25$, the profile is perfectly fitted by the Janssen law; for higher friction, the profile is sharper and saturates abruptly. The differences for simulated Janssen experiments observed on Fig. \[fig11\] are now identified as a consequence of the presence of a rigid bottom: we actually see on Fig. \[fig13\] that the decrease of pressure near the bottom is more important when the friction is higher i.e. the saturation mass is lower (and the effective Janssen constant is higher).
We now understand all these features: the parabolic part of asymptotic displacements is negligible for low friction; in this case, the flat displacement imposed by the rigid bottom matches the asymptotic displacement, and the pressure at the bottom is the saturation pressure: we thus obtain $K=K_{el}$. As the friction is increased (and the material leans on the walls), the parabolic part of asymptotic displacement becomes more important and the influence of the rigid bottom is to decrease pressure; we thus observe an increase in the effective Janssen constant $K$ with $\mu_{s}$.
### A unique parameter
In the Janssen picture, the Janssen coefficient $K$ and the friction at the walls $\mu_{s}$ are not independent parameters: the relevant parameter is $K\mu_{s}$. As regards the saturation mass $M_{sat}$, it remains true in elasticity, and we see on Fig. \[fig14\] that $K\mu_{s}$, extracted from the saturation mass value, is a function of $K_{el}\mu_{s}$ alone (i.e. data on Fig. \[fig9\] and \[fig12\] can be replotted on a single universal curve); we find a quadratic dependence of $K\mu_{s}$ on $K_{el}\mu_{s}$: $K\mu_{s}\approx
K_{el}\mu_{s}+0.29(K_{el}\mu_{s})^{2}$. This explains the dependence found before on $\nu_{p}$ at fixed $\mu_{s}$ or on $\mu_{s}$ at fixed $\nu_{p}$.
![Janssen constant $K$ extracted form 3D (triangles) simulations for various friction coefficient $\mu_{s}$ at constant $\nu_{p}=0.45$ (empty triangles) and various Poisson ratios $\nu_{p}$ at constant $\mu_{s}=0.5$ (squares); see Fig. \[fig8\] and \[fig11\] for $\nu_{p}$ and $\mu_{s}$ values. The dotted line is the $y=x$ line; the line is a polynomial fit with $K\mu
_{s}=K_{el}\mu_{s}+0.29(K_{el}\mu_{s})^{2}$.[]{data-label="fig14"}](fig14.eps){width="8cm"}
However, it is not true anymore for the whole Janssen profile: we see on Fig. \[fig15\] that there is not a universal curve $M_{a}/M_{sat}=f(M_{fill}/M_{sat})$: if most data may be fit by the Janssen curve (with $f(x)=1-\exp(-x)$, it is not true anymore for high (i.e. low $M_{sat}$ values); the data for high values of $K\mu_{s}$ (=0.65 here) fall above the Janssen curve. However, for most experimental conditions, $K\mu_{s}$ is not as high, and this effect will not be observable.
![$M_{a}/M_{sat}$ vs. $M_{fill}/M_{sat}$ for 3D simulations for various values of $\nu_{p}$ and $\mu_{s}$; the triangles are for $\nu_{p}=0.45$ and $\mu_{s}=0.8$.[]{data-label="fig15"}](fig15.eps){width="8cm"}
### Effect of walls elasticity
On Fig. \[fig16\], the Young modulus is varied.
![Simulation of a Janssen experiment for elastic materials of Young modulus $E=1$ MPa (squares), $E=200$ MPa (open triangles) et $E=4$ GPa (circles), of Poisson ratio $\nu _{p}=0.45$, in a steel column of radius $R=4$ cm; the friction coefficient at the walls is $\mu_{s}=0.5$. []{data-label="fig16"}](fig16.eps){width="8cm"}
We observe that the results are independent of the Young modulus $E$ value, as long as it is less than 500 MPa; we see on Fig. \[fig16\] that the curves obtained for the simulation of a Janssen experiment for $E=1$ MPa and $E=200$ MPa can be perfectly superposed. For higher $E$, the weighted mass is higher and does not seem to saturate anymore; as an example, for $E=4$ GPa on Fig. \[fig16\], $M_{a}$ seems to increase indefinitely.
Note that this Young modulus effect is present only because we take into account the walls elasticity; for rigid walls and bottom, there would not be any dependence on $E$. The values of $E$ presented here have a meaning only for a particular cylinder (same $E$ and same thickness).
Simulations with an overweight
------------------------------
In this section, we present simulations of a Janssen experiment, when an overweight corresponding to the saturation mass is added on top of the material.
### Effect of the Poisson ratio
On Fig. \[fig17\] and \[fig18\], we plot the apparent mass $M_{a}$ versus the filling mass $M_{fill}$, rescaled by the saturation mass $M_{sat}$, respectively in 2D and 3D, for various Poisson ratio.
![Weight at the bottom of a 2D elastic material, when an overweight equal to the saturation mass is added on top of the material, for various Poisson ratio $\nu_{p}$: 0.33 (open stars), 0.40 (diamonds), 0.48 (open down triangles), 0.57 (triangles), 0.67 (open circles) and 0.78 (squares). The friction coefficient at the walls is $\mu_{s}=1.0$. The data are scaled with the saturation mass obtained in a simulation without overweight.[]{data-label="fig17"}](fig17.eps){width="8cm"}
![Weight at the bottom of a 3D elastic material, when an overweight equal to the saturation mass is added on top of the material, for Poisson ratio $\nu_{p}=0.35$ (triangles), $\nu _{p}=0.4$ (circles) et $\nu_{p}=0.45$ (squares). The friction coefficient at the walls is $\mu_{s}=0.5$. The data are scaled with the saturation mass obtained in a simulation without overweight.[]{data-label="fig18"}](fig18.eps){width="8cm"}
We now observe that, contrary to the simulations without any overweight, the results depend strongly on $\nu_{p}$, i.e. on the elastic stress redirection constant $K_{el}$. The curves all have the same form: $M_{a}$ increases with $M_{fill}$, up to a maximum $M_{max}$, then decreases slowly towards the saturation mass $M_{sat}$. The relative maximum $M_{max}/M_{sat}$ increase with $\nu_{p}$, and takes its value for higher $M_{fill}/M_{sat}$.
The proposed rescaling law, similar to the Janssen one, is not correct. We did not find any rescaling law; we thus keep our rescaling as a practical representation of data.
### Effect of friction at the walls {#effect-of-friction-at-the-walls}
On Fig. \[fig19\] et \[fig20\], we plot the apparent mass $M_{a}$ versus the filling mass $M_{fill}$, rescaled by the saturation mass $M_{sat}$, respectively in 2D and 3D, for various friction coefficients at the walls.
![Weight at the bottom of a 2D elastic material, when an overweight equal to the saturation mass is added on top of the material, for various friction coefficients $\mu_{s}$: 0.2 (squares),0.3 (open circles)), 0.4 (triangles), 0.5 (open down triangles), 0.6 (diamonds), 0.7 (open left triangles), 0.8 (right triangles), 0.9 (open hexagons) and 1.0 (stars). The Poisson ratio is $\nu_{p}=0.78$. The data are scaled with the saturation mass obtained in a simulation without overweight.[]{data-label="fig19"}](fig19.eps){width="8cm"}
![Weight at the bottom of a 3D elastic material, when an overweight equal to the saturation mass is added on top of the material, pour friction coefficient $\mu_{s}=0.25$ (triangles), $\mu_{s}=0.5$ (squares) and $\mu_{s}=0.8$ (circles). The Poisson ratio is $\nu_{p}=0.45$. he data are scaled with the saturation mass obtained in a simulation without overweight.[]{data-label="fig20"}](fig20.eps){width="8cm"}
We observe that the relative maximum of the apparent mass $M_{max}/M_{sat}$ increases with the friction coefficient at the walls, and takes its value for higher $M_{fill}/M_{sat}$.
This result leads to an apparent paradox. As actually the maximum $M_{max}$ increases when $M_{sat}$ decreases (i.e. when $\mu_{s}$ and $\nu_{p}$ increase), we observe that *the more the weight of the grains is screened by the walls, the less the weight of the overload is screened*! We will propose an interpretation in the following.
### Study of the pressure profile
We can wonder if all these results remain true for the profile. These features could be due only to the presence of a bottom. On Fig. \[fig21\], we plot the mean pressure profile when an overweight of mass equal to the saturation mass obtained in the simulation of a Janssen experiment is added on top of the material.
![Integral of vertical stresses $F_{z}$ vs. depth $z$, for elastic materials of friction coefficient at the walls $\mu_{s}=0.5$, and Poisson ratios $\nu_{p}=0.35$ (squares) and $\nu_{p}=0.45$ (circles), when an overweight equal to the saturation mass is added on top of the material. We plot $F_{z}/F_{sat}$ vs. $K_{el}\mu_{s}\,z/R$ where $F_{sat}=\rho g\pi R^{3}/(2K_{el}\mu_{s})$ is the theoretical saturation value for $F_{z}$. $z/R=0$ corresponds to the top of the material.[]{data-label="fig21"}](fig21.eps){width="8cm"}
We observe the same features as for the simulation of a Janssen experiment. In order to go one step further, we now add an overweight which imposes on top of the material a mean pressure equal to the saturation pressure; the results are presented on Fig. \[fig22\].
![Integral of vertical stresses $F_{z}$ vs. depth $z$, for elastic materials of friction coefficient at the walls $\mu_{s}=0.5$, and Poisson ratios $\nu_{p}=0.35$ (squares) and $\nu_{p}=0.45$ (circles), when an overweight equal to the saturation value $F_{sat}$ of $F_{z}$. We plot $F_{z}/F_{sat}$ vs. $K_{el} \mu_{s}\,z/R$ where $F_{sat}=\rho g\pi R^{3}/(2K_{el}\mu_{s})$ is the theoretical saturation value for $F_{z}$. $z/R=0$ corresponds to the top of the material.[]{data-label="fig22"}](fig22.eps){width="8cm"}
Once again, we observe the same features as for the simulation of a Janssen experiment. The Janssen rescaling law is still incorrect, and the overshoot effect is more important for higher Poisson ratio and friction coefficient. Moreover, on the profile the amplitude of the maximum is more important: for friction coefficient $\mu_{s}=0.5$ and Poisson ratio $\nu_{p}=0.45$, we find that the maximum force on the profile is $1.06$ times the saturation force $F_{sat}$, whereas the maximum weighted mass is $1.03$ times the saturation mass $M_{sat}$.
![Study of the rescaling with the radius $R$ of the column. We plot $M_{a}/R^{3}$ vs. $M_{fill}/R^{3}$ for $R=1.9$ cm (squares) et $R=4$ cm (line), for an elastic material of Poisson ratio $\nu_{p}=0.45$ and friction coefficient at the walls $\mu_{s}=0.5$.[]{data-label="fig23"}](fig23.eps){width="8cm"}
### Rescaling with the radius
We finally verify the rescaling with the radius column for simulations with and without any overweight. This rescaling is perfect for both simulations (Fig. \[fig23\]) for the extreme radius employed in [@OvarlezSurp03] ($R=1.9$ cm et $R=4$ cm).
Observable features
-------------------
To summarize, several features are experimentally observable if the isotropic elastic theory is valid.
In a Janssen experiment with free top surface, for a same material, the Janssen constant $K$, deduced from the measured saturation mass $M_{sat}$, must be higher for higher friction coefficient at the walls.
In an experiment with $M_{sat}$ overweight on top of the material, the apparent mass $M_{a}$ must increase strongly with the filling mass $M_{fill}$, and then decrease slowly towards the saturation mass $M_{sat}$. The observed maximum $M_{max}$ must increase with friction at the walls and with Janssen coefficient $K$ (measured in a Janssen experiment with free top surface).
The saturation mass $M_{sat}$ deduced from the pressure profile must be higher than the one measured at the bottom, and a strong decrease in the pressure must be observed near the bottom. This kind of feature would not be observed for a hyperbolic theory such as OSL [@Cates]. Moreover, for experiments with an overweight on top of the material, the maximum of $M_{a}$ will be higher on the pressure profile.
Comparison with experimental results
------------------------------------
In this section, we compare the elastic theory predictions to the experimental results obtained in [@OvarlezSurp03]. The main features have already been presented in [@OvarlezSurp03]
### Classical Janssen experiment
The Janssen experiment data are perfectly fitted by the Janssen model. Therefore, they are also perfectly fitted by the elastic theory: we indeed showed in Sec. \[Nooverweight\] that for weak friction at the wall ($\mu
_{s}=0.25$ as in the experiment) the Janssen model and elastic theory predictions cannot be discerned.We also showed in Sec. \[Nooverweightvaryfriction\] that when the friction coefficient is increased, the elastic theory predict a sharper initial increase of $M_{a}$ with $M_{fill}$ and an more abrupt saturation. However, this cannot be observed experimentally for the small range of friction coefficient used in [@OvarlezSurp03] (from 0.22 to 0.28). In order to test this last prediction, it would be necessary to work with higher friction at the walls ($\mu_{s}\sim0.5$).
Note that for high packing fractions ($\nu=0.645\pm0.005$), we obtained Janssen coefficients higher than 1 ($K=1.2\pm0.1$): this cannot be obtained in the isotropic elastic theory, as $K_{el}=\nu_{p}/(1-\nu_{p})\leq1$. We saw that $K_{el}$ is less than $K$ (Fig. \[fig9\], \[fig12\]), due to presence of a rigid bottom, but for small friction at the walls, this cannot explain high $K$ values: we deduce from Fig. \[fig12\] that for $\mu _{s}=0.25$, $K$ must be less than 1.03; for $\mu_{s}=0.8$, $K$ could be as high as 1.2.We will see in Sec. \[anisotropy\] how to obtain higher $K$ values in the context of anisotropic elasticity.
It was observed in [@OvarlezSurp03] that for different preparations, characterized by different packing fractions, the saturation mass $M_{sat}$ is lower for higher packing fraction. Although there may be structural differences between the pilings other than the packing fraction, this can be interpreted as increase of the Janssen coefficient $K$ with packing fraction. Thus, in the context of isotropic elasticity, this leads to an effective increase of the Poisson ratio $\nu_{p}$ with packing fraction. In [@OvarlezSurp03], an effective relation between packing fraction $\bar\nu$ and Poisson ratio $\nu_{p}$ was derived: $\nu_{p}\simeq2.3(\nu-0.41)$ with a precision of $5\%$. Note that the largest packing fraction $\nu=0.645\pm0.005$ would give a Poisson ratio $\nu_{p}=0.54\pm0.03$ marginally larger than the limit value of $1/2$, as commented above.
### Overweight experiment
The elastic theory gives qualitatively the same behavior as the experimental results (see Fig. \[fig24\]): the apparent mass $M_{a}$ increases with the filling mass $M_{fill}$, up to a maximum $M_{max}$, then decreases slowly towards the saturation mass $M_{sat}$. Furthermore, all the features predicted by elasticity can be observed: $R^{3}$ rescaling, $M_{max}/M_{sat}$ increase with friction at the walls, and $M_{max}/M_{sat}$ increase with packing fraction (experiment) or Poisson ratio (theory). Note that $M_{max}/M_{sat}$ increases with packing fraction and Poisson ratio are equivalent because of the effective increase of Poisson ratio with packing fraction deduced from the classical Janssen experiment.
However, there is no quantitative agreement between isotropic elasticity predictions and experiments. Fig. \[fig24\]). The elastic curve is very similar to the experimental one, but the experimentally observed maxima are 30 to 40 times larger.
![a: Apparent mass $M_{a}$ vs. filling mass $M_{fill}$, rescaled by the saturation mass $M_{sat} $, for loose packing in medium-rough columns of 3 diameters (38 mm (squares), 56 mm (circles), 80 mm (triangles)) with an overweight equal to $M_{sat}$; the dotted line is the hydrostatic curve. b: Simulation of the experiment of Fig. \[fig24\]a, for an elastic medium characterized by the same saturation mass (Poisson ratio $\nu_{p}=0.46$) and the same friction at the walls ($\mu_{s}=0.25$). Inset: maximum mass $M_{max}$ rescaled by saturation mass $M_{sat}$ vs. static coefficient of friction at the walls $\mu_{s}$, in experiments made on loose (squares) and dense (circles) packing in 38 mm diameter columns, and in simulations for elastic media of Poisson ratios $\nu_{p}=0.45$ (open squares) and $\nu_{p}=0.49$ (open circles); the left vertical scale is used for the experimental data, the right vertical scale is used for the simulation data.[]{data-label="fig24"}](fig24.eps){width="8cm"}
An apparent paradox explained by elasticity
-------------------------------------------
Interestingly, we find in the elastic case the same qualitative phenomenology as in the experiment i.e. the computed values of the overshoot $M\max$ rescaled by the saturation mass $M_{sat}$ increases both with the friction at the walls and the Poisson coefficient (i.e. with the effective Janssen’s parameter). This features reads as a paradox: *the more the weight of the grains is screened by the walls, the less the weight of the overload is screened*, but we can now try to understand it at least qualitatively.
If we impose on the top surface of an elastic medium the asymptotic values for displacements and stresses (see eq. (\[stressasympt\]), (\[depasympt\])), these values then extend to the rest of the column. Thus, with such an overweight, a flat pressure profile along depth z $\sigma_{zz}(r,z)=-\frac
{\rho gR}{2K_{el}\mu_{s}}$ is obtained as in Janssen’s theory. Actually, with the overload, the displacement imposed experimentally on the surface is almost constant: $u_{z}(r)=u_{0}$ since the overweight is much more rigid than the material. Then, as the asymptotic displacement is parabolic, we must have a ”transition regime”, which is at the origin of the overshoot effect.
There are two limits in which this transition can disappear, i.e. when the Poisson coefficient $\nu$ or when friction at the walls $\mu_{s}$ are decreased to zero. Then the parabolic part of the asymptotic displacements (eq. (\[depasympt\])) becomes negligible and therefore, the imposed displacement on the surface is close to the asymptotic value. This results in a decrease of the overshoot amplitude.
Basically, we thus recover in the elastic case, the same paradox as the experimental situation. We now understand it as a consequence of the boundary condition imposed experimentally by the overweight i.e. an almost constant displacement on the surface.
It would be interesting to put overweights of different Young modulus on top of a granular material in order to see if the overshoot amplitude decreases when the overweight Young modulus is decreased.
Pushing experiment
------------------
In this section, we study the elastic theory predictions for the force needed to push an elastic material upwards at constant velocity in a column, and compare it to recent experimental observations [@OvarlezRheo01; @OvarlezRheo03] in the case of a slowly driven granular material. The main features of this analysis have been presented in [@OvarlezRheo03].
For a vertically pushed granular assembly [@OvarlezRheo01; @OvarlezRheo03] at constant velocity, the resistance force $\bar{F}$ increases very rapidly with the packing’s height $H$(see Fig. \[fig25\]).
![Mean resistance force to pushing as a function of the height $H$ of the packing scaled by the column diameter $D$, for 1.58 mm steel beads, of packing fraction 62.5%, in a 36 mm duralumin column, for $V=16\
\mu\mbox{m\,s}^{-1}$ (filled squares) and $V=100\
\mu\mbox{m\,s}^{-1}$ (open circles). The line and the dotted line are fits by eq. (\[jansseninv\]). The dashed line is the hydrostatic curve.[]{data-label="fig25"}](fig25){width="8cm"}
Following the standard Janssen screening picture, this strong resistance to motion is due to the leaning of the granular material on the walls (eq. (\[redirection\])) in association with solid friction at the side walls. At the walls, we suppose a sliding of the granular material at a velocity $V_{0}$ (the driving velocity); the shearing stress is then $$\sigma_{rz}(z)=-\mu_{d}(V_{0})\sigma_{rr}(z)$$ where $\mu_{d}(V_{0})$ is the dynamic coefficient of friction between the beads and the cylinder’s wall at a velocity $V_{0}$.
The force $\bar{F}$ exerted by the grains on the piston can be derived from equilibrium equations for all slices, thus we obtain: $$\bar{F}=\varrho g\lambda\pi R^{2}\times(\exp(\frac{H}{\lambda})-1)
\label{jansseninv}$$ where $\varrho$ is the mass density of the granular material, $R$ is the cylinder radius and $g$ the acceleration of gravity. The length $\lambda
=R/2K\mu_{d}(V_{0})$ is the effective screening length.
We see on Fig. \[fig25\] that the data are well fitted by eq. (\[jansseninv\]). We obtain $K\mu_{d}(V)=0.140\pm0.001$ at $V=100\ \mu \mbox{m\,s} ^{-1}$ and $K\mu_{d}(V)=0.146\pm0.001$ at $V=100\ \mu\mbox{m\,s} ^{-1}$.
In order to get the isotropic homogeneous elasticity prediction for the pushing experiment, we perform a series of numerical simulations using Finite Element Method [@Castem]. The condition $\sigma_{rz}=-\mu_{d}\sigma_{rr}$ is now imposed everywhere at the walls (for the pulling situation, we impose $\sigma_{rz}=+\mu_{d} \sigma_{rr}$). The cylinder is modelled as a duralumin elastic medium. As long as the Young modulus $E$ of the elastic medium is less than $500$ MPa, which is usually the case for granular media, we find no dependence of the results on $E$.
We find no appreciable difference between the elastic prediction (Fig. \[fig26\]) and the curve given by eq. (\[jansseninv\]) with $K=K_{el}$. Therefore, regarding the dependence of the stationary state force $\bar{F}$ on the height of beads, our system cannot be distinguished from an elastic medium.
![Comparison of the resistance force to pushing simulated for a homogeneous isotropic elastic medium (squares) of Poisson ratio $\nu=0.45$ and Young modulus $E=100$ MPa in a duralumin cylinder of radius $R=1.9\ $cm, with coefficient of friction $\mu_{d}=0.2$ at the walls, to the curve obtained with eq. (\[jansseninv\]) with Janssen coefficient $K=K_{el}=\nu/(1-\nu)=0.82$ (line).[]{data-label="fig26"}](fig26.eps){width="8cm"}
Pushing vs. pulling
-------------------
As a check of consistency, we performed the following dynamical experiment in [@OvarlezRheo03]. First, the granular column is pushed upwards in order to mobilize the friction forces downwards and far enough to reach the steady state compacity. Starting from this situation, the friction forces at the walls are reversed by moving the piston downwards at a constant velocity $V_{down}=16\
\mu\mbox{m\,s}^{-1}$, until a stationary regime is attained. Note that this stationary regime is characterized by the same compacity $\overset{\_}{\nu}\approx62.5\%$ as in the pushing situation. In Fig. \[fig27\] the pushing force $\bar{F}$ is measured for different packing heights $H$. The fit of experimental results with eq. (\[janssen\]) gives $K\mu_{d}(16\
\mu\mbox{m\,s}^{-1})=0.156\pm0.002$ which is $10\%$ larger than $K\mu_{d}(16\ \mu\mbox{m\,s}^{-1})$ extracted from the pushing experiment. This difference, though small, can be observed out of uncertainties, and is systematic. It cannot be due to a slight change in compacity $\overset{\_} {\nu}$ as from relation $\Delta
K/K\approx5\Delta\overset{\_}{\nu} /\overset{\_}{\nu}$, we would expect a 2% variation in compacity between the pushing and the pulling experiment, which would be observed; we actually measured $\Delta\overset{\_}{\nu}/\overset{\_}{\nu}=0\pm1\%$. According to Janssen’s picture, this would imply that vertical stress redirection is more efficient in the downward pulling situation. We believe this is a clear evidence of a granular structuring effects but its also shows that this effect is not dominant: it affects only $10\%$ of the average mechanical parameter $K$.
Note that finite element simulations show that the presence of a rigid bottom implies that the effective Janssen’s parameter $K_{eff}$ extracted from Janssen’s rescaling for the pulling situation is higher than $K_{el}$, whereas for the pushing $K_{eff}\approx K_{el}$ (as can be seen on Fig. \[fig26\]: the fit of the elastic curve with $K=K_{el}$ is good). Actually, if we adjust the elastic predictions for pushing and pulling experiments with an elastic material of Poisson coefficient $\nu_{p}=0.45$, eq. (\[jansseninv\]) yields a Janssen’s constant $K_{eff}$ for the pushing which is about $3\%$ lower than $K_{eff}$ for the pulling. This is qualitatively (though not quantitatively) in agreement with the experimental results.
Therefore isotropic elasticity can be a good framework only if we neglect the existence of bulk restructuring effects inducing differences in the effective Poisson coefficient of the material between the pulling and the pushing. Note that in this case, an isotropic modelling of the granular material is somehow questionable.
Towards anisotropy {#anisotropy}
------------------
We have seen in the preceding sections that isotropic elasticity reproduces qualitatively all the features observed in the experiments performed on granular materials. However, some problems remain: a Janssen constant $K$ of order 1.2 was observed experimentally [@OvarlezSurp03] whereas elasticity cannot provide Janssen constants higher than 1.03 with the same experimental parameters (i.e. $\mu_{s}<0.3$); moreover, the amplitude of the overshoot predicted by elasticity is 20 times lower than the one observed experimentally.
That is why we study here the predictions of the simplest extension of isotropic elasticity: transversely isotropic elasticity.
The stress-strain relations are now:
$$\begin{aligned}
\epsilon_{xx} &
=\frac{1}{E_{1}}\;\sigma_{xx}-\frac{\nu_{1}}{E_{1}}
\;\sigma_{yy}-\frac{\nu_{2}}{E_{2}}\;\sigma_{zz}\\
\epsilon_{yy} &
=-\frac{\nu_{1}}{E_{1}}\;\sigma_{xx}+\frac{1}{E_{1}}
\;\sigma_{yy}-\frac{\nu_{2}}{E_{2}}\;\sigma_{zz}\\
\epsilon_{zz} &
=-\frac{\nu_{2}}{E_{2}}\;\sigma_{xx}-\frac{\nu_{2}}{E_{2}
}\;\sigma_{yy}+\frac{1}{E_{2}}\;\sigma_{zz}\\
\epsilon_{yz} & =\frac{1}{2G}\;\sigma_{yz}\\
\epsilon_{xz} & =\frac{1}{2G}\;\sigma_{xz}\\
\epsilon_{xy} & =\frac{(1+\nu_{1})}{E_{1}}\;\sigma_{xy}\end{aligned}$$
Following the same calculation steps as in Sec. \[elastheory\] , one can show that, in the limit of high depths $z$, confinement imposes: $$\sigma_{rr}=\frac{E_{1}}{E_{2}}\frac{\nu_{2}}{1-\nu_{1}}\sigma_{zz}$$ i.e. we recover in the anisotropic case a Janssen-like relation between stresses, with a stress redirection constant $$\begin{aligned}
K_{anis.}=\frac{E_{1}}{E_{2}}\frac{\nu_{2}}{1-\nu_{1}}\end{aligned}$$
The constraints on the elastic parameters are now $$\begin{aligned}
\nu_{1} & >-1\\
\nu_{1} & <\frac{E_{1}}{E_{2}}+1\\
\nu_{2}^{2} & <(1-\nu_{1})\frac{E_{2}}{2E_{1}}\end{aligned}$$
We can see the consequences on $K_{anis.}$ through an example. If $\nu_{1}=0$, then the constraint is $\nu_{2}^{2}<\frac{E_{2}}{2E_{1}}$ which leads to $K_{anis.}<\sqrt{\frac{E_{1}}{2E_{2}}}$. By adjusting the modulus $E_{1}$ and $E_{2}$, one can give any value to $K_{anis.}$ which is not bounded anymore by a maximum value of 1 (the isotropic case). This means that the experimental values found in [@OvarlezSurp03] for dense packing, i.e. a Janssen constant $K\approx1.2$, which could not be reached by the isotropic elastic theory, can be understood in the framework of anisotropic elasticity. Note however, that we have now 5 independent parameters instead of 2.
The problem is now that there are several ways to give $K$ a value by adjusting independently (while satisfying the constraints) 4 parameters $\nu_{1}$, $\nu_{2}$, $E_{1}$, and $E_{2}$; another independent parameter is $G$, which may affect the Janssen profile shape; thus one has to find a physical justification for choosing these values.
We tried to vary independently most parameters while keeping $K$ and $\mu_{s}$ constant in order to see if for a given $M_{sat}$, there is a way to obtain the giant overshoot we observe experimentally.
Several tries are compared with the isotropic case on Fig. \[fig28\]. As far as we could see, there is only a rather small influence of anisotropy on the overshoot amplitude: the deviation from the isotropic value is at the most 15% for reasonable values of the modulus. Nevertheless, there are some general tendencies: the higher the shear modulus $G$, the smaller the overshoot amplitude is; the higher $\nu_{1}$, the higher the amplitude is, whatever the stiffer direction may be. However, the effect is far from sufficient to reproduce experimental results.
![Apparent mass $M_{a}$ vs. filling mass $M_{fill}$, rescaled by the saturation mass $M_{sat}$, for various transversely isotropic elastic media characterized by the same saturation mass $M_{sat}$. Squares: $G=0.345$ MPa, $E_{1}=E_{2}=1$ MPa, $\nu_{1}=\nu_{2}=0.45$ (isotropic medium). Open circles: $G=0.1$ MPa, $E_{1}=E_{2}=1$ MPa, $\nu_{1}=\nu_{2}=0.45$. Triangles: $G=1$ MPa, $E_{1}=E_{2}=1$ MPa, $\nu_{1}=\nu_{2}=0.45$. Open down triangles: $G=0.345$ MPa, $E_{1}=2$ MPa, $E_{2}=1$ MPa, $\nu_{1}=\nu_{2}=0.29$. Diamonds: $G=0.345$ MPa, $E_{1}=2$ MPa, $E_{2}=1$ MPa, $\nu_{1}=-0.34$, $\nu_{2}=0.55$. Open left triangles: $G=0.345$ MPa, $E_{1}=2$ MPa, $E_{2}=1$ MPa, $\nu_{1}=0.756$, $\nu_{2}=0.1$. Right triangles: $G=0.345$ MPa, $E_{1}=1$ MPa, $E_{2}=2$ MPa, $\nu_{1}=0.633$, $\nu_{2}=0.6$. Open stars: $G=0.345$ MPa, $E_{1}=1$ MPa, $E_{2}=2$ MPa, $\nu_{1}=0.815$, $\nu_{2}=0.3$. In all these simulations, the friction at the walls is $\mu_{s}=0.25$.[]{data-label="fig28"}](fig28.eps){width="8cm"}
A stress induced anisotropy? {#anisotropy}
----------------------------
In order to account for the height of the experimentally observed overshoot, we propose a toy model based on the idea of stress-induced anisotropy. It is possible that the overweight induces locally a change in the structure. We would expect an increase of the number of contacts i.e. of the young modulus $E_{2}$ in the vertical direction, if the granular material can be modelled as an effective elastic medium. From the relation $K_{anis.}=\frac{E_{1}}{E_{2} }\frac{\nu_{2}}{1-\nu_{1}}$, we then expect the Janssen coefficient to be lower near the overweight than in the bulk.
Let the Janssen constant be $K_{1}$ near the overweight, down to a depth $H_{1}$, and $K_{2}>K_{1}$ for any depth $z>H_{1}$. The saturation mass, which is the overweight mass, was measured with an homogeneous material of Janssen constant $K_{2}$ and is thus $M_{sat_{2}}=\frac{\rho\pi R^{3}}{2K_{2}\mu_{s}} $. Then, it underestimates the saturation mass $M_{sat_{1}}=\frac{\rho\pi
R^{3}}{2K_{1}\mu_{s}}$ of the material layer near the overweight: the measured mass $M_{a}$ then first increases with the filling mass $M_{fill}$ and would naturally tend to a higher value $M_{sat_{1}}$. But for a depth $H_{1}$ the material’s structure is no more affected by the overweight and is now characterized by a Janssen constant $K_{2}$: the mass imposed on the material 2 by the material 1 is then higher than its saturation mass $M_{sat_{2}}$, and the measured mass $M_{a}$ has to decrease with the filling mass $M_{fill}$ in order to reach its saturation value $M_{sat_{2}}$.
This simple idea can be easily formalized for an ideal Janssen material. The equilibrium equation on the vertical stress $\sigma_{zz}(z)$ in material $i$ characterized by Janssen constant $K_{i}$ reads $$\begin{aligned}
\frac{\partial\sigma_{zz}}{\partial z}+\frac{2K_{i}\mu_{s}}{R}\sigma
_{zz}=-\rho g\end{aligned}$$
In the material 1, for $0<z<H_{1}$, we solve this equation for the mass $M(z)=\sigma_{zz}(z)/(\pi R^{2}g)$ weighted at depth $z$, with the boundary condition $M(z\!=\!0)=M_{sat_{2}}$ and get $$\begin{aligned}
M(z)=M_{sat_{1}}\biggl(1-\exp\Bigl(-2K_{1}\mu_{s}\frac{z}{R}\Bigr)\biggr
)+M_{sat_{2}}\exp\Bigl(-2K_{1}\mu_{s}\frac{z}{R}\Bigr)\end{aligned}$$
In the material 2, for $H_{1}<z$ we solve this equation with the boundary condition $M(z\!=\!H_{1})=M_{sat_{1}}\bigl(1-\exp(-2K_{1}\mu_{s}\frac{H_{1}
}{R})\bigr)+M_{sat_{2}}\exp(-2K_{1}\mu_{s}\frac{H_{1}}{R})$ and get $$\begin{aligned}
M(z)=M_{sat_{2}}+(M_{sat_{1}}-M_{sat_{2}})\biggl(1-\exp\Bigl(-2K_{1}\mu
_{s}\frac{H_{1}}{R}\Bigr)\biggr)\exp\Bigl(-2K_{2}\mu_{s}\frac{(z-H_{1})}
{R}\Bigr)\end{aligned}$$
These equations can be rewritten using $M_{a}/M_{sat_{2}}$ and $M_{fill} /M_{sat_{2}}$ as variables: $$\begin{aligned}
0<z<H_{1} &
\rightarrow\frac{M_{a}}{M_{sat_{2}}}=\frac{M_{sat_{1}}
}{M_{sat_{2}}}\biggl(1-\exp\Bigl(-\frac{M_{fill}}{M_{sat_{2}}}\frac
{M_{sat_{2}}}{M_{sat_{1}}}\Bigr)\biggr)+\exp\Bigl(-\frac{M_{fill}}{M_{sat_{2}
}}\frac{M_{sat_{2}}}{M_{sat_{1}}}\Bigr)\\
H_{1}<z &
\rightarrow\frac{M_{a}}{M_{sat_{2}}}=1+\bigl(\frac{M_{sat_{1}}
}{M_{sat_{2}}}-1\bigr)\biggl(1-\exp\Bigl(-\frac{H_{1}}{\lambda_{2}}
\frac{M_{sat_{2}}}{M_{sat_{1}}}\Bigr)\biggr)\exp\Bigl(-\bigl(\frac{M_{fill}
}{M_{sat_{2}}}-\frac{H_{1}}{\lambda_{2}}\bigr)\Bigr)\end{aligned}$$ The two independent variables one can adjust are $\frac{M_{sat_{1}} }{M_{sat_{2}}}$ (which is the Janssen constant ratio $K_{2}/K_{1}$), and the ratio $H_{1}/\lambda_{2}=M_{fill}(H_{1})/M_{sat_{2}}$ where $\lambda_{2}
=\rho\pi R/(2K_{2}\mu_{s})$ is the Janssen screening length in the medium 2.
On a $M_{a}/M_{sat_{2}}=f(M_{fill}/M_{sat_{2}})$ plot, the parameter $H_{1}/\lambda_{2}$ is the X-axis value $M_{fill}/M_{sat_{2}}$ at which the weighted mass starts to decrease. The slope at the origin is $1-\frac
{M_{sat_{2}}}{M_{sat_{1}}}$. Typically (see Fig. \[fig29\] and Fig. \[fig30\]), the experimental slopes are of order 0.5 (it would be 1 for an hydrostatic pressure), i.e. the Janssen coefficient $K_{1}$ has to be about 2 times $K_{2}$; in the anisotropic elasticity framework, it would mean that the young modulus in the vertical direction is doubled by the overweight. The order of the extension $H_{1}/\lambda_{2}$ of the induced anisotropy is of order 0.4. The experimental data for dense and loose packing are compared on Fig. \[fig29\] and Fig. \[fig30\] with the model. It appears to reproduce correctly the data; however, the parameters cannot be determined accurately as they are obtained from the very beginning of the data (the slope at the origin and the X-axis value of the maximum) which cannot be measured with a high precision. Therefore, the parameters presented in these figures are just given as examples.
![Apparent mass $M_{a}$ vs. filling mass $M_{fill}$, rescaled by the saturation mass $M_{sat}$, for dense packing in medium-rough columns of 3 diameters (38 mm (squares), 56 mm (circles), 80 mm (triangles)) with an overweight equal to $M_{sat}$. The data are compared with our inhomogeneous Janssen model with $K_{1}/K_{2}=2.4$, and $H_{1}/\lambda_{2}=0.45$.[]{data-label="fig29"}](fig29){width="8cm"}
![Apparent mass $M_{a}$ vs. filling mass $M_{fill}$, rescaled by the saturation mass $M_{sat}$, for loose packing in medium-rough columns of 3 diameters (38 mm (squares), 56 mm (circles), 80 mm (triangles)) with an overweight equal to $M_{sat}$. The data are compared with our inhomogeneous Janssen model with $K_{1}/K_{2}=2.8$, and $H_{1}/\lambda_{2}=0.3$. []{data-label="fig30"}](fig30){width="8cm"}
Conclusion
==========
In conclusion, we performed an extensive study of the Janssen’s column problem. It is a mixture of numerical studies both in 2D and in 3D, in the case of frictional boundaries. The aim was to test thoroughly the classical and celebrated Janssen’s analysis in the context of an effective homogeneous elastic material and provide some meaning to the effective Janssen’s constant of stress redirection at it can be obtained experimentally. Note that here we do not make any assumption on a plastic threshold in the bulk as it is usually considered to provide bounds on the Janssen’s constant (active and passive limits). Also this analysis was performed in the context of an extensive experimental work where preparation was varied and were special care about friction at the wall was taken to be able to establish a precise comparison with theoretical modelling. Interestingly, we find that the Janssen’s approach is fully valid in the limit of low friction coefficients between the grains and the wall (up to a moderate value of 0.5). It means that an exponential saturation curve of the average normal stress at the bottom is an excellent approximation, which defines precisely an effective Janssen’s constant solely dependent on the Poisson ratio. We also derive ion the limit of an infinite column a value for an elastic Janssen’s constant ($K_{el}=\nu$ in 2D and $K_{el}=\nu/(1-\nu)$ in 3D). For a finite size column, the presence of a bottom diminish the average vertical stress such as to yield an effective Janssen’s constant if the bottom normal stress is measured with a value directly related to the elastic constant $K_{el}$. Therefore, in this context experiment data can be matched with isotropic elasticity if the packing fraction representing the preparation can be associated with an effective Poisson ratio.
Consequently, there is a need to provide a more strained test to the isotropic elastic theory. This was done experimentally by imposing on the top of the column an overweight equal to the saturation stress [@OvarlezSurp03] and here we propose the same test in the same conditions for an elastic material. The numerical simulations show that stresses at the bottom also exhibits an overshoot when the column depth is increased. The relative value of the overshoot is at most 7% . We propose a scaling relation of the overshoot amplitude when the wall fiction is varied. This result contrasts with a Janssen’s analysis where a flat profile should be observed. This overshoot effect is related to the deformation of the elastic medium below the overweight which sets a length where the deformation profile is not parabolic any more as in the rest (more like a flat profile as it is imposed by the overweight boundary condition). But at the quantitative level, the overshoot effect found experimentally has an amplitude about 20 to 30 times larger! This lack of quantitative agreement opens new questions on the modification on the influence of the medium due to the overload. This is the reason why we push further the investigation in the context of anisotropic elasticity. We consider an orthotropic elastic medium with the main stiff direction along the vertical. In this case the effective Janssen’s redirection coefficient can be changed according to the stiffness ratio but the overshoot test does not produce an overshoot value significantly larger than the isotropic situation.
Finally, we propose a qualitative model based on an extension of the Janssen’s approach where we assume that the overload has changed the medium within a given depth such as to yield a smaller Janssen’s constant. This would be consistent with the onset of strain induced anisotropy producing a stiffer medium in the vertical direction. The agreement of this simple model (with two fitting parameters) is satisfactory but more importantly it raises interesting questions and calls for new experimental work. In this frame of mind it would be very interesting to see differences of the stress saturation curve for two media prepared i) with a regular rain like pouring as before and ii) a rain like deposition process but where an overweight is imposed above each deposition step (a deposition step being of a height much smaller than the final height). In this case the stress induced texture changes could provide a rational explanation to the extension of the Janssen’s model that fits correctly the overweight experiments.
We thank Profs. R.P. Behringer and J. Socolar for many fruitful discussions.
[99]{} *Physics of Dry Granular Media*, ed. by H.J. Herrmann, J.-P. Hovi and S. Luding, Kluwer Acad. Publisher (1998).
M.Oda, Mechanics of Materials **16**, 35 (1993).
F. Radjai, D.E. Wolf, M. Jean and J.J. Moreau, Phys. Rev. Lett. **80**, 61 (1998).
J. Geng, G. Reydellet, E. Clément and R.P. Behringer, Physica D **182**, 274 (2003).
APF Attman, P. Brunet, J. Geng, G. Reydellet, G. Combe, P. Claudin, R.P. Behringer, E. Clément, *to be published in* J.Cond Mat A (2005); Preprint Cond-mat/0411734.
A.J. Liu, S.R. Nagel, Nature 396, 21 (1998).
M.E. Cates, J.P. Wittmer, J.-P. Bouchaud and P. Claudin, Phys. Rev. Lett. **81**, 1841 (1998).
D.M. Wood, *Soil Behaviour and Critical State Soil Mechanics* (Cambridge University, Cambridge, England, 1990).
L. Vanel et al., Phys. Rev. E **60**, R5040 (1999).
J.P. Wittmer, M.E. Cates, P. Claudin, J. Phys. I 7, 39 (1997); J.P. Wittmer et al., Nature **382** , 336 (1996).
G. Reydellet, E. Clément, Phys. Rev. Lett. **86**, 3308 (2001); J. Geng et al., Phys. Rev. Lett. **87**, 035506 (2001); D. Serero et al., Eur. Phys. J. E, **6**, 169 (2001).
H.A. Jannsen, Zeitschr. D. Vereines Deutscher Inginieure **39**, 1045 (1895).
L. Vanel, E. Clément, Eur. Phys. J. B **11**, 525 (1999). L. Vanel et al., Phys. Rev. Lett. **84**, 1439 (2000).
G. Ovarlez, E. Kolb, E. Clément, Phys. Rev. E **64**, 060302(R) (2001).
G. Ovarlez, E. Clément, Phys. Rev. E **68**, 031302 (2003).
G. Ovarlez, C. Fond, E. Clément, Phys. Rev. E **67**, 060302(R) (2003).
E. Kolb, T. Mazozi, E. Clement, and J. Duran, Eur. Phys. J. B, **8**, 483 (1999).
Y. Bertho, F. Giorgiutti-Dauphine, and J.-P. Hulin, Phys. Rev.Lett. **90**, 144301 (2002).
D. Arroyo-Cetto, G. Pulos, R. Zenit and M. A. Jimenez-Zapata, C. R. Wassgren, Phys. Rev. E **68**, 051301 (2003)
J.W. Landry and G.S. Grest, Phys. Rev. E **69**, 031303, (2004).
I. Bratberg, F. Radjai, and A. Hansen, *to be published in* Phys. Rev. E.
D.Senis, C.Allain, Phys. Rev. E **55**, 7797 (1997).
P. Evesque, P.-G. de Gennes, C. R. Acad. Sci. Paris, **326** IIb, 761 (1998).
with CAST3M, http://www-cast3m.cea.fr
C. Goldenberg, I. Goldhirsch, Phys. Rev. Lett. 89, 084302 (2002).
[^1]: author to whom correspondence should be addressed: ovarlez@lcpc.fr
|
---
abstract: |
We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0<s<1$ and $m>0$; on the other hand, it also generates an $L^1$-contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity.
After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.
author:
- |
[Juan Luis Vázquez]{}\
Universidad Autónoma de Madrid, Spain
date: November 2013
title: |
**Recent progress in the theory of Nonlinear\
Diffusion with Fractional Laplacian Operators**
---
Introduction {#sec.intro}
============
This is a follow-up to a previous survey by the author [@VazAbel], which reported on recent research on two models of nonlinear diffusion processes involving long-range diffusion, written on the occasion of the Abel Symposium held in Oslo in 2010. Such evolution processes are represented by nonlinear parabolic equations involving nonlocal operators of the fractional Laplacian type.
Recapitulating what was said there, the classical theory of diffusion is expressed mathematically by means of the heat equation, and more generally by parabolic equations of linear type; it has had an enormous success and is now a foundation stone in science and technology. The last half of the past century has witnessed intense activity and progress in the theories of nonlinear diffusion, examples being the Stefan Problem, the Porous Medium Equation, the $p$-Laplacian equation, the Total Variation Flow, evolution problems of Hele-Shaw type, the Keller-Segel chemotaxis system, and many others. Reaction diffusion has also attracted considerable attention. In the last decade there has been a surge of activity focused on the use of so-called fractional diffusion operators to replace the standard Laplace operator (and other kinds of elliptic operators with variable coefficients), with the aim of further extending the theory by taking into account the presence of the long range interactions that occur in a number applications. The new operators do not act by pointwise differentiation but by a global integration with respect to a singular kernel; in that way the nonlocal character of the process is expressed.
Fractional operators {#subs.frop}
--------------------
Though there is a wide class of interesting nonlocal operators under scrutiny, a substantial part of the current work deals diffusion modeled by the so-called fractional Laplacians. We recall that the fractional Laplacian operator is a kind of isotropic differentiation operator of order $2s$, for some $s\in (0,1)$, that can be conveniently defined through its Fourier Transform symbol, which is $|\xi|^{2s}$. Thus, if $g$ is a function in the Schwartz class in ${\mathbb{R}^N}$, $N\ge 1$, we write $(-\Delta)^{s} g=h$ if $$\widehat{h}\,(\xi)=|\xi|^{2s} \,\widehat{g}(\xi)\,,
\label{def-fourier}$$ so that for $s=1$ we recover the standard Laplacian. This definition allows for a wider range of parameters $s$. The interval of interest for fractional diffusion is $0<s\le 1$, and for $s< 1$ we can also use the integral representation $$\label{def-riesz}
(-\Delta)^{s} g(x)= C_{N,2s }\mbox{
P.V.}\int_{\mathbb{R}^N} \frac{g(x)-g(z)}{|x-z|^{N+{2s}
}}\,dz,$$ where P.V. stands for principal value and $C_{N,\sigma}$ is a normalization constant, with precise value $C_{N,2s}=2^{2s}s\Gamma((N+2s)/2)/( \pi^{N/2}\Gamma(1-s))$. In the limits $s\to 0$ and $s\to 1$ it is possible to recover respectively the identity or the standard minus Laplacian, $-\Delta$, cf. [@Brezis; @Mazya]. Remarkably, the latter one cannot be represented by a nonlocal formula of the type . It is also useful to recall that the operators $(-\Delta)^{-s}$, $0< s<1$, inverse of the former ones, are given by standard convolution expressions: $$(-\Delta)^{-s} g(x)= C_{N,-2s }\int_{\mathbb{R}^N}
\frac{g(z)}{|x-z|^{N-{2s} }}\,dz, \label{def-riesz2}$$ in terms of the usual Riesz potentials. Basic references for these operators are the books by Landkof [@Landkof] and Stein [@Stein]. A word of caution: in the literature we often find the notation $\sigma=2s$, and then the desired interval is $0<\sigma<2$. According to that practice, we will sometimes use $\sigma$ instead of $s$.
Another option is to use the classical way of defining the fractional powers of a linear self-adjoint nonnegative operator, and it is expressed in terms of the semigroup associated to such an operator. In the case of the standard Laplacian operator, it reads $$\label{sLapl.Rd.Semigroup}
\displaystyle(-\Delta)^{s}
g(x)=\frac1{\Gamma(-s)}\int_0^\infty
\left(e^{t\Delta}g(x)-g(x)\right)\frac{dt}{t^{1+s}}.$$ All the above definitions are equivalent when dealing with the Laplacian on the whole space ${\mathbb{R}}^N$.
The interest in these fractional operators has a long history in Probability for reasons we explain in the next paragraph. Motivation from Mechanics appears in the famous Signorini problem (with $\alpha=1/2$), cf. [@Signor; @CaffSign]. And there are applications in Fluid Mechanics, cf. [@CaffVass; @KisNazVol]. There is a wide literature on the subject, both for its relevance to Analysis, PDEs, Potential Theory, Stochastic Processes, and Finance, and for the growing number of practical applications. See e.g. [@Caffarelli-Silvestre; @DPV11; @Valdinoc; @VazAbel] where further references can be found.
The systematic study of the corresponding PDE models with fractional operators is relatively recent, and many of the results have been established in the last decade. A part of the current research concerns linear or quasilinear equations of elliptic type. This is a huge subject with well-known classical references that will not be discussed here.
**Linear evolution processes**
------------------------------
A significant part of the motivation and also a large part of the recent literature is related to evolution problems, that we will discuss in the sequel. Thus, the difference between the standard and the fractional Laplacian is best seen in the stochastic point of view (the stochastic theory of diffusion), and it consists in re-examining the conditions under which the Brownian motion is derived, and taking into account long-range interactions instead of the usual interaction driven by close neighbors. This change of model explains characteristic new features of great importance, like enhanced propagation with the appearance of fat tails at long distances (such tails are to be compared with the typical exponentially small tails of the standard diffusion, or the compactly supported solutions of porous medium flows). Moreover, the space scale of the propagation of the distribution is not proportional to $t^{1/2}$ as in the Brownian motion, but to another power of time, that can be adjusted in the model; this is known as [*anomalous diffusion.*]{} The fractional Laplacian operators of the form $(-\Delta)^{\sigma/2}$, $\sigma\in(0,2)$, are actually the infinitesimal generators of stable Lévy processes [@Applebaum; @Bertoin; @CKS2010].
We will be concerned with evolution partial differential equations that combine diffusion and fractional operators. As already mentioned, a great variety of diffusive problems in nature, namely those referred to as normal diffusion, are satisfactorily described by the classical Heat Equation or Fokker-Planck linear equation. However, anomalous diffusion is nowadays intensively studied, both theoretically and experimentally since it conveniently explains a number of phenomena in several areas of physics, finance, biology, ecology, geophysics, and many others, which can be briefly summarized as having non-Brownian scaling. This leads to linear anomalous diffusion equations, see e.g. [@Applebaum; @ContTankov2004; @Woy2001]. Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type represent a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion. These fractional equations are usually derived asymptotically from basic random walk models, cf. [@Jara0; @JKOlla; @MK2000; @MMM; @Valdinoc; @VIKH; @WZ] and their references.
The standard linear evolution equation involving fractional diffusion is $$\label{linfraceq}
\frac{\partial u}{\partial t}+(-\Delta)^{s}(u)=0\,,$$ a usual model for anomalous diffusion. The equation is solved with the aid of well-known functional analysis tools, like Fourier transform; for instance, it is proved that it generates a semigroup of ordered contractions in $L^1(\mathbb{R}^N)$. Moreover, in this setting it has the integral representation $$u(x,t)=\int_{\mathbb{R}^N}K_s(x-z,t)f(z)\,dz\,,
\label{lineal}$$ where $K_s$ has Fourier transform $\widehat K_s(\xi,t)=e^{-|\xi|^{2s} t}$. This means that, for $0<s<1$, the kernel $K_s$ has the form $$K_s(x,t)=t^{-N/2s}F(|x|/t^{1/2s})$$ for some profile function $F=F_s$ that is positive and decreasing, and it behaves at infinity like $F(r)\sim r^{-(N+2s)}$, [@Blumenthal-Getoor]. When $s=1/2$, $F$ is explicit: $$\label{kernel.lin}
F_{1/2}(r)={C}\,(a^2+r^2)^{-(N+1)/2}\,.$$ If $s=1$ the function $K_{s=1}$ is the Gaussian heat kernel, which has a negative square exponential tail, i.e., a completely different asymptotic behavior.
An integral representation of the evolution of the form is not available in the nonlinear models coming from the applications. This implies the need for new methods, thus motivating our work to be described below.
Nonlinear diffusion models {#sec.nd}
==========================
A main feature of current research in the area of PDEs is the interest in nonlinear equations and systems. There are a number of models of evolution equations that can be considered nonlinear counterparts of the linear fractional heat equation, and combine Laplace operators and nonlinearities in different ways. Let us mention some of the most popular in the recent PDE literature.
$\bullet$ [**Type I.**]{} A natural option is to consider the equation $$\label{fpme}
\partial_t u +(-\Delta)^{s}(u^m)=0$$ with $0<s<1$ and $m>0$. This is mathematically the fractional version of the standard Porous Medium Equation (PME) $$\partial_t u =\Delta(u^m)\,,$$ that is recovered as the limit $s\to 1$ and has been extensively studied, cf. [@ArBk86; @Vapme]. We will call equation the [*Fractional Porous Medium Equation, FPME,*]{} as proposed in our first works on the subject [@pqrv1; @pqrv2], in collaboration with de A. de Pablo, F. Quirós and A. Rodríguez.
Interest in studying the nonlinear model we propose is two-fold: on the one hand, experts in the mathematics of diffusion want to understand the combination of fractional operators with porous medium type propagation (which is described as degenerate parabolic). Models of this kind arise in statistical mechanics when modeling for instance heat conduction with anomalous properties. On the other hand, it is mentioned in heat control problems by [@ACld]. The rigorous study of such nonlinear models has been delayed until this time by the mathematical difficulties in treating at the same time the nonlinearity and fractional diffusion.
We will devote most of this paper to explain the main results obtained so far on the mathematical theory of this equation. Thus, Section \[sec.fpme\] contains the theory of existence, uniqueness and continuous dependence of the Cauchy problem posed in the whole space ${\mathbb{R}^N}$, developed in [@pqrv1; @pqrv2]. A main result is the property of infinite speed of propagation. Let us explain the result at this moment and point out the contrast to the PME. While one of the best known features of the PME for $m>1$ is the fact that compactly supported nonnegative initial data give rise to solutions that are also compactly supported as functions of $x$ for every fixed $t>0$, this property does not hold in equation if $s<1$. Indeed, the unique nonnegative solutions we have constructed for nontrivial data $u(x,0)\in L^1_+({\mathbb{R}^N})$ are positive everywhere for all $t>0$. This happens for all $m>0$ and all $0<s<1$ (for $m$ close to 0 attention must be paid to extinction of the solution in finite time, but that is another issue, see below). Some traits are common with the PME equation: an $L^1$-contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity.
The further investigation of the FPME has taken a number of directions. Thus, in Section \[sec.bar\] we discuss the existence of self-similar solutions with conserved finite mass (we call them [*fractional Barenblatt solutions*]{}), following the study made in [@VazBar2012]. The main properties are derived, allowing to construct the self-similar profiles. Generally, such profiles are not explicit. These solutions are then used to explain the asymptotic behavior of the whole class of nonnegative solutions with finite mass (a kind of fractional central limit theorem).
The study of a priori estimates (some of them universal), quantitative bounds on positivity and Harnack estimates is done in Section \[sec.mb\] following the results obtained in [@BV2012] in collaboration with M. Bonforte. This is done in the favorite setup, the Cauchy problem. But the fractional Laplace operators offer many novelties when posed in a bounded domain of the Euclidean space, say with Dirichlet boundary conditions. We report in Section \[sec.dir\] on recent work done with M. Bonforte on the issue and comment on related works.
As a natural extension of the equation, we have recently considered equations of the form $$\label{gfpme}
\partial_t u +(-\Delta)^{s}\Phi(u)=0$$ where $\Phi$ is a monotone increasing function. Collaboration with Pablo, Quirós and Rodríguez, [@pqrv3] and [@vpqr]; regularity is a main issue, we review the results in Section \[sec.gnl\]. Section \[sec.symm\] reports on the technique of Schwarz symmetrization of the solutions of the these problems and derives a priori estimates. This is the result of collaboration with B. Volzone [@VazVol; @VazVol2]. Finally, numerical methods for these equations are being investigated by our team and a number of authors, and we will make a comment on that issue.
$\bullet$ The combination of fractional diffusion with other effects leads to interesting behaviour. One of the most common combinations is reaction-diffusion. In that sense there has been interesting work on the possible extension of the well-known KPP behaviour to cover fractional diffusion, both in the linear and nonlinear case. Results by X. Cabré and J. M. Roquejoffre [@CabreRoquejoffre2] (for linear diffusion) and Diana Stan and the author [@StanVazquezKPP] (in the nonlinear case) are covered in Section \[sec.kpp\].
On the other hand, the combination of fractional diffusion convection appears in a number of models of the so-called geostrophic equations, like $$\partial_t u + {\bf v}\cdot \nabla u =(-\Delta)^s u,$$ where $0<s<1$ and $\bf v(x,t)$ is a divergence free vector field related to $u$ in different ways. Important works in this direction related to our research are due to Caffarelli and Vasseur [@CaffVass] and Kiselev et al. [@KisNazVol], but again, we will not deal with this topic here. This last model uses linear fractional diffusion. Nonlinear diffusion combined with convection has been studied by some authors, like Cifani and Jakobsen [@CJ11], where references to other models are given.
$\bullet$ [**Type II.**]{} We recall here the other model of diffusion with fractional operators where the author has been involved, mainly in collaboration with Luis Caffarelli. Since it was extensively reported in the survey paper [@VazAbel] we will only give a short reminder. This alternative model is derived in a more classical way from the Porous Medium Equation since it is based on the usual Darcy law, with the novelty that the pressure is related to the density by an inverse fractional Laplacian operator. This nonlinear fractional diffusion equation of porous medium type takes the form $$\label{PMEfp}
u_t=\nabla (u\, \nabla {\mathcal K} u),$$ where $\mathcal K$ is the Riesz operator that typically expresses the inverse to the fractional Laplacian, ${\mathcal K} u=(-\Delta)^{-s}u$. This has been studied by Caffarelli and the author in [@CV1; @CV2] and Biler, Karch and Monneau [@BKM; @BIK], where the equation is derived in the framework of the theory of dislocations. We recall that this model has some strikingly different properties, like lack of strict positivity and occurrence of free boundaries. Self-similar solutions also exist but their existence and properties are quite different from those of equation . Even the asymptotic behaviour is quite different. In order to distinguish both models the name [*porous medium equation with fractional pressure*]{} has been proposed for equation .
As for recent work, the boundedness and Hölder regularity of finite mass solutions is studied in joint work with L. Caffarelli and F. Soria, [@CSV]. As a limit case of this second model $s\to1$, one obtains a variant of the equation for the evolution of vortices in superconductivity derived heuristically by Chapman-Rubinstein-Schatzman [@CRS] and W. E [@WE] as the hydrodynamic limit of Ginzburg Landau, and studied by Lin and Zhang [@Liz], and Ambrosio and Serfaty [@AmSr]. The understanding of this limit has been done in collaboration with Sylvia Serfaty [@SerVaz], and is related to work by Bertozzi et al. on aggregation models [@Bertozzi1; @BLL].
The more general equation $u_t=\nabla (u\, \nabla {\mathcal K} (u^{m-1}))$, $m>0$, is considered in [@BIK2013], and self-similar solutions are found in explicit form, a very interesting fact. A very natural extension of these equations is $$u_t=\nabla (u^{m-1}\, \nabla {\mathcal K} (u^p)),\qquad$$ where we may take for simplicity $u\ge 0$, $m>1$ and $p>0$. We are working in such generalizations, cf. [@StanTesoVazquez] where the property of finite speed of propagation is examined.
The study of the fine asymptotic behavior, i.e., obtaining rates of convergence is not easy. Progress on this issue is under way in collaboration with Carrillo and Huang [@CH2013].
$\bullet$ [**Disclaimer.**]{} Since our objective is limited, we are leaving completely outside of the presentation many related topics. Thus, the anomalous diffusion that is encountered in many applications is often described by means of [*fractional time operators*]{}, using the so-called Riemann-Liouville derivatives. There is a huge literature on the subject in the case of linear equations. We will not enter into such topic in this document, though some nonlinear models are quite appealing.
The fractional porous medium equation {#sec.fpme}
=====================================
We now turn our attention to the nonlinear heat equation with fractional diffusion . Indeed, it is a whole family of equations with exponents $s\in (0,1)$ and $m>0$. They can be seen as fractional-diffusion versions of the PME described above, [@Vapme], [@JLVSmoothing]. The classical Heat Equation is recovered in this model in the limit $s=1$ when $m=1$, the PME when $m>1$, the Fast Diffusion Equation when $m<1$.
Some applied literature {#App.Motiv}
-----------------------
We gather here some updated information on the occurrence of the nonlinear fractional diffusion equation we propose and related models in the physical or probabilistic literature.
$\bullet$ Anomalous diffusion often takes a nonlinear form. To be more specific, there exist many phenomena in nature where, as time goes on, a crossover is observed between different diffusion regimes. Tsallis et al. [@BGT2000; @LMT2003] discuss the following cases: (i) a mixture of the porous medium equation, which is connected with non-extensive statistical mechanics, with the normal diffusion equation; (ii) a mixture of the fractional time derivative and normal diffusion equations; (iii) a mixture of the fractional space derivative, which is related with Lévy flights, and normal diffusion equations. In all three cases a crossover is obtained between anomalous and normal diffusions. This leads to models of nonlinear diffusion of porous medium or fast diffusion types with standard or fractional Laplace operators, cf. equation (4) of [@BGT2000].
$\bullet$ There have been many studies of hydrodynamic limits of interacting particle systems with long-range dynamics, which lead to fractional diffusion equations of our type, mainly linear like in [@JKOlla], but also nonlinear in the recent literature, cf. the works [@Jara1], [@Jara2]. Thus, in the last reference, Jara and co-authors study the non-equilibrium functional central limit theorem for the position of a tagged particle in a mean-zero one-dimensional zero-range process. The asymptotic behavior of the particle is described by a stochastic differential equation governed by the solution of the following nonlinear hydrodynamic (PDE) equation, $\partial_t \rho = a^2 \partial^2_x \Phi(\rho)$. When $\Phi$ is a power we recover equation .
$\bullet$ Equations like the last one (in several space dimensions) occur in boundary heat control, as already mentioned by Athanasopoulos and Caffarelli [@AC09], where they refer to the model formulated in the book by Duvaut and Lions [@DL1972], and use the so-called Caffarelli-Silvestre extension [@Caffarelli-Silvestre].
Mathematical problem and general notions {#sect-intro}
----------------------------------------
Let us present the main features and results in the theory we have developed. To be specific, the theory of existence and uniqueness as well the main properties are first studied by De Pablo, Quirós, Rodríguez, and Vázquez in [@pqrv1; @pqrv2] for the Cauchy problem $$\label{eq:main}
\left\{
\begin{array}{ll}
\dfrac{\partial u}{\partial t} + (-\Delta)^{\sigma/2}
(|u|^{m-1}u)=0, & \qquad x\in{\mathbb{R}^N},\; t>0,
\\ [4mm]
u(x,0) = f(x), & \qquad x\in{\mathbb{R}^N}.\end{array}
\right.$$ We have put $\sigma=2s$. The notation $|u|^{m-1}u$ is used to allow for solutions of two signs, but since it is a bit awkward we will often write $u^m$ instead of $|u|^{m-1}u$ even for signed solutions when no confusion is feared; the same happens with the power $u^{1/m}$. We take initial data $f\in L^1({\mathbb{R}^N})$, which is a standard assumption in diffusion problems on physical grounds. As for the exponents, we consider the whole fractional exponent range $0<\sigma<2$, and take porous medium exponent $m>0$. As we have said, in the limit $\sigma\to 2$ we want to recover the standard Porous Medium Equation (PME) $u_t-\Delta (|u|^{m-1}u)=0.$
The two papers contain a rather complete analysis of the problem. A semigroup of weak energy solutions is constructed for every choice of $m$ and $\sigma$, both the $L^\infty$ smoothing effect and $C^\alpha$ regularity work in most cases (with some restrictions if $m$ is near 0), and there is infinite propagation for all $m>0$ and $\sigma<1$. The results can be viewed as a nonlinear interpolation between the extreme cases $\sigma=2$: $u_t -\Delta(|u|^{m-1}u)=0$, and $\sigma=0$ which turns out to be a simple ODE: $u_t+|u|^{m-1}u=0$. It is to be noted that the critical exponent $m_c:= (N-\sigma)_+/N$ plays a role in the qualitative theory: the properties of the semigroup are more familiar when $m>m_c$. A similar exponent is well-known in the Fast Diffusion theory (putting $\sigma=2$). Note that such exponent is not considered when $N=1$ and $\sigma\ge 1$.
[**Preliminary notions.** ]{} If $\psi$ and $\varphi$ belong to the Schwartz class, the definition of the fractional Laplacian together with Plancherel’s theorem yield $$\int_{{\mathbb{R}^N}}(-\Delta)^{\sigma/2}\psi\,\varphi=\int_{{\mathbb{R}^N}}|\xi|^\sigma
\widehat\psi\,\widehat\varphi=\int_{{\mathbb{R}^N}}|\xi|^{\sigma/2} \widehat
\psi|\xi|^{\sigma/2}\,\widehat
\varphi=\int_{{\mathbb{R}^N}}(-\Delta)^{\sigma/4}\psi\,(-\Delta)^{\sigma/4}\varphi.$$ Therefore, if we multiply the equation in by a test function $\varphi$ and integrate by parts, we obtain $$\label{weak-nonlocal}
\displaystyle \int_0^T\int_{{\mathbb{R}^N}}u\dfrac{\partial
\varphi}{\partial
t}\,dxds-\int_0^T\int_{{\mathbb{R}^N}}(-\Delta)^{\sigma/4}(|u|^{m-1}u)(-\Delta)^{\sigma/4}\varphi\,d
xds=0.$$ This identity is the basis of our definition of a weak solution. The integrals in make sense if $u$ and $u^m$ belong to suitable spaces. The right space for $u^m$ is the fractional Sobolev space $\dot{H}^{\sigma/2}({\mathbb{R}^N})$, defined as the completion of $C_0^\infty({\mathbb{R}^N})$ with the norm $$\|\psi\|_{\dot{H}^{\sigma/2}}=\left(\int_{{\mathbb{R}^N}}
|\xi|^\sigma|\widehat{\psi}|^2\,d\xi\right)^{1/2}
=\|(-\Delta)^{\sigma/4}\psi\|_{L^2},$$ where $\widehat{\psi}$ denotes the Fourier transform of $\psi$. Note that $(-\Delta)^{\sigma/4}u^m\in L^2({\mathbb{R}^N})$ if $u^m\in\dot{H}^{\sigma/2}({\mathbb{R}^N})$.
[**Definition**]{}\[def:weak.solution.nonlocla\] A function $u$ is a [*weak solution*]{} to Problem if:
- $u\in
L^{1}({\mathbb{R}^N}\times (0,T))$ for all $T>0$, $u^m \in L^2_{\rm
loc}((0,\infty);\dot{H}^{\sigma/2}({\mathbb{R}^N}))$;
- identity holds for every $\varphi\in C_c^1({\mathbb{R}^N}\times(0,T))$;
- $u(\cdot,t)\in L^1({\mathbb{R}^N})$ for all $t>0$, $\lim\limits_{t\to0}u(\cdot,t)=f$ in $L^1({\mathbb{R}^N})$.
A drawback of this definition is that there is no convenient formula for the fractional Laplacian of a product or of a composition of functions. Moreover, we take no advantage in using compactly supported test functions since their fractional Laplacian loses this property. To overcome these and other difficulties, we will use the fact that our solution $u$ is the trace of the solution of a *local* problem obtained by extending $u^m$ to a half-space whose boundary is our original space.
[Extension Method.]{} In the particular case $\sigma=1$ studied in [@pqrv1], the problem is reformulated by means of the well-known representation of the half-Laplacian in terms of the Dirichlet-Neumann operator. This allowed us to transform the nonlocal problem into a local one (i.e., involving only derivatives and not integral operators). Of course, this simplification pays a prize, namely, introducing an extra space variable. The application of such an idea is not so simple when $\sigma\ne 1$; it involves a number of difficulties that we address in [@pqrv2]. We have to use the characterization of the Laplacian of order $\sigma$, $(-\Delta)^{\sigma/2}$, $0<\sigma<2$, described by Caffarelli and Silvestre in their famous paper [@Caffarelli-Silvestre], in terms of the so-called $\sigma$-harmonic extension, which is the solution of an elliptic problem with a degenerate or singular weight.
Let us explain this extension in some more detail. If $g=g(x)$ is a smooth bounded function defined in ${\mathbb{R}^N}$, its $\sigma$-harmonic extension to the upper half-space $\mathbb{R}^{N+1}_+$, $v=~{{\mathcal E}}(g)$, is the unique smooth bounded solution $v=v(x,y)$ to $$\left\{
\begin{array}{ll}
\nabla\cdot(y^{1-\sigma}\nabla v)=0,\qquad &x\in{\mathbb{R}^N},\, y>0,\\
v(x,0)=g(x),\qquad&x\in{\mathbb{R}^N}.
\end{array}
\right. \label{sigma-extension}$$ Then, $$-\mu_{\sigma}\lim_{y\to0^+}y^{1-\sigma}\frac{\partial v}{\partial
y}=(-\Delta)^{\sigma/2} g(x), \label{fract-lapla}$$ where the precise constant, which does not depend on $N$, is $\mu_{\sigma}=\frac{2^{\sigma-1}\Gamma(\sigma/2)}{\Gamma(1-\sigma/2)}$, see [@Caffarelli-Silvestre]. In the operator $\nabla$ acts in all $(x,y)$ variables, while in $(-\Delta)^{\sigma/2}$ acts only on the $x=(x_1,\cdots,x_N)$ variables. In the sequel we denote $$L_\sigma v\equiv \nabla\cdot(y^{1-\sigma}\nabla v),\qquad
\dfrac{\partial v}{\partial y^\sigma}\equiv
\mu_{\sigma}\lim_{y\to0^+}y^{1-\sigma}\frac{\partial v}{\partial
y}.$$
*Notation.* In this section we will use the notation $\Omega={\mathbb{R}^N}\times(0,\infty)$ for the upper half-space, with points $\overline x=(x,y)$, $x\in{\mathbb{R}^N}$, $y>0$; its boundary, which is identified to the original ${\mathbb{R}^N}$ with variable $x$, will be named $\Gamma$. Besides, we use the simplified notation $u^m$ for data of any sign, instead of the actual “odd power” $|u|^{m-1}u$, and we will also use such a notation when $m$ is replaced by $1/m$. The convention is not applied to any other powers.
[Extended problem. Weak solutions.]{} With the above in mind, we rewrite problem for $w=u^m$ as a quasi-stationary problem with a dynamical boundary condition $$\left\{
\begin{array}{ll}
L_\sigma w=0\qquad &\mbox{for } \overline x\in\Omega,\, t>0,\\
\dfrac{\partial w}{\partial y^\sigma}-\dfrac{\partial
w^{1/m}}{\partial
t}=0\qquad&\mbox{for } x\in\Gamma,\, t>0,\\
w(x,0,0)=f^m(x)\qquad&\mbox{for } x\in\Gamma.
\end{array}
\right. \label{pp:local}$$ This problem has been considered by Athanasopoulos and Caffarelli [@AC09], who prove that any bounded weak solution is Hölder continuous if $m>1$.
To define a weak solution of this problem we multiply formally the equation in by a test function $\varphi$ and integrate by parts to obtain $$\label{weak-local}
\displaystyle \int_0^T\int_{\Gamma}u\dfrac{\partial
\varphi}{\partial
t}\,dxds-\mu_\sigma\int_0^T\int_{\Omega}y^{1-\sigma}\langle\nabla
w,\nabla \varphi\rangle\,d\overline xds=0,$$ where $u=(\mathop{\rm Tr}(w))^{1/m}$ is the trace of $w$ on $\Gamma$ to the power $1/m$. This holds on the condition that $\varphi$ vanishes for $t=0$ and $t=T$, and also for large $|x|$ and $y$. We then introduce the energy space $X^\sigma(\Omega)$, the completion of $C_0^\infty(\Omega)$ with the norm $$\|v\|_{X^\sigma}=\left(\mu_\sigma\int_{\Omega} y^{1-\sigma}|\nabla
v|^2\,d\overline x\right)^{1/2}. \label{norma2}$$ The trace operator is well defined in this space, see below.
[**Definition**]{}\[def:weak.solution\] A pair of functions $(u,w)$ is a [*weak solution*]{} to Problem if:
- $w\in L^2_{\rm loc}((0,\infty);X^\sigma(\Omega))$, $u=(\mathop{\rm Tr}(w))^{1/m}\in
L^{1}(\Gamma\times (0,T))$ for all $T>0$;
- Identity holds for every $\varphi\in C_0^1(\overline\Omega\times(0,T))$;
- $u(\cdot,t)\in L^1(\Gamma)$ for all $t>0$, $\lim\limits_{t\to0}u(\cdot,t)=f$ in $L^1(\Gamma)$.
For brevity we will refer sometimes to the solution as only $u$, or even only $w$, when no confusion arises, since it is clear how to complete the pair from one of the components, $u=(\mathop{\rm Tr}(w))^{1/m}$, $w={{\mathcal E}}(u^m)$.
[Equivalence of weak formulations.]{} The key point of the above discussion is that the definitions of weak solution for our original nonlocal problem and for the extended local problem are equivalent. The main ingredient of the proof is that equation holds in the sense of distributions for any $g\in \dot H^{\sigma/2}(\Gamma)$.
[**Proposition**]{} [*A function $u$ is a weak solution to Problem if and only if $(u,{{\mathcal E}}(u^m))$ is a weak solution to Problem .*]{}
[Strong solutions.]{} Weak solutions satisfy equation in the sense of distributions. Hence, if the left hand side is a function, the right hand side is also a function and the equation holds almost everywhere. This fact allows to prove uniqueness and several other important properties, and hence motivates the following definition.
[**Definition**]{} We say that a weak solution $u$ to Problem is a strong solution if $u\in C([0,\infty):\,L^1(\Gamma))$ as well as $\partial_tu$ and $(-\Delta)^{\sigma/2}
(|u|^{m-1}u) \in
L^1_{\rm loc}(\Gamma\times(0,\infty))$.
Main results
------------
[Existence.]{} We prove existence of a suitable concept of (weak) solution for general $L^1$ initial data only in the restricted range $m>m_c$, which includes as a particular case the linear fractional heat equation, case $m=1$. If $0<m\le m_c$ (which implies that $0<\sigma<1$ if $N=1$, recall that $m_c=(N-\sigma)_+/N$) we need to slightly restrict the data to obtain weak solutions.
\[th:existence\] If either $f\in L^1({\mathbb{R}^N})$ and $m>m_c$, or $f\in
L^1({\mathbb{R}^N})\cap L^p({\mathbb{R}^N})$ with $p>p_*(m)=(1-m)n/\sigma$ and $ 0<m\le m_c $, there exists a weak solution to the Cauchy problem for the FPME.
[Uniqueness.]{} We first prove uniqueness of solutions in the range $m\ge m_{c}$. If $0<m<m_{c}$, we need to use the concept of strong solution, a concept that is standard in the abstract theory of evolution equations. This is no restriction in view of the next results proved in [@pqrv2].
\[th:strong\] The solution given by Theorem [\[th:existence\]]{} is a strong solution.
We state the uniqueness result in its simplest version.
\[th:uniqueness\] For every $f$ and $m>0$ there exists at most one strong solution to Problem .
[Qualitative behaviour.]{} The solutions to Problem have some nice properties that are summarized here.
\[th:properties\] Assume $f,f_1,f_2$ satisfy the hypotheses of Theorem [\[th:existence\]]{}, and let $u,u_1,u_2$ be the corresponding strong solutions to Problem .
*(i)* If $m\ge m_c$, the mass $\int_{{\mathbb{R}^N}}u(x,t)\,dx$ is conserved.
*(ii)* If $0<m< m_c$, then $u(\cdot,t)$ vanishes identically in a finite time $T(u_0)$.
*(iii)* A smoothing effect holds in the form: $$\|u(\cdot,t)\|_{L^\infty({\mathbb{R}^N})}\le C\,t^{-\alpha_p}
\|f\|_{L^p({\mathbb{R}^N})}^{\delta_p} \label{eq:L-inf-p}$$ with $\alpha_p=(m-1+{\sigma}p /N)^{-1}$, $\delta_p=\sigma
p\alpha_p/N$, and $C=C(m,p,N,\sigma)$. This holds for all $p\ge 1$ if $m>m_c$, and only for $p>p_*(m)$ if $0<m\le m_c$.
*(iv)* Any $L^p$-norm of the solution, $1\le p\le \infty$, is nonincreasing in time.
*(v)* There is an $L^1$-order-contraction property, $$\int_{{\mathbb{R}^N}}(u_1-u_2)_+(x,t)\,dx\le
\int_{{\mathbb{R}^N}}(u_1-u_2)_+(x,0)\,dx.$$
*(vi)* If $f\ge0$ the solution is positive for all $x$ and all positive times if $m\ge m_c$ (resp. for all $x$ and all $0<t<T$ if it vanishes in finite time $T(u_0)$ when $0<m<m_c$).
*(vii)* If either $m\ge 1$ or $f\ge0$, then $u\in
C^\alpha({\mathbb{R}^N}\times(0,\infty))$ for some $0<\alpha<1$.
In the linear case $m=1$ the above properties: conservation of mass, the smoothing effect with a precise decay rate, positivity and regularity, can be derived directly from the representation formula and the properties of the kernel $K_\sigma$.
[Continuous dependence.]{} We also show that the solution (i.e., the semigroup) depends continuously on the initial data and on both parameters $m$ and $\sigma$, in particular in the nontrivial limit $\sigma\to 2$, that allows to recover the standard PME, $\partial_t u-\Delta |u|^{m-1}u=0$, or the other end $\sigma\to
0$, for which we get the ODE : $\partial_t u+ |u|^{m-1}u=0$. Continuity will be true in general only in $L^1_{\rm loc}$, unless we stay in the region of parameters where mass is conserved.
\[th:contdep\] The strong solutions depend continuously in the norm of the space $C([0,T]:\,L^1_{\rm loc}({\mathbb{R}^N}))$ on the parameters $m$, $\sigma$, and the initial data $f$. If moreover $m\ge m_c$ and $0<\sigma\le2$, convergence also holds in $C([0,T]:\,L^1({\mathbb{R}^N}))$.
We will extend this theory in different ways. See in particular Section \[sec.gnl\].
Self-similar solutions and their role {#sec.bar}
=====================================
Several types of special solutions play a role in the theory of the FPME: fundamental solutions, very singular solutions, extinction solutions,...
**Fundamental solutions**
-------------------------
Let us consider the Cauchy problem for the FPME taking as initial data a Dirac delta, $$\label{eq.id}
u(x,0) = M\delta(x)\, \qquad M>0.$$ Solutions with such data are called [*fundamental solutions*]{} in the linear theory, and we will keep that name though their relevance is different in the nonlinear context. We use the concept of continuous and nonnegative weak solution introduced in [@pqrv1], [@pqrv2], for which there is a well-developed theory when the datum is a function in $L^1({\mathbb{R}^N})$ as we have outlined. We recall that the question under discussion has a well-known answer for the standard Porous Medium Equation, i.e., the limit case $s=1$, in the form of the usual Barenblatt solutions; they were actually discovered in the 1950’s by Zeldovich-Kompanyeets [@ZK50] and Barenblatt [@Bar52], cf. their use in applications in [@BarBook] and in theory of the equation in [@Vapme].
Let us describe the results of our paper [@VazBar2012], devoted to prove existence, uniqueness and main properties of the fundamental solutions for the FPME. The exponent $m$ varies in principle in the range $m>1$, but the methods extend to the linear case $m=1$ and even to the fast diffusion range (FDE) $m<1$ on the condition that $m> m_c=\max\{(N-2s)/N,0\}$. Such a type of restriction on $m$ from below carries over from the PME-FDE theory, [@JLVSmoothing]. The solutions we find are self-similar functions of the form $$\label{sss}
u^*(x,t)= t^{-\alpha} F(|x|\,t^{-\beta})$$ with suitable exponents $\alpha$ and $\beta$ and profile $F\ge 0$. Here is the main result.
\[thm.Bs\] For every choice of parameters $s\in(0,1)$ and $m>m_c$ where $m_c=\max\{(N-2s)/N,0\}$, and for every $M>0$, equation admits a unique fundamental solution $u^*_M(x,t)$; it is a nonnegative and continuous weak solution for $t>0$ and takes the initial data as a trace in the sense of Radon measures. It has the self-similar form for suitable $\alpha$ and $\beta$ that can be calculated in terms of $N$ and $s$ in a dimensional way, precisely $$\label{scale.expo}
\alpha=\frac{N}{N(m-1)+2s}, \qquad \beta=\frac{1}{N(m-1)+2s}\,.$$ The profile function $F_M(r)$, $r\ge 0$, is a bounded and continuous function, it is positive everywhere, it is monotone and it goes to zero at infinity.
Let us point out that the Brownian scaling ($\beta=1/2$ and $\alpha=N/2$) is recovered in the fractional case under the condition $N(m-1)=2(1-s)$, for instance for $m=(N+1)/N$ in the most common case $s=1/2$. Moreover, the profile $F_M$ can be obtained by a simple rescaling of $F_1$. $F$ is actually a smooth function by the regularity results of [@vpqr].
The initial data are taken in the weak sense of measures $$\lim_{t\to 0}\int_{{\mathbb{R}^N}} u(x,t)\phi(x)\,dx= M\phi(0)$$ for all $\phi\in C_b({\mathbb{R}^N})$, the space of continuous and bounded functions in ${\mathbb{R}^N}$. We will call these self-similar solutions of Problem - with given $M>0$ the [*Barenblatt solutions*]{} of the fractional diffusion model by analogy with the PME and other prominent studied cases. The form of the exponents explains the already mentioned restriction on $m$ from below.
Idea of the proofs: existence of the Barenblatt solutions is done by approximation of the Cauchy problem with smooth initial data and passing to the limit, using suitable a priori estimates. Uniqueness needs to go over to the potential equation as follows: we take the convolution of $u(x,t)$ with the Riesz kernel and define: $$U(x,t)= C_{n,s}\int \frac{u(y,t)}{|x-y|^{N-2s}}\,dy \,.$$ Then $(-\Delta )^s U= u$, and using the equation we get equation (PE): $$\label{eq.pot1}
U_t=((-\Delta )^{-s} u)_t=-(-\Delta )^{-s}(-\Delta )^{s}u^m=-u^m.$$ in other words, $U_t+((-\Delta )^s U)^m=0$. Though awkward-looking, this ‘dual equation’ admits a good uniqueness theorem, as shown in [@VazBar2012].
The properties of the profile function $F(r)$ are quite important, in particular the asymptotic behavior. This is why in [@VazBar2012] we derive the elliptic equation that it satisfies: $$\label{sss.form}
(-\Delta )^{s}F^m =\alpha F + \beta y\cdot \nabla F=\beta \nabla\cdot (yF)$$ Moreover, putting $ s'=1-s$ we have $$\nabla (-\Delta )^{-s'} F^m=-\beta \, y\, F\,,$$ which in radial coordinates gives $$L_{s'} F^m(r)=\beta \int_r^\infty rF(r)dr\,,$$ where $L_{s'}$ the radial expression of operator $(-\Delta)^{-{s'}}$. Using this 1D integral equation, the following characterization for the behavior of the fundamental solution is obtained. The critical exponents are $m_c=(N-2s)/N$ and $m_1=N/(N+2s)$, so that $m_c<m_1<1$.
For every $m>m_1$ we have the asymptotic estimate $$C_1\,M^{\mu}\le F_M(r)\,r^{N+2s} \le C_2\,M^{\mu},$$ where $M=\int F(x)\,dx$, $C_i= C_i(m,N,s)>0$, and $\mu=(m-m_1)(N+2s)\beta$. On the other hand, for $m_c<m<m_1$, there is a constant $C_\infty(m,N,s)$ such that $$F_M(r)\,r^{2s/(1-m)}=C_\infty.$$ The case $m=m_1$ is borderline and has a logarithmic correction.
[**Solutions with explicit form.**]{} As this survey is written, Y. Huang reports [@Huang2013] the explicit expression of the Barenblatt solution for the special value of $m$, $m_{ex} = (N+2-2s)/(N+2s)$. The profile is given by $$F_M(y) =\lambda\,(R^2 + |y|^2)^{-(N+2s)/2}$$ where the two constants $\lambda$ and $R$ are determined by the total mass $M$ of the solution and the parameter $\beta$. Note that for $s=1/2$ we have $m_{ex}=1$, and the solution is the one mentioned in the introduction for the linear case, . We always have $m_{ex}>m_1$; the range of $m$ that is covered for $0<s<1$ is $N/(N+2)<m_{ex}<(N+2)/N$.
**Asymptotic behavior**
-----------------------
As a main application of this construction, we prove in [@VazBar2012] that the asymptotic behaviour as $t\to\infty$ of the class of nonnegative weak solutions of with finite mass (i.e., $\int_{{\mathbb{R}^N}} u(x,t)\,dx<\infty$) is given in first approximation by the family $u_M^*(x,t)$, i.e., the Barenblatt solutions are the attractors in that class of solutions.
\[thm.exlimit\] Let $u$ be a nonnegative solution of the FPME with initial data $u_0=\mu \in {\mathcal M}_+({\mathbb{R}^N})$, and $m>m_c$. Let $M=\mu({\mathbb{R}^N})$ and let $u^* _M$ be the self-similar Barenblatt solution with mass $M$. Then as $t\to\infty$ the solutions $u(x,t)$ and $u^*_M(x,t)$ are increasingly similar, and more precisely we have $$\label{conver.express.1}
\lim_{t\to\infty} \|u(\cdot,t)-u^*_M(\cdot,t)\|_1=0\,,$$ and also $$\label{conver.express}
\lim_{t\to\infty} t^{\alpha}\,|u(x,t)-u^*_M(x,t)|=0\,, \qquad \alpha=N/(N(m-1)+2s)\,,$$ uniformly in $x\in {\mathbb{R}^N}$. It follows that for every $p\in (1,\infty)$ we have $$\label{conver.express.p}
\lim_{t\to\infty} t^{(p-1)\alpha/p}\|u(\cdot,t)-u^*_M(\cdot,t)\|_p=0\,.$$
This result is a clear example of the important role that the fundamental solutions and their properties can play in the applications.
**Very singular solutions**
---------------------------
Only in the range $m_c<m<m_1=N/(N+2s)$ we can pass to the limit $M\to\infty$ and obtain a special solution that has a fixed isolated singularity at $x=0$ that has separate-variables form and we call very singular solution (VSS) by analogy with the standard Fast Diffusion Equation. The result represents a marked difference with the case $s=1$ where VSS exist in the larger range $m_c<m<1$. It is another manifestation of the long-range interactions of the fractional Laplacian, that avoids some of the purely local estimates of the standard FDE with the classical Laplacian operator; indeed, extrapolation of the standard estimates would justify the existence of a VSS in cases where there is none.
Among other interesting qualitative properties of the equation, in [@VazBar2012] we prove an Aleksandrov reflection principle. This is related to symmetrization principles that we will discuss below.
**Other solutions**
-------------------
Self-similar solutions of the second kind, where the similarity exponents $\alpha$ and $\beta$ are only related by the compatibility condition $(m-1)\alpha +2s\beta=1$, but are otherwise free not to verify the constant-mass condition $\alpha= N\beta$ are constructed in the work with Volzone [@VazVol2].
The question of extinction of mass in finite time that was introduced in [@pqrv2] and happens for $m<m_c<1$ is also best understood in terms of explicit solutions, which were constructed in [@VazVol2] and have the form $$\label{ext.solut}
U(x,t)^{1-m}=C\,\frac{T-t}{|x|^{2s}}.$$
Estimates, better existence and positivity {#sec.mb}
==========================================
In collaboration with M. Bonforte we have studied the existence of quantitative a priori estimates of a local type for solutions of the FPME and we have then derived consequences for the theory. Thus, in [@BV2012] we have dealt with the properties of nonnegative solutions of the Cauchy problem. Such kind of estimates were obtained for the standard PME by Aronson-Caffarelli [@ArCaff] and by the present authors for the standard FDE [@BV; @BV-ADV]. The same local tools are not efficient for the present fractional model due to the nonlocal character of the diffusion operator, but then estimates occur in weighted spaces.
Weighted $L^1$ estimates in the fast diffusion range
----------------------------------------------------
We will concentrate on the case $m<1$. The use of suitable weight functions allows to prove crucial $L^1$-weighted estimates that enter substantially into the derivation of the main results. The results take different forms according to the value of the exponent $m$, a fact that is to be expected since it happens for standard FDE (i.e., the limit case $s=1$). When $s<1$ the equation is nonlocal, therefore we cannot expect purely local estimates to hold. Indeed, we will obtain estimates in weighted spaces if the weight satisfies certain decay conditions at infinity. We present first a technical lemma that shows the difference with the standard Laplacian.
\[Lem.phi\] Let $\varphi\in C^2({\mathbb{R}^N})$ and positive real function that is radially symmetric and decreasing in $|x|\ge 1$. Assume also that $\varphi(x)\le |x|^{-\alpha}$ and that $|D^2\varphi(x)| \le c_0 |x|^{-\alpha-2}$, for some positive constant $\alpha$ and for $|x|$ large enough. Then, for all $|x|\ge |x_0|>>1$ we have $$|(-\Delta)^s\varphi(x)|\le \dfrac{c}{|x|^{\gamma}}\,,$$ with $\gamma=\alpha+2s$ for $\alpha<N$ and $\gamma=N+2s$ for $\alpha>N$; the constant $c>0$ depends only on $\alpha,s,N$ and $\|\varphi\|_{C^2({\mathbb{R}^N})}$. If $\alpha>N$ the reverse estimate holds from below if $\varphi\ge0$: $|(-\Delta)^s\varphi(x)|\ge c_1 |x|^{-(N+2s)}$ for all $|x|\ge |x_0|>>1$.
A suitable particular choice is the function $\varphi$ defined for $\alpha>0$ as $\varphi(x)=1$ for $|x|\le 1$ and $\varphi(x)= \left(1+(|x|^2-1)^4\right)^{-\alpha/8}$ if $|x|\ge 1\,.$ We will use the following notations: $m_c=(N-2s)/N$, $m_1=N/(N+2s)$, $p_c=N(1-m)/2$,
These are the weighted $L^1$ estimates obtained in [@BV2012].
\[prop.HP.s\] Let $u\ge v$ be two ordered solutions to the FPME with $0<m<1$. Let $\varphi_R(x)=\varphi(x/R)$ where $R>0$ and $\varphi$ is as in the previous lemma with $0\le \varphi(x)\le |x|^{-\alpha}$ for $|x|>>1$ and $$N-\frac{2s}{1-m}<\alpha< N+\frac{2s}{m}\,.$$ Then, for all $0\le \tau,t <\infty$ we have $$\label{HP.s}
\begin{array}{l}
\left({\displaystyle\int}_{{\mathbb{R}^N}}\big(u(t,x)- v(t,x)\big)\varphi_R\,dx\right)^{1-m}\le \\
\left({\displaystyle\int}_{{\mathbb{R}^N}}\big(u(\tau,x)- v(\tau,x)\big)\varphi_R\,dx\right)^{1-m}
+ \dfrac{C_1 \,|t-\tau|}{R^{2s-N(1-m)}}
\end{array}$$ with $C_1(\alpha,m,N)>0$.
It is remarkable that the estimate holds for (very) weak solutions, even changing sign solutions. Also, it is worth pointing out that the estimate holds both for $\tau<t$ and for $\tau>t$. The estimate implies the conservation of mass when $m_c<m<1$, by letting $R\to \infty$. On the other hand, when $0<m<m_c$ solutions corresponding to $u_0\in L^1({\mathbb{R}^N})\cap L^p({\mathbb{R}^N})$ with $p\ge N(1-m)/2s$, extinguish in finite time $T(u_0)>0$, (see e.g. [@pqrv2]); the above estimates provide a lower bound for the extinction time in such a case, just by letting $\tau=T$ and $t=0$ in the above estimates: $$\label{HP.s.T}
\frac{1}{C_1\,R^{N(1-m)-2s}}\left(\int_{{\mathbb{R}^N}}u_0\,\varphi_R\,dx\right)^{1-m}\le T(u_0)\,.$$ Moreover, if the initial datum $u_0$ is such that the limit as $R\to+\infty$ of the right-hand side diverges to $+\infty$, then the corresponding solution $u(t,x)$ exists (and is positive) globally in time.
Existence of solutions in weighted $L^1$-spaces {#sect.exist.large}
-----------------------------------------------
As a first consequence of these estimates, we can extend the $L^1$ existence theory of [@pqrv1; @pqrv2] to an existence result for very weak solutions with non-integrable data in some weighted $L^1_{\varphi}$ space. In particular, bounded initial data or data with slow growth at infinity are allowed.
\[exist.large\] Let $0<m<1$ and let $u_0\in L^1({\mathbb{R}^N}, \varphi\,dx)$, where $\varphi$ is as in Theorem $\ref{prop.HP.s}$ with decay at infinity $|x|^{-\alpha}$, $d-[2s/(1-m)]<\alpha<d+(2s/m)$. Then there exists a very weak solution $u(t,\cdot)\in L^1({\mathbb{R}^N}, \varphi\,dx)$ to the FPME in $[0,T]\times {\mathbb{R}^N}$, in the sense that $$\int_0^T\int_{{\mathbb{R}^N}}u(t,x)\psi_t(t,x)\,dx{\,{\rm d}t}=\int_0^T\int_{{\mathbb{R}^N}}u^m(t,x)(-\Delta)^s\psi(t,x)\,dx{\,{\rm d}t}$$ for all $\psi\in C_c^\infty([0,T]\times{\mathbb{R}^N})\,.$ This solution is continuous in the weighted space, $u\in C([0,T]:L^1({\mathbb{R}^N}, \varphi\,dx))$.
Idea of the proof: We take $0\le u_{0,n}\in L^1({\mathbb{R}^N})\cap L^\infty({\mathbb{R}^N})$ be a non-decreasing sequence of initial data $u_{0,n-1}\le u_{0,n}$, converging monotonically to $u_0\in L^1({\mathbb{R}^N}, \varphi{\,{\rm d}x})$, i.e., such that $\int_{{\mathbb{R}^N}}(u_0- u_{n,0})\varphi{\,{\rm d}x}\to 0$ as $n\to \infty$. Consider the unique solutions $u_n(t,x)$ of the equation with initial data $u_{0,n}$. By the comparison results of [@pqrv2] we know that they form a monotone sequence. The weighted estimates show that the sequence is bounded in $L^1({\mathbb{R}^N}, \varphi{\,{\rm d}x})$ uniformly in $t\in[0,T]$. By the monotone convergence theorem in $L^1({\mathbb{R}^N}, \varphi{\,{\rm d}x})$, we know that the solutions $u_n(t,x)$ converge monotonically as $n\to \infty$ to a function $u(t,x)\in L^\infty ((0,T): L^1({\mathbb{R}^N}, \varphi\,dx))$. We then pass to the limit $n\to\infty$.
**Remark.** The solutions constructed above only need to be integrable with respect to the weight $\varphi$, which has a tail of order less than $d+2s/m$. Therefore, we have proved existence of solutions corresponding to initial data $u_0$ that can grow at infinity as $|x|^{(2s/m)-\varepsilon}$ for any $\varepsilon >0$. Note that for the linear case $m=1$ this exponent is optimal in view of the representation of solutions in terms of the fundamental solution, but this does not seem to be the case for $m<1$.
The solution constructed in Theorem $\ref{exist.large}$ by approximation from below is unique. We call it the minimal solution. In this class of solutions the standard comparison result holds, and also the estimates of Theorem $\ref{prop.HP.s}$.
Lower bounds for fractional fast diffusion
------------------------------------------
Section 3 of paper [@BV2012] studies the actual positivity of nonnegative solutions via quantitative lower estimates in the so called good fast diffusion range. We will use the previous notation: $\beta:=1/[2s-N(1-m)]$, which is positive if $m>m_c$.
The following is the main quantitative estimate from below for positive solutions.
\[thm.lower\] Let $R_0>0$, $m_c<m<1$ and let $0\le u_0\in L^1({\mathbb{R}^N}, \varphi{\,{\rm d}x})$, where $\varphi$ is as in Theorem $\ref{prop.HP.s}$ with decay at infinity $O(|x|^{-\alpha})$, with $N-[2s/(1-m)]<\alpha<N+(2s/m)$. Let $u(t,\cdot)\in L^1({\mathbb{R}^N}, \varphi{\,{\rm d}x})$ be a very weak solution to the FPME with initial datum $u_0$. Then can calculate a time $$\label{t*}
t_*:=C_* \,R_0^{2s-N(1-m)}\,\|u_0\|_{ L^1(B_{R_0})}^{1-m}$$ with quantified $C_*$ such that $$\label{low.1.thm}
\inf_{x\in B_{R_0/2}}u(t,x)\ge
K_1\,R_0^{-\frac{2s}{1-m}}\,t^{\frac{1}{1-m}}\quad \mbox{ if } \ 0\le t\le t_*\,,$$ while $$\inf_{x\in B_{R_0/2}}u(t,x)\ge K_2\, \|u_0\|_{L^1(B_{R_0})}^{2s\beta}
\,t^{-N\beta}\quad \mbox{ if } \ t\ge t_*\,.$$ The positive constants $C_*,K_1,K_2$ depend only on $m,s$ and $N\ge 1$.
We point out that the diffusion in the equation is nonlocal, but the estimates are local. These estimates can be combined with the $L^\infty$ bounds from above of the papers [@pqrv1; @pqrv2] to sandwich the solution from above and below. The paper continues to obtain sharp estimates on the behaviour as $|x|\to\infty$ (so-called tail behaviour).
**Estimates in other ranges**
-----------------------------
New local estimates are obtained in the paper either for $0<m<m_c$ or for $m\ge 1$. The question of Harnack inequalities is also discussed. We refrain from giving more details at this moment for lack of space, and refer to the paper [@BV2012] for further information.
An interesting result is the existence and uniqueness of an initial trace for nonnegative solutions that we state in complete detail for $m<1$.
\[thm.init.trace.m<1\] Let $0<m<1$ and let $u$ be a nonnegative weak solution of equation in $(0,T]\times{\mathbb{R}^N}$. Assume that $\|u(T)\|_{L^1({\mathbb{R}^N})}<\infty$. Then there exists a unique nonnegative Radon measure $\mu$ as initial trace, that is $$\label{eq.trace1}
\int_{{\mathbb{R}^N}}\psi\,{{\rm d}}\mu=\lim_{t\to 0^+}\int_{{\mathbb{R}^N}}u(t,x)\psi(x)\,{\,{\rm d}x}\,,\qquad\mbox{for all }\psi\in C_0({\mathbb{R}^N})\,.$$ Moreover, the initial trace $\mu$ satisfies the bound $$\label{intit.trace.bdd.lem}
\mu(B_{R}(x_0))\le \|u(T)\|_{L^1({\mathbb{R}^N})} + C_1 R^{N(1-m)-2s}\,T\,.$$ where $C_1=C_1(m,N,s)>0$ as in .
The result is indeed true for $m>0$, including of course the linear equation for $m=1$.
Problems in bounded domains {#sec.dir}
===========================
The investigation of the questions of existence of solutions for the FPME posed in a bounded domain with suitable boundary conditions was initiated in the papers [@pqrv1; @pqrv2]. The study of a priori estimates has been taken up in ongoing collaboration with M. Bonforte. These estimates are quite different from the problem in the whole space in statements and methods.
Fractional Laplacian operators on bounded domains {#ssect.Def.Fract.Lapl}
-------------------------------------------------
The first problem that we encounter is the possibility of various reasonable definitions of the concept of fractional Laplacian operator. This is in contrast with the situation in the whole Euclidean space ${\mathbb{R}}^N$, where there is a natural concept of fractional Laplacian that can be defined in several equivalent ways as we have mentioned in Subsection \[subs.frop\]. When we consider equation posed on bounded domains, the definition via the Fourier transform does not apply, and different choices appear as possible definitions of the fractional Laplacian.
$\bullet$ On one hand, starting from the classical Dirichlet Laplacian $\Delta_{\Omega}$ on the domain $\Omega$, the so-called spectral definition of the fractional power of $\Delta_{\Omega}$ uses the formula in terms of the semigroup associated to the Laplacian, namely $$\label{sLapl.Omega.Spectral}
\displaystyle(-\Delta_{\Omega})^{s}
g(x)=\frac1{\Gamma(-s)}\int_0^\infty
\left(e^{t\Delta_{\Omega}}g(x)-g(x)\right)\frac{dt}{t^{1+s}}.$$ We will call the operator defined in such a way the *spectral fractional Laplacian*, SFL for short, and denote as ${\mathcal{L}}_1=(-\Delta_{\Omega})^s$. In this case, the initial and boundary conditions associated to the fractional diffusion equation read $$\label{FPME.Dirichlet.conditions.Spectral}
\left\{
\begin{array}{lll}
u(t,x)=0\,,\; &\mbox{in }(0,\infty)\times\partial\Omega\,,\\
u(0,\cdot)=u_0\,,\; &\mbox{in }\Omega\,,
\end{array}
\right.$$ Let us list some properties of the operator: ${\mathcal{L}}_1=(-\Delta_{\Omega})^s$ is a self-adjoint operator on $ L^2(\Omega)$, with a discrete spectrum: its eigenvalues are the family of $s$-power of the eigenvalues of the Dirichlet Laplacian: $(\lambda_j)^s>0$, $j=1,2,\ldots$; the corresponding normalized eigenfunctions $\Phi_{j}$ are exactly the same as the Dirichlet Laplacian, therefore they are as smooth as the boundary allows, namely when $\partial\Omega$ is $C^k$, then $\Phi_j\in C^{\infty}(\Omega)\cap C^k(\overline{\Omega})$. We can thus write $$\label{sLapl.Omega.Spectral.2}
\displaystyle(-\Delta_{\Omega})^{s}
g(x)=\sum_{j=1}^{\infty}(\lambda_j)^s\, \hat{g}_j\, \phi_j(x)$$ with $
\hat{g}_j=\int_\Omega g(x)\Phi_j(x){\,{\rm d}x}$, and $\|\Phi_j\|_{ L^2(\Omega)}=1\,.$
The definition of the Fractional Laplacian via the Caffarelli-Silvestre extension [@Caffarelli-Silvestre] has been extended to the case of bounded domains by Cabré and Tan [@Cabre-Tan] by using as extended domain the cylinder ${\mathcal C}=(0,\infty)\times \Omega$ in ${\mathbb{R}}^{N+1}_+$, and by putting zero boundary conditions on the lateral boundary of that cylinder. This definition enables to understand the boundary conditions in an easy way. It is proved that this definition is equivalent to the SFL. See also [@capella-d-d-s].
$\bullet$ On the other hand, we can define a fractional Laplacian operator by using the integral representation in terms of hypersingular kernels and “restrict” the operator to functions that are zero outside $\Omega$: we will denote the operator defined in such a way as ${\mathcal{L}}_2=(-\Delta_{|\Omega})^s$, and call it the *restricted fractional Laplacian*, RFL for short. In this case, the initial and boundary conditions associated to the fractional diffusion equation read $$\label{FPME.Dirichlet.conditions.Restricted}
\left\{
\begin{array}{lll}
u(t,x)=0\; &\mbox{in }(0,\infty)\times ({\mathbb{R}}^N\setminus \Omega)\,,\\
u(0,\cdot)=u_0\; &\mbox{in }\Omega\,.
\end{array}
\right.$$ See more in [@BVdir2013]. We also refer to [@SV1] for a discussion and references about the differences between the Spectral and the Restricted fractional Laplacian. The authors of [@SV1] call the second type simply Fractional Laplacian, but we feel that the absence of descriptive name leads to confusion.
**Main results on the Dirichlet problem**
-----------------------------------------
In our paper [@BVdir2013] we choose to work with the Dirichlet spectral laplacian, ${\mathcal L}_1$. For a quite general class of nonnegative weak solutions to the above problem, we derive in absolute upper estimates up to the boundary of the form $$\label{Intro.Abs.Bdds}
u(t,x) \le K_2\, \dfrac{\Phi_1(x)^{\frac{1}{m}}}{t^{\frac{1}{m-1}}}\,,\qquad\forall t>0\,,\;\forall x\in \Omega\,.$$ In particular, we observe that the boundary behaviour is dictated by $\Phi_1$, the first positive eigenfunction of ${{\mathcal{L}}}_1$, which behaves like the distance to the boundary at least when the domain is smooth enough. We will also prove standard and weighted instantaneous smoothing effects of the type $$\label{Intro.smoothing}
\sup_{x\in \Omega}u(t,x)\le \dfrac{K_4}{t^{N\vartheta_{1,1}}}\left(\int_\Omega u(t,x)\Phi_1(x){\,{\rm d}x}\right)^{2s\vartheta_{1,1}}
\le\dfrac{K_4}{t^{N\vartheta_{1,1}}}\left(\int_\Omega u_0\Phi_1(x){\,{\rm d}x}\right)^{2s\vartheta_{1,1}}\,,$$ where $\vartheta_{1,1}=1/(2s+(N+1)(m-1))$. This is sharper than only for small times. As a consequence of the above upper estimates, we derive a number of useful weighted estimates and we also obtain backward in time smoothing effect of the form $$\label{Intro.smoothing.back}
\| u(t)\|_{ L^\infty(\Omega)} \le \frac{K_4}{t^{(N+1)\vartheta_{1,1}}}\left(1\vee \frac{h}{t}\right)^{\frac{2s\vartheta_{1,1}}{m-1}}\|u(t+h)\|_{ L^1_{\Phi_1}(\Omega)}^{2s\vartheta_{1,1}}\,,\qquad\mbox{$\forall t,h>0$}\,,$$ which is quite surprising and has not been observed before to our knowledge.
We then pass to the question of lower estimates, the main result being the estimate $$\label{Intro.lower}
u(t_0,x_0) \ge L_1\, \dfrac{\Phi_1(x_0)^{\frac{1}{m}}}{t^{\frac{1}{m-1}}}\qquad\mbox{$\forall t\ge t_* > 0\,,\;\forall x_0\in \Omega$}\,,$$ where the waiting time $t_*$ has the explicit form $$t_*= L_0\left(\int_{\Omega}u_0\Phi_1{\,{\rm d}x}\right)^{-(m-1)}\,.$$ Then we observe that the above estimates combine into global Harnack inequalities in the following form: for all $t\ge t_*$ $$\label{Intro.GHP}
H_0\,\frac{\Phi_1(x_0)^{\frac{1}{m}}}{t^{\frac{1}{m-1}}} \le \,u(t,x_0)\le H_1\, \frac{\Phi_1(x_0)^{\frac{1}{m}}}{t^{\frac{1}{m-1}}}\,.$$ We also provide as a corollary the local Harnack inequalities of elliptic type.
All the constants in the above results are universal, in the sense that may depend only on $N, m, s$ and $\Omega$, but not on $u$. They also have an almost explicit expression, usually given in the proof. Actually, we have tried to obtain quantitative versions of the estimates with indication of the dependence of the relevant constants. In some cases the estimates are absolute, in the sense that they are valid independently of the (norm of the) initial data.
[**New method.**]{} It is worth mentioning that the proofs are based on new ideas with respect to what was used in the standard nonlinear diffusion or in the previous section for fractional diffusion in the whole space (both in our papers and in the references). Thus, we exploit the functional properties of the linear operator as much as possible. More precisely, we use the Green function of the fractional operator even in the definition of solution, and we make use of estimates on its behaviour in the proofs. A key ingredient is thus the knowledge of good estimates for the Green function, that we discuss at length in the paper.
A careful inspection of the proofs shows that the presented method would allow to treat a quite wide class of linear operators, an issue that we shall discuss in a forthcoming paper [@BV-FPMEBDpaper2]. Here, we have written everything referring to the concrete case of the spectral fractional Laplacian in order to keep the exposition clear and to focus on the main ideas, but the arguments are devised in view of the wider applicability.
**Existence theory**
--------------------
The paper is complemented with a brief presentation of the theory of existence and uniqueness of the class of very weak solutions that we use. This complements the basic theory developed before in [@pqrv1; @pqrv2] and then extended in [@BV2012]. We have no space to enter this intriguing issue here.
The work on general linearities {#sec.gnl}
===============================
In collaboration with Pablo, Quirós and Rodríguez we have examined the question of regularity of the nonnegative solutions of the FPME. Since we had proved continuity and positivity, the equation is not more degenerate, at least at the formal level, and the solutions should be smooth if we recall what happens in the classical heat equation and fast diffusion case when $s=1$. Actually, in [@vpqr] we have decided to study the question for the more general class of equations $$\label{eq:mainphi}
\partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0 \,.$$ To be specific, we deal with the Cauchy problem posed in $Q={\mathbb{R}^N}\times(0,\infty).$ The constitutive function $\varphi$ is assumed to be at least continuous and nondecreasing. Further conditions will be introduced as needed. This generality was motivated by previous work on the model of logarithmic porous medium type equation with fractional diffusion $$\partial_tu+(-\Delta)^{1/2}\log(1+u)=0,$$ which is described in detail in [@pqrv3] and has interesting connections with drift-diffusion models.
The existence of a unique weak solution to the Cauchy problem for the FPME has been fully investigated in [@pqrv1; @pqrv2] for the case where $\varphi$ is a positive power. This theory is extended in [@vpqr] under suitable conditions on $\varphi$ that cover powers in particular.
Regarding regularity, for the linear fractional heat equation it follows from explicit representation with a kernel that solutions are $C^\infty$ smooth and bounded for every $t>0$, $x\in{\mathbb{R}^N}$, under the assumption that the initial data are integrable. In the nonlinear case such a representation is not available. Nevertheless, we will still be able to obtain that the solution is smooth if the equation is uniformly parabolic”, $0<c\le \varphi'(u)\le C<\infty$. Our first result establishes that if the nonlinearity is smooth enough, compared to the order of the equation, $\max\{1,\sigma\}$, then bounded weak solutions are indeed [*classical solutions*]{}.
\[th:main\] Let $u$ be a bounded weak solution to , and assume $\varphi\in C^{1,\gamma}(\mathbb{R})$, $0<\gamma<1$, and $\varphi'(s)>0$ for every $s\in\mathbb{R}$. If $1+\gamma>\sigma$, then $\partial_tu$ and $(-\Delta)^{\sigma/2}\varphi(u)$ are Hölder continuous functions and is satisfied everywhere.
The precise regularity of the solution is determined by the regularity of the nonlinearity $\varphi$; see the paper for the details. Notice that the condition $\varphi'>0$ together with the boundedness of $u$ implies that the equation is uniformly parabolic.
The idea of the proof is as follows: thanks to the results of Athanasopoulos and Caffarelli [@AC09], we know that bounded weak solutions are $C^\alpha$ regular for some $\alpha\in(0,1)$. In order to improve this regularity we write the equation as a fractional linear heat equation with a source term. This term is in principle not very smooth, but it has some good properties. To be precise, given $(x_0,t_0)\in Q$, we have $$\label{eq:nonlin.lin}
\partial_tu+(-\Delta)^{\sigma/2} u=(-\Delta)^{\sigma/2} f,$$ where $$\label{eq:linear-f}
f(x,t):=u(x,t)-\frac{\varphi(u(x,t))}{\varphi'(u(x_0,t_0))},$$ after the time rescaling $t\to t/\varphi'(u(x_0,t_0))$. A very delicate study of the linear theory, with special attention to the properties of the kernel of the fractional Laplacian, allows to show that solutions to the linear equation have the same regularity as $f$. Next, using the nonlinearity we observe that $f$ in the actual right-hand side is more regular than $u$ near $(x_0,t_0)$. We are thus in a situation that is somewhat similar to the one considered by Caffarelli and Vasseur in [@CaffVass], where they deal with an equation, motivated by the study of geostrophic equations, of the form $$\label{eq:caffa-vasseur}
\partial_tu+(-\Delta)^{1/2} u=\mathop{\rm div}( {\bf v}u),$$ where $\bf v$ is a divergence free vector. Comparing with , we see two differences: in their case $\sigma=1$, and the source term is local. These two differences will significantly complicate our analysis.
[**Singular and degenerate equations.**]{} The hypotheses made in Theorem \[th:main\] excludes all the powers $\varphi(u)=|u|^{m-1}u$ for $m>0$, $m\neq1$, since they are degenerate ($m>1$) or singular ($m<1$) at the level $u=0$. Nevertheless, a close look at our proof shows that we may in fact get a local” result, see the paper. Therefore, we get for these nonlinearities (and also for more general ones) a regularity result in the positivity (negativity) set of the solution that implies that bounded weak solutions which are either positive or negative are actually classical.
[**Higher regularity.**]{} If $\varphi$ is $C^\infty$ we prove that solutions are $C^\infty$. The result will be a consequence of the regularity already provided by Theorem \[th:main\] plus a special result for linear equations with variable coefficients. The case $\sigma<1$ is even more involved since we first have to raise the regularity in space exponent from $\sigma$ to 1.
\[th:main2\] Let $u$ be a bounded weak solution to equation . If $\varphi\in C^\infty(\mathbb{R})$, $\varphi'>0$ in $\mathbb{R}$, then $u\in C^\infty(Q)$.
[**Comments and extensions.**]{} Work on extending such results to problems in bounded domains, and to more general operators is in project. Regarding related literature, let us remark that Kiselev et al. [@KisNazVol] give a proof of $C^\infty$ regularity of a class of periodic solutions of geostrophic equations in 2D with $C^\infty$ data. Their methods are completely different to the ones used by us. On the other hand, Cifani and Jakobsen propose in [@CJ11] an alternative $L^1$ theory dealing with a more general class of nonlocal porous medium equations, including strong degeneration and convection. The quantitative study of continuous dependence has been taken up recently in work of Alibaud et al. [@acj2011; @acj2013].
Estimates via symmetrization {#sec.symm}
============================
In two papers [@VazVol; @VazVol2] in collaboration with B. Volzone we have used the techniques of Schwarz and Steiner symmetrization to obtain a priori estimates, in many cases with best constants, for the solutions of the FPME, or its more general version $u_t+(-\Delta)^{s}A(u)=0$, posed in the whole space or in a bounded domain. The main results concern the case of power nonlinearities, the equation is posed in the whole space and the exponent $m<1$. Elliptic results are obtained in [@VazVol] as a preliminary to the parabolic theory. In order to keep up with previous sections we will use the letter $\varphi$ for the nonlinearity instead of the $A$ in the paper.
[Motivation.]{} Symmetrization is a very old geometrical idea that has become nowadays a popular tool of obtaining a priori estimates for the solutions of different partial differential equations, notably those of elliptic and parabolic type. The application of Schwarz symmetrization to obtaining a priori estimates for elliptic problems is already described in [@Wein62]. The standard elliptic result refers to the solutions of an equation of the form $$Lu=f, \qquad Lu=-\sum_{i,j} \partial_i(a_{ij}\partial_j u)\,,$$ posed in a bounded domain $\Omega\subseteq {\mathbb{R}^N}$; the coefficients $\{a_{ij}\}$ are assumed to be bounded, measurable and satisfy the usual ellipticity condition; finally, we take zero Dirichlet boundary conditions on the boundary $\partial\Omega$. The classical analysis introduced by Talenti [@Talenti1] leads to pointwise comparison between (i) the symmetrized version (more precisely the spherical decreasing rearrangement) of the actual solution of the problem $u(x)$ and (ii) the radially symmetric solution $v(|x|)$ of some radially symmetric model problem which is posed in a ball with the same volume as $\Omega$. Sharp a priori estimates for the solutions are then derived. Extensions of this method to more general problems or related equations have led to a copious literature.
[Parabolic version.]{} This pointwise comparison fails for parabolic problems and the appropriate concept is comparison of concentrations, cf. Bandle [@Bandle] and Vázquez [@Vsym82]. The latter considers the evolution problems of the form $$\label{evol.pbm}
\partial_t u=\Delta \varphi(u), \quad u(0)=u_0,$$ where $\varphi$ a monotone increasing real function and $u_0$ is a suitably given initial datum which is assumed to be integrable. For simplicity the problem was posed for $x\in {\mathbb{R}^N}$, but bounded open sets can be used as spatial domains.
[Fractional operators.]{} Symmetrization techniques were first applied to PDEs involving fractional Laplacian operators in the paper [@BV], where the linear elliptic case is studied. In our first paper [@VazVol] we were able to improve on that progress and combine it with the parabolic ideas of [@Vsym82] to establish the relevant comparison theorems based on symmetrization for linear and nonlinear parabolic equations. To be specific, we deal with equations of the form $$\label{nolin.parab}
\partial_t u +(-\Delta)^{s}\varphi(u)=f, \qquad 0<s<1\,.$$ Let us describe the results in [@VazVol] concerning the idea of [*concentration comparison*]{} in the case of the solutions to the Cauchy problem $$\label{eq.1}
\left\{
\begin{array}
[c]{lll}
u_t+(-\Delta)^{\sigma/2}\varphi(u)=f & & x\in\mathbb{R}^{N}\,,t>0\\[6pt]
u(x,0)=u_{0}(x) & & x\in\mathbb{R}^{N}\,.
\end{array}
\right.$$ Here, the nonlinearity $\varphi(u)$ is a nonnegative function, smooth on $\mathbb{R}_{+}$, with $\varphi(0)=0$ and $\varphi'(u)>0$ for all $u>0$ (extended anti-symmetrically in the general two-signed theory). Special attention is paid to cases of the form $\varphi(u)=u^m$ with $m>0$.In [@VazVol] we have obtained that a concentration comparison for solutions to holds *only* when the nonlinearity $\varphi$ is a *concave* function, while for *convex* $\varphi$ a remarkable example is constructed, showing that a failure of concentration comparison occurs (see [@VazVol]).
\[Main comparison\] Let $u$ be the nonnegative mild solution to problem with $\sigma\in(0,2)$, initial data $u_0\in L^1({\mathbb{R}^N})$, $u_0\ge 0$, right-hand side $f\in L^1(Q)$ where $Q=\mathbb{R}^{N}\times (0,\infty)$, $f\ge 0$, and nonlinearity $\varphi(u)$ given by a concave function with $\varphi(0)=0$ and $\varphi'(u)>0$ for all $u>0$. Let $v$ be the solution of the symmetrized problem $$\label{eqcauchysymm.f}
\left\{
\begin{array}
[c]{lll}v_t+(-\Delta)^{\sigma/2}\varphi(v)=f^{\#}(|x|,t) & & x\in\mathbb{R}^{N}\,, \ t>0,\\[6pt]
v(x,0)=u_{0}^{\#}(x) & & x\in\mathbb{R}^{N},
\end{array}
\right. $$ where $f^{\#}(|x|,t)$ means the spherical rearrangement of $f(x,t)$ w.r. to $x$ for fixed time $t>0$. Then, for all $t>0$ we have $$u^\#(|x|,t)\prec v(|x|,t).\label{conccompa}$$ In particular, we have $\|u(\cdot,t)\|_p \le\|v(\cdot,t)\|_p$ for every $t>0$ and every $p\in [1,\infty]$.
Moreover, the following corollary justifies a reasonable consequence: if the data of problem are less concentrated that those of the symmetrized problem, so are the corresponding solutions.
\[corollarycomp\]With the same assumptions of Theorem [\[Main comparison\]]{}, suppose that $u$ is the solution to problem and $v$ solves $$\label{eqcauchysymm.f1}
\left\{
\begin{array}
[c]{lll}v_t+(-\Delta)^{\sigma/2}\varphi(v)=\widetilde{f}(|x|,t) & & x\in\mathbb{R}^{N}\,, \ t>0,\\[6pt]
v(x,0)=\widetilde{u}_{0}(x) & & x\in\mathbb{R}^{N},
\end{array}
\right. $$ where $\widetilde{f}\in L^{1}(Q)$, $\widetilde{u}_{0} \in L^{1}(\mathbb{R}^{N})$ are nonnegative, radially symmetric decreasing functions with respect to $|x|$. If $$u_{0}^{\#}(|x|)\prec\widetilde{u}_{0}(|x|),\quad f^{\#}(|x|,t)\prec\widetilde{f}(|x|,t)$$ for almost all $t>0$, then the conclusion $u^\#(|x|,t)\prec v(|x|,t)$ still holds.
[**A priori estimates and best constants.** ]{} We use the parabolic comparison results of [@VazVol] to obtain precise a priori estimates for the solutions of equation . One of these estimates is the so-called $L^1$ into $L^\infty$ smoothing effect in the following form: $$\|u(\cdot,t)\|_\infty\le C\,\|u_0\|_1^{2s\beta}\,t^{-\alpha}.$$ The estimates were obtained in [@pqrv1; @pqrv2] (and in the non-power case in [@vpqr]). In the paper [@VazVol2] we obtain the precise exponents and, what is important, the best constant $C$ in the decay inequality. The calculation of best constants in functional inequalities is a topic of continuing interest in the theory of PDEs, both in the elliptic and evolution settings. Classical references to the calculation of best constants by symmetrization methods are Aubin and Talenti’s computation of the best constants in the Sobolev inequality in [@Aubin76; @Talenti2]. Our calculation of a priori estimates with exact exponents and best constants is closely related to the sharp decay estimate for solutions of the porous medium/fast diffusion equation in [@Vsym82; @JLVSmoothing]. When treating the linear case $\varphi(u)=u$, the estimates are called ultra-contractivity, see the book [@Davies1] where the importance of best constants is stressed for the applications in Physics. As a further application of the comparison techniques, optimal estimates with initial data in Marcinkiewicz spaces are obtained.
An important critical exponent appears repeatedly in the paper as a lower bound, $$m_c:=(N-2s)/N\,.\label{supFFD}$$ Since we are assuming $m>0$ and $0<s<1$, it does not appear in dimension $N=1$ if $s\ge 1/2$. Thus, we study the question of deciding the possible extinction of solutions in the range $m<m_c$ in terms of some norm of the initial data, and estimating the extinction time. First of all, we construct an explicit extinction solution of the fractional fast diffusion equation in this range of $m$’s, formula . Then, we obtain optimal estimates by using comparison based on symmetrization. In this direction we improve significantly the results of the previous papers [@BV2012] and [@pqrv2], by obtaining optimal estimates on the extinction time for data in Marcinkiewicz spaces. Finally, all the results are stable under the limit $s\to 1$, where the standard diffusion case is recovered. See [@pqrv2] for details on such limit.
KPP propagation and fractional diffusion {#sec.kpp}
=========================================
The problem goes back to the work of Kolmogorov, Petrovskii and Piskunov, see [@KPP], that presents the most simple reaction-diffusion equation concerning the concentration $u$ of a single substance in one spatial dimension, $\partial_t u=D u_{xx} + f(u).$ The choice $f(u) = u(1-u)$ yields Fisher’s equation [@Fisher] that was originally used to describe the spreading of biological populations. The celebrated result says that the long-time behavior of any solution of $u(x,t)$, with suitable data $0\le u_0(x)\le 1$ that decay fast at infinity, resembles a traveling wave with a definite speed. In dimensions $N\geq 1$, the problem becomes $$\label{classicalKPP}
u_t-\Delta u=u(1-u) \quad \text{in }(0,+\infty)\times {\mathbb{R}^N}.$$ This case has been studied by Aronson and Weinberger in [@AronsonWeinberger], where they prove the same result as the one-dimensional case. The result is formulated in terms of linear propagation of the level sets of the solution. In the case of the more general model $$\label{PMEkpp}
u_t-\Delta u^m=u(1-u)$$ the same result as before holds in the case of slow diffusion $m>1$ (Vázquez and de Pablo [@dPJLVjde1991]). Departing from these results, King and McCabe examined in [@KingMcCabe] the case of fast diffusion $m<1$ of equation . For $(N-2)_+/N<m<1$, the authors showed that the problem does not admit traveling wave solutions by proving that level sets of the solutions of the initial-value problem with suitable initial data propagate exponentially fast in time.
On the other hand, and independently, Cabré and Roquejoffre ([@CabreRoquejoffre2]) studied the case of fractional linear diffusion $u_{t}(x,t) + (-\Delta)^s u(x,t)=f(u)$, where $(-\Delta)^s$ is the Fractional Laplacian operator with $s \in (0,1)$ and concluded in the same vein that there is no traveling wave behavior as $t\to\infty$, and indeed the level sets propagate exponentially fast in time. This came as a surprise since their problem deals with linear diffusion.
Motivated by these two examples of break of the asymptotic TW structure, we studied in [@StanVazquezKPP] the case of a diffusion that is both fractional and nonlinear. More exactly, we consider the following reaction-diffusion problem $$\label{KPP}
\left\{ \begin{array}{ll}
u_{t}(x,t) + (-\Delta)^s u^m(x,t)=f(u) &\text{for } x \in {\mathbb{R}^N}\text{ and }t>0, \\
u(x,0) =u_0(x) &\text{for } x \in {\mathbb{R}^N},
\end{array}
\right.$$ We are interested in studying the propagation properties of nonnegative and bounded solutions of this problem in the spirit of the Fisher-KPP theory. We assume that the reaction term $f(u)$ satisfies $$\label{propf}
f\in C^1([0,1])\text{ is a concave function with } f(0)=f(1)=0 , \quad f'(1)<0<f'(0).$$ For example we can take $f(u)=u(1-u).$ The initial datum $u_0(x) : {\mathbb{R}^N}\rightarrow [0,1]$ and satisfies a growth condition of the form $$\label{dataAssump}
0\le u_0(x) \leq C|x|^{-\lambda(N,s,m)}, \quad \forall x \in {\mathbb{R}^N},$$ where the exponent $\lambda(N,s,m)$ is stated explicitly in the different ranges of the exponent $m$. In our work we establish the negative result about traveling wave behaviour, more precisely, we prove that an exponential rate of propagation of level sets is true in all cases. Due to the nonlinearity, the solution of the diffusion problems involved in the proofs does not admit an integral representation as the case $m=1$. Instead, we use as an essential tool the behavior of the fundamental solution of the Fractional Porous Medium Equation, also called Barenblatt solution, recently studied in [@VazBar2012]. This allows us to explain the propagation mechanism in simple terms: the exponential rate of propagation of the level sets of solutions (with initial data having a certain minimum decay for large $|x|$) is a consequence of the power-like decay behaviour of the fundamental solutions of the FPME. To be precise, the decay rate of the tail of these solutions as $|x|\to\infty$ is the essential information we use to calculate the rates of expansion. This information is combined with more or less usual techniques of linearization and comparison with sub- and super-solutions. We also need accurate lower estimates for positive solutions of this latter equation, and a further self-similar analysis for the linear diffusion problem. The delicate details are explained in [@StanVazquezKPP].
[**Main results.**]{} The existence of a unique mild solution of problem follows by semigroup approach. The mild solution corresponding to an initial datum $u_0 \in
L^1({\mathbb{R}^N}),$ $0\leq u_0\leq1$ is in fact a positive, bounded, strong solution with $C^{1,\alpha}$ regularity. Let us introduce some notations. Once and for all, we put $\beta=1/(N(m-1)+2s)$ and $$\label{sigma}
\sigma_1 =\frac{1-m}{2s}f'(0), \quad \sigma_{2}=\frac{1}{N+2s}f'(0), \quad \sigma_3=\frac{1+2(m-1)\beta s}{N+2s}f'(0).$$ The value $\sigma_1$ appears for $m_c<m<m_1$ and then $\sigma_1>\sigma_2$. Notice also that $\sigma_{2}<\sigma_3$ for $m>1$. Here is the precise statement of our main results for the solutions of the generalized KPP problem .
\[mainThm1\] Let $N\geq 1$, $s\in (0,1)$, $f$ satisfying and $m_1<m \leq 1$. Let $u$ be a solution of , where $0\leq u_0(\cdot)\leq 1$ is measurable, $u_0 \neq 0$ and satisfies $$0\le u_0(x) \leq C|x|^{-(N+2s)}, \quad \forall x \in {\mathbb{R}^N}.$$ Then
1. if $\sigma>\sigma_{2}$, then $u(x,t) \rightarrow 0$ uniformly in $\{|x|\geq e^{\sigma t} \}$ as $t\rightarrow \infty$.
2. if $\sigma<\sigma_{2}$, then $u(x,t) \rightarrow 1$ uniformly in $\{|x|\leq e^{\sigma t} \}$ as $t\rightarrow \infty$.
\[mainThm2\] Let $N\geq 1$, $s\in (0,1)$, $f$ satisfying and $m_s<m<m_1$. Let $u$ be a solution of , where $0\leq u_0(\cdot)\leq 1$ is measurable, $u_0 \neq 0$ and satisfies $$0\leq u_0(x) \leq C |x|^{-2s/(1-m)}, \quad \forall x \in {\mathbb{R}^N}.$$ Then
1. if $\sigma>\sigma_{1}$, then $u(x,t) \rightarrow 0$ uniformly in $\{|x|\geq e^{\sigma t} \}$ as $t\rightarrow \infty$.
2. if $\sigma<\sigma_{1}$, then $u(x,t) \rightarrow 1$ uniformly in $\{|x|\leq e^{\sigma t} \}$ as $t\rightarrow \infty$.
\[mainThm3\] Let $N\geq 1$, $s\in (0,1)$, $f$ satisfying and $m>1$. Let $u$ be a solution of , where $0\leq u_0(\cdot)\leq 1$ is measurable, $u_0 \neq
0$ and satisfies $$0\le u_0(x) \leq C|x|^{-(N+2s)}, \quad \forall x \in {\mathbb{R}^N}.$$ Then
1. if $\sigma>\sigma_{3}$, then $u(x,t) \rightarrow 0$ uniformly in $\{|x|\geq e^{\sigma t} \}$ as $t\rightarrow \infty$.
2. if $\sigma<\sigma_{2}$, then $u(x,t) \rightarrow 1$ uniformly in $\{|x|\leq e^{\sigma t} \}$ as $t\rightarrow \infty$.
Our main conclusion is that exponential propagation is shown to be the common occurrence, and the existence of traveling wave behavior is reduced to the classical KPP cases mentioned at the beginning of this discussion
Equations similar to present interest for various researcher groups, like the groups led by X. Cabré, J.M. Roquejoffre, H. Berestycki, F. Hamel, and P. Felmer.
Current work and comments
=========================
A number of related models, issues and perspectives on elliptic and parabolic equations involving fractional Laplacians and more general integral operators is under way.
$\bullet $ [**Numerics.**]{} The numerical analysis of the solutions of the FPME was started by Teso [@Teso] where the case $s=1/2$ was studied. The extension method that we use to implement the fractional Laplacian has a number of specific difficulties for $s\ne 1/2$ that we have addressed in a subsequent paper [@TesoVaz]. One of the main differences of our work is that we do not directly deal with the integral formulation of the fractional Laplacian; instead of this, we pass through the Caffarelli-Silvestre extension, mentioned above.
Previous works dealing with the numerical analysis of nonlocal equations of this type are due to Cifani, Jakobsen, and Karlsen in [@cjakobk]. In particular, they formulate some convergent numerical methods for entropy and viscosity solutions. The numerical analysis of the elliptic PDE $(-\Delta)^s u =f$ in a bounded domain with zero boundary data via the extension method has been recently studied by Nochetto and collaborators using finite elements, [@NOS13].
$\bullet$ [**Disclaimer.**]{} We have not covered the extensive work on stationary states, i.e., elliptic equations of fractional type. Neither did we go into the probabilistic approach that comes from long ago and has been very actively pursued up to these days. Evolution equations involving the fractional Laplacian appear in many applied models and some of this directions are now actively pursued, like quasi-geostrophic flows, a topic that we have touched in the paper [@pqrv3], but has not been addressed here. Another interesting direction concerns geometric flows with fractional operators. This list does not aim at being representative. Finally, we have not discussed the nonlocal diffusion evolution model treated in the monograph [@AMRT] which uses integral operators with kernels that do not resemble the fractional Laplacians and lead to a different theory.
Work partially supported by Spanish Project MTM2011-24696. The author is very grateful to his main collaborators in the recent part of this effort: Matteo Bonforte, Arturo de Pablo, Fernando Quirós, Ana Rodríguez, Diana Stan, Félix del Teso, and Bruno Volzone.
[10]{}
, SIAM J. Math. Anal. [**44**]{}, 2 (2012), 603–632.
, arXiv:1307.1218.
. [Annales IHP, Analyse non linéaire]{} [**28**]{} (2011), no. 2, 217–246.
Comm. Pure Appl. Math. [**61**]{} (2008), no. 11, 1495–1539.
. [“Nonlocal Diffusion Problems”]{}. AMS Mathematical Surveys and Monographs, vol. 165, 2010.
. Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009.
Nonlinear diffusion problems (Montecatini Terme, 1985), 1–46, Lecture Notes in Math., 1224, Springer, Berlin, 1986.
. [*The initial trace of a solution of the porous medium equation*]{}, [Trans. Amer. Math. Soc.]{} **280** (1983), 351–366.
, **30** (1978), no. 1, 33–76.
. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 \[34\], 49–66, 226; translation in J. Math. Sci. (N. Y.) 132 (2006), no. 3, 274–284.
. Advances in Mathematics 224 (2010), no. 1, 293–315.
J. Diff. Geom. [**11**]{} (1976), 573–598.
. [*Isoperimetric inequalities and applications.*]{} Monographs and Studies in Mathematics, 7. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.
. [*On some unsteady motions of a liquid or a gas in a porous medium.*]{} [Prikl. Mat. Mekh.]{} [**16**]{}, 1 (1952), 67–78 (in Russian).
. “Scaling, self-similarity, and intermediate asymptotics”, volume 14 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1996.
Lévy processes”. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. ISBN: 0-521-56243-0.
Nonlinearity [**22**]{} (2009), no. 3, 683–710.
, M3AS [**22**]{}, Supp. 1, (2012), special issue.
Comptes Rendus Mathematique [**349**]{}, 11 (2011), 641–645.
arXiv preprint arXiv:1302.7219, 2013.
. Comm. Math. Phys. [**294**]{} (2010), no. 1, 145–168. MR2575479.
Trans. Amer. Math. Soc. [**95**]{} (1960), no. 2, 263–273.
, *Physical Review E* **62**, (2000).
, [J. Funct. Anal.]{} **240**, no.2, (2006) 399–428.
, [Advances in Math.]{} **223 (2010), 529–578.**
. Proceedings Natl. Acad. Sci. USA, [**107**]{}, no. 38 (2010) 16459–16464.
. [*Quantitative Local and Global A Priori Estimates for Fractional Nonlinear Diffusion Equations.*]{} Advances in Math. [**250**]{} (2014) 242–284. In arXiv:1210.2594.
. [*A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains*]{}. Preprint arXiv:1311.6997 \[math.AP\].
. [*A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains. Part II*]{}, In preparation.
. [*Limiting embedding theorems for $W^{s,p}$ when $s\to 1$ and applications.*]{}
, Adv. in Math. **224** (2010), no. 5, 2052–2093.
, **347** (2009), no. 23-24, 1361–1366.
, **320** (2013) no.3, 679–722.
, Comm. Partial Differential Equations [**4**]{} (1979), 1067–1075.
, J. Amer. Math. Soc [**24**]{} (2011), 849–869.
, Invent. Math. [**171** ]{} (2008), no. 2, 425–461.
. Comm. Partial Differential Equations [**32**]{} (2007), no. 7-9, 1245–1260.
J. Eur. Math. Soc. (JEMS) [**15**]{} 5 (2013), 1701–1746. [arXiv 1201.6048v1]{}, 2012.
. Arch. Rational Mech. Anal. [**202**]{} (2011), 537–565.
, Discrete Cont. Dyn. Systems-A [**29**]{}, no. 4 (2011), 1393–1404. Ann. of Math. [**171**]{} (2010), 1903–1930.
, Comm. Partial Diff. Eq. **36** (2011), no. 8, 1353–1384
In preparation.
. J. Eur. Math. Soc. (JEMS) [**12**]{} (2010), no. 5, 1307–1329.
. Ann. Inst. H. Poincaré Anal. Non Linéaire [**28**]{}, no 3 (2011), 413–441.
BIT [**51**]{} (4) (2011), 809–844.
, Eur. J. Appl. Math. [**7**]{} (1996), no. 2, 97–111.
Chapman & Hall/CRC, 2004.
. [ “Heat kernels and spectral theory”,]{} Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1990. x+197 pp. ISBN: 0-521-40997-7
, **93** (1991), no. 1, 19–61.
, Advances in Mathematics 226 (2011), no. 2, 1378–1409.
. Comm. Pure Appl. Math. [**65**]{} (2012), no. 9, 1242–1284.
. [*Classical solutions for a logarithmic fractional diffusion equation*]{}. Journal de Math. Pures Appliquées, to appear. Posted as arXiv:1205.2223v2 \[math.AP\].
. [*Hitchhiker’s Guide to the Fractional Sobolev Spaces*]{}, Preprint, 2011.
, Travaux et Recherches Mathématiques, No. 21. Dunod, Paris, 1972.
, Phys. Rev. B [**50**]{} (1994), no. 3, 1126–1135.
, **7** (1937), 355–369.
, Trans. Amer. Math. Soc. [**101**]{} (1961), 75–90.
Phil. Mag. [**26**]{} (1972), 65–72.
, Preprint, 2013.
, Comm. Pure Applied Math. [**62**]{}. no. 2 (2009), 198–214.
. [*Hydrodynamic Limit Of Particle Systems With Long Jumps*]{},\
`http://arxiv.org/abs/0805.1326v2`
. *Ann. Appl. Probab.* **19** (2009), no. 6, 2270–2300.
Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes, *Probab. Theory Relat. Fields* [**145** ]{}(2009), 565–590.
, Calc. Var. [**34**]{} (2009) 1–21.
, **459** (2003), no. 2038, 2529–2546.
Invent. Math. [**167**]{} (2007), no. 3, 445–453.
, **17** (1937), 1–26
Die Grundlehren der mathematischen Wissenschaften, Band 180. Translated from the Russian by A. P. Doohovskoy. Springer, New York, 1972.
Crossover in diffusion equation: Anomalous and normal behaviors, *Physical Review E* **67**, 031104 (2003).
(2000), 121–142.
. [*On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces*]{}. Journal Funct. Anal. [*195*]{}(2002) 230–238.
. Preprint, http://arxiv.org/abs/0809.2455.
, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, [*Physics Reports*]{} [**339**]{} (2000) 1-77.
. Calc. Var. PDEs [**526**]{}, online. In arXiv: 1205.632229 \[math.AP\].
. to appear. `http://www.ma.utexas.edu/mparc/c/12/12-123.pdf`
, Rendiconti di Matematica e delle sue applicazioni [**18**]{} (1959), 95–139.
Indiana Univ. Math. J. [**55**]{} (2006), no. 3, 1155–1174.
. Comm. Pure Appl. Math. [**60**]{} (2007), no. 1, 6–112.
Siam Journal Mathematical Analysis, to appear; [arXiv:1303.6823]{}.
Comptes Rendus Mathématique (Comptes Rendus Acad. Sci. Paris), online; arXiv:1311.7007 \[math.AP\].
. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
Ann. Scuola Norm. Sup. (4) 3 (1976), 697–718.
Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.
To appear in Calcolo, [arXiv:1301.4349]{}, (2013).
In arxiv 1307.2474v1.
, Bol. Soc. Esp. Mat. Apl. [**49**]{} (2009), 33–44.
. [*Symétrisation pour $u_t=\Delta\varphi(u)$ et applications,*]{} C. R. Acad. Sc. Paris [**295**]{} (1982), pp. 71–74.
. Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type”. Oxford Lecture Series in Mathematics and its Applications, 33. Oxford University Press, Oxford, 2006. . [“The Porous Medium Equation. Mathematical Theory”]{}, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007). . [*Nonlinear Diffusion with Fractional Laplacian Operators.*]{} in “Nonlinear partial differential equations: the Abel Symposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer, 2012. Pp. 271–298. Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, [*To appear in J. Europ. Math. Soc.,* ]{}(2013). Preprint: `http://arxiv.org/abs/1205.6332`
. [*Classical solutions and higher regularity for nonlinear fractional diffusion equations*]{}, arXiv:1311.7427 \[math.AP\].
. [*Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type,*]{} arXiv:1303.2970. To appear in J. Math. Pures Appl.
. [*Optimal estimates for Fractional Fast diffusion equations*]{}. Submitted. ArXiv:1310.3218v1 \[math.AP\].
. In Order and chaos”, 10th volume, T. Bountis (ed.), Patras University Press (2008).
. [*Symmetrization in uniformly elliptic problems*]{}, Studies in Math. Anal., Stanford Univ. Press, 1962, pp. 424–428.
. Commun. Nonlinear Sci. Numer. Simul. 8 (2003), no. 3-4, 273–281.
, in [Lévy Processes – Theory and Applications]{}, T. Mikosch, O. Barndorff-Nielsen and S. Resnick, Eds., Birkhäuser, Boston 2001, pp. 241–266.
. In [“Collection of papers dedicated to 70th Anniversary of A. F. Ioffe”]{}, Izd. Akad. Nauk SSSR, Moscow, 1950, pp. 61–72.
2010 *Mathematics Subject Classification.* 26A33, 35K55, 35K65, 35S10.
*Keywords and phrases.* Nonlinear diffusion, fractional Laplacian operator.
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abstract: 'Within 2+1-dimensional cosmological new massive gravity, we consider thin-shell and thin-shell wormhole construction. For this, we introduce first, the junction conditions apt for the fourth order terms in the action of the theory. Then, by employing some specific static solutions in new massive gravity, we study the characteristics of associated thin-shells and thin-shell wormholes. Our finding suggests that, firstly, there cannot exist any thin-shells regarding our chosen solutions of cosmological new massive gravity, and secondly, the constructed thin-shell wormhole does not need to be symmetric. More importantly, the thin-shell wormhole, if ever forms, possesses null energy density and null angular pressure on its throat which preferable to their negative-valued counterparts.'
author:
- 'S. Danial Forghani'
- 'S. Habib Mazharimousavi'
- 'M. Halilsoy'
title: 'Thin-Shells and Thin-Shell Wormholes in New Massive Gravity'
---
Introduction
============
Spherically symmetric thin-shells (TSs) have been the subject of many interesting studies in the literature. For instance, long ago, in 1965, Arnowitt, Deser and Misner showed that the self-energy of a charged dust shell is finite [@Arnowitt]. In 1973 Boulware in his short paper [Boulware]{} studied a charged TS whose inside and outside spacetimes are flat Minkowski and Reissner-Nordström, respectively. In there, using the Israel formalism, the dynamics of the shell was studied and it was shown that if the matter energy density of the shell is negative then the shell may collapse to form a naked singularity. A general spherical shell, however, was studied by Lake in 1978 [@Lake]. Lake showed that, imposing positive definite total proper mass for the shell implies the impossibility of the merge of the inner and black-hole horizons. In [@Heusler], Heusler *et al.* calculated the self-energy of a charged TS with flat spacetime inside and Reissner-Nordström outside. While their work for a dust shell gives the results of [@Arnowitt], some of their findings were already found in other papers [@Kuchar; @Chase]. Later on, spherically symmetric TS in Brans-Dicke theory of gravity was considered by Letelier and Wang in [@Wang]. Furthermore, the linear stability analysis of spherically symmetric timelike shells and bubbles attracted attentions [Simeone]{}. For instance, in [@Brady], the stability of a TS surrounding a Schwarzschild black hole was studied. It was shown that such a TS may be stable against a radial perturbation if its radius is larger than the radius of the circular photon orbit. Finally we would like to mention the collapsing of the higher-dimensional spherically symmetric TS with rotation which has been studied recently [@Delsate]. Such kind of study in $3+1-$dimensions with different symmetry, prior to [@Delsate], had already been studied in [@Nolan; @Mena; @Prisco].
Wormholes appeared in general relativity as special solutions of the Einstein’s field equations. These bizarre structures can be visualized as tunnels connecting remote points within a single spacetime or points of two distinct spacetimes. The main problem of wormholes is that they are supported by exotic matter; a kind of matter which does not satisfy known energy conditions. In 1988, Morris and Thorne published a paper and discussed traversability of wormholes [@Morris1]. The paper soon came to focus of researchers and caused a new wave of studies over structural characteristics of wormholes. In 1989, in one of these attempts, Visser developed a method nowadays known as the cut-and-paste procedure, with which a new class of traversable wormholes came into existence [@Visser1]. The idea of these so-called thin-shell wormholes (TSWs) gained much popularity since its original construction. The principal aim was how to tackle with the exotic matter that necessarily gave life to such an object. Visser’s recipe was to confine the exotic matter to a very narrow surface, i.e. the thin-shell, so that its existence can be justified in some way. Another advantage of TSWs is that they can be constructed by a vast variety of spacetimes, as there are already many TSWs appearing in the literature [Richarte1,Varela1,Sharif1,Sharif2,Montelongo1,Dias1,Eiroa1,Thibeault1,Lobo1,Eiroa2,Eiroa3,Ishak1,Poisson1]{}. However, almost all TSWs suffer from two serious drawbacks: the exotic matter as source, and fragile/limited and/or non-physical stability characteristics. This point is precisely the reason that the challenge of finding stable TSWs is still going on, hopefully, with no exotic matter on them. In this regard, we make our search of TSWs in a special, modified massive theory of gravity. (For more details on wormholes and TSWs the reader is advised to consult [@Visser2] and references therein.)
The mass of the quantum gravity’s fundamental particle, the graviton, has been one of the most disputable subjects of modern physics. In massive theories of gravity, the spin-2 graviton is destined to move inside the local light cone, not on it. To point out the importance of massive theories of gravity let us draw a rough analogy with the Standard Model (SM) of particle physics by recalling that neutrino shared a similar history. In the SM, neutrino was also thought to be massless for a long time. With the advent of experimental neutrino physics, the picture has changed: neutrino has a very small but non-zero mass [@Mertens1]. This amounts to changing much of the rules in the SM, leading even to revise certain proportion of the textbooks. In analogy, with massive gravity a certain revision in the fundamental physics of gravitational waves is expected.
To prove the mass of a graviton, however, is more challenging than the neutrino. In the gravity side, even at a classical level, we had to wait until very recently when LIGO and Virgo detected gravitational waves from the merger of massive black holes in distant past [@LIGO1; @Virgo1]. Assuming that the gravity waves are massive, their speed in vacuum will be less than the speed of light and this will naturally cause a delay in their arrival to the Earth. Nonetheless, an observation by LIGO and Virgo in 2017, put a constraint on this time delay, limiting the difference between the speed of gravitons and the speed of photons to the interval $\left( -3\times
10^{-15},+7\times 10^{-16}\right) $-times the speed of light [@Virgo1]. This means, that there still is an uncertainty in the mass of gravitons, and until the time the precision of our measurements lets us decisively confirm or reject the existance of massive gravitons, theoretical physics will keep contributing to the concept. See [@Hinterbichler1; @de; @Rham1] as review studies, and references therein for more details on massive gravity.
In literature, it was Fierz and Pauli who added a mass dependent term to the gravity action for the first time in 1939 [@Fierz1]. Since then, there has been so many attempts to establish a consistent quantum gravity theory, especially in 3+1 dimensions. However, most of these theories suffer from a common disadvantage: absence of renormalizable theory. While renormalization is a big problem in 3+1 dimensions, things are different in 2+1. From simple power counting method of field theory, the lower dimension expectedly has natural advantages over the higher dimension [@Gavela1]. Since 2+1-dimensional Weyl curvature vanishes identically, it is well-known that there are no pure gravitational degrees of freedom. For this reason, in order to create a theory, source must be supplied in the lower dimension to make a gravitationally feasible theory. This is done by different methods, among which, two received more endorsements.
In an attempt to construct such a theory in 2+1 dimensions, Deser *et al.* established the theory of Topological Massive Gravity (TMG) in 1982 by adding a Chern-Simons term to the 3-dimensional Einstein-Hilbert (EH) action [@Deser1; @Deser2]. In 2009, Bergshoeff *et al.* suggested a different theory which later was known as New Massive Gravity (NMG) [Bergshoeff1,Bergshoeff2]{}. This theory, which at the linearized level is the 2+1 dimensional equivalent of Pauli-Fierz theory, has the advantage over TMG that preserves parity symmetry [@Bergshoeff1; @Bergshoeff2; @Bergshoeff4]. Furthermore, it is shown in [@Oda1] and [@Nakasone1] that the theory is unitary in tree level and renormalizable. de Rham *et al.* [@de; @Rham2] go further to indicate that NMG is unitary even beyond the tree level. In [@Accioly1] the authors prove that NMG at tree level is actually the only 3-dimensional unitary system which can be constructed by adding quadratic curvature terms to EH action. Add to all these remarkable characteristics, its invariance under general coordinate transformations is also manifest. NMG, therefore, gained much attentions right after its introduction, for being a promising candidate for a renormalizable theory of quantum gravity. The theory also exhibits features like gravitational time dilation and time delay which the usual 3-dimensional general relativity is not subject to [@Accioly1; @Accioly2]. Soon after the first publication, it was shown that it admits exact black hole solutions. In [@Clement1] warped AdS$_{3}$ black holes and AdS$_{2}\times $S$^{1}$ solutions, in [Bergshoeff2]{} Bañados-Teitelboim-Zanelli (BTZ) [@Banados1], new type black holes, and warped dS$_{3}$ and dS$_{2}\times $S$^{1}$ solutions, in [@Clement2] extreme BTZ and a family of massive ‘log’ black holes, and in [@Ayon-Beato2] and [@Clement2] AdS waves are discussed. Another important contribution is made by Oliva *et al.* who investigated exact black hole and non-black hole solutions of a special case with negative, positive and vanishing cosmological constant in [@Oliva1]. Moreover, the Lifshitz metrics have been shown to be solutions of NMG for generic values of the dynamical exponent $z$ (with an exact - asymptotically Lifshitz - black hole solution at $z=3$) [@Ayon-Beato1]. Ahmedov and Aliev in a series of exquisite papers [@Ahmedov1; @Ahmedov2; @Ahmedov3] discuss algebraic type D and type N solutions of NMG by employing a first-order differential Dirac-type operator. They indicate that the NMG field equations can be considered as square of TMG field equations, and accordingly, argue the possibility of mapping all types D and N solutions of TMG into NMG. Besides, they find new types D and N solutions in NMG with no counterparts in TMG. The stability of BTZ black holes in NMG are classically studied in [@Myung1]. Some of these aforementioned solutions will be considered here in this study.
Among all these, AdS$_{3}$ solutions are of greater importance for an obvious reason: Where there is a quantum-gravity consistent theory along with AdS solutions, there exists the AdS/CFT correspondence [@Maldacena1]. However, it was shown that on the boundary of the dual CFT, the unitarity of the AdS vacuum connote a negative central charge [@Bergshoeff2; @Liu1]. Also, for logarithmic CFT correspondence (AdS$_{3}$/LCFT$_{2}$) see [Grumiller1]{}. Setare and Kamali in [@Setare2] show that there is a perfect agreement between their results using 2-dimensional Galilean conformal algebra on the boundary of NMG with the Bekenstein-Hawking entropy (in nonrelativistic limit) for warped AdS$_{3}$ and contracted BTZ black hole solutions of NMG. Phase transitions between BTZ black hole solutions and thermal solitons within NMG are studied in [@Eune1; @Myung2; @Zhang1]. It is also worth-mentioning that later, the new type black holes initially appeared in [@Bergshoeff2], came to the attention of Kwon *et al.,* who obtained their quasi-normal modes [@Kwon1], and Gecim and Sucu, who studied the properties of relativistic spin-1/2 and spin-0 particles in this background [@Gecim1].
NMG has also been generalized to 4th [@Bergshoeff3] and higher arbitrary dimensions [@Dalmazi2]. Although, it is summed up in [@Nakasone1] that the higher dimensional generalizations are not unitary at the tree level.
Furthermore, there have been attempts to extend NMG. Among these, we point out the novel works by Güllü *et al.* [@Gullu1; @Gullu2], which extend NMG to a 3-dimensional Born-Infeld theory of gravity (BI-NMG). There, the authors discuss that the cubic order extension of their augmented action duplicates the deformation of the NMG gained from AdS/CFT correspondence. Exact black hole solutions of BI-NMG are discussed in [Ghodsi1]{}, where properties such as mass, angular momentum, entropy and CFT dual central charges of the solutions are also determined. Extensions to higher curvature theories (R$^{3}$-NMG) and their exact solutions are considered in [@Sinha1; @Nam1; @Anastasiou1; @Setare1]. In [@Dalmazi1] even higher derivative kinetic terms are discussed. Algebraic type N spacetime solutions to BI-NMG and their higher order curvature corrections of NMG are studied in [@Ahmedov4]. An extension of the theory by scalar matter with Higgs-like self-interaction is investigated in [@Louzada1] with exact asymptotically dS$_{3}$ solutions which qualifies as an eternally accelerated non-singular bounce-like 3D Universe. Another extension by scalar matter is discussed in [@Camara1], where the authors study a family of flat static domain walls as solutions. In [@Ghodsi2], the NMG action is coupled to Maxwell’s electromagnetic and Chern–Simons actions to give rise to charged black holes in both warped AdS$_{3}$ and log forms. Generalized Massive Gravity (GMG), whose action contains quadratic terms of both TMG and NMG along with coupling constants, and all its homogenous solutions are studied in [@Bakas1]. Exploiting the NMG action, a new bi-gravity model is constructed in 3 dimensions in [@Akhavan1]. Finally, a novel work by Dereli and Yetişmişoğlu suggests a model (new improved massive gravity (NIMG)) which includes TMG, NMG and minimal massive gravity (MMG) as subclasses of the theory [@Dereli1; @Dereli2].
In this paper we particularly consider the cosmological new massive gravity (CNMG) in 2+1-dimensions [@Bergshoeff2] and construct TSs and TSWs in such a theory.
Our findings are interesting from physics point of view. On one hand, we find that the propounded metrics admit no TS. This TS-nonexistence is particularly important when it holds for AdS metrics. On the other hand, we find TSWs within CNMG which can be stable, but come to exist only when are asymmetric, in the sense that the bulk spacetimes on the two sides are different in geometry and nature. These rather new type of TSWs are called asymmetric TSWs (ATSWs). As spherical and cylindrical ATSWs in general relativity, these are explicitly considered in [Forghani1,Forghani2,Forghani3]{}, while spherical ATSWs in $F(R)$ gravity are studied implicitly in [@Eiroa4]. The tidbit is that for the cases we are studying here, such ATSWs do not need matter (neither ordinary nor exotic) as support, and hence, provide a smooth passage from one universe to the other. Stated otherwise, our TS with zero energy-momentum acts as vacuum as long as it is not perturbed. Once perturbed the perturbation energy will accumulate a non-zero energy-momentum on the TS.
To achieve this aim, we choose a class of static solutions in CNMG and introduce the necessary junction conditions (JCs) apt for a higher order theory. This amounts to revision of Darmois-Israel JCs (DI-JCs) that were designed for Einstein’s general relativity [@Israel1]. The qualified JCs in quadratic gravity are mentioned in [@Deruelle1], and later in 2016, revised in [@Reina1]. It is worth mentioning that Eiroa *et al.* successfully applied the latter JCs to establish TSWs with a double layer [@Eiroa4], pure double-layer bubbles [@Eiroa5], and spherical TS [@Eiroa6] in $F\left( R\right) $ theory of gravity.
The paper is organized as follows. In section $II$ we briefly introduce the CNMG theory and a class of static solutions of the theory. Construction of TS and TSWs are generally discussed in section $III$. Section $IV$ is devoted to introducing the proper JCs in CNMG, which are applied to legitimize the details of the TS and TSW’s constructions in subsections. Finally, section $V$ completes the paper with our conclusion.
Cosmological New Massive Gravity Solutions
==========================================
The NMG theory is based on the $2+1$-dimensional cosmological new massive gravity (CNMG) action [@Bergshoeff2]$$\begin{gathered}
I_{\text{CNMG}}=\frac{1}{2\kappa }\int d^{3}x\sqrt{-g} \\
\left( \varsigma R+\frac{1}{m^{2}}\left( R_{\mu \nu }R^{\mu \nu }-\frac{3}{8}%
R^{2}\right) -2\lambda m^{2}\right) ,\end{gathered}$$in which $\kappa ^{-1}$ is the three dimensional reduced Planck mass, $m$ is the mass of the graviton, $R_{\mu \nu }$ and $R$ define the Ricci tensor and the Ricci scalar, respectively, and $\lambda $ is a dimensionless cosmological parameter. The factor $\varsigma $ in Einstein-Hilbert term is merely a convention dependent factor which takes on either $1$ or $-1$.
In this piece of work we consider the solutions of the theory which can be cast into the generic circularly symmetric form$$ds^{2}=-f\left( r\right) dt^{2}+\frac{1}{f\left( r\right) }%
dr^{2}+H^{2}\left( r\right) d\theta ^{2},$$where$$f\left( r\right) =c_{0}+c_{1}r+\frac{1}{2}c_{2}r^{2}$$and $H\left( r\right) $ are functions of the radial coordinate $r$. Herein, $%
c_{0}$ and $c_{1}$ are integration constants to be interpreted as the mass parameter and gravitational hair of the given spacetime, respectively. The cosmological-like parameter $c_{2}$ can be reparametrized by the cosmological parameter $\lambda $ as $$c_{2}=4m^{2}\left( \varsigma \pm \sqrt{1+\lambda }\right) .$$Setting $H\left( r\right) =r$ comprises BTZ, warped (A)dS$_{3}$ and new type black hole solutions, while setting $H\left( r\right) =1$ recovers non-black hole (A)dS$_{2}\times $S$^{1}$ solutions. These are explained below in more details. Note that, in general, for $\lambda <-1$ there is no solution, and for $\lambda =0$ the metric represents a flat spacetime.
$H\left( r\right) =r$:
a\) For $\lambda >-1$ we must have $c_{1}=0$. These solutions for $\varsigma
=1$ are locally isometric to AdS$_{3}$ and for $\varsigma =-1$ are locally isometric to dS$_{3}$. In the special case of $c_{0}=1$ one recovers (A)dS$%
_{3}$ vacua. Furthermore, in AdS$_{3}$ case $c_{0}<1$ admits static BTZ black holes with mass parameter $\mu =-c_{0}$. Also, for $\lambda =0$ the solution is trivially flat.
b\) For $\lambda =-1$ the solutions are called new type black holes. These special vacuum solutions for $\varsigma =1$ ($c_{2}>0$) exhibit asymptotically AdS$_{3}$ unique vacua, while they are asymptotically dS$_{3}$ for $\varsigma =-1$ ($c_{2}<0$). In this case, the metric function $f\left(
r\right) $ , provided $c_{1}^{2}-8\varsigma m^{2}c_{0}\geq 0$, has a real double root at$$r_{1,2}=\frac{1}{4m^{2}}\left( -\varsigma c_{1}\pm \sqrt{c_{1}^{2}-8%
\varsigma m^{2}c_{0}}\right) .$$In AdS$_{3}$ case, when $r_{1}>0$ we have an asymptotically AdS black hole with its horizon at $r=r_{1}$. There will also be an inner horizon at $%
r=r_{2}$ in case $r_{2}>0$. For dS$_{3}$, on the other hand, there potentially exist two horizons. When $r_{1}>0$, the surface $r=r_{1}$ is similar to the cosmological horizon of dS spacetime. If also $r_{2}>0$, we will have a black hole with an event horizon at $r=r_{2}$. In this case, the occurrence of double roots implies that in between the roots, $r_{2}<r<r_{1}$, the static spacetime remains static, whereas for $r<r_{2}$ and $r>r_{1}$becomes dynamic with $t\longleftrightarrow r$.
$H\left( r\right) =1$:
For $\lambda =-1$ the metric is Kaluza-Klein (KK) type vacuum solution, i.e. locally isometric to AdS$_{2}\times $S$^{1}$ when $\varsigma =1$ ($c_{2}>0$) [@Clement1], and to dS$_{2}\times $S$^{1}$ when $\varsigma =-1$ ($%
c_{2}<0 $) [@Bergshoeff2].
Let us note that $c_{0}$ and $c_{1}$ are parameters of the solution, while $%
c_{2}$ is related to the essential parameters of the theory, $m$ and $%
\lambda $.
Thin-Shell and Thin-Shell Wormhole Construction
===============================================
Since the construction procedure of TSWs has been given extensively and repeatedly in the literature [@Poisson1], we shall make our presentation in this section very brief.
In general, construction of TS and TSW have similarities and differences. To construct a TS, consider two distinct Lorentzian spacetimes denoted by $%
\left( \Sigma ,g\right) ^{-}=\{x^{\mu }|r\leq a\}$ as the inner and $\left(
\Sigma ,g\right) ^{+}=\{x^{\mu }|r\geq a\}$ as the outer spacetimes, which are distinguished by their common hypersurface $\partial \Sigma =\{x^{\mu
}|r=a\}$ such that $\partial \Sigma \subset $ $\left( \Sigma ,g\right) ^{\pm
}$. These two necessarily non-symmetric spacetimes represent smooth manifolds and contain no singularities, event horizons or kinks. The coordinates of these two spacetimes are not necessarily the same and will be denoted by $x_{\pm }^{\mu }$. The line element on the hypersurface $\partial
\Sigma $ (the TS) is given by$$ds^{2}=h_{ij}^{\pm }d\xi ^{i}d\xi ^{j},$$where $\xi ^{i}$ are the local coordinates on the shell and $h_{ij}^{\pm }=%
\frac{\partial x_{\pm }^{\mu }}{\partial \xi ^{i}}\frac{\partial x_{\pm
}^{\nu }}{\partial \xi ^{j}}g_{\mu \nu }^{\pm }$ is the localized metric of $%
\partial \Sigma $. In the next section we will discuss that in fact $%
h_{ij}^{-}=h_{ij}^{+}$ must hold on the shell. The unit normal to the surface is also given by $n_{\mu }^{\pm }\frac{\partial x_{\pm }^{\mu }}{%
\partial \xi ^{i}}=0$; $n_{\mu }^{\pm }n_{\pm }^{\mu }=1$.
In case of a TSW, we cut a region of each spacetime, such that $\left(
\Sigma ,g\right) ^{\pm }=\{x^{\mu }|r\geq a>r_{e}\}$, in which $r_{e}$ is any existed event horizon. Then we glue the two regions at their common hypersurface $\partial \Sigma \subset $ $\left( \Sigma ,g\right) ^{\pm }$ which usually is referred to as the throat of the TSW. Note that in most cases the two separated regions $\left( \Sigma ,g\right) ^{\pm }$ are copies of each other but this is not compulsory [@Forghani1]. Similar to the TS case, one finds the metric on the throat unique as approaching it, no matter from which side. The line element and the normal can be introduced in the same way as for a TS. The key difference between a TS and a TSW is that in a TS only one of the normals is chosen (for instance the one going into $%
\left( \Sigma ,g\right) ^{+}$ and out of $\left( \Sigma ,g\right) ^{-}$), while for a TSW both normals are considered independently. Therefore, while passing across the shell, the normal vector is continuous in one case (TS) and discontinuous in the other (TSW). In TSW, this distinction between normals is transmitted through all the extrinsic properties of the throat, for the normals play parts in them, by definition. Hence, one must be careful to hold the ($\pm $) signs of the normals for a TSW, while they can be dropped casually for the case of a TS. For the intrinsic properties (such as Riemann or Ricci tensors and Ricci scalar), of course, this is not the case.
The extrinsic curvature tensor of the TS(W) is given by$$K_{ij}^{\pm }=-n_{\lambda }^{\pm }\left( \frac{\partial ^{2}x_{\pm
}^{\lambda }}{\partial \xi ^{i}\partial \xi ^{j}}+\Gamma _{\alpha \beta
}^{\lambda \pm }\frac{\partial x_{\pm }^{\alpha }}{\partial \xi ^{i}}\frac{%
\partial x_{\pm }^{\beta }}{\partial \xi ^{j}}\right) ,$$where $\Gamma _{\alpha \beta }^{\lambda \pm }$ are the Christoffel symbols of the bulk spacetimes, compatible with $g_{\alpha \beta }^{\pm }$.
In the following section we shall introduce the proper JCs for the static solutions within NMG framework.
The Junction Conditions
=======================
In this section, we introduce two distinct sets of JCs, independently derived in [@Deruelle1] and [@Reina1] qualified for quadratic theories of gravity in arbitrary dimensions. In the latter, Reina, Senovilla and Vera (RSV) take advantage of the standard distributional analysis, while in the former, Deruelle, Sasaki and Sendouda (DSS) simplify the problem by using Gaussian coordinates at the joint hypersurface. In [@Reina1] RSV argue that using Gaussian coordinates often causes ignoring some important subtleties, and therefore, the reliability extend of the method is ambiguous (specially, when it comes to double layers). Nevertheless, since we are not considering double layers, for the sake of curiosity we will apply both methods, independently, and count similarities and differences, if there is any. In what follows we particularly concentrate on timelike hypersurfaces which make more physical sense.
THE RSV Junction Conditions
---------------------------
According to [@Reina1], a general quadratic Lagrangian density in $n+1$ dimensions has the form$$\begin{gathered}
\mathcal{L}_{\text{RSV}}=\sqrt{-g}\times \\
\left( R+a_{1}R^{2}+a_{2}R_{\mu \nu }R^{\mu \nu }+a_{3}R_{\alpha \beta \mu
\nu }R^{\alpha \beta \mu \nu }-2\Lambda \right) ,\end{gathered}$$where $a_{n}$ and $\Lambda $ are constants with physical dimensions of $%
\left[ L^{2}\right] $. A quick comparison with the CNMG action given in Eq. (1), reveals that $a_{1}=\frac{-3\varsigma }{8m^{2}}$, $a_{2}=\frac{%
\varsigma }{m^{2}}$ and $a_{3}=0$. Similar to the normal ID-JCs in general relativity, the diffeomorphism of the two spacetimes to be joined at the junction requires the first fundamental form to be continuous at their common hypersurface. This is the first JC and guarantees the identification of a global metric in the sense of distributions. To avoid non-physical distributional terms, in the case either $a_{2}$ or $a_{3}$ is nonzero, the second JCs are identified as the continuity of the second fundamental form at the junction. There are also other JCs which basically insure that the other fundamental generalized functions are well-defined, as well. RSV also introduce the parameters $\kappa _{1}=2a_{1}+a_{2}/2$ and $\kappa
_{2}=2a_{3}+a_{2}/2$ and classify their JCs in the case of a *TS without double layer* based on the values of $\kappa _{2}$ and $n\kappa _{1}+\kappa _{2}$. With regard to the coefficients $a_{1}$, $a_{2}$ and $a_{3}$, the proper JCs for 2+1-dimensional NMG are the ones for the case $\kappa
_{2}\neq 0$* and* $n\kappa _{1}+\kappa _{2}=0$.This is very interesting in the sense that even RSV would not think of this case as a serious one:
> *Nevertheless, the relevance of this exceptional case is probably marginal, as the coupling constants satisfy a dimensionally dependent condition.*
### TS construction with RSV-JCs
Accordingly, for investigating a TS, the proper JCs for $n=2$ will be
$$\left[ g_{\alpha \beta }\right] _{-}^{+}=0;$$
$$\left[ K_{\alpha \beta }\right] _{-}^{+}=0;$$
$$\begin{gathered}
\left[ R_{\alpha \beta \mu \nu }\right] _{-}^{+}=\frac{\left[ R\right]
_{-}^{+}}{4}\times \\
\left( n_{\alpha }n_{\mu }h_{\beta \nu }-n_{\beta }n_{\mu }h_{\alpha \nu
}-n_{\alpha }n_{\nu }h_{\beta \mu }-n_{\beta }n_{\nu }h_{\alpha \mu }\right)
;\end{gathered}$$
$$\left[ R_{\alpha \beta }\right] _{-}^{+}=\frac{\left[ R\right] _{-}^{+}}{2}%
\left( \frac{1}{2}h_{\alpha \beta }+n_{\alpha }n_{\beta }\right) ;$$
$$\begin{gathered}
\left[ \nabla _{\mu }R_{\alpha \beta }\right] _{-}^{+}=n^{\nu }\left[ \nabla
_{\nu }R_{\alpha \beta }\right] _{-}^{+}n_{\mu } \\
+\frac{1}{2}\left( \frac{1}{2}h_{\alpha \beta }+n_{\alpha }n_{\beta }\right)
\overline{\nabla }_{\mu }\left[ R\right] _{-}^{+} \\
-\frac{1}{4}\left[ R\right] _{-}^{+}\left( n_{\alpha }K_{\beta \mu
}+n_{\beta }K_{\alpha \mu }\right) ;\end{gathered}$$
$$S_{\alpha }^{\alpha }=0;$$
$$\begin{gathered}
\kappa \left[ S_{\alpha \beta }\right] _{-}^{+}=-\left( \kappa _{1}+\kappa
_{2}\right) \left[ R\right] _{-}^{+}K_{\alpha \beta } \\
+\kappa _{1}n^{\nu }\left[ \nabla _{\nu }R\right] _{-}^{+}+2\kappa
_{2}\delta _{\alpha }^{\rho }\delta _{\beta }^{\sigma }n^{\nu }\left[ \nabla
_{\nu }R_{\rho \sigma }\right] _{-}^{+}.\end{gathered}$$
Here, the $S_{\alpha }^{\alpha }$ is the trace of the energy-momentum tensor of the shell, $S_{\alpha \beta }$, in its distribution form. Also, $\nabla $ and $\overline{\nabla }$ are the covariant derivatives compatible with the metric $g$ of the bulks and $h$ of the shell, respectively. Furthermore, $%
\left[ \Psi \right] _{-}^{+}\equiv \Psi ^{+}-\Psi ^{-}$ denotes a jump in the function $\Psi $, passing across the thin-shell. Remark that although all the indices are in Greek, for the shell quantities they only take on the coordinates on the shell, i.e. $\left\{ t,\theta \right\} $.
For the metric defined in Eq. (2) we calculate the nonzero independent components of Riemann and Ricci tensors and Ricci scalar as follows
$$R_{trtr}=\frac{1}{2}f^{\prime \prime };$$
$$R_{t\theta t\theta }=\frac{1}{2}ff^{\prime }HH^{\prime };$$
$$R_{r\theta r\theta }=-\frac{H}{2f}\left( 2fH^{\prime \prime }+f^{\prime
}H^{\prime }\right) ;$$
$$R_{tt}=\frac{f}{2H}\left( f^{\prime \prime }H+f^{\prime }H^{\prime }\right) ;$$
$$R_{rr}=-\frac{1}{2fh}\left( f^{\prime \prime }+2fH^{\prime \prime
}+f^{\prime }H^{\prime }\right) ;$$
$$R_{\theta \theta }=-H\left( f^{\prime }H^{\prime }+fH^{\prime \prime
}\right) ;$$
$$R=-\frac{1}{H}\left( f^{\prime \prime }H+2f^{\prime }H^{\prime }+2fH^{\prime
\prime }\right) .$$
Herein, a prime ($^{\prime }$) denotes a total derivative with respect to the radial coordinate $r$. Now we are at the position to study the JCs in Eqs. (9).
As discussed in the previous section, the first JCs (Eq. (9a)) require the continuity of the metric on the shell. Therefore they admit
$$\left\{
\begin{array}{c}
f^{+}\left( a\right) =f^{-}\left( a\right) =f_{a} \\
H^{+}\left( a\right) =H^{-}\left( a\right) =H_{a}%
\end{array}%
\right. .$$
The second JCs require the first derivatives of the metric functions to be continuous on the shell, as well; i.e.$$\left\{
\begin{array}{c}
f^{+\prime }\left( a\right) =f^{-\prime }\left( a\right) =f_{a}^{\prime } \\
H^{+\prime }\left( a\right) =H^{-\prime }\left( a\right) =H_{a}^{\prime }%
\end{array}%
\right. \text{ }.$$The jump in the Ricci scalar is a degree of freedom and considering Eqs. (10g), (11) and (12) is given by$$\begin{gathered}
\left[ R\right] _{-}^{+}=\frac{1}{H_{a}}\times \\
\left[ H_{a}\left( f_{a}^{-\prime \prime }-f_{a}^{+\prime \prime }\right)
+2f\left( H_{a}^{-\prime \prime }-H_{a}^{+\prime \prime }\right) \right] .\end{gathered}$$However, it amounts to$$\left[ R\right] _{-}^{+}=0,$$since the third JCs (Eq. (9c)) urge$$\left\{
\begin{array}{c}
f^{+\prime \prime }\left( a\right) =f^{-\prime \prime }\left( a\right)
=f_{a}^{\prime \prime } \\
H^{+\prime \prime }\left( a\right) =H^{-\prime \prime }\left( a\right)
=H_{a}^{\prime \prime }%
\end{array}%
\right. .$$With the results obtained so far, the fourth and fifth JCs (Eqs. (9d) and (9e)) are self-satisfied.
For a perfect fluid on a 1+1 hypersurface, the energy-momentum tensor is $%
S_{\alpha }^{\beta }=diag\left( -\sigma ,p\right) $, with $\sigma $ and $p$ being the circumferential energy density and the angular pressure on the shell, respectively. Hence, the sixth JC (Eq. (9f)) simply implies$$\sigma =p.$$This, of course, is nothing but the static equation of state (EoS) of the matter on the throat.
Finally, the seventh JCs (Eq. (9g)) amount to the equations$$\begin{gathered}
\sigma =\frac{\varsigma }{4\kappa m^{2}}\sqrt{f_{a}}\times \\
\left[ 3\left( f_{a}^{+\prime \prime \prime }-f_{a}^{-\prime \prime \prime
}\right) +\frac{2f_{a}}{H_{a}}\left( H_{a}^{+\prime \prime \prime
}-H_{a}^{-\prime \prime \prime }\right) \right]\end{gathered}$$and$$\begin{gathered}
p=-\frac{\varsigma }{4\kappa m^{2}}\sqrt{f_{a}}\times \\
\left[ \left( f_{a}^{+\prime \prime \prime }-f_{a}^{-\prime \prime \prime
}\right) +\frac{6f_{a}}{H_{a}}\left( H_{a}^{+\prime \prime \prime
}-H_{a}^{-\prime \prime \prime }\right) \right] .\end{gathered}$$However, together with Eq. (16), these two equations result in a strong condition as follows$$f_{a}^{+\prime \prime \prime }-f_{a}^{-\prime \prime \prime }-\frac{2f_{a}}{%
H_{a}}\left( H_{a}^{+\prime \prime \prime }-H_{a}^{-\prime \prime \prime
}\right) =0.$$For both choices $H\left( r\right) =r$ and $H\left( r\right) =1$, the condition above leads to$$f^{+\prime \prime \prime }\left( a\right) =f^{-\prime \prime \prime }\left(
a\right) =f_{a}^{\prime \prime \prime }.$$This in turn makes both $\sigma $ and $p$ null, i.e.$$\sigma =p=0$$according to Eqs. (17) and (18). Hence, not only for the solutions we have reviewed in section $II$, but also for any other solutions in the form of Eq. (2), with an $H\left( r\right) $ function less than (at least) cubic in $%
r$, the existence of the TS is jeopardized as if it had never existed. This also could be confirmed taking into account the Eqs. (11), (12) and (15), explicitly. These equations amount to$$c_{2}^{+}=c_{2}^{-},$$$$c_{1}^{+}=c_{1}^{-},$$and$$c_{0}^{+}=c_{0}^{-},$$which imply that the inner and outer regions are in fact one spacetime, with no TS, whatsoever.
### TSW construction with RSV-JCs
In the literature of TSWs, the two spacetimes on the sides of the throat are traditionally considered to be exact copies of each other. However, it is shown that this mirror symmetry can be broken by assigning different spacetimes to $\left( \Sigma ,g\right) ^{\pm }$, and develop ATSWs [Forghani1,Forghani3]{}. In this section we consider this rather generalized type of TSWs.
The JCs in Eqs. (9) can also be applied to TSWs with some slight modifications to deal with the discontinuity in the normal vector. The proper JCs are therefore
$$\left[ g_{\alpha \beta }\right] _{-}^{+}=0;$$
$$\left[ K_{\alpha \beta }\right] _{-}^{+}=0;$$
$$\begin{gathered}
\left[ R_{\alpha \beta \mu \nu }\right] _{-}^{+}=\frac{\left[ R\right]
_{-}^{+}}{4}\times \\
\left( n_{\alpha }n_{\mu }h_{\beta \nu }-n_{\beta }n_{\mu }h_{\alpha \nu
}-n_{\alpha }n_{\nu }h_{\beta \mu }-n_{\beta }n_{\nu }h_{\alpha \mu }\right)
;\end{gathered}$$
$$\left[ R_{\alpha \beta }\right] _{-}^{+}=\frac{\left[ R\right] _{-}^{+}}{2}%
\left( \frac{1}{2}h_{\alpha \beta }+n_{\alpha }n_{\beta }\right) ;$$
$$\begin{gathered}
\left[ \nabla _{\mu }R_{\alpha \beta }\right] _{-}^{+}=\left[ n^{\nu }\nabla
_{\nu }R_{\alpha \beta }\right] _{-}^{+}n_{\mu } \\
+\frac{1}{2}\left( \frac{1}{2}h_{\alpha \beta }+n_{\alpha }n_{\beta }\right)
\overline{\nabla }_{\mu }\left[ R\right] _{-}^{+} \\
-\frac{1}{4}\left[ R\right] _{-}^{+}\left( n_{\alpha }K_{\beta \mu
}+n_{\beta }K_{\alpha \mu }\right) ;\end{gathered}$$
$$S_{\alpha }^{\alpha }=0;$$
$$\begin{gathered}
\kappa \left[ S_{\alpha \beta }\right] _{-}^{+}=-\left( \kappa _{1}+\kappa
_{2}\right) \left[ RK_{\alpha \beta }\right] _{-}^{+} \\
+\kappa _{1}\left[ n^{\nu }\nabla _{\nu }R\right] _{-}^{+}+2\kappa
_{2}\delta _{\alpha }^{\rho }\delta _{\beta }^{\sigma }\left[ n^{\nu }\nabla
_{\nu }R_{\rho \sigma }\right] _{-}^{+}.\end{gathered}$$
Imposing the first JCs (Eq. (25a)), analogous to the TS case, necessitates
$$\left\{
\begin{array}{c}
f^{+}\left( a\right) =f^{-}\left( a\right) =f_{a} \\
H^{+}\left( a\right) =H^{-}\left( a\right) =H_{a}%
\end{array}%
\right. .$$
However, the second JCs (Eq. (25b)) compel a different result as$$\left\{
\begin{array}{c}
f^{+\prime }\left( a\right) =-f^{-\prime }\left( a\right) \\
H^{+\prime }\left( a\right) =-H^{-\prime }\left( a\right)%
\end{array}%
\right. .$$It is convenient to construct the ATSW with two spacetimes with same $%
H\left( r\right) $ functions on the sides. Hence, we require $H^{+}\left(
r_{+}\right) =H^{-}\left( r_{-}\right) $, which with Eq. (27) admits$$H^{+}\left( r_{+}\right) =H^{-}\left( r_{-}\right) =H_{0},$$where $H_{0}$ is an arbitrary constant. Accordingly, the second condition in Eq. (26) is also self-satisfied. With this assumption, exploiting Eq. (13) for $\left[ R\right] _{-}^{+}$ and the JCs for Riemann tensor in Eq. (25c), result in$$f^{+\prime \prime }\left( a\right) =f^{-\prime \prime }\left( a\right)
=f_{a}^{\prime \prime },$$and consequently$$\left[ R\right] _{-}^{+}=0.$$While the JCs for the Ricci tensor components and their covariant derivatives (Eqs. (25d) and (25e)) are automatically satisfied, the JC for the trace of the energy-momentum tensor of the throat (Eq. (25f)) implies$$\sigma =p;$$similar to the TS case (Eq. (16)). Finally, the last of JCs (Eq. (25g))give explicit terms for energy density and tangential pressure as$$\sigma =\frac{3\varsigma }{4\kappa m^{2}}\sqrt{f_{a}}\left( f_{a}^{+\prime
\prime \prime }+f_{a}^{-\prime \prime \prime }\right)$$and$$p=-\frac{\varsigma }{4\kappa m^{2}}\sqrt{f_{a}}\left( f_{a}^{+\prime \prime
\prime }+f_{a}^{-\prime \prime \prime }\right) ,$$respectively. However, simultaneous consideration of Eqs. (31), (32) and (33) suggests$$f_{a}^{+\prime \prime \prime }=-f_{a}^{-\prime \prime \prime },$$which in turn leads to the static EoS$$\sigma =p=0.$$Considering all the results above, imposes conditions on the metric function coefficients $c_{1}^{\pm }$ and $c_{2}^{\pm }$, as well as the radius of the ATSW, as follows
$$c_{2}^{+}=c_{2}^{-},$$
$$c_{1}^{-}=-c_{1}^{+}-2c_{2}^{+}a,$$
and$$\begin{gathered}
a=\frac{-c_{1}^{+}\pm \sqrt{c_{1}^{+2}-2c_{2}^{+}\left(
c_{0}^{+}-c_{0}^{-}\right) }}{2c_{2}^{+}} \\
=\frac{-c_{1}^{-}\pm \sqrt{c_{1}^{-2}-2c_{2}^{-}\left(
c_{0}^{-}-c_{0}^{+}\right) }}{2c_{2}^{-}},\end{gathered}$$respectively. Obviously, the radius is real only for $c_{1}^{+2}-2c_{2}^{+}%
\left( c_{0}^{+}-c_{0}^{-}\right) \geq 0$. The TSW radius for the maximally symmetric case $c_{0}^{+}=c_{0}^{-}$ amounts to the non-trivial result
$$a=-\frac{c_{1}^{+}}{c_{2}^{+}}=-\frac{c_{1}^{-}}{c_{2}^{-}},$$
which is positive only when $c_{1}^{\pm }$ and $c_{2}^{\pm }$ have different signs. Since $c_{2}^{+}=c_{2}^{-}$ this alludes $c_{1}^{+}=c_{1}^{-}$, and the TSW is symmetric. For $H\left( r\right) =1$ and $\lambda =-1$ this explicitly becomes$$a=-\frac{c_{1}}{4\varsigma m^{2}}.$$This is a strong condition which dictates on the radius of the TSW. Note that the signs of the parameters included must eventually be set such that $%
a>0$. Depending on the sign of the quadratic term in the metric function $f$, this can be the maximum or the minimum of $f$. Here we emphasize that one may consider both possibilities $m^{2}>0$ and $m^{2}<0$, for the plus sign behind the quadratic terms in the CNMG action in Eq. (1) is more of a convention.
The above results imply, that firstly, the TSW is generally an asymmetric one, except for the maximally symmetric case where $c_{0}^{+}=c_{0}^{-}$, $%
c_{1}^{+}=c_{1}^{-}$ and $c_{2}^{+}=c_{2}^{-}$, and secondly, depending on the values of the metric coefficients, the TSW’s radius can actually be real and positive. Providing the tuned up coefficients support the TSW’s existence, it will have null energy density and pressure on its throat, providing a vacuum condition. Unexpected as it is, now the TSW indeed satisfies all the energy conditions. This represents a natural wormhole [Kim1]{} with no matter on its throat. The two spacetimes are joined smoothly and the result is a complete Riemannian manifold with no exotic matter, no discontinuity or singularity of any sort.
DSS Junction Conditions
-----------------------
In [@Deruelle1] DSS have investigated the JCs for the quadratic Lagrangian density$$\mathcal{L}_{\text{DSS}}=\sqrt{-g}\left( R-2\Lambda -4\beta R_{\mu \nu
}R^{\mu \nu }+\alpha R^{2}\right) ,$$where $\alpha $ and $\beta $ are two free parameters and $\Lambda $resembles a bare cosmological constant. To do so, they considered the Gaussian-normal coordinates to express the bulk metrics as $$ds^{2}=dy^{2}+h_{ij}dx^{i}dx^{j},$$in which there exists a thin-shell located at $y=0$, and $h_{ij}$represents the metric tensor of the 2-dimensional sub-spacetime. The proper JCs are found to be$$4\left[ -\beta \mathcal{H}_{ij}+\left( \alpha -\beta \right) h_{ij}\mathcal{H%
}\right] _{-}^{+}=S_{ij}$$where $$\mathcal{H}_{ij}\equiv -\frac{1}{2}\frac{\partial ^{3}h_{ij}}{\partial y^{3}}%
,$$$$\mathcal{H}\equiv h^{ij}\mathcal{H}_{ij},$$and $S_{ij}$ is the total energy-momentum tensor on the shell.
With a brief comparison between the Lagrangian density of Eq. (39) and the action of CNMG given in Eq. (1), one finds $\alpha =-\frac{3\varsigma }{%
8m^{2}}$, $\beta =-\frac{\varsigma }{4m^{2}}$, and of course $\Lambda
=\varsigma \lambda m^{2}$. Therefore, the JCs for CNMG can be written as
$$\frac{\varsigma }{m^{2}}\left[ \mathcal{H}_{i}^{j}-\frac{1}{2}\delta _{i}^{j}%
\mathcal{H}\right] _{-}^{+}=S_{i}^{j}.$$
However, note that prior to checking for the JCs in Eq. (44a) one must check for the continuity of the metric, and its first and second derivatives with respect to the normal coordinate $y$ at the shell’s position; i.e. we must have $$\left[ h_{ij}\right] _{-}^{+}=0,$$$$\left[ \frac{\partial h_{ij}}{\partial y}\right] _{-}^{+}=0,$$and$$\left[ \frac{\partial ^{2}h_{ij}}{\partial y^{2}}\right] _{-}^{+}=0.$$The JCs in Eq. (44a) can explicitly be determined as
$$\sigma =p=\frac{\varsigma }{2m^{2}}\left[ \mathcal{H}_{\theta }^{\theta }-%
\mathcal{H}_{t}^{t}\right] _{-}^{+}.$$
Comparing the bulk metrics in Eqs. (2) and (40) admits $$dy^{2}=\frac{1}{f}dr^{2}$$and so$$\frac{dr}{dy}=\sqrt{f}.$$This also casts the metric of the TS(W) as$$h_{ij}dx^{i}dx^{j}=-fdt^{2}+H^{2}d\theta ^{2},$$In case of a TS, the continuity of the metric and its first and second derivatives with respect to the normal coordinate $y$ (JCs in Eqs. (44b-d)), necessitate$$\left\{
\begin{array}{c}
f^{+}\left( a\right) =f^{-}\left( a\right) =f_{a} \\
H^{+}\left( a\right) =H^{-}\left( a\right) =H_{a}%
\end{array}%
\right. ,$$$$\left\{
\begin{array}{c}
f^{+\prime }\left( a\right) =f^{-\prime }\left( a\right) =f_{a}^{\prime } \\
H^{+\prime }\left( a\right) =H^{-\prime }\left( a\right) =H_{a}^{\prime }%
\end{array}%
\right. ,$$and$$\left\{
\begin{array}{c}
f^{+\prime \prime }\left( a\right) =f^{-\prime \prime }\left( a\right)
=f_{a}^{\prime \prime } \\
H^{+\prime \prime }\left( a\right) =H^{-\prime \prime }\left( a\right)
=H_{a}^{\prime \prime }%
\end{array}%
\right. .$$Accordingly, one obtains$$c_{2}^{+}=c_{2}^{-},$$$$c_{1}^{+}=c_{1}^{-},$$and$$c_{0}^{+}=c_{0}^{-}.$$These amount to$$\sigma =p=-\frac{\varsigma }{4m^{2}}\frac{f_{a}^{3/2}}{H_{a}}\left(
H_{a}^{+\prime \prime \prime }-H_{a}^{-\prime \prime \prime }\right) ,$$in consideration of the JCs in Eq. (44a) and the explicit form$$\mathcal{H}_{ij}=-\frac{1}{2}\left( \left( \left( \frac{\partial h_{ij}}{%
\partial r}\right) \sqrt{f}\right) ^{\prime }\sqrt{f}\right) ^{\prime }\sqrt{%
f}.$$Here the prime ($^{\prime }$) indicates a total derivative with respect to the radial coordinate $r$ and the chain rule is applied. This leads to$$\sigma =p=0,$$for both cases, $H\left( r\right) =r$ and $H\left( r\right) =1$, we have considered here. Hence, our results using DSS-JCs, analogous to the previous section, suggest nonexistence of any TS within the framework of CNMG for special solutions in Eqs. (2) and (3).
For a general ATSW at $r=a$ (where $r_{2}<a<r_{1}$ in case $\varsigma =-1$, and $r_{1}<a$ in case $\varsigma =1$), the last three JCs give rise to the exact same results as the previous section’s for the metric functions and their derivatives as$$H^{+}\left( r_{+}\right) =H^{-}\left( r_{-}\right) =H_{0},$$$$f^{+}\left( a\right) =f^{-}\left( a\right) =f_{a}\text{ },$$$$f^{+\prime }\left( a\right) =-f^{-\prime }\left( a\right) \text{,}$$and$$f^{+\prime \prime }\left( a\right) =f^{-\prime \prime }\left( a\right)
=f_{a}^{\prime \prime },$$if again the rather physical assumption $H^{+}\left( r_{+}\right)
=H^{-}\left( r_{-}\right) $ holds. Note that, each time a derivative with respect to the Gaussian normal coordinate $y$ is taken, the discontinuity in the normal vector to the throat must be considered, which emerges as the minus sign in the right-hand-side of Eq. (59). So far, it has been cleared up that only for the special choice $H(r)=1$ the TSW can be constructed, and the circumstances and discussions after Eq. (38) in the previous section are also valid here. Finally, the original JCs in Eq. (44a) impose $$\sigma =p=0,$$which is in full agreement with the previous results.
It appears to us, that using DSS-JCs, the same analysis can be applied to a wider range of massive quadratic Lagrangian densities and their solutions. To clear things up, let us consider a massive Lagrangian density of the form$$\mathcal{L=}\sqrt{-g}\left( \varsigma R+\frac{1}{m^{2}}\left( R_{\mu \nu
}R^{\mu \nu }+\gamma R^{2}\right) -2\Lambda \right) .$$Any solution to this Lagrangian density with the form in Eq. (2) and $%
H\left( r\right) =r$ to be used to construct a TSW will suffer from a discontinuity in the first derivative of the angular component of the TSW metric. Hence, the natural unsatisfactory behavior of such solutions to the JCs affects the occurrence of TSW, as if it had never existed.
On the contrary, solutions of the same type with $H\left( r\right) =1$ satisfy all the boundary conditions, but identically lead to$$\sigma =p=0.$$The same analysis, however, is not applicable to RSV-JCs, since for a general quadratic Lagrangian density such as the one in Eq. (62), the value of coefficient $\gamma $ alters the JCs accordingly.
Conclusion
==========
It has been a long-standing challenge to obtain TSWs with physical, i.e. non-exotic matter in Einstein’s general relativity. This was overcome in the past in particular models by changing the topology of the shell from spherical (in 3+1-dimensions) [@Mazhari1] and from circular (2+1-dimensions) forms [@Mazhari2; @Mazhari3]. Giving up those symmetric topologies, however, gave rise to different problems in connection with their stability analysis. In the present study we have shown that the exotic matter problem is overcome for TSWs in CNMG. In the meantime, the JCs are modified and they are distinct from those of Einstein’s general relativity, i.e. the DI-JCs. The new JCs are redefined and applied to some static solutions of CNMG. Our results show no indications of major difference between the two distinct sets of JCs we have used independently. However, this is mostly for the quadratic nature of the metric function $f\left(
r\right) $, and the specific selection of the gauge function $H\left(
r\right) $. This can be seen the best in the structural differences between the expressions found for $\sigma $ and $p$ in Eqs. (32) and (33) using RSV-JCs, and Eq. (55) using DSS-JCs. More noticeably, the exotic matter nightmare gets resolved for TSWs in this theory, in the sense that the energy density and lateral pressure on the shell become zero (better than negative!), hence no known energy condition is violated anymore. Nevertheless, these TSWs could only be constructed for the gauge selection $%
H\left( r\right) =1$. It was observed that for $H\left( r\right) =r$, which specifically comprises AdS solutions, no TSWs can be established. The existed TSWs, however, could be symmetric as well as asymmetric. On the other hand, it was shown that for the specific class of static solutions we considered here, there cannot be a TS. We leave the profound question of “how this nonexistence for AdS bulk translates into its CFT correspondence” for further studies. As a next step, investigating TS and/or TSWs’ constructions under naturally different geometries, such as Lifshitz black holes [@Ayon-Beato1], is in order. Studying thin-shells with double layers may also be of interest, as for these ones demand some other JCs [Reina1]{}. As another subject for further studies one may have a look into extended theories of NMG and solutions therein [Gullu1,Gullu2,Ghodsi1,Sinha1,Nam1,Anastasiou1,Setare1,Dalmazi1,Ahmedov4,Louzada1,Camara1,Ghodsi2,Bakas1,Akhavan1,Dereli1,Dereli2]{}.
[99]{} R. Arnowitt, S. Deser, and C. W. Misner, Ann. Phys. (N.Y.) **33**, 88 (1965).
D. G. Boulware, Phys. Rev. D **8**, 2363 (1973).
K. Lake, Phys. Rev. D **19**, 2847 (1978).
M. Heusler, C. Kiefer, and N. Straumann, Phys. Rev. D **42**, 4254 (1990).
K. Kuchař, Czech. J. Phys. B **18**, 435 (1968).
J. E. Chase, Nuovo Cimento **867**, 136 (1970).
P. S. Letelier, and A. Wang, Phys. Rev. D **48**, 631 (1993).
E. F. Eiroa, and C. Simeone, Phys. Rev. D 83, 104009 (2011).
P. R. Brady, J. Louko, and E. Poisson, Phys. Rev. D **44**, 1891 (1991).
T. Delsate, J. V. Rocha, and R. Santarelli, Phys. Rev. D **89**, 121501(R) (2014).
B. C. Nolan, Phys. Rev. D **65**, 104006 (2002).
F. C. Mena, J. Natário, and P. Tod, Class. Quantum Grav. **25**, 045016 (2008).
A. Di Prisco, L. Herrera, M. A. H. MacCallum, and N. O. Santos, Phys. Rev. D **80**, 064031 (2009).
M. S. Morris, and K. S. Thorne, Am. J. Phys. **56**, 395 (1988).
M. Visser, Phys. Rev. D **39,** 3182 (1989).
M. G. Richarte, I. G. Salako, J. P. Morais Graca *et al.*, Phys. Rev. D **96**, 084022 (2017).
V. Varela, Phys. Rev. D **92**, 044002 (2015).
M. Sharif, and Z. Yousaf, Astrophys. Space Sci **351**, 351 (2014).
M. Sharif, and M. Azam, Eur. Phys. J. C **73**, 2407 (2013).
N. Montelongo Garcia, F. S. N. Lobo, and M. Visser, Phys. Rev. D **86**, 044026 (2012).
G. A. S. Dias, and J. P. S. Lemos, Phys. Rev. D **82**, 084023 (2010).
E. F. Eiroa, Phys. Rev. D **78**, 024018 (2008).
M. Thibeault, C. Simeone, and E. F. Eiroa, Gen. Rel. Gravit. **38**, 1593 (2006).
F. S. N. Lobo, and P. Crawford, Class. Quantum Grav. **21**, 391 (2004).
E. F. Eiroa, and G. E. Romero, Gen. Rel. Gravit. **36**, 651 (2004).
E. F. Eiroa, and C. Simeone, Phys. Rev. D **70**, 044008 (2004).
M. Ishak, and K. Lake, Phys. Rev. **D** 65, 044011 (2002).
E. Poisson, and M. Visser, Phys. Rev. D **52**, 7318 (1995).
M. Visser, *Lorentzian Wormholes: From Einstein to Hawking* (American Institute of Physics, New York, 1995).
S. Mertens, J. Phys.: Conf. Ser. **718,** 022013 (2016).
B.P. Abbott et al., Phys. Rev. Lett. **119**, 161101 (2017).
B. P. Abbott et al., Astrophys. J. **848**, 2 (2017).
K. Hinterbichler, Rev. Mod. Phys. **84**, 671 (2012).
C. de Rham, Living Rev. Relativ. **17**, 7 (2014).
M. Fierz, and W. Pauli, Proc. Roy. Soc. Lond. A **173**, 211 (1939).
B. Gavela, E. E. Jenkins, A. V. Manohar, and L. Merlo, Eur. Phys. J. C **76**, 485 (2016).
S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. **140**, 372 (1982); **185**, 406(E) (1988).
S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. **281**, 409 (2000).
E. A. Bergshoeff, O. Hohm, and P. K. Townsend, Phys. Rev. Lett. **102**, 201301 (2009).
E. A. Bergshoeff, O. Hohm, and P. K. Townsend, Phys. Rev. D **79**, 124042 (2009).
E. A. Bergshoeff, O. Hohm, and P. K. Townsend, J. Phys. Conf. Ser. **229,** 012005 (2010).
I. Oda, J. High Energy Phys. **05**, 064 (2009).
M. Nakasone, and I. Oda, Prog. Theor. Phys. **121**, 6 (2009).
C. de Rham, G. Gabadadze, D. Pirtskhalava, A. J. Tolley, and I. Yavin, J. High Energy Phys. **06,** 028 (2011).
A. Accioly, J. Helayël-Neto, E. Scatena *et al.*, Class. Quantum Grav. **28**, 225008 (2011).
A. Accioly, J. Helayël-Neto, J. Morais *et al.*, Phys. Rev. D **83**, 104005 (2011).
G. Clement, Class. Quantum Grav. **26**, 105015 (2009).
M. Bañados, C. Teitelboim, and Jorge Zanelli, Phys. Rev. Lett. **69**, 1849 (1992).
G. Clement, Class. Quantum Grav. **26**, 165002 (2009).
E. Ayon-Beato, G. Giribet and M. Hassaine, J. High Energy Phys. **05,** 029 (2009).
J. Oliva, D. Tempoa, and R. Troncoso, J. High Energy Phys. **07,** 011 (2009).
E. Ayón-Beato, A. Garbarz, G. Giribet, and M. Hassaine, Phys. Rev. D **80**, 104029 (2009).
H. Ahmedov, and A. N. Aliev, Phys. Rev. Lett. **106**, 021301 (2011).
H. Ahmedov, and A. N. Aliev, Phys. Rev. D **83**, 084032 (2011).
H. Ahmedov, and A. N. Aliev, Phys. Lett. B **694**, 143 (2010).
Y. S. Myung, Y. W. Kim, T. Moon *et al.*, Phys. Rev. D **84**, 024044 (2011).
J.M. Maldacena, Adv. Theor. Math. Phys. **2,** 231 (1998); Int. J. Theor. Phys. **38**, 1113 (1999).
Y. Liu, and Y. W. Sun, J. High Energy Phys. **04**, 106 (2009).
D. Grumiller, and O. Hohm, Phys. Lett. B **686**, 264 (2010).
M. R. Setare, and V. Kamali, Adv. High Energy Phys. 472718 (2011).
M. Eune, W. Kim, S.-H. Yi, J. High Energy Phys. **1303**, 020 (2013).
Y. S. Myung, Adv. High Energy Phys. 478273 (2015).
S. J. Zhang, Phys. Lett. B **747**, 158 (2015).
Y. Kwon, S. Nam, J. D. Park, and S. H. Yi, Class. Quantum Grav. **28,** 145006 (2011).
G. Gecim, and Y. Sucu, J. Cosmol. Astropart. Phys. **02**, 023 (2013).
E. Bergshoeff, J. J. Fernandez-Melgarejo, J. Rosseel, and P. K. Townsend, J. High Energy Phys. **04**, 070 (2012).
D. Dalmazi, and R. C. Santos, Phys. Rev. D **87**, 085021 (2013).
I. Güllü, T. C. Şişman, and B. Tekin, Class. Quantum Grav. **27**, 162001 (2010).
I. Güllü, T. C. Şişman, and B. Tekin, Phys. Rev. D **82**, 024032 (2010).
A. Ghodsi, and D. Mahdavian Yekta, Phys. Rev. D **83**, 104004 (2011).
A. Sinha, J. High Energy Phys. **06**, 061 (2010).
S. Nam, J.-D. Park, and S.-H. Yi, J. High Energy Phys. **07**, 058 (2010).
G. G. Anastasiou, M. R. Setare, and E. C. Vagenas, Phys. Rev. D **88**, 064054 (2013).
M. R. Setare, and M. Sahraee, Int. J. Mod. Phys. A **28**, 14 (2013).
D. Dalmazi, and E. L. Mendonca, Eur. Phys. J. C **76**, 373 (2016).
H. Ahmedov, and A. N. Aliev, Phys. Lett. B **711**, 117 (2012).
H. L. C. Louzada, U. Camara dS, and G. M. Sotkov, Phys. Lett. B **686**, 268 (2010).
U. Camara dS, and G. M. Sotkov, Phys. Lett. B **694**, 94 (2010).
A. Ghodsi, and M. Moghadassi, Phys. Lett. B **695**, 359 (2011).
I. Bakas, and C. Sourdis, Class. Quantum Grav. **28**, 015012, (2011).
A. Akhavan, M. Alishahiha, A. Naseh *et al.*, J. High Energy Phys. **05**, 006 (2016).
T. Dereli, and C. Yetişmişoğlu, EPL **114**, 60004 (2016).
T. Dereli, and C. Yetişmişoğlu, Phys. Rev. D **94**, 064067 (2016).
S. D. Forghani, S. H. Mazharimousavi, and M. Halilsoy, Eur. Phys. J. C **78**, 469 (2018).
S. D. Forghani, S. H. Mazharimousavi, and M. Halilsoy, Eur. Phys. J. Plus **133**, 497 (2018).
S. D. Forghani, S. H. Mazharimousavi, and M. Halilsoy, arXiv:1807.05080 (2018).
E. F. Eiroa, and G. F. Aguirre, Phys. Rev. D **94**, 044016 (2016).
W. Israel, Nuovo Cimento **44B,** 1 (1966).
N. Deruelle, M. Sasaki, and Y. Sendouda, Prog. Theor. Phys. **119**, 2, 237-251 (2008).
B. Reina, J. M. M. Senovilla, and R. Vera, Class. Quantum Grav. **33**, 105008 (2016).
E. F. Eiroa, G. F. Aguirre, and J. M. M. Senovilla, Phys. Rev. D **95**, 124021(2017).
E. F. Eiroa, and G. F. Aguirre, Eur. Phys. J. C **78**, 54 (2018).
J. Y. Kim, C. O. Lee, and M. Park, Eur. Phys. J. C **78**, 990 (2018).
S. H. Mazharimousavi, and M. Halilsoy, Eur. Phys. J. C **75**, 271 (2015).
S. H. Mazharimousavi, and M. Halilsoy, Eur. Phys. J. C **75**, 81 (2015).
S. H. Mazharimousavi, and M. Halilsoy, Eur. Phys. J. C **75**, 540 (2015).
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---
abstract: |
In this paper, we reduce the logspace shortest path problem to biconnected graphs; in particular, we present a logspace shortest path algorithm for general graphs which uses a logspace shortest path oracle for biconnected graphs. We also present a linear time logspace shortest path algorithm for graphs with bounded vertex degree and biconnected component size, which does not rely on an oracle. The asymptotic time-space product of this algorithm is the best possible among all shortest path algorithms.
[**Keywords:**]{} logspace algorithm, shortest path, biconnected graph, bounded degree
author:
- Boris Brimkov
title: A reduction of the logspace shortest path problem to biconnected graphs
---
Introduction
============
The logspace computational model entails algorithms which use a read-only input array and $O(\log n)$ working memory. For general graphs, there is no known deterministic logspace algorithm for the shortest path problem. In fact, the shortest path problem is NL-complete, so the existence of a logspace algorithm would imply that L=NL [@jakoby1]. In this paper, we reduce the logspace shortest path problem to biconnected graphs, and present a linear time logspace shortest path algorithm for parameter-constrained graphs.
An important result under the logspace computational model which is used in the sequel is Reingold’s deterministic polynomial time algorithm [@reingold] for the undirected $st$-connectivity problem (USTCON) of determining whether two vertices in an undirected graph belong to the same connected component. There are a number of randomized logspace algorithms for USTCON (see, for example, [@barnes_feige; @feige; @kosowski]) which perform faster than Reingold’s algorithm but whose output may be incorrect with a certain probability. There are also a number of logspace algorithms for the shortest path problem and other graph problems on special types of graphs (see [@asano6; @brimkov2; @ktrees; @jakoby1; @munro_ramirez]). As a rule, due to time-space trade-off, improved space-efficiency is achieved on the account of higher time-complexity. Often the trade-off is rather large, yielding time complexities of $O(n^c)$ “for some constant $c$ significantly larger than 1" [@jakoby2]. In particular, the time complexity of Reingold’s USTCON algorithm remains largely uncharted but is possibly of very high order. The linear time logspace shortest path algorithm presented in this paper avoids this shortcoming, at the expense of some loss of generality. In fact, its time (and space) complexity is the best possible, since a hypothetical sublinear-time algorithm would fail to print a shortest path of length $\Omega(n)$.
This paper is organized as follows. In the next section, we recall some basic definitions and introduce a few concepts which will be used in the sequel. In Section 3, we present a reduction of the logspace shortest path algorithm to biconnected graphs. In Section 4, we present a linear time logspace algorithm for parameter-constrained graphs. We conclude with some final remarks in Section 5.
Preliminaries
=============
A *logspace algorithm* is an algorithm which uses $O(\log n)$ working memory, where $n$ is the size of the input. In addition, the input and output are respectively read-only and write-only, and do not count toward the space used. The *shortest path problem* requires finding a path between two given vertices $s$ and $t$ in a graph $G$, such that the sum of the weights of the edges constituting the path is as small as possible. In general, if $s$ and $t$ are not in the same connected component, or if the connected component containing $s$ and $t$ also contains a negative-weight cycle, the shortest path does not exist. For simplicity, we will assume there are no negative-weight cycles in $G$, although the proposed algorithms can be easily modified to detect (and terminate at) such cycles without any increase in overall complexity. We will also assume that $G$ is encoded by its adjacency list, where vertices are labeled with the first $n$ natural numbers. The $j^{\text{th}}$ neighbor of vertex $i$ is accessed with $Adj(i,j)$ in $O(1)$ time, and *degree*$(i)=|Adj(i)|$.
An *articulation point* in $G$ is a vertex whose deletion increases the number of connected components of $G$. A *block* is a maximal subgraph of $G$ which has no articulation points; if $G$ has a single block, then $G$ is *biconnected*. The *block tree* $T$ of $G$ is the bipartite graph with parts $A$ and $B$, where $A$ is the set of articulation points of $G$ and $B$ is the set of blocks of $G$; $a\in A$ is adjacent to $b \in B$ if and only if $b$ contains $a$. We define the *id* of a block in $G$ to be $(\emph{largest, smallest})$, where *largest* and *smallest* are the largest and smallest vertices in the block with respect to their labeling from 1 to $n$. Clearly, each block in $G$ has a unique *id*. Note also that it is possible to lexicographically compare the $id$s of two or more blocks, i.e., if $id_1=(\ell_1,s_1)$ and $id_2=(\ell_2,s_2)$, then $id_1>id_2$ if $\ell_1> \ell_2$ or if $\ell_1= \ell_2$ and $s_1>s_2$.
Given numbers $a_1$, $a_2$, and $p$, we define the *next* number after $p$ as follows:
$$next(a_1,a_2,p)=\begin{cases}
a_1&\text{ if }a_2\leq p<a_1\text{ or }p<a_1\leq a_2\text{ or }a_1 \leq a_2 \leq p\\
a_2&\text{ otherwise.}
\end{cases}$$
We extend this definition to a list $L$ of not necessarily distinct numbers by defining the *next* number in $L$ after $p$ to be a number in $L$ larger than $p$ by the smallest amount, or if no such number exists, to be the smallest number in $L$. The *next* number in $L$ can be found with logspace and $O(|L|)$ time, given sequential access to the elements of $L$, by repeatedly applying the *next* function.
Reducing the logspace shortest path problem to biconnected graphs
=================================================================
Let *connected*$(H;v_1,v_2)$ be an implementation of Reingold’s USTCON algorithm which takes in two vertices of a graph $H$ and returns *true* if they belong to the same connected component, and *false* otherwise. Let *pathInBlock*$(H;v_1,v_2)$ be a polynomial time, logspace oracle which takes in two vertices of a biconnected graph $H$ and prints the shortest path between them.
Clearly, the encoding of a graph $H$ can be reduced with logspace and polynomial time to the encoding of some induced subgraph $H[S]$. Thus, by transitivity and closure of reductions, the functions *connected*$(H[S];v_1,v_2)$ and *pathInBlock*$(H[B];v_1,v_2)$ can be used with logspace and polynomial time, where $S$ and $B$ are sets of vertices computed at runtime and $H[B]$ is biconnected.
The *connected* function reduces the logspace shortest path problem to connected graphs. In this section, we will further reduce this problem to biconnected graphs, by presenting a logspace algorithm for finding the shortest path between two vertices in an arbitrary graph using the oracle *pathInBlock*.
Constructing a logspace traversal function
------------------------------------------
Let $G$ be a graph of order $n$, and $v_1$ and $v_2$ be two vertices that belong to the same block; the set of all vertices in this block will be referred to as *block*$(v_1,v_2)$. Using the *connected* function, is easy to construct a logspace function *isInBlock*$(v_1,v_2,v)$ which returns *true* when $v$ is part of *block*$(v_1,v_2)$ and *false* otherwise; see Table 1 for pseudocode. This procedure can be used to access the vertices in $block(v_1,v_2)$ sequentially. A similar procedure *areInBlock*$(u,v)$ can be defined which returns *true* when $u$ and $v$ are in the same block, and *false* otherwise.
A vertex of $G$ is an articulation point if and only if two of its neighbors are not in the same block. Thus, using the *isInBlock* function, we can construct a function *isArticulation*$(v)$ which returns *true* when $v$ is an articulation point and *false* otherwise; see Table 1 for pseudocode. We also define the function $id(v_1,v_2)$, which goes through the vertices of *block*$(v_1,v_2)$ and returns (*largest, smallest*), where *largest* and *smallest* are respectively the largest and smallest vertices in *block*$(v_1,v_2)$ according to their labeling.
Let $p$ be an articulation point[^1] in *block*$(v_1,v_2)$. To find the *next* articulation point in *block*$(v_1,v_2)$ after $p$, we can create a function *nextArticulation*$(v_1,v_2,p)$ which uses each articulation point in *block*$(v_1,v_2)$ as a member of list $L$ and applies the *next* function. Note that the vertices in $L$ do not have to be stored, but can be generated one at a time; see Table 1 for pseudocode. Similarly, to identify the block containing $p$ and having the *next* $id$ after $id(v_1,v_2)$, we can create a function *nextBlock*$(v_1,v_2,p)$ which uses the *id*s of the blocks identified by $p$ and each of its neighbors as members of a list $L$ and applies the *next* function. Note that the *id*s in $L$ do not have to be stored but can be computed one at a time; see Table 1 for pseudocode.
Finally, given articulation point $p$ and vertex $v$ in the same block, we will call the component of $G-\{block(v,p)\backslash\{p\}\}$ which contains $p$ the *subgraph of G rooted at block$(v,p)$ containing p*, or *subgraph*$(v,p)$. This subgraph can be traversed with logspace by starting from $p$ and repeatedly moving to the *next* block and to the *next* articulation point until the starting block is reached again. This procedure indeed gives a traversal, since it corresponds to visiting the *next* neighbor in the block tree $T$ of $G$, which generates an Euler subtour traversal (cf. [@tarjan_vishkin]). In addition, during the traversal of *subgraph*$(v,p)$, each vertex can be compared to a given vertex $t$, in order to determine whether the subgraph contains $t$. Thus, we can create a function *isInSubgraph*$(v,p,t)$ which returns *true* if $t$ is in *subgraph*$(v,p)$ and *false* otherwise; see Table 1 for pseudocode.
Main Algorithm
--------------
Using the subroutines outlined in the previous section and the oracle *pathInBlock*, we propose the following logspace algorithm for finding the shortest path in a graph $G$. The main idea is to print the shortest path one block at a time by locating $t$ in one of the subgraphs rooted at the current block.
Algorithm 1 finds the correct shortest path between vertices $s$ and $t$ in graph $G$ with logspace and polynomial time, using a shortest path oracle for biconnected graphs.
Let $p_0=s$ and $p_{\ell+1}=t$; the shortest path between $p_0$ and $p_{\ell+1}$ is $P=p_0 P_0 p_1 P_1\ldots p_{\ell}P_{\ell} p_{\ell+1}$, where $p_1,\ldots,p_{\ell}$ are articulation points and $P_0,\ldots,P_{\ell}$ are (possibly empty) subpaths which contain no articulation points. Let $b_i=block(p_i,p_{i+1})$ for $0\leq i \leq \ell$, so that *pathInBlock*$(G[b_i];p_i,p_{i+1})=p_iP_ip_{i+1}$.
Suppose the subpath $p_0 P_0 \ldots p_i$, $i\geq 0$, has already been printed and that the vertex $p_i$ is stored in memory. In each iteration of the main loop, the function *isInSubgraph*$(p_i,p,t)$ returns *true* only for $p=p_{i+1}$ when run for all articulation points $p$ in all blocks containing $p_i$. The function *pathInBlock*$(G[b_i],p_i,p_{i+1})$ is then used to print $P_{i+1}$ and $p_{i+1}$. Finally, $p_i$ is replaced in memory by $p_{i+1}$, and this procedure is repeated until $p_{\ell+1}$ is reached. Since the main loop is entered only if the shortest path is of finite length, the algorithm terminates, and since each subpath printed is between two consecutive articulation points of $P$, the output of Algorithm 1 is the correct shortest path between $s$ and $t$.
Since the *connected* function is logspace, the *isInBlock*, *isArticulation* and *isInSubgraph* functions are each logspace. Only a constant number of variables, each of size $O(\log n)$, are simultaneously stored in Algorithm 1, and every function call is to a logspace function (assuming the *pathInBlock* oracle is logspace); thus, the space complexity of Algorithm 1 is $O(\log n)$. Note that since the vertices in *block*$(v_1,v_2)$ cannot be stored in memory simultaneously, a call to the function *pathInBlock*$(G[block(v_1,v_2)],v_1,v_2)$ needs to be realized by a logspace reduction, i.e., the vertices $v_1$ and $v_2$ are stored, and whenever the function *pathInBlock* needs to access an entry of the adjacency list of $G[V(block(v_1,v_2))]$, it recomputes it by going through the vertices of $G$ and using the function *isInBlock*.
Similarly, since the *connected* function uses polynomial time, the *isInBlock*, *isArticulation* and *isInSubgraph* functions each use polynomial time. The main loop is executed at most $O(n)$ times, and each iteration calls a constant number of polynomial time functions (assuming the *pathInBlock* oracle uses polynomial time); thus, the time complexity of Algorithm 1 is $O(n^c)$ for some constant $c$. $\square$
Linear time logspace algorithm for parametrically constrained graphs
====================================================================
Let *BellmanFord*$(H;v_1,v_2)$ be an implementation of the Bellman-Ford shortest path algorithm [@bellman_ford] which takes in two vertices of a graph $H$ and prints out the shortest path between them. Let *HopcroftTarjan*$(H)$ be an implementation of Hopcroft and Tarjan’s algorithm [@hopcroft_tarjan] which returns all blocks and articulation points of a graph $H$. If the size of $H$ is bounded by a constant, *BellmanFord* and *HopcroftTarjan* can each be used with constant time and a constant number of memory cells.
Let $G$ be a graph of order $n$ with maximum vertex degree $\Delta$ and maximum biconnected component size $k$. We will regard $\Delta$ and $k$ as fixed constants, independent of $n$. Using these constraints and some additional computational techniques, we will reformulate Algorithm 1 as a linear-time logspace shortest path algorithm which does not rely on an oracle. Asymptotically, both the time and space requirements of this algorithm are the best possible and cannot be improved; see Corollary 1 for more information.
Constructing a linear time logspace traversal function
------------------------------------------------------
By the assumption on the structure of $G$, the number of vertices at distance at most $k$ from a specified vertex $v$ is bounded by $\lfloor \frac{\Delta^{k+1}-1}{\Delta-1}\rfloor$. Thus, any operations on a subgraph induced by such a set of vertices can be performed with constant time and a constant number of memory cells, each with size $O(\log n)$; note that since each vertex of $G$ has a bounded number of neighbors, $G[S]$ can be found in constant time for any set $S$ of bounded size. In particular, we can construct a function *blocksContaining*$(v)$ which uses *HopcroftTarjan* to return all blocks containing a given vertex $v$ and all articulation points in these blocks; see below for pseudocode.
Using the set of blocks and articulation points given by the *blocksContaining* function, we can define functions *isInBlock*$(v_1,v_2,v)$, *areInBlock*$(u,v)$, *isArticulation*$(v)$, *id*$(v_1,v_2)$, *nextArticulation*$(v_1,v_2,p)$, and *nextBlock*$(v_1,v_2,p)$ analogous to the ones described in Section 3, each of which uses $O(\log n)$ space and $O(1)$ time. We can also construct an analogue of *isInSubgraph*$(v,p,t)$, which uses time proportional to the size of *subgraph*$(v,p)$; in particular, the time for traversing the entire graph $G$ via an Euler tour of its block tree is $O(n)$ (provided $G$ is connected) since there are $O(n)$ calls to the *nextArticulation* function and $O(n)$ calls to the *nextBlock* function.
Finally, it will be convenient to define the following functions: *adjacentPoints*$(v_1,v_2)$ which returns the set of articulation points belonging to blocks containing $v_2$ but not $v_1$ if $v_1\neq v_2$ and the set of articulation points belonging to blocks containing $v_2$ if $v_1=v_2$ (this function is slight modification of *blocksContaining*); *last*$(L)$ which returns the last element of a list $L$; *traverseComponent*$(s,t)$ which traverses the component containing a vertex $s$ and returns *true* if $t$ is in the same component and *false* otherwise (this function is identical to *isInSubgraph*, with a slight modification in the stopping condition).
Linear time logspace shortest path algorithm
--------------------------------------------
We now present a modified version of Algorithm 1, which uses the subroutines outlined in the previous section as well as some additional computational techniques such as “simulated parallelization" (introduced by Asano et al. [@asano6]) aimed at reducing its runtime.
Algorithm 2 finds the correct shortest path between vertices $s$ and $t$ in graph $G$ with bounded degree and biconnected component size using logspace and linear time.
Using the notation in the proof of Theorem 1, suppose the subpath $p_0P_0\ldots p_i$ has already been printed; $p_{\ell+1}$ cannot be in *subgraph*$(p_i,p_{i-1})$, so there is no need to run *isInSubgraph*$(p_i,p_{i-1},t)$. Thus, *adjacentPoints*$(p_{i-1},p_i)$ is the set of feasible articulation points. Moreover, if $p_{\ell+1}$ is not in *subgraph*$(p_i,p)$ for all-but-one feasible articulation points, then the last of these must be $p_{i+1}$ and there is no need to run *isInSubgraph*$(p_i,p_{i+1},t)$. Finally, two subgraphs rooted at $block(p_{i-1},p_i)$ can be traversed concurrently with the technique of simulated parallelization: instead of traversing the feasible subgraphs one-after-another, we maintain two copies of the *isInSubgraph* function and use them to simultaneously traverse two subgraphs. We do this in serial (without the use of a parallel processor) by iteratively advancing each copy of the function in turn; if one subgraph is traversed, the corresponding copy of the function terminates and another copy is initiated to traverse the next unexplored subgraph. Thus, Algorithm 2 is structurally identical to Algorithm 1[^2] and prints the correct shortest path between $s$ and $t$.
Only a constant number of variables, each of size $O(\log n)$, are simultaneously used in Algorithm 2, and every function call is to a logspace function; moreover, keeping track of the internal states of two logspace functions can be done with logspace, so the space complexity of Algorithm 2 is $O(\log n)$.
Finally, to verify the time complexity, note that by traversing two subgraphs at once, we can deduce which subgraph contains $t$ in the time it takes to traverse all subgraphs which do *not* contain $t$ or $s$. Thus, each subgraph rooted at *block*$(p_i,p_{i+1})$, $0\leq i\leq \ell$, which does not contain $t$ or $s$ will be traversed at most once, so the time needed to print the shortest path is of the same order as the time needed to traverse $G$ once. $\square$
The time and space complexity of Algorithm 2 is the best possible for the class of graphs considered.
Let $G$ be a graph of order $n$; the shortest path between two vertices in $G$ may be of length $\Omega(n)$ so any shortest path algorithm will require at least $\Omega(n)$ time to print the path. Moreover, a pointer to an entry in the adjacency list of $G$ has size $\Omega(\log n)$, so printing each edge of the shortest path requires at least $\Omega(n)$ space. $\square$
Conclusion
==========
We have reduced the logspace shortest path problem to biconnected graphs using techniques such as computing instead of storing, transitivity of logspace reductions, and Reingold’s USTCON result. We have also proposed a linear time logspace shortest path algorithm for graphs with bounded degree and biconnected component size, using techniques such as simulated parallelization and constant-time and -space calls to functions over graphs with bounded size.
Future work will be aimed at further reducing the logspace shortest path problem to triconnected graphs using SPQR-tree decomposition, and to $k$-connected graphs using branch decomposition or the decomposition of Holberg [@decomposition]. Another direction for future work will be to generalize Algorithm 2 by removing or relaxing the restrictions on vertex degree and biconnected component size.
Acknowledgements {#acknowledgements .unnumbered}
================
This material is based upon work supported by the National Science Foundation under Grant No. 1450681.
[99]{}
Asano, T., Mulzer, W., Wang, Y., Constant work-space algorithms for shortest paths in trees and simple polygons. J. of Graph Algorithms and Applications 15(5), 569–586 (2011)
Barnes, G., Feige, U., Short random walks on graphs. Proc. 25th Annual ACM Symposium of the Theory of Computing (STOC), 728–737 (1993)
Bellman, R., On a routing problem. Quarterly of Applied Mathematics 16 87–90 (1958)
Brimkov, B., Hicks, I.V., Memory efficient algorithms for cactus graphs and block graphs. Discrete Applied Math. In Press, doi:10.1016/j.dam.2015.10.032 (2015)
Das, B., Datta, S., Nimbhorkar, P., Log-space algorithms for paths and matchings in k-trees. Theory of Computing Systems 53 (4) 669–689 (2013)
Feige, U., A randomized time-space trade-off of $\tilde{O}(mR)$ for USTCON. Proc. 34th Annual Symposium on Foundations of Computer Science (FOCS), 238–246 (1993)
Holberg, W., The decomposition of graphs into $k$-connected components. Discrete Mathematics 109(1), 133–145 (1992)
Hopcroft, J. and Tarjan, R., Algorithm 447: efficient algorithms for graph manipulation. Communications of the ACM 16(6), 372–378 (1973)
Jakoby, A., Tantau, T., Logspace algorithms for computing shortest and longest paths in series-parallel graphs. FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science. Springer Berlin Heidelberg, 216–227 (2007)
Jakoby, A., Liskiewicz, M., Reischuk, R., Space efficient algorithms for series-parallel graphs. J. of Algorithms 60, 85–114 (2006)
Kosowski, A., Faster walks in graphs: a $O(n^2)$ time-space trade-off for undirected st connectivity. Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM (2013)
Munro, I.J., Ramirez, R. J., Technical note–reducing space requirements for shortest path problems. Operations Research 30.5, 1009-1013 (1982)
Reingold, O., Undirected connectivity in log-space. J. ACM 55(4), Art. 17 (2008)
Tarjan, R.E. and Vishkin, U., Finding biconnected components and computing tree functions in logarithmic parallel time. Proceedings of FOCS, 12–20 (1984)
[^1]: The subsequent definitions and functions remain valid when $p$ is not an articulation point, and can be used in special cases, e.g., when $G$ only has one block.
[^2]: Indeed each of the described modifications can be implemented in Algorithm 1 as well, but would not make a significant difference in its time complexity.
|
---
abstract: 'Let $\Pi$ be the étale fundamental group of a smooth affine curve over an algebraically closed field of characteristic $p>0$. We establish a criterion for profinite freeness of closed subgroups of $\Pi$. Roughly speaking, if a closed subgroup of $\Pi$ is “captured” between two normal subgroups, then it is free, provided it contains most of the open subgroups of index $p$. In the proof we establish a strong version of “almost $\omega$-freeness” of $\Pi$ and then apply the Haran-Shapiro induction.'
address:
- 'Instituts für Experimentelle Mathematik, Universität Duisburg-Essen'
- 'Universität Duisburg-Essen, Fakultät für Mathematik'
author:
- 'Lior Bary-Soroker'
- Manish Kumar
title: Subgroup structure of fundamental groups in positive characteristic
---
Introduction
============
The étale fundamental group of a variety over a field of characteristic $p>0$ is a mysterious group. Even in the case of smooth affine curves it is not yet completely understood. The prime-to-$p$ part of the group is understood because of Grothendieck’s Riemann Existence Theorem [@SGA1 XIII, Corollary 2.12]. The main difficulty arises from the wildly ramified covers.
Let $C$ be a smooth affine curve over an algebraically closed field $k$ of characteristic $p>0$. Let $X$ be the smooth completion of $C$, $g$ the genus of $X$ and $r=\operatorname*{card}(X\smallsetminus C)$. Let $\pi_1(C)$ be the étale fundamental group of $C$. This is a profinite group and henceforth the terminology should be understood in the category of profinite groups, e.g. subgroups are closed, homomorphisms are continuous, etc.
Abhyankar’s conjecture which was proved by Raynaud [@Raynaud1994] and Harbater [@Harbater1994] classifies the finite quotients of $\pi_1(C)$, namely a finite group $G$ is a quotient of $\pi_1(C)$ if and only if $G/p(G)$ is generated by $2g+r-1$ elements. Here $p(G)$ is the subgroup of $G$ generated by all the $p$-sylow subgroups of $G$. In particular, $\pi_1(C)$ is not finitely generated, hence it is not determined by the set of its finite quotients.
Recently people have tried to understand the structure of $\pi_1(C)$ by studying its subgroups. In [@Kumar2008] the second author shows that the commutator subgroup is free of countable rank, provided $k$ is countable. In [@PachecoStevensonZalesskii2009] Pacheco, Stevenson and Zalesskii make an attempt to understand the normal subgroups $N$ of $\pi_1(C)$, again when $k$ is countable. In [@HarbaterStevenson2011] Harbater and Stevenson show that $\pi_1(C)$ is almost $\omega$-free, in the sense that every finite embedding problem has a solution after restricting to an open subgroup (see also [@Jarden2011]). If $k$ is uncountable, the result of [@Kumar2008] is extended in [@Kumar2009]. We prove a diamond theorem for $\pi_1(C)$ when $k$ is countable, Theorem \[thm:main\] below, by using Theorem \[thm:pi\_1\] below which strengthens [@HarbaterStevenson2011 Theorem 6].
Let us explain the main result of this paper in detail. We say that a subgroup $H$ of a profinite group $\Pi$ lies in a $\Pi$-diamond if there exist $M_1,M_2\lhd \Pi$ such that $M_1\cap M_2\leq H$, but $M_1\not\leq H$ and $M_2\not\leq H$. Haran’s diamond theorem states that a subgroup $H$ of a free profinite group of infinite rank that lies in a diamond is free [@Haran1999a]. In [@Bary-Soroker2006] the first author extends the diamond theorem to free profinite groups of finite rank $\geq 2$. Many subgroups lie in a diamond, notably, the commutator and any proper open subgroup of a normal subgroup.
The diamond theorem is general in the sense that most of the other criteria follow from it, and it applies also to non-normal subgroups, in contrast to Melnikov’s theorem. Moreover, the diamond theorem can be extended to Hilbertian fields [@Haran1999b] and to other classes of profinite groups. In [@Bary-SorokerHaranHarbater2010] Haran, Harbater, and the first author prove a diamond theorem for semi-free profinite groups, and in [@Bary-SorokerStevensonZalesskii2010] Stevenson, Zalesskii and the first author establish a diamond theorem for $\pi_1(C)$, where $C$ is a projective curve of genus at least $2$ over an algebraically closed field of characteristic $0$.
For each $g\ge 0$, we define a subgroup of $\pi_1(C)$, $$P_g(C)=\bigcap\{\pi_1(Z)\mid Z\to C \text{ is \'etale ${\mathbb{Z}}/p{\mathbb{Z}}$-cover and the genus of }Z\ge g \}.$$ We note that $P_0(C)$ is the intersection of all open normal subgroups of index $p$ and we have $P_{g+1}(C)\geq P_g(C)$.
Our main result is the following diamond theorem for subgroups of $\pi_1(C)$ that are contained in $P_g(C)$ for some $g\geq 0$.
\[thm:main\] Let $k$ be a countable algebraically closed field of characteristic $p>0$, let $C$ be a smooth affine $k$-curve, let $\Pi=\pi_1(C)$, let $g$ be a non-negative integer, and let $M$ be a subgroup of $\Pi$. Assume $M\leq P_g(C)$ and there exist normal subgroups $M_1,M_2$ of $\Pi$ such that $M$ contains $M_1\cap M_2$ but contain neither $M_1$ or $M_2$. Then:
1. For every finite simple group $S$ the direct power $S^\infty$ is a quotient of $M$. \[part:main\_a\]
2. \[part:main\_b\] Assume further that $[M M_i:M] \neq p$ for $i=1,2$. Then $M$ is free of countable rank.
This generalizes [@Kumar2008 Theorem 4.8]. Let $\Pi'$ be the commutator subgroup of $\Pi$. Then $\Pi'\leq P_0(C)$, so $\Pi/\Pi'$ is a non finitely generated abelian profinite group, hence $\Pi/\Pi' = A\times B$ with $A,B$ infinite. So $\Pi'$ is the intersection of the preimages of $A$ and $B$ in $\Pi$, hence lies in a diamond as in the above theorem. Since $A$ and $B$ are infinite, the hypothesis of Theorem \[thm:main\](\[part:main\_b\]) also holds for $\Pi'$.
We note that Theorem \[thm:main\] implies that for every $g\geq 0$ the subgroup $P_g(C)$ is free (cf. [@Kumar2009]). However we do not know whether Theorem \[thm:main\] follows from this latter assertion, because it is not clear that $M$ lies in a $P_g(C)$-diamond even if it contained in a $\pi_1(C)$-diamond.
Although a normal subgroup $N$ of $\pi_1(C)$ of infinite index is not necessarily free, every proper open subgroup of $N$ is free, provided it is contained in $P_g(C)$ for some $g$ (cf. Lubotzky-v.d. Dries’ theorem [@FriedJarden2008 Proposition 24.10.3]).
\[cor:LD\] Let $k$ be a countable algebraically closed field of characteristic $p>0$, let $C$ be a smooth affine $k$-curve, let $\Pi=\pi_1(C)$. Let $N$ be a normal subgroup of $\Pi$ of infinite index and let $M$ be a proper open subgroup of $N$ that is contained in $P_g(C)$ for some $g\geq 0$. Then $M$ is free.
A stronger form of this corollary appears in [@PachecoStevensonZalesskii2009]. There instead of assuming $M\leq P_g(C)$, it is assumed that $\Pi/N$ has a ’big’ $p$-sylow subgroup. However there is a gap in [@PachecoStevensonZalesskii2009]. In a private communication the authors of [@PachecoStevensonZalesskii2009] renounced their main result. The gap can be fixed under the assumption that $N\leq P_g(C)$. Note that under the assumption $N\leq P_g(C)$, the assertion of Corollary \[cor:LD\] follows from Lubotzky-v.d. Dries’ theorem for free groups because $P_g(C)$ is free. But in general, it seems that one cannot apply the group theoretical theorem directly.
If $[M:N]\neq p$ the corollary follows directly from Theorem \[thm:main\] Part . Indeed, since $M$ is open in $N$, there exists an open normal subgroup $\tilde{N}$ of $\Pi$ such that $N\cap \tilde{N}\leq M$. Thus $M$ is in a diamond with the extra properties, as needed. For the general case we need the diamond theorem for free groups, Melnikov’s theorem on normal subgroups of free groups, and Theorem \[thm:main\] Part. See §\[pf:cor\].
The proof of Theorem \[thm:main\] contains two key ingredients. We prove a geometric result on solvability of embedding problems for $\pi_1(C)$. For this we need a piece of notation. An open subgroup $\Pi^{o}$ of $\Pi$ is the étale fundamental group of an étale cover $D$ of $C$. We say $\Pi^{o}$ *corresponds to a curve of genus ${\mathrm{g}}$* if the genus of the smooth completion of $D$ is ${\mathrm{g}}$.
\[thm:pi\_1\] Let $k$ be an algebraically closed field of characteristic $p > 0$, let $C$ be a smooth affine $k$-curve, $\Pi = \pi_1(C)$, $g$ a positive integer, and $\mathcal{E}=(\mu\colon \Pi \to G, \alpha \colon \Gamma\to G)$ a finite embedding problem for $\Pi$. Then there exists an open normal index $p$ subgroup $\Pi^o$ of $\Pi$ such that:
1. $\mu(\Pi^o) = G$.
2. $\Pi^o$ corresponds to a curve of genus at least ${\mathrm{g}}$.
3. The restricted embedding problem $(\mu|_{\Pi^o}: \Pi^o \to G, \alpha: \Gamma\to G)$ is properly solvable.
Harbater and Stevenson prove that $\Pi=\pi_1(C)$ is almost $\omega$-free, i.e., there exists an open normal subgroup $\Pi^{o}$ satisfying (1) and (3) [@HarbaterStevenson2011 Theorem 6], however no bound on the index of $\Pi^{o}$ is given. The proof of [@HarbaterStevenson2011 Theorem 6] is based on “adding branch points”. In [@Jarden2011] Jarden proves that a profinite group $\Gamma$ is almost-$\omega$ free if the mild condition that $A_n^m$ is a quotient of $\Gamma$ for all sufficiently large $n$ and $m$ (here $A_n$ is the alternating group). Since $A_n^m$ is quasi-$p$, if $n\geq p$, By Abhyankar’s conjecture, $A_n^m$ is a quotient of $\pi_1(C)$, hence Jarden’s result implies [@HarbaterStevenson2011 Theorem 6]. We note that both Harbater-Stevenson’s and Jarden’s methods do not give any bound on the index of $\Pi^{o}$ in $\Pi$, which is essential for applications, e.g. to the proof of Theorem \[thm:main\]. Our proof relies on the constructions in [@Kumar2008].
The second ingredient is the Haran-Shapiro induction, see [@Bary-SorokerHaranHarbater2010].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank David Harbater and Kate Stevenson for useful discussions.
The first author is a Alexander von Humboldt fellow.
Solutions of embedding problems for the fundamental group
=========================================================
An *embedding problem* $\mathcal{E}=(\mu: \Pi \to G, \alpha: \Gamma\to G)$ for a profinite group $\Pi$ consists of profinite group epimorhisms $\mu$ and $\alpha$. We call $H=\ker\alpha$ the *kernel* of the embedding problem. $$\xymatrix{
& & & \Pi\ar@{->>}[d]^{\mu}\ar@{-->}[ld]_{{\varphi}}\\
1\ar[r]&H\ar[r]& \Gamma\ar[r]^{\alpha}& G\ar[r]&1}$$ The embedding problem is *finite* (resp. *split*) if $\Gamma$ is finite (resp. $\alpha$ splits). A *weak solution* of $\mathcal{E}$ is a homomorphism ${\varphi}\colon \Pi\to \Gamma$ such that $\alpha\circ{\varphi}=\mu$. A *(proper) solution* is a surjective weak solution ${\varphi}$.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $C$ be a smooth affine $k$-curve, $\Pi = \pi_1(C)$, and $ (\mu: \Pi \to G, \alpha: \Gamma\to G)$ a finite embedding problem for $\Pi$. Let $H$ be the kernel of $\alpha$.
Prime-to-$p$ kernel
-------------------
In the rest of the section we shall prove the existence of solutions of finite embedding problems with prime-to-$p$ kernels when restricting to a normal subgroup of index $p$:
\[pro:prime-to-p\] Let $g$ be a positive integer and let $\mathcal{E}=(\mu: \Pi \to G, \alpha: \Gamma\to G)$ be a finite split embedding problem for $\Pi$ with $H=\ker\alpha$ prime-to-$p$. Then there exists an open normal subgroup $\Pi^o$ of $\Pi$ of index $p$ such that
1. $\mu(\Pi^o) = G$;
2. $\Pi^o$ corresponds to a curve of genus at least ${\mathrm{g}}$;
3. The restricted embedding problem, $ (\mu|_{\Pi^o}: \Pi^o \to G, \alpha: \Gamma\to G)$, is properly solvable.
Let $K^{un}$ denote the compositum (in some fixed algebraic closure of $k(C)$) of the function fields of all finite étale covers of $C$. Let $X$ be the smooth completion of $C$. By a strong version of Noether normalization theorem ([@Eis Corollary 16.18]), there exists a finite surjective $k$-morphism $\Phi_0: C\to {\mathbb{A}}^1 _x$ which is generically separable. Here $x$ denotes the local coordinate of the affine line. The branch locus of $\Phi_0$ is of codimension $1$, hence $\Phi_0$ is étale away from finitely many points. By translation we may assume none of these points map to $x=0$. $\Phi_0$ extends to a finite surjective morphism $\Phi_X:X\rightarrow {\mathbb{P}}^1 _x$. Let $\{r_1,\ldots,r_N\}=\Phi_X^{-1}(\{x=0\})$, then $\Phi_X$ is étale at $r_1,\ldots,r_N$.
We use the construction in [@Kumar2008 Section 6], which we recall for the reader’s convenience. Let $\Phi_Y:Y\to {\mathbb{P}}^1_y$ be a $p$-cyclic cover between smooth curves, locally given by $Z^p-Z-y^{-r}=0$, for some $r$ that is prime to $p$. The genus of $Y$ is $$\label{eq:gY}
g_Y = (p-1)(r-1)/2$$ and $\Phi_Y$ is totally ramified at $y=0$ and étale elsewhere (see [@Pries2006]). Let $F$ be the zero locus of $t-xy$ in ${\mathbb{P}}^1_x\times_{\operatorname*{Spec}(k)}{\mathbb{P}}^1_y\times_{\operatorname*{Spec}(k)}\operatorname*{Spec}(k[[t]])$. Let $X_F$ and $Y_F$ be the normalization of an irreducible dominating component of the product $X\times_{{\mathbb{P}}^1_x} F$ and $Y\times_{{\mathbb{P}}^1_y} F$, respectively. Let $T$ be the normalization of an irreducible dominating component of $X_F\times_F Y_F$. We summarize the situation in Figure \[defnofT\].
\[htbp\]
\[defnofT\] $$\xymatrix {
& & T\ar@<1ex>[dl] \ar@<1ex>[dr]\\
&X_F\ar@<1ex>[dl]\ar@<1ex>[dr] & &Y_F\ar@<1ex>[dl]\ar@<1ex>[dr]\\
X\ar@<1ex>[dr]& &F\ar@<1ex>[dl]\ar@<1ex>[dr]& &Y\ar@<1ex>[dl] \\
&{\mathbb{P}}^1 _x & &{\mathbb{P}}^1 _y \\
}$$
\[lm:p-cyclic\] The fiber of the morphism $T\to X_F$ over $\operatorname*{Spec}(k((t))$ induces a ${\mathbb{Z}}/p{\mathbb{Z}}$-cover $T\times_{\operatorname*{Spec}(k[[t]])} \operatorname*{Spec}(k((t)))\to X\times_{\operatorname*{Spec}(k)}\operatorname*{Spec}(k((t)))$ which is étale over $C$.
Note that $k(X_F)=k((t))k(X)$ because generically $t\ne 0$ and over $t\ne 0$ the morphism $F\to {\mathbb{P}}^1_x$ is the base change ${\mathbb{P}}^1_x\times_{\operatorname*{Spec}(k)}\operatorname*{Spec}(k((t)))\to {\mathbb{P}}^1_x$. Also note that over $t \ne 0$ the local coordinates $x$ and $y$ satify the relation $y=t/x$ on $T$. Therefore, since $Y_F$ is the base change of $Y\to {\mathbb{P}}^1_y$, it follows that $k(Y_F)=k((t))(x)[Z]/(Z^p-Z-(x/t)^r)$. Since $T$ is a dominating component of $X_F\times_F Y_F$, we have $$k(T) = k(X_F)k(Y_F) = k((t))k(X)(k(x)[Z]/(Z^p-Z-(x/t)^r) ).$$ But $Z^p-Z-(x/t)^r$ is irreducible over $k(X)k((t))$, hence $k(T)/k(X_F)$ is a Galois extension with Galois group ${\mathbb{Z}}/p{\mathbb{Z}}$. Finally since the cover $T\times_{\operatorname*{Spec}(k[[t]])} \operatorname*{Spec}(k((t)))\to X\times_{\operatorname*{Spec}(k)}\operatorname*{Spec}(k((t)))$ is given by the equation $Z^p-Z-(x/t)^r=0$, it is étale away from $x=\infty$. In particular, it is étale over $C$.
Since $\operatorname*{Gal}(K^{un}/k(C))=\pi_1(C)$, the surjection $\mu\colon\pi_1(C)\to G$ induces an irrdeucible normal $G$-cover $\Psi_X:W_X\to X$ which is étale over $C$. By , we can choose a prime-to-$p$ integer $r$ to be sufficiently large so that $g_Y \geq \max\{|H|,g\}$. Then there exist an irreducible smooth étale $H$-cover $\Psi_Y:W_Y\to Y$, since $H$ is a prime-to-$p$ group ([@SGA1 XIII,Corollary 2.12]).
\[many-covers\] There exist an irreducible normal $\Gamma$-cover $W\to T$ of $k[[t]]$-curves, a regular finite type $k[t,t^{-1}]$-algebra $B$ contained in $k((t))$ and an open subset $S$ of $\operatorname*{Spec}(B)$ such that the $\Gamma$-cover $W\to T$ descends to a $\Gamma$-cover of connected normal projective $\operatorname*{Spec}(B)$-schemes $W_B\to T_B$ and for any closed point $s$ in $S$ we have the following:
1. The fiber $T_s$ of $T_B$ at $s$ is a smooth irreducible ${\mathbb{Z}}/p{\mathbb{Z}}$-cover of $X$ étale over $C$.
2. The fiber $W_s\to T_s$ of $W_B\to T_B$ at $s$ is a smooth irreducible $\Gamma$-cover such that the composition $W_s\to T_s \to X$ is étale over $C$.
3. The quotient $W_s/H$ is isomorphic to an irreducible dominating component of $W_X\times_X T_s$ as $G$-covers of $T_s$. In particular, $k(W_X)$ and $k(T_s)$ are linearly disjoint over $k(X)$.
By [@Kumar2008 Proposition 6.4] there exists a normal irreducible $\Gamma$-cover $W\to T$ of $k[[t]]$-curves with prescribed properties. We claim that this cover is étale away from the points lying above $x=\infty$ in $T$.
Indeed, by Conclusion $(1)$ of [@Kumar2008 Proposition 6.4], we have $$\label{eqqq}
W\times_T \widetilde{T_X} = {\rm Ind}_G^\Gamma (\widetilde{W_{XT} \times_T T_X}),$$ where tilde denotes the $(t)$-adic completion, $T_X = T\setminus \{x=0\}$, $W_{XT}$ is the normalization of an irreducible dominating component of $W_X\times_X T$, and for any variety $V$, ${\rm Ind}_G^\Gamma V= V^{\Gamma}/\sim$, where $(\gamma,v)\sim (\gamma',v')$ if and only if $\gamma'=\gamma g^{-1}$ and $v'= gv$, for some $g\in G$. Note that the left hand side of is the $(t)$-adic completion of $W\times_T T_X$ hence the branch locus of $W\times_T T_X \to T_X$ and $W\times_T \tilde T_X\to \tilde T_X$ are same. On the other hand, the right hand side of is the disjoint union of copies of $\widetilde{W_{XT} \times_T T_X}$, so the branch locus of $W\times_T \tilde T_X \to \tilde T_X$ maps to the branch locus of $W_X\to X$ under the morphism $T_X\to X$. Thus, since $W_X\to X$ is ètale away from $\{x=\infty\}$, so is $W\times_T T_X \to T_X$. Similarly, using (1’) of [@Kumar2008 Proposition 6.4] and the fact that $W_Y\to Y$ is étale everywhere, we get that over $T_Y=T\setminus \{y=0\}$ the cover $W\to T$ is étale. It follows from Conclusion (2) of [@Kumar2008 Proposition 6.4] that $W\to T$ is étale over $\{x=y=0\}$, as needed.
By Conclusion (5) of [@Kumar2008 Proposition 6.4] $W/H \cong W_{XT}$ as $G$-covers of $T$, hence we identify them. By Lemma \[lm:p-cyclic\], away from $t=0$, $T\to X\times_k \operatorname*{Spec}(k[[t]])$ is a ${\mathbb{Z}}/p{\mathbb{Z}}$-cover.
Now applying “Lefschetz’s type principle”, [@Kumar2008 Proposition 6.9], to the proper surjective morphisms of projective $k[[t]]$-curves $$W\to W_{XT} \to T \to X\times_k\operatorname*{Spec}(k[[t]])$$ and the points $r_1,\ldots,r_N$ in $X$ lying over $x=\infty$ it can be seen that there exists an open subset $S$ of the spectrum of a $k[t, t^{-1}]$-algebra $B$ such that the above morphisms descend to the morphisms of $\operatorname*{Spec}(B)$-schemes $$W_B\to W_{XT,B} \to T_B \to X\times_k\operatorname*{Spec}(B)$$ and the fiber over every point $s$ in $S$ leads to covers of smooth irreducible curves $W_s\to T_s$ and $T_s\to X$ with the desired ramification properties and Galois groups. This proves Conclusions (1) and (2). For (3), we note that $W_s\to T_s$ factors through $W_{XT,s}$ and since $W/H$ is isomorphic to $W_{XT}$, $W_s/H$ is isomorphic to $W_{XT,s}$. So $W_{XT,s}$ is an irreducible $G$-cover of $T_s$. Also $W_{XT,s}$ is an irreducible dominating component of $W_X\times_X T_s$. Finally, the compositum $k(W_X)k(T_s)$ is the function field of $W_{XT,s}$ and it is a Galois extension of $k(T_s)$ with Galois group $G$. Also $\operatorname*{Gal}(k(W_X)/k(X))=G$, so $k(T_s)$ and $k(W_X)$ are linearly disjoint over $k(X)$.
Note that the statement of [@Kumar2008 Proposition 6.9] only asserts the existence of the fibers $W_s$, $T_s$, etc. and covering morphisms between them with the desired Galois group and ramification properties. Since the morphisms of $k[[t]]$-curves $W\to T$ and $T\to X_F$ are finite, they descend to morphisms of a regular finite type $k[t,t^{-1}]$-algebra $B$ contained in $k((t))$. In the proof of [@Kumar2008 Proposition 6.9] it is further shown that there exist an open subset of $\operatorname*{Spec}(B)$ such that fibers over any point in this open subset have the desired properties.
Using the notation of the above proposition, let $\Sigma \subset T_s$ be the preimage of $X\setminus C$ under the covering $T_s\to X$, let $T_s^o=T_s\setminus \Sigma$, and let $\Pi^o=\pi_1(T_s^o)$. By $(1)$ of Proposition \[many-covers\], $k(T_s^o)/k(C)$ is a Galois extension with Galois group ${\mathbb{Z}}/p{\mathbb{Z}}$. If $Z\to T_s^o$ is an étale cover of $T_s^o$ then the composition with $T_s^o\to C$ gives an étale cover $Z\to C$. Conversely, if $Z\to C$ is an étale cover of $C$ which factors through $T_s^o\to C$, then $Z\to T_s^o$ is also étale. So $\Pi^o=\operatorname*{Gal}(K^{un}/k(T_s))=\pi_1(T_s^o)$ is an index $p$ normal subgroup of $\operatorname*{Gal}(K^{un}/k(C))=\pi_1(C)$.
For (2), note that by construction of $T$ (see Figure \[defnofT\]), $T$ dominates $Y_F$, so over the generic point, the genus of the $k((t))$-curve $T\times_{\operatorname*{Spec}(k[[t]]}\operatorname*{Spec}(k((t))$ is at least the genus of $Y$. But $g_Y\ge {\mathrm{g}}$, so this genus is at least $g$. Since $T_s$ is a smooth fiber of $T_B$ which in turn descends from $T$, the genus of $T_s$ is also at least ${\mathrm{g}}$.
Since the $G$-cover $W_X\to X$ is induced by the epimorphism $\mu\colon \Pi\to G$, we have $\ker\mu=\operatorname*{Gal}(K^{un}/k(W_X))$. So $(\ker\mu)\cap \Pi^o=\operatorname*{Gal}(K^{un}/k(W_X)k(T_s))$. Hence $$\mu(\Pi^o)=\Pi^o/((\ker\mu)\cap\Pi^o)=\operatorname*{Gal}(k(W_X)k(T_s)/k(T_s)).$$ But $\operatorname*{Gal}(k(W_X)k(T_s)/k(T_s))=G$ by $(3)$ of Proposition \[many-covers\], so we get $\mu(\Pi^o) = G$, as needed.
Also from Proposition \[many-covers\], we get a $\Gamma$-cover $W_s$ of $T_s$ such that $k(W_X)\subset k(W_s)$ and we obtain the following tower of field extensions: $$\xymatrix{
& K^{un}\ar@{-}[d]\ar@/_2pc/@{-}[dddl]_{\Pi^o} \\
& k(W_s)\ar@{-}[d]^H\ar@/_/@{-}[ddl]_{\Gamma}\\
& k(T_s)k(W_X)\ar@{-}[dl]^G\ar@{-}[dr]\\
k(T_s)\ar@{-}[dr]_{{\mathbb{Z}}/p{\mathbb{Z}}}& &k(W_X)\ar@{-}[dl]^{G}\\
& k(X) }$$
Hence there is a surjection from $\Pi^o {\twoheadrightarrow}\Gamma=\operatorname*{Gal}(k(W_s)/k(T_s))$ which dominates $\mu|_{\Pi^o}\colon\Pi^o
{\twoheadrightarrow}G$. Hence $W_s$ provides a solution to the embedding problem restricted to $\Pi^o$.
Proof of Theorem \[thm:pi\_1\]
------------------------------
Let $k$ be an algebraically closed field of characteristic $p > 0$, let $C$ be a smooth affine $k$-curve, $\Pi = \pi_1(C)$, $g$ a positive integer, and $\mathcal{E}=(\mu\colon \Pi \to G, \alpha \colon \Gamma\to G)$ a finite embedding problem for $\Pi$. Put $H=\ker\alpha$.
Since $\Pi$ is projective ([@Serre1990 Proposition 1]), $\mathcal{E}$ has a weak solution. It is well known that $\mathcal{E}$ can be dominated by a finite split embedding problem $\mathcal{E}'$, in the sense that a solution of $\mathcal{E}'$ induces a solution of $\mathcal{E}$ (even when restricted to a subgroup $\Pi^o$ with $\mu(\Pi^o)=G$), see for example [@HarbaterStevenson2005 Discussion after Lemma 2.3]. Hence, without loss of generality, we can assume that $\mathcal{E}$ splits.
Let $p(H)$ be the subgroup of $H$ generated by all $p$-sylow subgroups. Since $p(H)$ is characteristic in $H$, it is normal in $\Gamma$. Then there is $\alpha'\colon \Gamma/p(H)\to G$ such that $\alpha = \alpha'\circ \alpha''$, where $\alpha''\colon \Gamma\to \Gamma/p(H)$ is the natural quotient map.
Since $\ker \alpha'$ is prime-to-$p$, by Proposition \[pro:prime-to-p\], there exists an index $p$ normal subgroup $\Pi^o$ of $\Pi$ such that $\mu(\Pi^{o})=G$, $\Pi^{o}$ corresponds to a curve of genus at least $g$, and the embedding problem $$(\mu|_{\Pi^o}: \Pi^o \to G, \alpha'\colon \Gamma/p(H)\to G)$$ has a solution, say $\mu'\colon \Pi^{o}\to \Gamma/p(H)$.
The kernel of $\alpha''$ is $p(H)$ which by definition is quasi-$p$. So using Pop’s result ([@pop Theorem B]) the embedding problem $$(\mu': \Pi^o \to \Gamma/p(H), \alpha''\colon \Gamma \to\Gamma/p(H))$$ has a solution, say ${\varphi}$. Then $$\alpha\circ {\varphi}= \alpha'\circ \alpha''\circ {\varphi}=\alpha'\circ \mu'=\mu|_{\Pi^o}.$$ Hence ${\varphi}$ is also a solution of $ (\mu|_{\Pi^o}: \Pi^o \to G, \alpha\colon \Gamma\to G)$, as needed.
Proof of Theorem \[thm:main\]
=============================
Let $k$ be a countable algebraically closed field of characteristic $p>0$, let $C$ be an affine $k$-curve, let $\Pi=\pi_1(C)$, let $g$ be a non-negative integer, and let $M\leq P_g(C)$ be a subgroup of $\Pi$. In particular $[\Pi:M]=\infty$. Assume there exist normal subgroups $M_1,M_2$ of $\Pi$ such that $M_1\cap M_2\leq M$ and $M_1,M_2\not\leq M$. In Part we have to prove that for every finite simple group $S$, the direct power $S^\infty$ is a quotient of $M$. In Part we further assume that $[MM_i:M]\neq p$, and we have to prove that $M$ is free profinite group of countable rank.
First reduction {#first-reduction .unnumbered}
---------------
We can assume that $[MM_1:M] = \infty$ and $M\neq MM_2$. For Part we can further assume that $[MM_2:M]\neq p$.
Indeed, if $[M_1:M_1\cap M]= [MM_1:M]<\infty$, then there exists an open normal subgroup $U$ of $\Pi$ such that $U\cap M_1\leq M\cap M_1 \leq M$. Since $[\Pi:M]=\infty$, we have $[U:U\cap M]=[UM:M]=\infty$. Thus we can replace $M_1$ by $U$ and $M_2$ by $M_1$.
Second reduction {#second-reduction .unnumbered}
----------------
For Part it suffices to solve an arbitrary finite split embedding problem $$\mathcal{E}_1 = (\mu \colon M \to G_1, \alpha_1\colon A\rtimes G_1 \to G_1)$$ for $M$ and for Part it suffices to solve $\mathcal{E}_1$ when $G_1=1$ and $A=S^n$, for arbitrary $n\geq 1$.
Indeed, since $\Pi$ is projective [@Serre1990 Proposition 1], $M$ is also projective [@FriedJarden2008 Proposition 22.4.7]. Since $k$ is countable, $\Pi$ is of rank $\aleph_0$, hence ${\rm rank}(M)\leq \aleph_0$. Thus to prove that $M$ is free of countable rank it suffices to properly solve any finite split embedding problem for $M$ [@HarbaterStevenson2005 Theorem 2.1]. In order to prove that $S^\infty$ is a quotient of $M$, it suffices to prove that $S^n$ is a quotient for every positive integer $n$. The latter is equivalent to properly solve any $\mathcal{E}_1$ with $G_1=1$ and $A=S^n$.
Solving $\mathcal{E}_1$ under the assumption $\boldsymbol{[MM_2:M]\neq p}$ {#solving-mathcale_1-under-the-assumption-boldsymbolmm_2mneq-p .unnumbered}
--------------------------------------------------------------------------
Let $L$ be an open normal subgroup of $\Pi$ such that $M\cap L \leq \ker\mu$ and the following conditions hold:
1. \[it:1\] $[M_1 M L : M L ] = [M_1 : M_1\cap ( ML) ] \geq 3 p$ (recall $[M_1M:M]=\infty$).
2. \[it:2\] $[M_2ML:ML] = [ M_2 : M_2\cap ( ML)] \neq 1,p$ (recall $[M_2M:M]\neq 1,p$) .
3. \[it:3\] $[\Pi:ML] \geq 3p$ (recall $[\Pi:M]=\infty$).
Let $G = \Pi/L$, ${\varphi}\colon \Pi\to G$ the natural epimorphism, and $G_0 = {\varphi}(M)= ML/L$. Since $M\cap L\leq \ker \mu$, the map $\mu$ factors as $\bar \mu\circ {\varphi}|_M $, $\bar\mu \colon G_0\to G_1$. Hence $G_0$ acts on $A$ via $\bar\mu$, namely $a^g := a^{\bar\mu(g)}$, $a\in A$, $g\in G_0$. Let $I= \{ f\colon G\to A \mid f(\sigma\rho)=f(\sigma)^\rho, \ \forall \sigma\in G, \rho\in G_0\}$. Then $I\cong A^{(G:G_0)}$ and $G$ acts on $I$ by the formula $$f^\sigma(\tau) = f(\sigma \tau), \qquad \sigma,\tau\in G.$$ We denote the corresponding semidirect product by $A\wr_{G_0} G := I\rtimes G$, and we refer to it as the twisted wreath product. It comes with a canonical projection map $\alpha\colon A\wr_{G_0} G \to G$. Thus $$\mathcal{E} = ({\varphi}\colon \Pi\to G, \alpha\colon A\wr_{G_0} G\to G)$$ is a split embedding problem for $\Pi$. Theorem \[thm:pi\_1\] gives an index $p$ open normal subgroup $U$ of $ \Pi$ containing $P_g(C)$, hence $M\leq U$, such that ${\varphi}(U) = G$, hence $UL=\Pi$; and such that the restricted embedding problem $$\label{eq:mathcalEU}
\mathcal{E}|_U = ({\varphi}|_U\colon U\to G, \alpha\colon A\wr_{G_0} G\to G)$$ is solvable. Let $\psi\colon U \to A\wr_{G_0} G$ be a solution of $\mathcal{E}|_U$.
$$\xymatrix{
M_i\cap U \ar@{-}[r]^{\leq p}\ar@{-}[d]\ar@<-7ex>@/_10pt/@{--}[dd]_{A_i} & M_i\ar@{-}[d]\\
M_i\cap (ML)\cap U\ar@{-}[r]\ar@{-}[d] &M_i\cap (ML)\ar@{-}[dd]\\
(M_i\cap U) \cap (M (L\cap U))\ar@{-}[d]\\
M_i\cap U \cap M \ar@{=}[r] &M_i\cap M
}$$
Let $A_i = [M_i\cap U :(M_i\cap U)\cap (M(L\cap U))]$, for $i=1,2$. Then $A_1 \geq \frac{[M_1:M_1\cap (ML)]}{p} \geq 3$. If $M_2\leq U$, then $M_2 \cap U = M_2$, and we have $$A_2=[M_2 :M_2\cap (M(L\cap U))] \geq [M_2:M_2\cap ML] >1.$$ If $M_2\not\leq U$, then $[M_2:M_2\cap U] = p$. Then we have $$\begin{aligned}
p\cdot [M_2\cap U:M_2\cap (ML)\cap U] &=&[M_2:M_2\cap U] [M_2\cap U:M_2\cap (ML)\cap U] \\
&=&[M_2:M_2\cap (ML)\cap U] \\
&=& [M_2:M_2\cap(ML)][M_2\cap(ML):M_2\cap(ML)\cap U]. \end{aligned}$$ By we get that the right hand side does not equal $p$, hence $[M_2\cap U:M_2\cap (ML)\cap U] > 1$, hence $A_2>1$. Since $UL = \Pi$, we have $$[U:M (L\cap U)] \geq [U:U\cap ML] = [UML:ML] = [\Pi:ML]\geq 3p\geq 3.$$
The last paragraph implies that if we set $\tilde{\Pi}=U$, $\tilde{M}_i=M_i\cap U$, $\tilde{M}=M\cap U = M$, $\tilde{L}=L\cap U$, and $\tilde{\mathcal{E}}=\mathcal{E}|_{U}$, then we have
1. \[it:1tilde\] $[\tilde{M}_1 \tilde M \tilde L : \tilde M \tilde L ] = [\tilde M_1 : \tilde M_1\cap ( \tilde M \tilde L) ] \geq 3$.
2. \[it:2tilde\] $[\tilde M_2 \tilde M\tilde L:\tilde M\tilde L] = [ \tilde M_2 : \tilde M_2\cap ( \tilde M\tilde L)] >1$ .
3. \[it:3tilde\] $[\tilde \Pi:\tilde M\tilde L] \geq 3$.
and a proper solution $\psi$ of $\tilde{\mathcal{E}}$.
Let $K_i = {\varphi}(\tilde{M}_i)=\tilde{M}_i\tilde{L}/\tilde{L}$, $i=1,2$. Then - can be reformulated as
1. $[K_1G_0:G_0]\geq 3$;
2. $[K_2 G_0:G_0]\geq 2$;
3. $[G:G_0]\geq 3$.
We apply [@Bary-SorokerHaranHarbater2010 Proposition 4.6]. Let $D=\tilde{L}$, $\tilde{\Pi}_0 = \tilde{M}\tilde{L}$, and $\tilde{N}=\tilde{L}\cap \tilde{M_1}\cap \tilde{M_2}$. Let $A_0$ be a proper subgroup of $A$ that is normal in $A\rtimes G_0$. Then $G_0$ acts on the nontrivial group $\bar{A}$. It suffices to prove by [@Bary-SorokerHaranHarbater2010 Proposition 4.6] that in this setting the embedding problem $$(\bar {\varphi}\colon \tilde{\Pi}/N\to G, \bar{\alpha} \colon \bar{A}\wr_{G_0} G\to G)$$ is not solvable. Here $\bar{\varphi}$ is the quotient map (recall that $G\cong \tilde{\Pi}/\tilde{L}$ and $N\leq \tilde{L}$).
Put $H = \bar{A} \wr_{G_0} G$. Assume by negation that there is an epimorphism $\rho\colon \tilde{\Pi}\to H$ with $\bar\alpha \circ \rho = {\varphi}$ and $\rho(N) = 1$. For $i=1,2$, let $H_i = \rho(\tilde{M}_i)$. Then $H_i$ is normal in $H$ and $\alpha(H_i) = {\varphi}(\tilde{M_i}) = K_i$. By [@FriedJarden2008 Lemma 13.7.4(a)] there is $h_1\in H_1$ and $h_2\in H_2$ such that $\alpha(h_1) = 1$ and $[h_1, h_2]\neq 1$.
Let $\gamma_i \in \tilde{M}_i$ with $\rho(\gamma_i) = h_i$. Then ${\varphi}(\gamma_1) = \alpha(h_1) = 1$, so $\gamma_1\in L$. Then $$[\gamma_1,\gamma_2]\in [\tilde{L},\tilde{M}_2]\cap [\tilde{M}_1,\tilde{M}_2]\leq \tilde{L}\cap (\tilde{M}_1\cap \tilde{M}_2) = N.$$ (Recall that if $A,B$ are normal subgroups of a group $C$, then $[A,B]\leq A\cap B$, because $a^{-1}b^{-1} a b = a^{-1} a^{b} = b^{-a} b$.) We thus get that $[h_1,h_2] = [\rho(\gamma_1),\rho(\gamma_2)] \in \rho(N) = 1$. This contradiction implies there is no proper solution $\rho$ of $(\bar{\varphi}, \bar\alpha)$, and hence by [@Bary-SorokerHaranHarbater2010 Proposition 4.6] $\psi$ induces a proper solution $\nu$ of $\mathcal{E}_1$. This finishes the proof of Part by the reduction steps.
Solving $\mathcal{E}_1$ with $G_1=1$ and $A=S^n$ {#solving-mathcale_1-with-g_11-and-asn .unnumbered}
------------------------------------------------
We separate the proof into two cases.
### Case A {#case-a .unnumbered}
Assume $p\mid |S|$. Then $S$ is a quasi-$p$ group and hence $A=S^n$ is quasi-$p$. Then by Pop’s result [@pop Theorem B] we get that the embedding problem $(\Pi\to \Pi/L, \alpha\colon A\wr_{G_0} G \to G)$ is properly solvable; let $\psi\colon \Pi \to A\wr_{G_0} G$ be a proper solution. If we take $U=\Pi$ above instead of the index $p$ subgroup $\tilde{\Pi}$ of $\Pi$, then we have $\tilde{\mathcal{E}}=\mathcal{E}$, $\tilde{M}=M$, etc. In particular - follows directly from -. Then the rest of the proof can be applied mutatis mutandis to get that $\psi$ induces a solution $\nu$ of $\mathcal{E}_1$, as needed by the reduction steps.
### Case B {#case-b .unnumbered}
Assume $p\nmid |S|$. We note that the above proof uses the assumption $[M_2:M_2\cap M] =[M_2 M:M]\neq p$ only to ensure that is satisfied which in turn is only used when $M_2\not\leq U$. Thus if $M_2\leq U$, the above can be applied to get that $\mathcal{E}_1$ is properly solvable, and we are done.
We thus assume that $[M_2:M_2\cap M] =[M_2 M:M]= p$ and $M_2\not\leq U$. Thus, since $[\Pi:U]=p$ we have $[M_2:M_2\cap U]=p$. Since $M\leq P_g(C)\leq U$ and since $[M_2:M_2\cap M]=p$ we have $M_2\cap M =M_2\cap U$. In particular $U/(M_2\cap U) \cong \Pi/M_2$ hence $U/M_2$ is a quotient of $\Pi$.
Let $X$ be a completion of $C$, let $g_{\Pi}$ be the genus of $X$ and $r+1 = |X\smallsetminus C|$. Let $\nu$ be the maximal positive integer such that $S^\nu$ is a generated by $2g+r$ elements. Then $\nu$ is the maximal integer such that $S^\nu$ is a quotient of $\Pi$, as a prime-to-$p$ group.
We note that in the choice of $U$, the genus $g_U$ of the completion of the corresponding curve can be taken such that $S^{\nu+n}$ is generated by $2g_U$ elements. Since the prime-to-$p$ quotient of $U$ is free of rank at least $2 g_U$, there exists an epimorphism $\psi \colon U \to S^{\nu+n}$. Let $\Gamma = \psi(M_2\cap U)$.
Since $M_2\cap U\lhd U$ it follows that $\Gamma \normal S^{\nu+n}$. Thus $\Gamma\cong S^{\nu_1}$ and $S^{\nu+n} = \Gamma\times \Lambda$, where $\Lambda\cong S^{\nu_2}$ and $\nu_1+\nu_2=\nu+n$. We have $\Lambda \cong \psi(U/(M_2\cap U))$, hence $\Lambda$ is a quotient of $U/(M_2\cap U)\cong \Pi/M_2$, and hence of $\Pi$. Thus $\nu_2\leq \nu$, so $\nu_1\geq n$.
Let ${\varphi}\colon U \to \Gamma$ be the composition of $\psi$ with the projection onto $\Gamma$. Then ${\varphi}(M_2\cap U)=\psi(\Gamma\times 1)=\Gamma$. Since $M_2\cap U \leq M \leq U$ we get that ${\varphi}(M) = \Gamma$. This finishes the proof because $\Gamma\cong S^{\nu_1}$ with $\nu_1\geq n$.
Proof of Corollary \[cor:LD\] {#pf:cor}
-----------------------------
Let $k$ be a countable algebraically closed field of characteristic $p>0$, let $C$ be an affine $k$-curve, let $\Pi=\pi_1(C)$. Let $N$ be a normal subgroup of infinite index and let $M$ be a proper open subgroup of $N$ of index $\neq p$ that is contained in $P_g(C)$ for some $g\geq 0$. We have to prove that $M$ is free.
Let $U$ be an open normal subgroup of $\Pi$ such that $N\cap U\leq M$. Then $M$ is in the $\Pi$-diamond determined by $N$ and $U$. Note that $[\Pi:M]=\infty$, so $[UM:M]=\infty$. Thus if $[N:M]\neq p$, then $M$ is free by Part of Theorem \[thm:main\] and we are done.
If $N\leq P_g(C)$, then $M$ is in the $P_{g}(C)$-diamond determined by $U\cap P_{g}(C)$ and $N$. Hence $M$ is free by the diamond theorem for free profinite groups.
If $N\not\leq P_g(C)$ and $[N:M]=p$, then $M=P_g(C)\cap N$ (because $M\leq P_g(C)\cap N \lneqq N$). Thus $M$ is a normal subgroup of the free profinite group $P_g(C)$. By Part of Theorem \[thm:main\], $S^\infty$ is a quotient of $M$, for every finite simple group $S$. Thus by Melnikov’s theorem $M$ is free.
[1]{}
L. Bary-Soroker, *Diamond theorem for a finitely generated free profinite group*, Math. Ann. **336** (2006), no. 4, 949–-961
L. Bary-Soroker, D. Haran, and D. Harbater, *Permanence criteria for semi-free profinite groups*, Math. Ann. **348** (2010), no. 3, 539–-563
L. Bary-Soroker, Katherine Stevenson, and Pavel Zalesskii, *Subgroups of profinite surface groups*, arXiv:1011.1419, preprint
Eisenbud, David [*Commutative Algebra with a view towards algebraic geometry*]{} Book.
M. D. Fried and M. Jarden, “Field arithmetic”, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics \[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics\], vol. 11, Springer-Verlag, Berlin, 2008, Revised by Jarden
D. Haran, *Free subgroups of free profinite groups*, J. Group Theory **2** (1999), no. 3, 307–-317
D. Haran, *Hilbertian fields under separable algebraic extensions*, Invent. Math. **137** (1999), no. 1, 113–-126
D. Harbater, *Abhyankar’s conjecture on Galois groups over curves*, Invent. Math. **117**, no. 1, 1-–25
D. Harbater and K. Stevenson, *Local Galois theory in dimension two* Advances in Math., **198** (2005), 623–653
D. Harbater and K. Stevenson, *Embedding problems and open subgroups*, Proc. Amer. Math. Soc., **139** (2011), 1141–1154
M. Jarden, *Almost locally free fields* , J. of Algebra, **335**, no. 1, (2011), 171–176
M. Kumar, *Fundamental group in positive characteristic*, J. Algebra **319** (2008), no. 12, 5178–5207
M. Kumar, *Fundamental group of affine curves in positive characteristic*, arXiv:0903.4472, Preprint, 2009
A. Pacheco, K. Stevenson, and P. Zalesskii, *Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic*, Math. Ann. **343** (2009), no. 2, 463–-486
F. Pop, *Ètale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar’s conjecture.* Invent. Math. **120** (1995), no. 3, 555–-578.
E.-J. Pries, *Wildly ramified covers with large genus.* J. Number Theory **119** (2006), no. 2, 194-–209.
M. Raynaud, *Revêtements de la droite affine en caractéristique $p > 0$*, Invent. Math. **116** (1994), no. 1, 425-–462
J.-P. Serre, *Construction de Revtements etale de la droite affine en caractéristique $p$*, Comptes Rendus **311** (1990), 341–346
A. Grothendieck, “Revêtements étales et groupe fondamental” (SGA 1), Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-Heidelberg-New York, 1971
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abstract: 'The photon asymmetry in the reaction $p(\vec{\gamma},\pi^{0})p$ close to threshold has been measured for the first time with the photon spectrometer TAPS using linearly polarized photons from the tagged–photon facility at the Mainz Microtron MAMI. The total and differential cross sections were also measured simultaneously with the photon asymmetry. This allowed determination of the $S$–wave and all three $P$-wave amplitudes. The values obtained at threshold are $E_{0+} = (-1.33 \pm 0.08_{stat} \pm 0.03_{sys})10^{-3}/m_{\pi^+}$, $P_1 = (9.47 \pm 0.08_{stat} \pm 0.29_{sys}) 10^{-3} q/m^2_{\pi^+}$, $P_2 = (-9.46 \pm 0.1_{stat} \pm 0.29_{sys}) 10^{-3} q/m^2_{\pi^+}$ and $P_3 = (11.48 \pm 0.06_{stat} \pm 0.35_{sys}) 10^{-3} q/m^2_{\pi^+}$. The low–energy theorems based on the parameter–free third–order calculations of heavy–baryon chiral perturbation theory for $P_1$ and $P_2$ agree with the experimental values.'
address: |
$^1$Institut für Kernphysik, Universität Mainz, 55099 Mainz, Germany\
$^2$Departement of Physics and Laboratory for Nuclear Science, MIT, Boston, MA, USA\
$^3$Institut für Kernphysik, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany\
$^4$II. Physikalisches Institut, Justus–Liebig–Universität Gie[ß]{}en, 35392 Gie[ß]{}en, Germany\
$^5$Department für Physik und Astronomie, Universität Basel, 4056 Basel, Switzerland\
$^6$II. Physikalisches Institut, Georg–August–Universität Göttingen, 37073 Göttingen, Germany\
$^7$Department of Physics and Astronomy, Glasgow University, Glasgow G128QQ, UK\
$^8$Physikalisches Institut, Eberhard-Karls-Universität Tübingen, 72076 Tübingen, Germany
author:
- 'A. Schmidt$^1$, P. Achenbach$^3$, J. Ahrens$^1$, H. J. Arends$^1$, R. Beck$^1$, A. M. Bernstein$^2$, V. Hejny$^3$, M. Kotulla$^4$, B. Krusche$^5$, V. Kuhr$^6$, R. Leukel$^1$, I. J. D. MacGregor$^7$, J. C. McGeorge$^7$, V. Metag$^4$, V. M. Olmos de León$^1$, F. Rambo$^6$, U. Siodlaczek$^8$, H. Str[ö]{}her$^3$, Th. Walcher$^1$, J. Wei[ß]{}$^4$, F. Wissmann$^6$ and M. Wolf$^4$'
title: 'Test of low–energy theorems for $\mathbf{p(\vec{\gamma},\pi^{0})p}$ in the threshold region'
---
In the early 70’s, low–energy theorems (LETs) were derived for the amplitudes of pion photoproduction from the nucleon at threshold [@Bae70; @Vai72]. Based on fundamental principles, like gauge invariance and the partially conserved axial current, the LETs predict the value of the $S$–wave threshold amplitude $E_{0+}$ in a power series in $\mu=m_\pi/m_N$, the ratio of the masses of the pion and nucleon. The LETs represent tests of effective degrees of freedom in the non–perturbative domain of QCD and, therefore, their investigation is of considerable interest for an understanding of QCD at low momentum transfers. Only the development of high duty factor accelerators enabled first precise measurements of the photoproduction of neutral pions from the proton at Saclay [@Maz86] and Mainz [@Bec90]. The experimental values for $E_{0+}$ at threshold were in conflict with the LET prediction. Most calculations also failed to predict the strong dependence of $E_{0+}$ on the photon energy between the $\pi^0$–threshold (144.7 MeV) and 160 MeV, where a unitary cusp due to the two–step process $\gamma p \rightarrow \pi^+ n \rightarrow \pi^0 p$ [@Fae80] was seen in the Mainz measurement [@Bec90]. These disagreements motivated several theoretical and experimental investigations. New experiments were performed at Mainz [@Fuc96] and Saskatoon [@Berg96], measuring the total and differential cross sections close to threshold. The extracted values of $E_{0+}$ confirmed the strong energy dependence and were again nearly a factor of two smaller than the LET prediction at threshold. This discrepancy was explained by Bernard, Kaiser and Mei[ß]{}ner [@Ber91], who investigated threshold pion photoproduction in the framework of heavy–baryon chiral perturbation theory (ChPT), which showed that additional contributions due to pion loops in $\mu^2$ have to be added to the old LET.
In the following years, refined calculations within heavy–baryon ChPT [@Ber96a] led to descriptions of the four relevant amplitudes at threshold by well–defined expansions up to order $p^4$ in the $S$–wave amplitude $E_{0+}$ and $p^3$ in the $P$–wave combinations $P_1$, $P_2$ and $P_3$, where $p$ denotes any small momentum or pion mass, the expansion parameters in heavy–baryon ChPT. To that order, three low–energy constants (LEC) due to the renormalization counter terms appear, two in the expansion of $E_{0+}$ and an additional LEC $b_P$ for $P_3$, which have to be fitted to the data or estimated by resonance saturation. However, two combinations of the $P$–wave amplitudes, $P_1$ and $P_2$, are free of low–energy constants. Their expansions in $\mu$ converge rather well leading to new LETs for these combinations. Therefore, the P–wave LETs offer a significant test of heavy–baryon ChPT.
However, for this test the S–wave amplitude $E_{0+}$ and the three P–wave combinations $P_1$, $P_2$ and $P_3$ have to be separated. This separation can be achieved by measuring the photon asymmetry using linearly polarized photons, in addition to the measurement of the total and differential cross sections. The $p(\vec{\gamma},\pi^{0})p$ experiment [@Sch01], reported in this letter, was performed at the Mainz Microtron MAMI [@Her90] using the Glasgow/Mainz tagged photon facility [@Ant91; @Hal95] and the photon spectrometer TAPS [@Nov91]. The MAMI accelerator delivered a continuous wave beam of 405 MeV electrons. Linearly polarized photons were produced via coherent bremsstrahlung in a 100 $\mu m$ thick diamond radiator [@Loh94; @Sch95] with degrees of polarization of up to 50%. The diamond radiator was mounted on a goniometer [@Sch95], which was adjusted so that the linearly polarized photons lay in the energy region between $\pi^0$–threshold and 166 MeV. The energy of the photons was determined by measuring the energy of the electron after the bremsstrahlung process with the tagging spectrometer. The resolution was approximately at intensities of up to 5$\times 10^5$ photons s$^{-1}$ MeV$^{-1}$.
Neutral pions were produced in a liquid hydrogen target of cylindrical shape with a length of 10 cm and a diameter of 4 cm. The neutral pion decay photons were detected in TAPS, consisting of six blocks of hexagonally shaped $\rm BaF_2$ scintillation crystals each arranged in a matrix of $8\times8$ detectors. The blocks were mounted in a horizontal plane around the target at polar angles of $\pm50$, $\pm100$, and $\pm150$ degrees with respect to the photon beam direction. A forward wall, consisting of 120 phoswich telescopes [@Nov96], covered polar angles between 5 and 20 degrees. Further details of the experimental set–up are found in Ref. [@Hej00].
The identification of neutral pions relies on the coincident detection of the two photons from $\pi^0$–decay in the TAPS detector (the $\pi^0 \rightarrow \gamma\gamma$ branching ratio is $\approx$ 99.8%). The photons were identified with the help of charged–particle veto detectors, a pulse shape and a time–of–flight analysis. An invariant mass analysis was performed to identify neutral pions and a resolution of $\simeq$ 19 MeV (FWHM) was achieved. Accidental coincidences between TAPS and the tagging spectrometer were subtracted using scaled distributions of background events outside the prompt coincidence time window. For each event a missing energy analysis was performed for an unambiguous identification of neutral pions in the threshold region. The missing energy resolution for $\pi^0$–mesons close to threshold was approximately 5 MeV (FWHM). The acceptance of TAPS for neutral pions and the analysing efficiency were determined by a Monte Carlo simulation using the GEANT3 code [@Bru86] in which all relevant properties of the setup and the TAPS detectors were taken into account.
The differential cross sections can be expressed in terms of the $S$– and $P$–wave multipoles, assuming that close to threshold neutral pions are only produced with angular momenta $l_\pi$ of zero and one. Due to parity and angular momentum conservation only the $S$–wave amplitude $E_{0+}$ ($l_\pi=0$) and the $P$–wave amplitudes $M_{1+}$, $M_{1-}$ and $E_{1+}$ ($l_\pi=1$) can contribute and it is convenient to write the differential cross section and the photon asymmetry in terms of the three $P$–wave combinations $P_1=3E_{1+}+M_{1+}-M_{1-}$, $P_2=3E_{1+}-M_{1+}+M_{1-}$ and $P_3=2M_{1+}+M_{1-}$. The c.m. differential cross section is $$\frac{d\sigma(\theta)}{d\Omega} = \frac{q}{k}(A+B~cos(\theta)+C~cos^2(\theta))\,\,,$$ where $\theta$ is the c.m. polar angle of the pion with respect to the beam direction and $q$ and $k$ denote the c.m. momenta of pion and photon, respectively. The coefficients $A=|E_{0+}|^2+|P_{23}|^2$, $B=2Re(E_{0+}P_1^\ast)$ and $C=|P_1|^2-|P_{23}|^2$ are functions of the multipole amplitudes with $P_{23}^{\,2}=\frac{1}{2}(P_2^{\,2}+P_3^{\,2})$. Earlier measurements of the total and differential cross sections already allowed determination of $E_{0+}$, $P_1$ and the combination $P_{23}$. In order to obtain $E_{0+}$ and all three $P$–waves separately, it is necessary to measure, in addition to the cross sections, the photon asymmetry $\Sigma$, $$\Sigma = \frac{d\sigma_{\perp} -d\sigma_{\parallel}}
{d\sigma_{\perp} + d\sigma_{\parallel}}\,\,,$$ where $d\sigma_{\perp}$ and $d\sigma_{\parallel}$ are the differential cross sections for photon polarizations perpendicular and parallel to the reaction plane defined by the pion and proton. The asymmetry is proportional to the difference of the squares of $P_3$ and $P_2$: $$\Sigma(\theta)=\frac{q}{2k}(P_3^2-P_2^2)\cdot sin^2(\theta)/\frac{d\sigma(\theta)}{d\Omega}\,.$$ Thus, the measurement of the total and differential cross sections together with $\Sigma$ allows a separate determination of $P_2$ and $P_3$ and hence a test of the new LET of ChPT [@Ber96a].
In the present work the total and differential cross sections were measured over the energy range from $\pi^0$–threshold to 168 MeV. Fig. \[totwq\] shows the results for the total cross section which agrees with the data of Ref. [@Berg96]; the results of Ref. [@Fuc96] are systematically lower, at least in the incident photon energy range of 153–162 MeV. This discrepancy may be due to a better elimination of pions produced in the target cell windows, performed in the analysis of the present data, combined with the improved detector acceptance for forward and backward angles. The different slope in the total cross section of [@Fuc96] compared to the other experiments results in a steeper energy dependence for the real part of $E_{0+}$ and slightly smaller values for $P_1$ and $P_{23}$ (see Table \[mult\]). The results for the photon asymmetry are shown in Fig. \[asym\] in comparison to the values of ChPT [@Ber96a] and to a prediction of a dispersion theoretical calculation (DR) by Hanstein, Drechsel and Tiator [@Han97]. The photon asymmetry was determined from all the data between threshold and 166 MeV for which the mean energy was 159.5 MeV. The theoretical predictions are shown for the same energy. The energy dependence of the ChPT prediction for the photon asymmetry was explored in the range threshold to 166 MeV and found to have a very small effect on the average, eg. $<\,2\%$ at $90^\circ$.
The values for the real and imaginary part of $E_{0+}$ and the three $P$–wave combinations were extracted via two multipole fits to the cross sections and the photon asymmetry simultaneously using the following minimal model assumptions. The parameterizations of $ReE_{0+}$ and $ImE_{0+}$ take into account the strong energy dependence of $E_{0+}$ below and above $\pi^+$–threshold ($E_{thr}^{n\pi^+}=$151.4 MeV) due to the two–step process $\gamma p \rightarrow \pi^+ n \rightarrow \pi^0 p$ [@Bern97]: $$E_{0+}(E_\gamma)=A^{p\pi^0}(E_\gamma)+i\,\beta\,q_{\pi^+}\,,$$ where $q_{\pi^+}$ is the $\pi^+$ c.m. momentum. $E_{0+}$ is a sum of two parts, $A^{p\pi^0}$ due to the direct process and a second part, arising from the two–step process. Below $\pi^+$–threshold, one must analytically continue $q_{\pi^+}\rightarrow i|q_{\pi^+}|$. Thus $E_{0+}$ is purely real and has the value $E_{0+}=A^{p\pi^0}-\beta |q_{\pi^+}|$, where $\beta$ is the product of the S–wave amplitude $E_{0+}^{n\pi^+}$ for $\pi^+$–production and the scattering length $a_{n\pi^+ \rightarrow p\pi^0}$. Above $\pi^+$–threshold, $E_{0+}$ is complex with $E_{0+}=A^{p\pi^0}+i\,\beta |q_{\pi^+}|$ and $ImE_{0+}=\,\beta |q_{\pi^+}|$, the cusp function. In the threshold region the imaginary parts of the P-waves are negligible because of the small $\pi N$–phase shifts. The two multipole fits differ in the energy dependence of the real parts of the P–wave combinations. For the first fit the usual assumption of a behaviour proportional to the product of $q$ and $k$ was adopted ($q k$–fit, $\chi^2/dof=1.28$). The assumption made for the second fit is an energy dependence of the P–wave amplitudes proportional to $q$ ($q$–fit, $\chi^2/dof=1.29$). This is the dependence which ChPT predicts for the P–wave amplitudes in the near–threshold region, but at higher energies the prediction is in between the $q$ and $qk$ energy dependence.
The results of both multipole fits for $ReE_{0+}$ as a function of the incident photon energy are shown in Fig. \[reeop\] and compared with the predictions of ChPT and of DR. The results for the threshold values of $ReE_{0+}$ (at the $\pi^0$– and $\pi^+$–threshold), for the parameter $\beta$ of $ImE_{0+}$ and for the values of the threshold slopes of the three P–wave combinations of the $q k$–fit and the $q$–fit are summarized in Table \[mult\] together with the results of [@Berg96] and [@Fuc96; @Bern97]. To obtain the threshold slope of the $qk$–fits the values of the P–wave combinations of these fits (unit: $q k\cdot 10^{-3}/m_{\pi^+}^3$) must be multiplied by the threshold value of the photon momentum $k$. In addition the results are compared to the ChPT and DR predictions, where the errors of ChPT refer to a 5% theoretical uncertainty.
The extracted value for $\beta$ and thus $ImE_{0+}$ of the $q$–fit is larger than the value of $\beta$ obtained with the $q k$–fit. This result can be explained by the observation that A is the best measured of the three coefficients of the differential cross section, and by noting that this determines the absolute value of $E_{0+}$ in addition to the dominant P–wave contribution. Since $ReE_{0+}$ is determined from the B coefficient this gives $ImE_{0+}$ after a subtraction of the P–wave contribution to the A coefficient. If one assumes a smaller energy dependence in the P–wave amplitudes ($q$–fit), a stronger energy dependence for $ImE_{0+}$ will result. However, the values of both fits for $ReE_{0+}$ and the values of the three P–wave combinations at threshold are in remarkable agreement. Assuming for $E_{0+}^{n\pi^+}$ the prediction of ChPT, which agrees with the result of [@Kor99], taking for $a_{n\pi^+ \rightarrow p\pi^0}$ the measured value of [@Schr99] and thus fixing the parameter $\beta$ to the expected unitary value of $3.61\cdot10^{-3}/m_{\pi^+}^2$, the values of the P–wave combinations for both fits change by less than 3%.
The main experimental uncertainty is the value of $\beta$. The systematic error of $\beta$ in Table \[mult\] includes the experimental uncertainty in the energy dependence of the P–wave amplitudes. The average value for $\beta = (3.8 \pm 1.4)\cdot 10^{-3}/m_{\pi^+}^2$ of the two fit results, obtained in this experiment, is consistent with the unitary value. To determine $\beta$ more accurately will require a direct measurement of $ImE_{0+}$ in the $\vec{p}(\gamma,\pi^{0})p$ reaction with a polarized target [@Bern98].
For both fits the low–energy theorems of ChPT (${\cal{O}}(p^3)$) for $P_1$ and $P_2$ agree with the measured experimental results within their systematic and statistical errors. The experimental value for $P_3$ is higher than the value of ChPT, which can be explained by the smaller total and differential cross sections of Ref. [@Fuc96], used by ChPT to determine the dominant low–energy constant $b_P$ for this multipole.
A new fourth–order calculation in heavy–baryon ChPT by Bernard et al., introduced in [@Ber01] and compared to the new MAMI data presented in this letter, shows, that the potentially large $\Delta$–isobar contributions are cancelled by the fourth–order loop corrections to the P–wave low–energy theorems. This gives confidence in the third–order LET predictions for $P_1$ and $P_2$, which are in agreement with the present MAMI data. With the new value of $b_P$ [@Ber01], fitted to the present MAMI data, the ChPT calculation is in agreement with the measured photon asymmetry.
To summarize, the total and differential cross sections and the photon asymmetry for the reaction $p(\vec{\gamma},\pi^{0})p$ have been measured simultaneously for the first time in the threshold region. Using a multipole fit to the physical observables the threshold values of the $S$–wave amplitude $E_{0+}$ and all three $P$–wave amplitudes were extracted. The main conclusion is that the calculations of heavy–baryon ChPT for $P_1$ and $P_2$ are in agreement with the experimental results.
The authors wish to acknowledge the excellent support of K.H. Kaiser, H. Euteneuer and the accelerator group of MAMI, as well as many other scientists and technicians of the Institut für Kernphysik at Mainz. We would like to thank also D. Drechsel, O. Hanstein, L. Tiator and U. Mei[ß]{}ner for very fruitful discussions and comments. A.M. Bernstein is grateful to the Alexander von Humboldt Foundation for a Research Award. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 443) and the UK Engineering and Physical Sciences Research Council.
[99]{} P. de Baenst, Nucl. Phys. B24, 633 (1970) I. A. Vainshtein and V. I. Zakharov, Nucl. Phys. B36, 589 (1972) E. Mazzucato et al., Phys. Rev. Lett. 57, 3144 (1986) R. Beck et al., Phys. Rev. Lett. 65, 1841 (1990) G. Fäldt, Nucl. Phys. A333, 357 (1980) M. Fuchs et al., Phys. Lett. B368, 20 (1996) J. C. Bergstrom et al., Phys. Rev. C53, R1052 (1996); Phys. Rev. C55, 2016 (1997) V. Bernard, J. Gasser, N. Kaiser, and U.–G. Mei[ß]{}ner, Phys. Lett. B268, 291 (1991) V. Bernard, N. Kaiser, and U.–G. Mei[ß]{}ner, Z. Phys. C70, 483 (1996) A. Schmidt, Ph. D. thesis, University Mainz (2001) H. Herminghaus, K.H. Kaiser, and H. Euteneuer, Nucl. Instr. and Meth. A [138]{} 1 (1976) I. Anthony et al., Nucl. Instrum. Meth. A [301]{} 230 (1991) S. J. Hall et al., Nucl. Instrum. Meth. A [368]{}, 698 (1996) R. Novotny, IEEE Trans. Nucl. Sci 38 (1991) 379 D. Lohmann, and J. Peise et al., Nucl. Instr. and Meth. A343 494 (1994) A. Schmidt, Diplomarbeit, University Mainz (1995). R. Novotny, IEEE Trans. Nucl. Sci 43, 1260 (1996) V. Hejny et al., Eur. Phys. J. A 6, 83 (2000) R. Brun et al., GEANT3, Cern/DD/ee/84–1 (1986) O. Hanstein, D. Drechsel, and L. Tiator, Phys. Lett. B399, 13 (1997) A. M. Bernstein et al., Phys. Rev. C55, 1509 (1997) E. Korkmaz et al., Phys. Rev. Lett. 83, 3609 (1999) H.–Ch. Schröder et al., Phys. Lett. B 469, 25 (1999) A. M. Bernstein, Phys. Lett. B442, 20 (1998) V. Bernard, N. Kaiser, and U.–G. Mei[ß]{}ner, Phys. Lett. B378, 337 (1996) V. Bernard, N. Kaiser, and U.–G. Mei[ß]{}ner, hep–ph/0102066 (2001), to be published in Eur. Phys. J. A
Bergstrom$^a$ Fuchs$^a$ ChPT DR$^a$
----------------------------- ------------------------ ------------------------ ----------------- ----------------- -------------- --------
$qk$–fit$^a$ $q$–fit $qk$–fit $qk$–fit
$ E_{0+}(E_{thr}^{p\pi^0})$ $-1.23\pm0.08\pm0.03$ $-1.33\pm0.08\pm0.03$ $ -1.32\pm0.05$ $-1.31\pm 0.2$ -1.16 -1.22
$ E_{0+}(E_{thr}^{n\pi^+})$ $-0.45\pm0.07\pm0.02 $ $-0.45\pm0.06\pm0.02 $ $ -0.52\pm0.04$ $-0.34\pm 0.03$ -0.43 -0.56
$\beta$ $2.43\pm0.28\pm1.0$ $5.2\pm0.2\pm1.0$ 3.0–3.8 $2.82\pm0.32$ 2.78 3.6
$ {P_1}$ $ 9.46\pm0.05\pm0.28$ $9.47\pm0.08\pm0.29$ $ 9.3\pm0.09$ $ 9.08\pm 0.14$ $9.14\pm0.5$ 9.55
$ {P_2}$ $-9.5\pm0.09\pm0.28 $ $ -9.46\pm0.1\pm0.29 $ $-9.7\pm0.5$ -10.37
$ {P_3}$ $11.32\pm0.11\pm0.34$ $11.48\pm0.06\pm0.35$ $10.36$ 9.27
$ P_{23}$ $ 10.45\pm 0.07$ $ 10.52\pm0.06$ $10.53\pm 0.07$ $10.37\pm0.08$ 11.07 9.84
: Results of both fits ($qk$–fit and $q$–fit) for $ReE_{0+}$ at the $\pi^0$– and $\pi^+$–threshold (unit: $10^{-3}/m_{\pi^+}$), for the parameter $\beta$ of $ImE_{0+}$ (unit: $10^{-3}/m_{\pi^+}^2$) and for the three combinations of the $P$–wave amplitudes (unit: $q \cdot 10^{-3}/m_{\pi^+}^2$) with statistical and systematic errors in comparison to the results of previous experiments ( [@Berg96] and [@Fuc96; @Bern97], only with statistical errors) and to the predictions of ChPT [@Ber96a; @Ber96b] (${\cal{O}}$($p^3$)) and of a dispersion theoretical approach (DR, [@Han97]). ($^a$ Values of the P–wave combinations converted into the unit $q \cdot 10^{-3}/m_{\pi^+}^2$.)[]{data-label="mult"}
|
---
abstract: 'We present Gemini spectroscopy for 21 candidate optical counterparts to X-ray sources discovered in the Galactic Bulge Survey (GBS). For the majority of the 21 sources, the optical spectroscopy establishes that they are indeed the likely counterparts. One of the criteria we used for the identification was the presence of an H$\alpha$ emission line. The spectra of several sources revealed an H$\alpha$ emission line only after careful subtraction of the F or G stellar spectral absorption lines. In a sub-class of three of these sources the residual H$\alpha$ emission line is broad ($\simgt400$ km s$^{-1}$) which suggests that it is formed in an accretion disk, whereas in other cases the line width is such that we currently cannot determine whether the line emission is formed in an active star/binary or in an accretion disk. GBS source CX377 shows this hidden-accretion behaviour most dramatically. The previously-identified broad H$\alpha$ emission of this source is not present in its Gemini spectra taken $\sim$1 year later. However, broad emission is revealed after subtracting an F6 template star spectrum. The Gemini spectra of three sources (CX446, CX1004, and CXB2) as well as the presence of possible eclipses in light curves of these sources suggest that these sources are accreting binaries viewed under a high inclination.'
author:
- |
Jianfeng Wu,$^1$[^1] P. G. Jonker,$^{1,2,3}$ M. A. P. Torres,$^{2,3}$ C. T. Britt,$^{4,5}$[^2] C. B. Johnson,$^{4}$R. I. Hynes,$^{4}$S. Greiss,$^{6}$ D. T. H. Steeghs,$^{1,6}$ T. J. Maccarone,$^{5}$ C. O. Heinke,$^{7,8}$ T. Wevers$^{3}$\
$^1$Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA\
$^2$SRON, Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA, Utrecht, The Netherlands\
$^3$Department of Astrophysics/IMAPP, Radboud University, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands\
$^4$Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA\
$^5$Department of Physics, Texas Tech University, Box 41051, Lubbock TX, 79409-1051, USA\
$^6$Department of Physics, University of Warwick, Coventry, CV4 7AL, UK\
$^7$Department of Physics, University of Alberta, CCIS 4-183, Edmonton, AB T6G 2E1, Canada\
$^8$Max Planck Institute for Radio Astronomy, Auf dem Hugel 69, 53121 Bonn, Germany\
title: 'Gemini spectroscopy of Galactic Bulge Sources: a population of hidden accreting binaries revealed?[^3]'
---
binaries: close — stars: emission line, Be — Galaxy: Bulge — X-rays: binaries
Introduction {#intro}
============
Previous surveys of faint sources have been focused on the Galactic Center or globular clusters. While the Galactic Center Survey (e.g., Muno et al. 2003) benefits from a high source density, the crowding and significant extinction make the optical/infrared follow-up necessary for classification difficult (e.g., Mauerhan et al. 2009).
The Galactic Bulge Survey (GBS; Jonker et al. 2011; Jonker et al. 2014; Paper I & II hereafter) is a multiwavelength project that is designed to allow optical/infrared classification of X-ray sources detected in the Galactic Bulge. The GBS consists of [*Chandra*]{} and multiwavelength observations of two $6^\circ\times1^\circ$ strips centered $1.5^\circ$ above and below the Galactic plane (see Fig. 1 of Paper I), thus avoiding the $|b|<1^\circ$ regions with serious crowding and extinction problems, while still maintaining a relatively high source density. The GBS utilizes observations with an exposure of 2 ks for each pointing; the exposure time is chosen to maximize the relative numbers of low-mass binaries (LMXBs) to cataclysmic variables (CVs). The completed observations have detected 1640 unique sources (Paper II), agreeing well with the estimation in Paper I which also gave a break-down of the expected numbers of various kinds of objects based on source density, expected flux limit, etc. Among the 1640 sources, $\sim600$ are expected to be CVs, including both intermediate polars (IPs) and non-magnetic CVs, while the number of LMXBs is expected to be $\sim250$ (see Table 2 of Paper I). We also expect $\sim$600 chromospherically active stars or binaries, e.g., RS Canum Venaticorum variables (RS CVn systems; Walter et al. 1980).
The GBS combines a large sky coverage with the good sensitivity to faint X-ray sources and the excellent positional accuracy possible with [*Chandra*]{}. There are two main science goals of the GBS (see §1 of Paper I for more details): 1) constraining the nature of the common-envelope phase in binary evolution by comparing the observed number of sources with model predictions in each class, e.g., CVs and LMXBs; 2) measuring the mass of the compact objects in X-ray binaries, e.g., eclipsing quiescent black hole (BH) and neutron star (NS) LMXBs, to investigate the Galactic BH mass distribution (e.g., Özel et al. 2010) and to constrain the NS equation of state (EoS).
Both of the science goals rely on the multiwavelength identification and classification of this large sample of faint sources. A variety of optical/infrared follow-up campaigns have been conducted. Hynes et al. (2012) identified 69 sources in the GBS using the [*Tycho*]{}-2 catalogue. These sources are coincident with or very close to the bright stars in that catalogue, most of which are likely to be the real optical counterparts to the sources. This sample is a mix of objects with a broad range of spectral types, including both late-type stars with coronal emission and early-type stars with wind emission. Many sources are foreground objects instead of residing in the Galactic Bulge. Britt et al. (2014) reported on an optical photometric survey of three quarters of the sky area covered by the GBS, and presented the light curves of variable objects consistent with the positions of GBS sources catalogued in Paper I. About a quarter of the optical counterparts are variable, and they are expected to be a mix of IPs, non-magnetic CVs, LMXBs, and RS CVns. Greiss et al. (2014) provided likely near-infrared identification of GBS sources using current near-infrared sky surveys. Maccarone et al. (2012) found 12 candidate radio matches to the GBS sources using the archival NRAO VLA Sky Survey (NVSS; Condon et al. 1998). The majority of them appear to be background active galactic nuclei.
Optical/infrared spectroscopy of the detected sources is an essential tool to investigate their nature. Accreting binaries can be identified by the emission features in their optical spectra. The only firm way to distinguish white dwarf (WD), NS, and BH as the primaries of the X-ray binaries is via measurements of the accretor masses, which requires high-quality optical/infrared spectroscopy. Britt et al. (2013) presented five accreting binaries identified in the GBS based on the strong emission lines in their spectra, including three likely IPs, one CV undergoing a dwarf nova outburst, and one likely quiescent LMXB (qLMXB). Torres et al. (2014) identified 22 new accreting binaries via the H$\alpha$ emission lines in their optical spectra. They developed criteria of accreting binaries based on the equivalent width (EW) of H$\alpha$ emission line (EW$>18$ Å), the breadth of the H$\alpha$ emission line (FWHM $>400$ km s$^{-1}$), or the strength of He [i]{} $\lambda$5876,6678 in case of narrow and weak H$\alpha$ lines. There are also several extensive spectroscopic studies on individual GBS sources. Ratti et al. (2013) presented a dynamical analysis of GBS source CX93 in which they measured the mass of the compact primary and the companion star, and concluded that the source is a long orbital period CV. Hynes et al. (2014) identified a symbiotic binary with a carbon star companion in the Galactic Bulge based on the spectra of its optical counterpart.
In this work, we present Gemini spectroscopy of 21 GBS sources with a better positional accuracy than the median (due to low off-axis angles). These 21 sources are listed in Table \[log\_table\]. We will refer to these sources with their labels, i.e. CX (or CXB) IDs, introduced in the GBS source catalogue (Paper I and II). GBS sources listed in Paper I have the prefix of “CX”, while the remaining GBS sources, detected in the last quarter of the coverage published in Paper II, have the prefix of “CXB”. The sources in each catalogue were ranked by their counts, where CX1 has the most counts among the CX sources. The majority of the sources in this work are CXB sources, while previous works were focused on CX sources. This paper is structured as follows. In §\[obs\] we describe the Gemini observations and data reduction. In §\[ana\] we present the analysis of the Gemini spectroscopy, including spectral classification and radial velocity analysis. In §\[discuss\] we give results and discuss each interesting object. Overall conclusions are summarized in §\[conclusion\].
Observation & Data Reduction {#obs}
============================
Gemini Spectroscopy {#obs:spec}
-------------------
The list of 21 GBS sources of which we obtained Gemini spectroscopy consists of six CX sources and 15 CXB sources. The six CX sources were proven to be interesting on the basis of earlier spectra and/or photometric variability. For example, three of the CX sources (CX377, CX446, and CX1004) have shown H$\alpha$ emission lines in their previous spectra (Torres et al. 2014) obtained by the VIsible Multi-Object Spectrograph (VIMOS) mounted on the Very Large Telescope (VLT). Based on our follow-up strategy (i.e., prioritizing sources with higher positional accuracy and brighter in optical/infrared), the 15 CXB sources are chosen to have off-axis angles less than 5 arcmin in their observations and also have sufficient counts to allow for an accurate position ($<1^{\prime\prime}$). Optical/infrared brightness, colour and photometric variability are also among the factors of sample selection.
The finding charts of our sources are shown in Appendix A (see Fig. \[fc\_fig\]–\[fc\_fig2\]). The coordinates listed in Table \[log\_table\] are for the candidate optical counterparts, i.e., the objects for which we took Gemini spectroscopy. The astrometry was performed on images from VLT/VIMOS (CX377, CX446, and CX1004; Torres et al. 2014), Gemini (CXB117), and Mosaic-II/Dark Energy Camera (DECam; other sources; see §\[obs:photo\]). All of them have a RMS accuracy of $<0.2^{\prime\prime}$, while some sources (CX377, CX1004, and CXB117) have $<0.1^{\prime\prime}$ positional accuracy (Britt et al. 2014; Torres et al. 2014).
Optical spectroscopy of the 21 GBS sources was obtained with GMOS (Gemini Multi-Object Spectrograph; Davies et al. 1997) mounted on the Gemini-South Telescope in Chile. All the observations were taken between 2012 Apr 20 and 2013 May 4 under programmes GS-2012A-Q-44 and GS-2012A-Q-67 (see Table \[log\_table\] for an observation log). Nine objects have multi-epoch spectroscopy, while other objects have one epoch; each epoch has 900–3600 s integration time. The seeing of each spectroscopic observation was measured from the corresponding acquisition image (see the last column of Table \[log\_table\]). Typical seeing was around $0.7^{\prime\prime}$ (with a range of 0.5–1.3$^{\prime\prime}$). GMOS was operated in long-slit mode. We used the R400\_G5325 grating (400 line mm$^{-1}$) and a $0.75^{\prime\prime}$ slit. The two-dimensional spectra were binned by a factor of two in both spatial and spectral dimensions, resulting in a spatial dispersion of $0.15^{\prime\prime}$/pixel and a spectral dispersion of $1.36$ Å/pixel. The spectral resolution is estimated to be $\approx5$ Å FWHM for the sources that had filled the whole slit during the observation (i.e., the seeing was greater than the slit width 0.75$^{\prime\prime}$), while spectral resolution should be better than $\approx5$ Å FWHM for the sources with seeing less than 0.75$^{\prime\prime}$. All the Gemini/GMOS spectra were split into three equal parts in the wavelength dimension by detector gaps. The bluest part of the spectra was ignored in the analysis because of the low signal-to-noise ratio ($S/N$) in the spectra due to the extinction towards our sources and the lack of arc lines in this part of the spectra. The middle part (wavelength range $\sim4800$–6100 Å) and the red part (wavelength range $\sim6200$–7600 Å) of the spectra were reduced and analyzed separately. The results presented in this work are mainly based on the analyses of the red part of the spectra because it is least affected by extinction.
The Gemini/GMOS data were reduced using the [figaro]{} package implemented in the [starlink]{} software suite and the packages of [pamela]{} and [molly]{} developed by T. Marsh.The two-dimensional spectra were bias-corrected and flat-fielded. The bias was corrected using the overscan areas of the detectors. We utilized the flat fields taken directly following each target observation for flat fielding. We fit the background on both sides of the spectra with a second-order polynomial and determined the background at the position of the spectra. The object spectra were optimally extracted using the `optexp` procedure in the [pamela]{} package (Marsh 1989). The spectra were wavelength-calibrated in [molly]{} using the helium-argon arc spectra which were taken either right after observing the target or at the end of the night. The average arc spectrum was used in case of multiple arc spectra in one night. The resulting root mean square (RMS) scatter on the wavelength calibration is $\simlt0.3$ Å. The wavelength calibration was examined using the skyline O [i]{} $\lambda 6300.303$. Offsets relative to the wavelength of this line have been corrected. For sources with seeing less than the slit width ($0.75^{\prime\prime}$), the centroiding uncertainty (i.e., if the source is not placed in the middle of the slit) may introduce a small wavelength shift (Bassa et al. 2006). This wavelength shift cannot be corrected by examining the sky-line wavelength as those fill the whole slit. However, it can potentially be assessed by checking the wavelengths of diffuse interstellar bands (DIBs; Herbig 1995). Three such sources (CX84, CXB149, and CXB174) have strong DIBs at $\lambda5780$ for which the line profiles do not deviate from Gaussians. We checked these features and found they have minor shifts relative to the rest-frame wavelength (65 km s$^{-1}$ for CX84, $-30$ km s$^{-1}$ for CXB149, and $-75$ km s$^{-1}$ for CXB174 in heliocentric frame). We corrected the wavelength scale for these small shifts. Each spectrum was normalized by dividing it by the result of fitting a 5th-order polynomial fit to the continuum.
Optical Photometry from Mosaic-II & DECam {#obs:photo}
-----------------------------------------
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Time-resolved optical photometry for all six CX sources was obtained with the Mosaic-II imager mounted on the Blanco 4-meter telescope at the Cerro Tololo Inter-American Observatory (CTIO) in 2010 July 8–15. Nineteen exposures in the Sloan $r^\prime$-band with an integration time of 120 s were taken on 45 overlapping fields to cover a nine square degree area, which contain all but seven of the X-ray sources identified in Paper I (see Britt et al. 2014 for the full description and results of this variability campaign). Typical seeing for these imaging observations was around $1^{\prime\prime}$ (with a range of 0.8–3.0$^{\prime\prime}$). The data were reduced using the NOAO Mosaic Pipeline (Shaw 2009). We performed differential photometry using Alard’s image subtraction ISIS (Alard & Lupton 1998; Alard 2000) to obtain the changes in flux with respect to reference images. The zero-point flux in the reference images was measured with either aperture photometry or DAOPHOT-II (Stetson 1987). The number of variable interlopers within the 95% error circles of the position is $\sim40$ ($\sim3\%; $see §3.1 of Britt et al. 2014 for details).
The optical photometry information of four CX sources (CX84, CX377, CX446, CX1004) in this paper is shown in Table \[rmag\_table\], while their light curves have been presented in literature (CX84 in Fig. 6 of Britt et al. 2014; CX377, CX446 and CX1004 in Fig. A2 of Torres et al. 2014). For CX138, there are two blended sources in the Mosaic-II image at its position; neither of the possible counterparts is variable. The counterpart to CX139 is saturated in the Mosaic-II imaging. Among the four CX sources with light curves, only CX84 shows possibly periodic behaviour with a period of 4.67 days. However, it is worth noting that it is currently not possible to confirm the periodic nature of this variability because the baseline of our monitoring was only 8 days, which is less than two full cycles. The light curves of the other three sources only have random flickering with an RMS$\sim0.05$–0.1 magnitude, although CX446 possibly experiences eclipse events.
Optical photometry for five of the CXB sources in our sample (CXB2, CXB64, CXB82, CXB99, and CXB113) was obtained using the Dark Energy Camera (DECam) instrument mounted on the Blanco 4-meter telescope at CTIO on the nights of June 10 and 11 of 2013. The average seeing for both nights was $1.3^{\prime\prime}$ with a range of 0.9–1.9$^{\prime\prime}$. DECam provides a $2.2\times2.2$ square degree field of view combining 62 science CCDs, 8 focus CCDs, and 4 guiding CCDs with a scale of 0.27$^{\prime\prime}$ per pixel. For all of our images, the SDSS $r^\prime$ filter was used with exposure times of either $2\times90$ s or $2\times1$ s for the faint and bright sources, respectively. The DECam pipeline reduction provided the resampled images with cross-talk corrections, overscan, trimmed sections, bias subtraction, flat-fielding and saturation masks.[^4] We then used standard IRAF tasks (including `wcsctran`, `digiphot`, and `apphot`) to extract magnitudes and fluxes and to generate light curves through the use of differential photometry. Calibration of the target stars was achieved by using reference stars in the field of view that were contained in the Carlsberg Meridian 14 catalogue (Evans et al. 2002) and VizieR catalogue: I/304. The photometry of these five sources are included in Table \[rmag\_table\] while their light curves are shown in Fig. \[lcb\_fig\]. Among the five GBS sources, CXB2 shows possible eclipsing/dipping events and an “outburst”. Although the outburst was towards the end of the night with higher airmass, visual inspection of the images confirms that the brightening is real. This outburst is possibly the reprocessed emission from a Type I burst, which is similar to the case of NS LMXB EXO 0748$-$676 (Hynes et al. 2006). The light curve shows a likely periodic modulation of $P\approx0.447$ day (see the phase-folded light curve). CXB82 and CXB99 both appear to be long period variables with periods longer than our 2-night observing run. For both sources, there is a steady increase in brightness by about $\sim0.1$ and $\sim0.25$ magnitude, respectively. Due to the lack of $r^\prime$ standard stars in their fields, CXB64 and CXB113 only have approximate magnitudes without absolute calibration. CXB64 has a counterpart USNO B1.0 star 0578$-$0732346 with the optical magnitude of $R\sim18.7$ and $I=16.9$. CXB113 shows a sinusoidal modulation with period of $P = 0.58791(12)$ day. The phase-folded light curve for this source is also shown in Fig. \[lcb\_fig\]. The optical counterpart to CXB113 was also identified by OGLE (Optical Gravitational Lensing Experiment; Field: BUL\_SC37, StarID: 9614; Udalski et al. 1992). The OGLE source has optical magnitudes of $V =
19.135(154)$ and $I=15.847(49)$. It also shows a periodic modulation of $P\approx0.588$ day.
Near-Infrared Photometry {#obs:irphoto}
------------------------
![Gemini/GMOS spectra for six sources with H$\alpha$ emission lines (CX446, CX1004, CXB2, CXB64, CXB99, and CXB113). The broad H$\alpha$ emission of CX446, CX1004, and CXB2 likely originate in the accretion disk, while the narrow H$\alpha$ and H$\beta$ emission of CXB64 and CXB113 are an indication of chromospherically active binaries. The H$\alpha$ emission of CXB99 is narrow and weak. All the spectra are normalized to unity. The positions of the H$\alpha$ and the H$\beta$ emission lines are labeled by the dotted lines.[]{data-label="spec_fig1"}](allspec_e.ps){width="3in"}
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![Gemini/GMOS spectra for the remaining 6 sources presented in this paper. All the spectra are normalized to unity. The positions of the H$\alpha$ and the H$\beta$ emission lines are labeled by the dotted lines.[]{data-label="spec_fig3"}](allspec_n.ps){width="3in"}
The likely near-infrared matches to the GBS sources detected by (Greiss et al. 2014) were mainly obtained from the public variability survey VISTA Variables in the Vía Láctea (VVV; Minniti et al. 2010). The VVV survey provides the most complete near-infrared coverage of the GBS area with consistent depth. Complementary coverage from the Two-Micron All Sky Survey (2MASS; Skrutskie et al. 2006) and UKIDSS Galactic Plane Survey (GPS; Lucas et al. 2008) were also utilized for bright ($K_s < 12.5$) and faint ($K_s > 16$) sources, respectively. Greiss et al. (2014) developed a method to estimate the likelihood (by calculating the false alarm probability) of a near-infrared source to be the counterpart of the X-ray source. However, for the optical sources where we have obtained Gemini/GMOS spectra, we are able to search for the near-infrared counterparts that matches our optical sources by comparing the optical and near-infrared images. We set a $0.2^{\prime\prime}$ astrometric error circle, and searched for any matches within that error circle between the optical images and VVV images. Then we visually examined both images and selected the true near-infrared counterparts. A visual inspection is crucial given that the GBS fields are crowded.
Table \[vvv\_table\] lists the near-infrared counterparts to 13 of the 21 optical sources with Gemini/GMOS spectroscopy. The other eight sources do not have reliable near-infrared counterparts. Three of them (CX446, CXB26, and CXB137) are too faint in the VVV frames and there are no near-infrared sources at the optical position. For the other five sources (CX1004, CXB64, CXB117, CXB189, and CXB201), there could be near-infrared matches but they are blended with nearby sources, for which the current VVV data release does not provide photometry.
The infrared colours can be used to estimate the distance of the source by calculating the needed extinction to match the source infrared colours to those of a standard star with the same spectral type (e.g., Ratti et al. 2013). Some of the GBS sources have an infrared excess, i.e., their infrared colours are redder than expected based on optical colours (see §4 of Hynes et al. 2012), which is possibly for cases where the optical classification may represent the hotter component of a binary (or of a blend, if another star is serendipitously along the same line of sight) while the infrared colours are from the cooler component. Accreting systems (CVs and LMXBs) and Be stars with a circumstellar disk may also have colours that do not match those of single stars. Be stars could also have significant colour variations due to the formation/dissolution of the circumstellar disks.
Data Analysis {#ana}
=============
The Gemini/GMOS spectra of the 21 GBS sources are shown in Fig. \[spec\_fig1\]–\[spec\_fig3\]. Three sources (CX446, CX1004, and CXB2) show apparent broad H$\alpha$ emission lines, while another three (CXB64, CXB99, and CXB113) have narrow H$\alpha$ emission (see Fig. \[spec\_fig1\]). We fit Gaussian profiles to these broad H$\alpha$ emission lines to measure their width and velocity separations (if double-peaked) using the `mgfit` procedure in the [molly]{} package. The equivalent widths (EWs) of these H$\alpha$ lines were measured using the `light` procedure in [molly]{}. Nine other objects have H$\alpha$ absorption features (Fig. \[spec\_fig2\]). The stellar absorption features shown in the Gemini/GMOS spectra can be utilized to perform spectral classification, and radial velocity analysis. The spectra of the remaining six sources are shown in Fig. \[spec\_fig3\]; they appear at first sight to have neither H$\alpha$ emission features nor H$\alpha$ absorption features.
Some of the objects in our sample have strong DIBs in their spectra. We estimated the reddening for these sources via the equivalent width (EW) of the DIB at $\lambda5780$ with the calibration in Table 3 of Herbig (1993). We also compared the measured reddening to the Bulge reddening along the line of sight provided by Gonzalez et al. (2011,2012), which utilized the Red Clump stars in the Bulge.[^5]
Spectral Classification {#ana:class}
-----------------------
We utilize the optimal subtraction technique following previous works (e.g., Marsh et al. 1994; Ratti et al. 2013) to classify the spectra. A set of template star spectra was chosen from the library of the Ultraviolet and Visual Echelle Spectrograph Paranal Observatory Project (UVES POP; Jehin et al. 2005), covering spectral types from A0 to M6 with luminosity class of V (see Table \[uves\_table\]). The UVES POP template spectra provide coverage over the wavelength range of 3000–10000 Å with a spectral resolution of $\sim80,000$. The templates were re-sampled and Gaussian-smoothed to match the spectral resolution of the object spectra. The object spectra were Doppler-shifted into the same rest frame and averaged. Each stellar template was optimally subtracted from the object spectrum, while a $\chi^2$ test was performed on the residuals. All the emission lines, DIBs, and telluric features (e.g., Kurucz 2006; Wallace et al. 2011) were masked during the procedure. The resulting $\chi^2$ values for each template were compared with each other. The template with the minimum $\chi^2$ value provides our best estimate of the spectral classification of each GBS source in our sample. These spectral classification procedures were first performed on the red part of the Gemini spectra. We verified our results by performing the same procedures to the middle part of the Gemini spectra and the results are consistent with each other. The results of spectral classifications are listed in Table \[vsini\_table\]. Note we are not controlling luminosity class in the spectral classification. Nine of the 21 GBS sources in our sample were spectrally classified. The uncertainty of spectral classification is estimated to be one or two spectral sub-classes. The other three sources listed in Table \[vsini\_table\] are classified as early–mid M-type based on their spectral features. We have also tried to measure the rotational broadening of the absorption features. However, our limited spectral resolution of $\sim200$ km s$^{-1}$ precluded the determination of rotational velocities.
Radial Velocity Analysis {#ana:vel}
------------------------
The radial velocities of the optical counterpart of each GBS source are measured by cross-correlating the object spectra using the `xcor` procedure in [molly]{}. For each source, all the object spectra and the template spectrum were rebinned to the same velocity dispersion with the `vbin` procedure. We also masked all the emission lines, interstellar/telluric features during the cross-correlation procedure. We performed two sets of the cross-correlation analysis. The first set uses the first source spectrum as the cross-correlation template, i.e., we derive the radial velocity (RV1) relative to the first source spectrum. For the GBS sources that were spectrally classified using the procedure in §\[ana:class\], we also performed a second set of cross-correlation analyses, taking the UVES POP standard star with the best-fit spectral type as the template. Thus the second set of radial velocity (RV2) is relative to the template star. All spectra had been shifted to a heliocentric frame.
Radial velocity values were derived by fitting a Gaussian profile to the cross-correlation function. The results are listed in Table \[rvc\_table\]. Seven sources have non-zero radial velocities relative to the standard star template. Three sources (CX84, CX138, CX139) have radial velocity variations of $\sim100$ km s$^{-1}$ on timescales of days.
Results & Discussion {#discuss}
====================
Objects with H$\alpha$ Emission Lines {#discuss:emission}
-------------------------------------
There are six sources with apparent H$\alpha$ emission in their Gemini/GMOS spectra (Fig. \[spec\_fig1\]). Three of them (CX446, CX1004, and CXB2) have broad H$\alpha$ emission lines (FWHM$\simgt800$ km s$^{-1}$), while those of the other three sources (CXB64, CXB99, and CXB113) are narrow (FWHM$\simlt200$ km s$^{-1}$). The strong, broad H$\alpha$ emission likely originates from an accretion disk, indicating CX446, CX1004, and CXB2 are likely accreting binaries (§\[discuss:cx446\]).
CXB64 and CXB113 are possibly chromospherically active stars or binaries because of their narrow and weak H$\alpha$ emission lines (Torres et al. 2014; also see Fig. \[spec\_fig1\]). The $H-K$ colour ($0.300\pm0.033$; see Table \[vvv\_table\]) of CXB113 agrees well with that of a M4V–M5V star ($H-K\approx0.30$; see Table 5 of Pecaut & Mamajek 2013), while its $J-H$ colour ($0.509\pm0.030$) appears slightly bluer than that of a M4V–M5V type star ($J-H\approx0.57$).
CXB99 has a weak, narrow H$\alpha$ emission. After the optimal subtraction with the best-fit K2V template, the H$\alpha$ emission appears to be stronger, indicating that it partially fills in the H$\alpha$ absorption line in the spectrum.
### Potential Quiescent Accreting Binaries: CX446, CX1004, and CXB2 {#discuss:cx446}
![The H$\alpha$ region of both epochs of Gemini/GMOS spectra for CX446, acquired on 2012 Jun 22 (blue line) and 2013 May 4 (red line), respectively, showing evidence for strong variability in the EW of the H$\alpha$ between the two epochs. The VLT/VIMOS spectrum of CX446 is also overlaid (black line).[]{data-label="cx446_fig"}](cx446.ps){width="3.4in"}
![The H$\alpha$ region of the four epochs of Gemini/GMOS spectra for CX1004 (from top to bottom in chronological order; see Table \[log\_table\]). All four spectra show a broad, double-peaked H$\alpha$ profile which varied in strength between the observations.[]{data-label="cx1004_fig"}](cx1004.ps){width="3.4in"}
![The H$\alpha$ region of the four epochs of Gemini/GMOS spectra for CXB2 (from top to bottom in chronological order; see Table \[log\_table\]). The spectra of the first and third epochs show broad, weak H$\alpha$ emission, whereas it is absent in the bottom spectrum. Both the emission and absorption features have substantial negative velocity offset which may indicate the high space velocity of this object.[]{data-label="cxb2_fig"}](cxb2.ps){width="3.4in"}
Both epochs of the Gemini/GMOS spectra of CX446 show broad H$\alpha$ emission (see Fig. \[cx446\_fig\]). The strength of the H$\alpha$ line is weaker in the spectra taken on 2013 May 4 than that on 2012 Jun 22. The line width ($1250\pm50$ km s$^{-1}$) is smaller than that of the H$\alpha$ emission lines in the VLT/VIMOS spectra of CX446 ($2200\pm50$ km s$^{-1}$; Torres et al. 2014), while the line EW is bigger. No absorption lines from the companion star are visible in the Gemini spectra. We also see no evidence of He [i]{} $\lambda$ 6678. The light curve of CX446 does show a possible eclipse event with a depth of 0.4 magnitude (see Fig. A2 of Torres et al. 2014; on HJD$=2455387.82$). No significant periodicity is found in the light curve. CX446 is a candidate eclipsing CV or qLMXB.
The Gemini/GMOS spectra of CX1004 show double-peaked H$\alpha$ emission in all four epochs (see Fig. \[cx1004\_fig\]). We measure the H$\alpha$ emission line properties using the spectra of the last epoch because of its best $S/N$. The line width is FWHM = $2500\pm100$ km s$^{-1}$, which is broader than that in the VLT/VIMOS spectra of CX1004 (FWHM = $2100\pm20$ km s$^{-1}$; see Torres et al. 2014). The line strength (EW$=38.0\pm0.6$ Å) has also slightly increased compared to that in the VLT/VIMOS spectra (EW$=32.9\pm0.4$ Å). The velocity separation between the red and blue peaks is $\Delta v = 1160\pm30$ km s$^{-1}$, which is consistent with the result in Torres et al. (2014). The centroid of the H$\alpha$ line does not have significant radial velocity (velocity offset $-15\pm20$ km s$^{-1}$), while that of the VLT/VIMOS spectra shows substantial radial velocity ($-170\pm20$ km s$^{-1}$; Torres et al. 2014). One possible scenario for this is that the accretion disk of CX1004 is precessing. The double-peaked profile of the H$\alpha$ emission line is asymmetric in three of the four epochs; the relative strength of the blue peak and the red peak varies with time (see Fig. \[cx1004\_fig\]). This behaviour can be explained by the presence of an S-wave originating in a hot-spot or the donor star (see §IV of Johnston et al. 1989).
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The prominent molecular absorptions in the wavelength range of 6300–7300 Å are signatures of an M-type companion star. However, the detailed rotational broadening analysis and spectral classification are not feasible for the Gemini/GMOS spectra of CX1004. As discussed in Torres et al. (2014), the line width and the double-peak velocity separation are consistent with that of an eclipsing quiescent CV or qLMXB. However, possibly owing to the faintness of the source, the light curve of CX1004 (see Fig. A2 of Torres et al. 2014) does not show significant periodic behaviour. Torres et al. (2014) suggested CX1004 to be a nearby source due to the lack of diffuse interstellar bands.
The four epochs of Gemini/GMOS spectra of CXB2 are shown in Fig. \[cxb2\_fig\]. The spectra of the first and third epochs appear to have weak, broad H$\alpha$ emission. Both of the lines have large velocity offsets (which are also visible in the figure) with $v_{\rm peak} = -260\pm60$ km s$^{-1}$ and $-200\pm30$ km s$^{-1}$, respectively. The widths of the two lines are FWHM = $825\pm175$ km s$^{-1}$ and $725\pm100$ km s$^{-1}$, respectively. The spectrum of the fourth epoch shows a narrow absorption feature (FWHM = $180\pm40$ km s$^{-1}$), which also has a significant velocity offset ($v_{\rm peak} =
-175\pm15$ km s$^{-1}$). These large negative offsets could originate in a high space velocity of this source, which may imply a NS or a BH as the primary star. CXB2 also shows the H$\alpha$ emission variation as we have seen in CX377. No photospheric line from the companion star is detected from the spectra of CXB2. CXB2 is brighter in than most of the GBS sources. It has 147 ACIS-I counts in 0.3–8.0 keV band in a 2 ks exposure. CXB2 was serendipitously detected by an archival Suzaku observation with 53 ks exposure time (ObsID: 507031010). Full details of the analysis will be presented elsewhere, but we briefly summarize relevant results here. The Suzaku XIS0 and XIS1 data were reduced and analysed using standard `FTOOLS`. A good spectral fit ($\chi^2/\nu$=1.0) was found for CXB2 with an absorbed power-law model (photon index $\Gamma=0.96\pm0.11$, $N_{\rm H}=1.5^{+1.4}_{-1.0}\times10^{21}$ cm$^{-2}$). The flux is $1.5\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$, which is consistent with that from our Chandra observation ($1.3\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$, assuming the same spectral model). This source is probably associated with the [*ASCA*]{} source AX J1754.0$-$2929 which is 7.2$^{\prime\prime}$ away (see catalogue in Sakano et al. 2002). The flux in the [*ASCA*]{} observation is $1.4\times10^{-12}$ cm$^{-2}$ s$^{-1}$ in 0.7–10 keV band, which is similar to the flux in the observation. This source was, however, not detected in and is classified as a transient (Paper II). The $N_{\rm H}$ value is smaller than that predicted to the Galactic Bulge ($\sim10^{22}$ cm$^{-2}$), hence we infer that CXB2 probably lies in the 1–4 kpc distance range. Its near-infrared colours also indicate low reddening. The luminosity would be 1–40$\times10^{32}$ erg s$^{-1}$ (assuming the above distance range), which is consistent with either a high-inclination CV or a qLMXB.
Objects with H$\alpha$ in Absorption {#discuss:cx377}
------------------------------------
![The comparison of the H$\alpha$ region of CX377 optical spectra, acquired by VLT/VIMOS (upper line) and Gemini/GMOS (lower line), respectively. The broad H$\alpha$ emission line in the VLT/VIMOS spectrum is absent in the Gemini/GMOS spectrum, indicating strong spectral variability even though the source remains in quiescence.[]{data-label="cx377_fig"}](cx377.ps){width="3.4in"}
Nine sources (CX138, CX377, CXB26, CXB73, CXB117, CXB149, CXB174, CXB189, and CXB201) in our sample only show H$\alpha$ absorption lines in their Gemini/GMOS spectra (Fig. \[spec\_fig2\]). We are able to obtain a spectral classification for five of them (CX138, CX377, CXB73, CXB149, and CXB174). The residual spectra of these sources after optimally subtracting their best-fit stellar templates are shown in Fig. \[resid\_fig\]. Three of them (CX138, CX377, and CXB73) have unambiguous H$\alpha$ emission features, which partially fill in the stellar H$\alpha$ absorption features in our Gemini/GMOS spectra. We measured the strength (EW) and width (FWHM) of the H$\alpha$ emission shown in these residual spectra (see Table \[line\_table\]). The broad H$\alpha$ emission (FWHM$\simgt400$ km s$^{-1}$; see Table \[line\_table\] and Fig. \[resid\_fig\]) is most likely due to the accretion disk. Therefore, for the first time, we may have discovered a hidden population of accreting binaries where the accretion disk contribution to the optical light is small, and the H$\alpha$ emission lines become apparent only after subtracting the stellar contribution to the spectra. In what follows, we provide further details on the spectroscopic and/or photometric properties of several individual objects.\
[*(a)*]{} CX138 has a broad H$\alpha$ emission line (FWHM$=1350\pm75$ km s$^{-1}$) in its residual spectrum. It also has a radial velocity variation of $\sim100$ km s$^{-1}$ between its Gemini spectra. Therefore, CX138 is a likely candidate of the hidden accreting binaries. The reddening measured via the strength of DIB $\lambda5780$ (EW = 1971$\pm$131 mÅ) is $A_K
= 1.29 \pm 0.08$, which also puts the object in the Galactic Bulge, if not farther. For the Galactic Bulge distance ($\sim8$ kpc), the absolute $V$-band magnitude of CXB138 would be $M_V=-2$, suggesting a giant companion star.\
[*(b)*]{} CX377 is also a candidate hidden accreting binary. The Gemini/GMOS spectra of CX377 do not show any notable emission features (see Fig. \[spec\_fig2\] and Fig. \[cx377\_fig\]). However, the optical spectra of CX377 taken $\sim1$ year before (2011 May 28 and 2011 July 23) by VLT/VIMOS show strong broad H$\alpha$ emission with an asymmetric double-peaked profile (Torres et al. 2014; see Fig. \[cx377\_fig\]). The intrinsic width of the H$\alpha$ emission is $\sim1200$ km s$^{-1}$, while the double-peak separation is $\sim700$ km s$^{-1}$. This apparent dramatic change of the spectrum is not caused by incorrect targeting. The finding charts of CX377 for our Gemini/GMOS spectroscopy and for the VLT/VIMOS spectroscopy (Fig. \[fc\_cx377\_fig\]; also see Fig. 9 in Torres et al. 2014) show the exact same targets. The slit was placed during our Gemini/GMOS observations (see Fig. \[fc\_cx377\_fig\]) to contain both the target and the adjacent object so that we are able to distinguish their spectra. The seeing during the Gemini/GMOS observations ($\sim0.6^{\prime\prime}$) was better than that during the VLT/VIMOS observations ($\sim1^{\prime\prime}$). Thus, the contamination of the nearby bright sources is less of a problem for the Gemini/GMOS spectra. Therefore, the H$\alpha$ emission feature of CX377 indeed has significant variation within one year. Although the Gemini/GMOS spectra of CX377 only show H$\alpha$ absorption lines, its residual spectrum after optimally subtracting the F6V standard star template shows broad H$\alpha$ emission (see Fig. \[resid\_fig\]). The H$\alpha$ emission has not completely disappeared, but it is significantly weaker during our Gemini/GMOS observation than during the VLT/VIMOS observations of Torres et al. (2014). It has also become narrower (FWHM = $660\pm30$ km s$^{-1}$ in the Gemini/GMOS residual spectrum). This rare behaviour of H$\alpha$ emission line variations indicates significant accretion disk variations for CX377. The soft transient GRO J1655$-$40 has shown similar strong variability of its H$\alpha$ emission line (e.g., Soria et al. 2000). GRO J1655$-40$ has a period of 2.62 days; the spectral type of its companion star had been classified as F3–F6 (Orosz & Bailyn 1997), similar to that of CX377 (see below). However, the H$\alpha$ emission line variability of GRO J1655$-$40 occurred when this object was going through an outburst cycle, while CX377 has remained in quiescence. It is possible that CX377 experienced a faint outburst ending before June 1, 2012 when our Gemini spectra were taken, as long as the flux remained below the detection threshold of all sky monitors such as Monitor of All-sky Image (MAXI; Matsuoka et al. 2009). Another possibility is that CX377 has variable accretion rate during quiescence.
![The finding charts of CX377 for our Gemini/GMOS spectroscopy (left panel) and the VLT/VIMOS spectroscopy in Torres et al. (2014; right panel). The two short thick bars in each panel indicate the position of CX377, while the thin lines show the slit position. The sky area shown in both panels is $15^{\prime\prime}\times15^{\prime\prime}$. North is up and east is left in both panels.[]{data-label="fc_cx377_fig"}](fc_cx377_2.ps){width="3.4in"}
CX377 is spectrally classified as F6 type. The relative radial velocities between the Gemini spectra of CX377 (see Table \[rvc\_table\]) are consistent with zero or less than 10 km s$^{-1}$, which is not surprising given that the three object spectra were only 15 minutes apart, and the orbital period of CX377 is expected to be much longer than 15 minutes for an F6 main sequence star filling its Roche lobe. Then we cross-correlated the object spectra to the F6V star template (HD 16673). The radial velocities relative to the star template range from $-50$ to $-20$ km s$^{-1}$. The radial velocity of the star template itself is consistent with zero ($-4\pm5$ km s$^{-1}$). The light curve of CX377 obtained in July 2010 only shows non-periodic flickering with a RMS scatter of 0.06 magnitude (see Fig. A2 of Torres et al. 2014). The strength of DIB $\lambda5780$ (EW $= 1093\pm86$ mÅ) in the spectra of CX377 corresponds to $A_K=0.73\pm0.06$. Extinction map from Gonzalez et al. (2012) gives $A_K=0.72\pm0.15$ for CX377, which is consistent with the extinction derived above. Therefore, it is likely that CX377 resides in the Galactic Bulge.
The neutral hydrogen column density, $N_H=(1.44\pm0.14)\times10^{22}$ cm$^{-2}$, was estimated from the extinction following the relation in G[ü]{}ver & [Ö]{}zel (2009). CX377 has 7 counts in the 0.3–8 keV band in the 2 ks exposure. Assuming an absorbed power-law spectrum with photon index of 1.6, the unabsorbed flux of CX377 is $\approx1\times10^{-13}$ erg cm$^{-2}$ s$^{-1}$. At a distance of $\sim8$ kpc, the luminosity would be $L_{\rm X}=8\times10^{32}$ erg s$^{-1}$. With the estimated extinction and the apparent $r^\prime$-band magnitude (see Table \[rmag\_table\]), we calculate the absolute magnitude of $M_{r^\prime}=-1.3$ and the to optical flux ratio $f_{\rm X}/f_{r^\prime}\sim0.005$; both values are consistent with those for a giant companion star. However, the disk luminosity at the time of the VIMOS observation should have been larger than that of the F6 giant ($M_{r^{\prime}}=-1.3$), implying that the source might have experienced a faint outburst (cf. Wijnands & Degenaar 2013).\
[*(c)*]{} CXB149 and CXB174 were spectrally classified as G6 and F4, respectively. Their residual spectra after optimal subtraction do not show unambiguous H$\alpha$ emission or absorption features. The Gemini/GMOS spectra could possibly be from interlopers instead of the true optical counterparts of the sources. The H$\alpha$ absorption features of these two sources are significantly blueshifted from the laboratory wavelength ($-109\pm7$ km s$^{-1}$ for CXB149 and $-179\pm8$ km s$^{-1}$ for CXB174). This shift is not introduced by the wavelength calibration process since offsets to the sky lines were small and they have been corrected. It cannot be explained by the wavelength shifts caused by the centroiding uncertainty within the slit (which are $-30$ km s$^{-1}$ and $-75$ km s$^{-1}$, respectively; see §\[obs:spec\]). These velocities are consistent with that for stars residing in the Galactic Bulge (e.g., Zoccali et al. 2014). Regarding the reddening measurements, CXB149 has EW(DIB $\lambda5780$) = 1099$\pm$28 mÅ; the corresponding reddening is E(B-V) $=$ 2.11$\pm$0.05 and $A_K=0.73\pm0.02$, which put this object in the Galactic Bulge or farther ($A_K=0.30\pm0.07$ from the map of Gonzalez et al. 2012). For CXB174, EW(DIB $\lambda5780$) = 616$\pm$34 mÅ; the corresponding reddening is E(B-V) $=$ 1.21$\pm$0.06 and $A_K=0.42\pm0.02$, which is consistent with the Bulge reddening ($A_K=0.35\pm0.08$) from Gonzalez et al. (2012). The absolute magnitudes for CXB149 and CXB174 residing in the Bulge would be $M_V=0$ and $M_V=-0.7$, respectively, which are consistent with those of the giant stars with their spectral types.\
[*(d)*]{} CXB26 and CXB189 were not spectrally classified using our method. CXB26 is considered to be likely associated with an OGLE source $1^{\prime\prime}$ away from the position (OGLE BUL-SC3 6033) which is identified as a CV (see Paper II). However, the Gemini/GMOS spectra of CXB26 do not show apparent features of an accretion disk. It is possibly a hidden accreting binary, which could be verified after it is spectrally classified with future spectroscopy. The physical nature of CXB189 remains unclear.
Objects without Apparent H$\alpha$ Emission/Absorption Features
---------------------------------------------------------------
![The residual spectra of CX84 after subtracting the best-fit star template (G9), and other templates with spectra types (F6, G6, K2) bracketing that of the best-fit template. The H$\alpha$ emission feature exists in all four residual spectra.[]{data-label="cx84test_fig"}](cx84_resid_red_test.ps){width="3.4in"}
The remaining six sources show neither clear H$\alpha$ emission nor absorption features in their Gemini/GMOS spectra (see Fig. \[spec\_fig3\]). Three of them (CX84, CX139, CXB82) were spectrally classified (see Table \[vsini\_table\]). The residual spectra of these sources after optimally subtracting their best-fit stellar templates are shown in Fig. \[resid\_fig\]. Similar to CX138, CX377, and CXB73, they all have H$\alpha$ emission features, which fill in the stellar H$\alpha$ absorption features in our Gemini/GMOS spectra. To test the robustness of our results, we also subtracted other non-best-fit stellar template from the object spectra to see how the residual H$\alpha$ emission would vary. Fig. \[cx84test\_fig\] shows one example of this test. The H$\alpha$ emission feature remains in the residual spectra of CX84 after subtracting several stellar templates with spectral types bracketing that of the best-fit template, which supports the detection of a residual accretion disk. Note that the choice of templates with non-best-fit spectral types could affect the profile of the residual H$\alpha$ emission. For example, the mis-matched templates generate broader wings of the H$\alpha$ emission which may result in a higher FWHM value (e.g., FWHM increases from $225$ km s$^{-1}$ to $320$ km s$^{-1}$ if using an F6 template spectrum).
It is worth noting that even in the case that the residual H$\alpha$ emission features are real, there could be alternative explanations. For example, it is possible that the observed optical light is dominated by a foreground, physically unrelated star (instead of the companion star in the above scenario) so that we can only see the accretion after subtracting the foreground stellar emission. However, based on the analysis in Britt et al. (2014), we expect $\sim0.7$ interloper in our sample of 21 GBS sources (also see §\[obs:photo\]). In §\[discuss:cx377\], we already discussed two possible interlopers (for CXB149 and CXB174) for which the residual spectra do not have clear H$\alpha$ emission. Therefore, it is not likely that the six optical sources that only have H$\alpha$ emission in their residual spectra are all chance interlopers. Another possibility is that these sources are active stars/binaries, where the H$\alpha$ emission fills in the stellar absorption features. Sources with narrower H$\alpha$ emission (e.g., $\simlt200$ km s$^{-1}$) in their residual spectra could fit into this scenario. However, it is difficult to explain the broad residual H$\alpha$ emission lines of CX138 and CX377. The periodic modulation in the light curves of CX84, CXB82, and CXB99 could originate from either binary motion or stellar activity (hotspots). Alternative explanations for the narrowness of their H$\alpha$ emission lines are that the binaries are relatively face-on, or that the disks are highly recessed.
CX84 is spectrally classified as G9. The reddening estimated from the strength of interstellar band $\lambda5780$ (EW = 2499$\pm$218 mÅ) is $A_K = 1.62 \pm 0.14$, which means the object is in the Bulge or farther. Assuming CX84 resides in the Galactic Bulge, the absolute $V$-band magnitude would be $M_V=-1$, which indicates a G9 giant star. The luminosity is $\sim2.6\times10^{33}$ erg s$^{-1}$, which appears higher than that for G9 giants (see Fig. 2 of G[ü]{}del 2004). However, it is typical of long orbital period qLMXBs (e.g., Jonker et al. 2012). The optical light curve of CX84 indeed shows a modulation with a tentative period of 4.67 days. If it is a qLMXB, the orbital period should be twice that value ($\approx9.3$ days) because of the ellipsoidal modulation. In summary, the luminosity of CX84 suggests the qLMXB nature, while the narrow residual H$\alpha$ emission line favors the chromospherically active star/binary scenario. Similarly, the reddening of CXB82 derived from its near-infrared colour ($A_K\approx0.32$) is also consistent with the reddening expected if it resides in the Galactic Bulge. The absolute $V$-band magnitude ($M_V=-0.3$) suggests a G9 giant star, while its luminosity ($\sim1.1\times10^{33}$ erg s$^{-1}$) indicates that it is more likely to be a qLMXB. The H$\alpha$ emission line in the residual spectrum of CXB82 is also narrow. Based on current data, we are not able to determine whether CX84 and CXB82 are chromospherically active stars/binaries or qLMXBs.\
Among the seven potential accreting binaries identified in our sample with a spectral classification, one source (CX377) has an F-type companion star, another four (CX84, CX138, CXB73, CXB82) have G-type companion stars, while the other two (CX139 and CXB99) have K-type companion stars. Given the mass of F/G-type companion stars, the mass ratios ($q=M_2/M_1$ here) for the sources with F/G-type companion stars should be higher than those of the more typical binaries with later spectral types (K, M). In contrast to LMXBs, very few of these “intermediate” mass binaries (IMXBs) have been found. Existing examples include Cyg X-2 (e.g., Casares et al. 2010) and Her X-1 (e.g., Reynolds et al. 1997). Our Gemini/GMOS spectroscopy indicates that the GBS may be able to provide an example of this type of X-ray binary.
Conclusions {#conclusion}
===========
The GBS is a multiwavelength survey project where one of the goals is to identify binaries in the Galactic Bulge area. In this work, we present optical and infrared photometry, and Gemini/GMOS spectroscopy of 21 GBS sources detected in the GBS. One prime goal of the GBS is to identify eclipsing qLMXBs. CX446, CX1004 and CXB2 are promising candidates to be eclipsing qLMXBs. CX446 has broad H$\alpha$ emission, and its optical light curve shows likely eclipsing events. CX1004 has broad, double-peaked H$\alpha$ emission. The light curve of CXB2 also contains eclipsing events suggesting a high systemic inclination.
We may have discovered a population of hidden accreting binaries. After optimally subtracting the stellar templates with matched spectral type, the residual spectra of seven sources show H$\alpha$ emission. Three of them (CX138, CX377, and CXB73) have broad ($\simgt400$ km s$^{-1}$) H$\alpha$ lines, which are likely produced by an accretion disk. CX84 and CXB82 are also possibly hidden accreting binaries based on their luminosity, while their residual H$\alpha$ emission lines are narrow ($\simlt200$ km s$^{-1}$). Previous VLT/VIMOS spectra of CX377 showed a strong, double-peaked H$\alpha$ emission line, while our Gemini/GMOS spectra of CX377 only contains an H$\alpha$ absorption line. This may indicate the strong variability of the accretion disk of this source. The residual spectrum of CX377 after optimal subtraction supports this scenario by showing weak, broad H$\alpha$ emission. In summary, based on the emission features, the spectral classification, luminosity, and the residual spectra after optimal subtraction, we are able to constrain the likely nature for eight sources in our sample: CX446, CX1004, and CXB2 are accreting binaries (CVs or qLMXBs) while CX446 and CXB2 are likely also eclipsing; CXB64 and CXB113 are chromospherically active stars or binaries; CX138, CX377, and CXB73 could be hidden accreting binaries.
Appendix A: Finding Charts {#appendix-a-finding-charts .unnumbered}
==========================
The $r^\prime$-band finding charts for the optical counterparts of 18 of the 21 GBS sources presented in this paper are shown in Fig. \[fc\_fig\] and Fig. \[fc\_fig2\] . The sky area in each chart is 20$^{\prime\prime}\times$20$^{\prime\prime}$ (except for CXB113 which is 20$^{\prime\prime}\times$15$^{\prime\prime}$) with the source in the center and indicated by the thick horizontal and vertical short bars. North is up and East is to the left. CXB189 is blended with a nearby bright source because the field is crowded. Part of the emission from the nearby bright source may leak into the slit. The finding chart for CX377 is shown in Fig. \[fc\_cx377\_fig\]. The charts for CX446 and CX1004 can be found in Torres et al. (2014).
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the referee, Prof. P. Charles, for his comments which substantially improved the paper. The software packages `pamela`, and `molly` utilized in this work were developed by T. Marsh. R.I.H., C.B.J., and C.T.B., acknowledge support from the National Science Foundation under Grant No. AST-0908789 and from the National Aeronautics and Space Administration through Chandra Award Number AR3-14002X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. D.S. acknowledges the support of the Science and Technology Facilities Council, grant number ST/L000733/1. C.O.H. is supported by NSERC, an Ingenuity New Faculty Award, and an Alexander von Humboldt fellowship.
Alard C., 2000, A&AS, 144, 363
Alard C., Lupton R. H., 1998, ApJ, 503, 325
Bassa C. G., van Kerkwijk M. H., Koester D., Verbunt F. 2006, A&A, 456, 295
Britt C. T. et al., 2013, ApJ, 769, 120
Britt C. T. et al., 2014, ApJS, 214, 10
Casares J., Gonz[á]{}lez Hern[á]{}ndez J. I., Israelian G., Rebolo R., 2010, MNRAS, 401, 2517
Condon J. J., Cotton W. D., Greisen E. W., Yin Q. F., Perley R. A., Taylor G. B., Broderick J. J., 1998, AJ, 115, 1693
Davies R. L. et al., 1997, Proc SPIE, 2871,1099
Evans D. W., Irwin M. J., Helmer, L., 2002, A&A, 395, 347
Gonzalez O. A., Rejkuba M., Zoccali M., Valenti E., Minniti D., 2011, A&A, 534, A3
Gonzalez O. A., Rejkuba M., Zoccali M., Valenti E., Minniti D., Schultheis M., Tobar R., Chen B., 2012, A&A, 543, A13
Greiss S. et al., 2014, MNRAS, 438, 2839
G[ü]{}del M., 2004, A&AR, 12, 71
G[ü]{}ver T., & [Ö]{}zel F., 2009, MNRAS, 400, 2050
Herbig G. H., 1993, ApJ, 407, 142
Herbig G. H., 1995, ARA&A, 33, 19
Hynes R. I., Horne Keith, O’Brien K., Haswell C. A., Robinson E. L., King A. R., Charles P. A., Pearson K. J., 2006, ApJ, 648, 1156
Hynes R. I. et al., 2012, ApJ, 761, 162
Hynes R. I. et al., 2014, ApJ, 780, 11
Jehin E., Bagnulo S., Melo C., Ledoux C., Cabanac R. 2005, in Hill V., François P., Primas F., eds, IAU Symp. 228, From Lithium to Uranium: Elemental Tracers of Early Cosmic Evolution. Cambridge University Press, p. 261
Johnston H. M., Kulkarni S. R., Oke J. B., 1989, ApJ, 345, 492
Jonker P. G. et al., 2011, ApJS, 194, 18 (Paper I)
Jonker P. G., Miller-Jones J. C. A., Homan J., Tomsick J., Fender R. P., Kaaret P., Markoff S., Gallo E., 2012, MNRAS, 423, 3308
Jonker P. G. et al., 2014, ApJS, 210, 18 (Paper II)
Kurucz R. L., 2006, preprint (arXiv:astro-ph/0605029)
Lucas P. W. et al., 2008, MNRAS, 391, 136
Maccarone T. J. et al., 2012, MNRAS, 426, 3057
Marsh T. R., 1989, PASP, 101, 1032
Marsh T. R., Robinson E. L., Wood J. H., 1994, MNRAS, 266, 137
Matsuoka M. et al., 2009, PASJ, 61, 999
Mauerhan J. C., Muno M. P., Morris M. R., Bauer F. E., Nishiyama S., Nagata T., 2009, ApJ, 703, 30
Minniti D. et al., 2010, New Astron., 15, 433
Muno M. P. et al., 2003, ApJ, 589, 225
Orosz J. A., Bailyn C. D., 1997, ApJ, 477, 876
zel F., Psaltis D., Narayan R., McClintock J. E., 2010, ApJ, 725, 1918
Pecaut M. J., Mamajek E. E., 2013, ApJS, 208, 9
Ratti E. M. et al., 2013, MNRAS, 428, 3543
Reynolds A. P., Quaintrell H., Still M. D., Roche P., Chakrabarty D., Levine S. E., 1997, MNRAS, 288, 43
Sakano M., Koyama K., Murakami H., Maeda Y., Yamauchi S., 2002, ApJS, 138, 19
Shaw R. A., ed. 2009, NOAO Data Handbook, Version 1.1. National Optical Astronomical Observatory, Tucson.
Skrutskie M. F., et al., 2006, AJ, 131, 1163
Soria R., Wu K., Hunstead R. W., 2000, ApJ, 539, 445
Stetson P. B., 1987, PASP, 99, 191
Torres M. A. P. et al., 2014, MNRAS, 440, 365
Udalski A., Szymanski M., Kaluzny J., Kubiak M., Mateo M., 1992, Acta Astron., 42, 253
Wallace L., Hinkle K. H., Livingston W. C., Davis S. P., 2011, ApJS, 195, 6
Walter F. M., Cash W., Charles P. A., Bowyer C. S., 1980, ApJ, 236, 212
Wijnands R., Degenaar N., 2013, MNRAS, 434, 1599
Zoccali M. et al., 2014, A&A, 562, A66
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{width="1.5in"} {width="1.5in"} {width="1.5in"}\
{width="1.5in"} {width="1.5in"} {width="1.5in"}
{width="1.5in"} {width="1.5in"} {width="1.5in"}\
{width="1.6in"} {width="1.5in"} {width="1.5in"}\
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[lcccllc]{} CX84 & $17:38:12.84$ & $-29:06:12.4$ & 67-48 & 2012-05-31 &1200 & 0.5\
& & & 67-9 & 2012-06-01 &1200 & 0.5\
CX138 & $17:46:23.14$ & $-25:49:30.3$ & 67-52 & 2012-06-01 &1200 & 0.5\
& & & 67-15 & 2012-06-14 &1200 & 1.1\
CX139 & $17:45:22.15$ & $-25:50:48.2$ & 67-3 & 2012-05-31 &900 & 0.6\
& & & 67-44 & 2012-06-01 &900 & 0.6\
CX377 & $17:43:16.54$ & $-27:45:37.0$ & 67-70 & 2012-06-01 &$900\times3$ & 0.6\
& & & 67-72 & 2012-06-09 &$900\times3$ & 1.1\
CX446 & $17:46:27.17$ & $-25:49:52.6$ & 67-96 & 2012-06-22 &$900\times3$ & 0.8\
& & & 67-109 & 2013-05-04 &$900\times3$ & 0.6\
CX1004 & $17:46:23.47$ & $-31:05:49.8$ & 44-63 & 2012-05-14 &$900\times3$ & 0.8\
& & & 44-59 & 2012-05-17 &$900\times3$ & 0.8\
& & & 44-3 & 2012-05-17 &$900\times3$ & 0.8\
& & & 44-61 & 2012-05-17 &$900\times3$ & 0.7\
CXB2 & $17:53:59.86$ & $-29:29:06.5$ & 44-53 & 2012-04-20 &$900\times3$ & 0.9\
& & & 44-55 & 2012-04-20 &$900\times3$ & 0.9\
& & & 44-57 & 2012-04-20 &$900\times3$ & 0.8\
& & & 44-80 & 2012-05-19 &$900\times3$ & 0.9\
CXB26 & $17:53:47.89$ & $-29:44:37.4$ & 44-49 & 2012-05-17 &$900\times3$ & 0.7\
CXB38 & $17:58:33.81$ & $-27:30:22.9$ & 67-21 & 2012-06-09 &$900\times3$ & 1.3\
CXB64 & $17:46:02.87$ & $-32:08:11.3$ & 67-74 & 2012-05-29 &$900\times4$ & 0.7\
CXB73 & $17:52:29.07$ & $-30:03:21.7$ & 67-68 & 2012-06-22 &$900\times2$ & 1.1\
CXB82 & $17:56:54.57$ & $-28:12:34.4$ & 67-66 & 2012-06-14 &$900\times2$ & 1.0\
& & & 67-99 & 2012-06-22 &$900\times2$ & 1.0\
CXB92 & $17:34:25.02$ & $-30:41:04.6$ & 44-47 & 2012-05-03 &$900\times3$ & 0.7\
& & & 44-85 & 2012-05-19 &900 & 1.1\
CXB99 & $17:54:59.36$ & $-29:10:21.1$ & 67-64 & 2012-06-09 &$900\times2$ & 1.2\
CXB113 & $17:52:05.53$ & $-30:19:31.8$ & 67-62 & 2012-06-22 &900 & 1.1\
CXB117 & $17:56:02.42$ & $-28:24:45.4$ & 67-60 & 2012-06-01 &$900\times2$ & 0.6\
CXB137 & $17:57:10.01$ & $-27:57:54.9$ & 44-45 & 2012-05-17 &$900\times3$ & 0.8\
CXB149 & $17:53:06.92$ & $-29:51:13.1$ & 67-58 & 2012-05-31 &$900\times2$ & 0.5\
CXB174 & $17:56:18.52$ & $-28:45:40.5$ & 44-43 & 2012-04-19 &$900\times3$ & 0.6\
CXB189 & $17:52:55.98$ & $-29:29:50.7$ & 44-41 & 2012-04-19 &$900\times3$ & 1.0\
CXB201 & $17:33:26.08$ & $-30:40:24.4$ & 44-39 & 2012-04-19 &$900\times3$ & 0.9\
[lcccc]{} CX84 & 19.018 & 0.007 & 0.029 & 35\
CX377 & 18.885 & 0.009 & 0.060 & 36\
CX446 & 21.162 & 0.062 & 0.166 & 37\
CX1004 & 20.761 & 0.019 & 0.040 & 35\
CXB2 & 20.350 & 0.003 & 0.14 & 110\
CXB64 & $\sim18.63$ & 0.011 & 0.014 & 56\
CXB82 & 16.894 & 0.006 & 0.023 & 52\
CXB99 & 16.516 & 0.021 & 0.073 & 110\
CXB113 & $\sim19.13$ & 0.018 & 0.041 & 55\
[lccccccc]{} CX84 & $17:38:12.84$ & $-29:06:12.5$ & $0.12$ & $14.566\pm0.018$ & $13.570\pm0.019$ & $13.108\pm0.021$\
CX138 & $17:46:23.12$ & $-25:49:30.5$ & $0.36$ & $13.122\pm0.004$ & $12.134\pm0.004$ & $11.715\pm0.004$\
CX139 & $17:45:22.15$ & $-25:50:48.3$ & $0.22$ & $12.555\pm0.002$ & $11.914\pm0.003$ & $11.658\pm0.004$\
CX377 & $17:43:16.55$ & $-27:45:37.2$ & $0.20$ & $14.832\pm0.022$ & $13.870\pm0.026$ & $13.443\pm0.027$\
CXB2 & $17:53:59.85$ & $-29:29:06.4$ & $0.13$ & $17.251\pm0.344$ & $16.408\pm0.314$ & $16.251\pm0.336$\
CXB38 & $17:58:33.82$ & $-27:30:22.7$ & $0.25$ & $14.182\pm0.019$ & $13.264\pm0.016$ & $12.958\pm0.016$\
CXB73 & $17:52:29.08$ & $-30:03:21.5$ & $0.18$ & $13.978\pm0.017$ & $13.143\pm0.016$ & $12.796\pm0.015$\
CXB82 & $17:56:54.57$ & $-28:12:34.3$ & $0.10$ & $13.514\pm0.011$ & $12.732\pm0.010$ & $12.536\pm0.011$\
CXB92 & $17:34:25.02$ & $-30:41:04.5$ & $0.10$ & $14.866\pm0.016$ & $13.967\pm0.018$ & $13.372\pm0.018$\
CXB99 & $17:54:59.34$ & $-29:10:21.1$ & $0.19$ & $14.069\pm0.022$ & $13.356\pm0.022$ & $13.206\pm0.024$\
CXB113 & $17:52:05.52$ & $-30:19:31.8$ & $0.10$ & $14.108\pm0.020$ & $13.599\pm0.023$ & $13.299\pm0.024$\
CXB149 & $17:53:06.93$ & $-29:51:13.0$ & $0.15$ & $14.242\pm0.022$ & $13.720\pm0.027$ & $13.527\pm0.029$\
CXB174 & $17:56:18.53$ & $-28:45:40.5$ & $0.10$ & $14.388\pm0.029$ & $13.552\pm0.026$ & $13.268\pm0.025$\
[cl]{} A0V & HD 162305\
A1V & HD 65810\
A2V & HD 60178\
A3V & HD 211998\
A4V & HD 145689\
A5V & HD 39060\
A7V & HD 187642\
A9V & HD 26612\
F0V & HD 109931\
F1V & HD 40136\
F2V & HD 33256\
F3V & HD 18692\
F4V & HD 37495\
F6V & HD 16673\
F8V & HD 45067\
F9V & HD 10647\
G0V & HD 105113\
G1V & HD 20807\
G2V & HD 14802\
G3V & HD 211415\
G4V & HD 59967\
G5V & HD 59468\
G6V & HD 140901\
G9V & HD 25069\
K2V & HD 22049\
K5V & HD 10361\
M0V & HD 156274\
M6V & HD 34055\
[lc]{} CX84 & G9\
CX138 & G9\
CX139 & K2\
CX377 & F6\
CX1004 & M0–M5\
CXB64 & M0–M5\
CXB73 & G9\
CXB82 & G9\
CXB99 & K2\
CXB113 & M0–M5\
CXB149 & G6\
CXB174 & F4\
[lccc]{} CX84 & $2456078.811$ & $0$ & $-104.3\pm3.3$\
& $2456079.677$ & $74.0\pm3.0$ & $-17.3\pm6.0$\
CX138 & $2456079.760$ & $0$ & $-110.7\pm2.1$\
& $2456092.650$ & $75.6\pm6.5$ & $-44.2\pm5.4$\
CX139 & $2456078.665$ & $0$ & $-96.2\pm10.6$\
& $2456079.783$ & $69.0\pm2.5$ & $-17.4\pm4.5$\
CX377 & $2456079.714$ & $0$ & $-20.5\pm9.2$\
& $2456079.725$ & $-14.6\pm12.9$ & $-43.6\pm11.9$\
& $2456079.736$ & $19.0\pm8.8$ & $-17.2\pm6.6$\
CXB73 & $2456100.751$ & $0$ & $-30.0\pm6.2$\
& $2456100.762$ & $-1.7\pm5.1$ & $-31.0\pm6.0$\
CXB82 & $2456092.798$ & $0$ & $-36.6\pm4.7$\
& $2456092.809$ & $-11.3\pm6.3$ & $-41.0\pm7.9$\
& $2456100.806$ & $-2.9\pm2.2$ & $-39.7\pm3.8$\
& $2456100.817$ & $-6.2\pm2.3$ & $-41.7\pm2.7$\
CXB99 & $2456087.659$ & $0$ & $21.5\pm7.0$\
& $2456087.670$ & $-11.0\pm3.8$ & $-1.7\pm10.2$\
[clccc]{} & CX377 & $3.1\pm0.6$ & $6.3\pm0.2$ & $233\pm9$\
& CX446 & $-67.6\pm5.4$ & $27.5\pm1.2$ & $1250\pm50$\
& CX1004 & $-38.0\pm0.6$ & $55.1\pm1.9$ & $2500\pm100$\
& CXB2 & $-4.4\pm0.7$ & $18.6\pm3.7$ & $800\pm160$\
& CXB64 & $-2.0\pm0.1$ & $4.9\pm0.1$ & $63\pm5$\
& CXB99 & $-1.3\pm0.1$ & $6.3\pm0.6$ & $185\pm25$\
& CXB113 & $-5.3\pm0.1$ & $5.3\pm0.1$ & $110\pm5$\
& CX84 & $-2.8\pm0.1$ & $6.0\pm0.3$ & $225\pm15$\
& CX138 & $-2.6\pm0.1$ & $30.0\pm1.5$ & $1350\pm75$\
& CX139 & $-2.2\pm0.1$ & $6.4\pm0.6$ & $250\pm25$\
& CX377 & $-2.8\pm0.1$ & $15.0\pm0.7$ & $660\pm30$\
& CXB73 & $-1.3\pm0.1$ & $9.0\pm1.0$ & $350\pm50$\
& CXB82 & $-1.3\pm0.1$ & $6.1\pm0.2$ & $180\pm10$\
& CXB99 & $-4.1\pm0.1$ & $6.1\pm0.3$ & $180\pm15$\
[^1]: E-mail: jianfeng.wu@cfa.harvard.edu
[^2]: Visiting astronomer, Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which are operated by the Association of Universities for Research in Astronomy, under contract with the National Science Foundation.
[^3]: Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina).
[^4]: See DECam Data Handbook at http://www.noao.edu/meetings/decam/ media/DECam\_Data\_Handbook.pdf.
[^5]: See the Bulge Extinction And Metalicity (BEAM) calculator at http://mill.astro.puc.cl/BEAM/calculator.php.
|
---
abstract: 'The role of antiferromagnetic spin correlations in high-temperature superconductors remains a matter of debate. We present inelastic neutron scattering evidence that gapless spin fluctuations coexist with superconductivity in La$_{1.905}$Ba$_{0.095}$CuO$_4$. Furthermore, we observe that both the low-energy magnetic spectral weight and the spin incommensurability are enhanced with the onset of superconducting correlations. We propose that the coexistence occurs through intertwining of spatial modulations of the pair wave function and the antiferromagnetic correlations. This proposal is also directly relevant to sufficiently underdoped and .'
author:
- Zhijun Xu
- 'C. Stock'
- Songxue Chi
- 'A. I. Kolesnikov'
- Guangyong Xu
- Genda Gu
- 'J. M. Tranquada'
title: 'Neutron Scattering Evidence for a Periodically-Modulated Superconducting Phase in the Underdoped Cuprate La$_{1.905}$Ba$_{0.095}$CuO$_4$'
---
It is commonly accepted that cuprate superconductors have a spatially-uniform $d$-wave pair wave function [@tsue00]. It has also become a paradigm that antiferromagnetic spin fluctuations are gapped in the superconducting state, with a pile up of excitations in the magnetic “resonance” peak above the gap [@scal12a; @esch06; @yu09; @ross91; @mook93]. A number of neutron scattering studies of underdoped have found evidence for incommensurate spin fluctuations that remain gapless at temperatures far below the superconducting transition temperature, $T_c$ [@lee00; @chan07; @lips09; @kofu09]. Theoretical analyses have tended to view such spin-density-wave correlations as soft fluctuations of an order that competes with spatially-uniform superconductivity [@deml01] and that may be locally pinned by disorder [@ande10]. As a consequence, researchers have crafted interpretations of the low-energy spin fluctuations that maintain consistency with the spin-gap paradigm [@chan07; @kofu09].
In an alternative approach, the superconductivity and antiferromagnetism are both treated as spatially modulated and intimately intertwined [@berg09b]. Such a state, which variational calculations indicate to be energetically competitive with uniform superconductivity [@corb14], has been invoked [@hime02; @berg07] to explain the depression of superconducting order in certain stripe-ordered cuprates [@taji01; @li07]. While the poorly-superconducting phase is fascinating on its own, it leaves open the question of whether a modulated pair wave function might be relevant to the case of a good bulk superconductor.
In this paper, we present neutron scattering measurements of the low-energy spin fluctuations in La$_{1.905}$Ba$_{0.095}$CuO$_4$, a bulk superconductor with $T_c=32$ K. Rather than developing a spin gap on cooling below $T_c$, the lowest-energy excitations are actually enhanced. By putting the measurements on an absolute scale, we show that the strength of the spin response is comparable to that of spin waves in antiferromagnetic La$_2$CuO$_4$. To generate this large a response, we conclude that all parts of the sample must contribute to the signal, ruling out macroscopic phase separation. A previous optical conductivity study has shown that the superfluid density of this sample is consistent with the trend established for bulk superconductivity in all cuprate families in the form of Homes’ law [@home12]. It thus appears that there must be local coexistence of the spin fluctuations and superconductivity. This view is supported by changes in the low-energy magnetic spectral weight and incommensurability that correlate with the onset of superconductivity. Given the empirical observation that commensurate antiferromagnetism and superconductivity do not coexist, at least in single-layer cuprates [@birg06], the best option to reconcile the new results is to have a superconducting state that is spatially modulated to minimize overlap with the amplitude-modulated antiferromagnetic correlations.
The single crystal of La$_{1.905}$Ba$_{0.095}$CuO$_{4}$ used here, a cylinder of size $\diameter 8\mbox{\ mm}\times35$ mm and mass $\sim 11$ g, was grown by the floating-zone technique at Brookhaven [@huck11]. Previous neutron scattering measurements provided evidence for weak charge and spin stripe order [@wen12]. The signal is averaged over the sample, so one cannot distinguish between uniformly weak order and macroscopic phase separation, such as occurs in oxygen-doped [@udby13]. Here we focus on the spin fluctuations in order to deduce the bulk behavior.
{width="5.in"}
The low-energy (1 to 6 meV) inelastic neutron scattering measurements were performed on the Multi-Axis Crystal Spectrometer (MACS)[@macs08] at the NIST Center for Neutron Research (NCNR). We used a fixed final energy of 5 meV, with Be filters after the sample, and horizontal collimations of $100'$-open-S-$90'$-open, where S $=$ sample. The middle-energy (6 to 12 meV) data were collected on the triple-axis spectrometer BT-7 at NCNR [@lynn12]. There we used horizontal collimations of open-$80'$-S-$50'$-$50'$ with fixed final energy of 14.7 meV and two pyrolytic graphite filters after the sample. The high-energy (10 to 110 meV) experiments were performed on the SEQUOIA time-of-flight spectrometer at the Spallation Neutron Source (SNS), Oak Ridge National Laboratory [@sequoia10]. Incident energies of 50, 100, and 180 meV were used to measure excitations from 10–34 meV, 35–70 meV, and above 70 meV, respectively.
Constant energy slices through the dynamical structure factor $S({\bf Q},\omega)$ are shown in Fig. \[fg:slices\]. The wave vectors [**Q**]{} are specified in reciprocal lattice units (rlu), $(a^*, b^*, c^*) = (2\pi/a, 2\pi/b, 2\pi/c)$, where the lattice constants are $a \approx b = 3.79$ Å, and $c\approx 13.2$ Å. $S({\bf Q},\omega)$ is the Fourier transform of the spin-spin correlation function. To extract it from the measured scattering intensity, it was necessary to divide out the square of the magnetic form factor [@xu13]. We used a recent determination of the Cu form factor that takes account of hybridization [@walt09]. To put the scattering data in absolute units, the BT7 data were normalized to measurements of incoherent elastic scattering from the sample [@xu13]. There the integration of the magnetic peaks was evaluated from scans along $(H,0.5,0)$, taking account of the calculated spectrometer resolution along $K$ of 0.087 rlu. The SEQUOIA (MACS) data were cross normalized with the BT7 data through integrated magnetic peak intensities at 32 K and 10 meV (6 meV). Examples of line cuts comparing the data and fits are given in [^1].
The constant-energy slices in Fig. \[fg:slices\](a)–(f) illustrate the dispersion of the magnetic excitations. At low energy, we see incommensurate peaks at positions $(0.5\pm\delta,0.5)$ and $(0.5,0.5\pm\delta)$. With increasing energy, they disperse inwards towards ${\bf Q}_{\rm AF}$ near 35 meV, and then outwards again at higher energies, following the common hour-glass dispersion [@fuji12a]. From slices such as those in Fig. \[fg:slices\](g)–(i), one can see that the peak positions $\delta$ and widths $\kappa$ (full width at half maximum) change with temperature. To parametrize the data, we have performed least-squares fitting with four symmetrically-positioned, normalized Gaussian peaks. We have expressed the amplitude in terms of the imaginary part of the dynamical spin susceptibility, given by [@xu13] $$\chi''({\bf Q},\omega) = g^2\mu_{\rm B}^2\frac{\pi}{\hbar}\big(1-e^{-\hbar\omega/k_{\rm B}T}\big) S({\bf Q},\omega),
\label{eq:S}$$ where $g\approx2$ is the gyromagnetic factor. Since we use normalized Gaussians, the fitted amplitude parameter can be expressed in terms of the [**Q**]{}-integrated local susceptibility, $\chi''(\omega)$, which, at low temperature, is essentially the magnetic spectral weight. Examples of fits are shown in Fig. \[fg:slices\](j)–(l); the results for the fitted parameters are summarized in Fig. \[fg:sum\].
Focusing on excitations below 10 meV, one can see in Fig. \[fg:sum\](a) that cooling leads to an enhancement of $\chi''(\omega)$ that saturates by $T_c$—except for $\hbar\omega<3$ meV. At $T\le5$ K, $\chi''(\omega)$ exhibits a quasi-elastic peak associated with spin-stripe correlations [@huck11]. For the quasi-elastic energies, there is also a temperature-dependent shift in the incommensurability $\delta$, as shown in Fig. \[fg:sum\](b). Near $T_c$, we find $\delta\approx0.075$ rlu, but at 5 K there is a substantial upward shift towards 0.09 for $\hbar\omega < 3$ meV, effectively dispersing [@Note1] towards the elastic peak centered at $\delta=0.105$ rlu, which develops below $\sim T_c$ [@huck11].
A complementary picture is given by the temperature dependence of $\delta$ for $\hbar\omega=1$ meV, as shown in Fig. \[fg:sum\](e). The strong shift in $\delta$ begins slightly above $T_c$, at $\sim40$ K, which corresponds with the onset of strong superconducting correlations [@steg13]. The growth of $\chi''(1{\rm\ meV})$ also takes off below 40 K, and the line width $\kappa$ decreases.
![(color online) Summary of fitted parameters: (a) $\chi''(\omega)$, (b) $\delta$, and (c) $\kappa$ vs. $\hbar\omega$, for temperatures of $\le5$ K (violet circles), 32 K (red diamonds), 60 K (orange squares), 120 K (yellow triangles); (d) $\chi''(\omega)$, (e) $\delta$, and (f) $\kappa$ vs. $T$ for energies of 1 meV (blue circles), 6 meV (green squares). Dashed line indicates $T_c=32$ K. []{data-label="fg:sum"}](figure2.pdf){width="\columnwidth"}
The magnetic incommensurability is a consequence of the charge carriers forming intertwined stripes [@frad14]. Changes in the stripe spacing with temperature are reflected in $\delta$ and are tied to the behavior of the charge carriers; for example, in La$_{1.875}$Ba$_{0.125}$CuO$_4$ there is an abrupt jump in $\delta$ at a structural transition associated with pinning of the charge stripes [@fuji04]. In the present case, where there is also a structural transition [@wen12a], the quasi-static stripe correlations appear to develop in a cooperative fashion with the superconductivity.
Figure \[fg:chi\] presents the results for $\chi''(\omega)$ over a broader energy range at select temperatures above, below, and at $T_c$. In optimally- and over-doped cuprates, the spin gap for $T<T_c$ is generally observed to be comparable to the superconducting gap $\Delta$, with a resonance peak appearing at $E_r\approx 1.3\Delta$ [@sidi04; @yu09]. Measurements such as Andreev reflection indicate that $\Delta\approx 2.5k_{\rm B}T_c$ [@deut99; @gonn01], yielding a prediction of $\Delta\approx 7$ meV for our sample, consistent with the gap measured on the Fermi arc by angle-resolved photoemission [@he09]. Correspondingly, one predicts $E_r\sim 9$ meV.
It is quite clear from Fig. \[fg:chi\] that there is no meaningful spin gap for $T<T_c$ on the predicted scale of 7 meV. While there is a depression of $\chi''(\omega)$ below 6 meV at $T_c$, it should be noted that $\chi''$ must decrease to zero at $\hbar\omega=0$. There is a definite enhancement of the signal below 3 meV at low temperature. Similarly, there is no obvious resonance feature. While $\chi''(\omega)$ does exhibit a peak near 18 meV, that peak is already present at 100 K, and there is no significant correlation between the peak intensity and the development of superconductivity. The conclusion of an independent neutron scattering study is that the peak is a consequence of spin-phonon hybridization [@wagm14].
![(color online) (a) Wave-vector-integrated local susceptibility $\chi''(\omega)$ measured at temperatures of $\le 6$ K (violet), 32 K (red), and 100 K (orange). (b) Difference in $\chi''(\omega)$ between $\sim6$ and 32 K (violet), and between 32 and 100 K (red). In (a) and (b), diamonds were measured on MACS (base $T=1.5$ K); circles on BT7 (base $T=5$ K); squares on SEQUOIA (base $T=6$ K). []{data-label="fg:chi"}](figure3.pdf){width="0.85\columnwidth"}
To put the absolute magnitude of $\chi''(\omega)$ in context, we can compare with the signal from spin waves in La$_2$CuO$_4$ [@cold01; @head10]. Using the results of spin-wave theory [@mano91], we find that $\chi''(\omega) = (Z_\chi/J_{\rm eff})\,\mu_{\rm B}^2/{\rm Cu}$, where $Z_\chi\approx0.5$. From experiment, the value of the effective superexchange energy, $J_{\rm eff}$, describing the low-energy dispersion is 128 meV [@cold01; @head10]. From this we obtain $\chi''(\omega)=3.9\,\mu^2_{\rm B}\,{\rm eV}^{-1}{\rm Cu}^{-1}$, which is indicated by the gray bar in Fig. \[fg:chi\]. It is strikingly similar to the magnetic spectral weight found for our sample of with $x=0.095$. The degree of similarity may be coincidental, as parameter and data normalization uncertainties are on the order of 20%. The point is that the strong magnetic response cannot come from only a small fraction of the sample. In combination with the evidence for bulk superconductivity [@home12], it appears inescapable that superconductivity and antiferromagnetic spin correlations must coexist locally.
From the perspective of competing orders [@deml01], coexistence of superconductivity and spin-density-wave order (SDW) requires one or both these orders to be weak. While the true SDW order is weak in our sample, the presence of the strong magnetic spectral weight at energies far below the superconducting gap, together with the optical evidence for a substantial superfluid density, is problematic. Similarly, it would be difficult to rationalize the experimental observations in terms of disorder effects alone [@ande10]. A different approach is necessary.
A way to reconcile the coexisting spin fluctuations and superconductivity is to relax the expectation of spatial uniformity. We know from their incommensurability that the locally-antiferromagnetic spin correlations are spatially modulated with a period of roughly 9 lattice spacings. It is also possible for the pair-wave function to be spatially modulated and phase shifted so as to minimize overlap with the low-energy spin correlations. If the pair wave function is sinusoidally modulated with the same period as the spin correlations, so that its amplitude varies from positive to negative, then it represents a pair density wave (PDW) [@berg09b]; such a state has been proposed to explain the decoupling of superconducting layers in with $x=\frac18$ and related systems [@berg07; @hime02]. It is also possible to have the amplitude modulated but without a sign change, in which case it should have half the period of the spin correlations. Recent variational calculations applied to the $t$-$J$ model have found that the energies of the PDW and the in-phase striped superconductor are very close, and both are competitive with the uniform $d$-wave state [@corb14].
Previous studies [@wen12; @steg13] have shown that application of a strong $c$-axis magnetic field to with $x=0.095$ causes a decoupling of the superconducting layers in a manner consistent with the PDW scenario for the $x=\frac18$ composition in zero field. Given that the PDW state is quite sensitive to disorder [@berg09b], the robust superconductivity found for $x=0.095$ in zero field may favor an in-phase striped superconductor.
As already noted, gapless spin fluctuations have also been detected by neutron scattering in underdoped [@lee00; @chan07; @lips09; @kofu09]. In the case of La$_{1.875}$Sr$_{0.125}$CuO$_4$, where charge stripe order has recently been reported [@chri14; @tham14; @crof14], Kofu [*et al.*]{} [@kofu09] proposed that the spin excitations below 4 meV come from different spatial regions than the excitations above 4 meV, thus invoking large scale phase separation to maintain consistency with the spin-gap paradigm. We actually share the concept of phase separation, but on a much shorter length scale. We argue that the low-energy spin excitations come from the spin stripes coexisting with the superconductivity. Regarding the possibility of large-scale phase separation in , we note that, for superconducting samples with $0.06<x<0.10$, it has been concluded from muon spin rotation ($\mu$SR) studies that there is static, inhomogeneous magnetic order throughout the volume at $T<1\ {\rm K}<<T_c$, with any non-magnetic regions being smaller in size than 20 Å [@nied98]. These materials are also believed to be bulk superconductors, which again is consistent with intertwined coexistence.
To rationalize the differences between superconducting samples with and without spin gaps, we suggest the following scenario. At optimal doping and above, where the pair wave function is spatially uniform, it is favorable to gap out any residual spin fluctuations at $\hbar\omega<\Delta$. At lower doping, when strong low-energy spin-stripe correlations are present in the normal state, it may be too energetically costly to gap the spin excitations. Instead, it may be favorable to modulate the pair wave function to avoid the antiferromagnetic spin correlations by intertwining with them [@hime02; @berg09b]. This scenario is consistent with the idea that antiferromagnetism and superconductivity are closely associated [@scal12a; @norm11], but it suggests the need for a pairing mechanism [@emer97] that goes beyond the conventional concept of “pairing glue” [@scal12a].
Finally, we note that spin-gapless superconductivity is not limited to “214” cuprates. $\mu$SR measurements also indicate static magnetic fields in superconducting Y$_{1-x}$Ca$_x$Ba$_2$Cu$_3$O$_{6.02}$ for hole concentrations similar to those in [@nied98]. Furthermore, neutron and $\mu$SR results for superconducting with hole concentrations $p\lesssim0.08$ indicate coexisting gapless spin correlations [@hink08; @stoc08; @haug10]. More generally there have been theoretical and experimental papers proposing the relevance of a PDW state to understanding the pseudogap in cuprates such as , especially at high magnetic field and low temperature [@lee14; @yu14].
We are grateful for helpful comments from E. Fradkin, S. A. Kivelson, and T. M. Rice. Work at Brookhaven was supported by the Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering, U.S. Department of Energy (DOE), through Contract No. DE-AC02-98CH10886. This work utilized facilities at the NCNR supported in part by the National Science Foundation under Agreement No. DMR-0944772. The experiments at ORNL’s SNS were sponsored by the Scientific User Facilities Division, BES, U.S. DOE.
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[**Supplemental Material for “Indirect Evidence for Periodically-Modulated Superconductivity in Underdoped Cuprates”**]{}
Zhijun Xu,$^{1,*}$ C. Stock,$^{2,\dagger}$ Songxue Chi,$^{2,\ddagger}$ A. I. Kolesnikov,$^3$ Guangyong Xu,$^1$ Genda Gu,$^1$ and J. M. Tranquada$^1$\
@noop [“,” ]{} ()
The constant-energy slices of the inelastic neutron scattering data for La$_{1.905}$Ba$_{0.095}$CuO$_4$, examples of which are shown in Fig. 1 of the main paper, were fit as a function of wave vector $(H,K)$. This procedure makes good use of the available data; however, two-dimensional representations of the data and fits can be difficult to compare at a detailed level. Here we present some representative line cuts that allow a direct comparison of the data and the fits. Figure S1 shows cuts along $(H,0.5)$ and $(0.5,K)$, corresponding to sample temperatures of 1.5 K and 32 K. Each slice has a with of 0.02 rlu in the transverse direction. Note that the range over which the actual fitting was performed was within 0.2 rlu of ${\bf Q} = (0.5,0.5)$.
{width="\columnwidth"} {width="\columnwidth"}
In Fig. S2, we present data from Fig. 2(b) in the form of an effective dispersion of the magnetic excitations. The rapid inward dispersion of the excitations at low energy, followed by a vertical rise is quite unusual. We do not know of any models that directly describe such behavior. Note that the unusual dispersion occurs below 4 meV, the same energy range in which Kofu [*et al.*]{} \[10\] observed distinct behavior in the superconducting phase of La$_{1.875}$Sr$_{0.125}$CuO$_4$.
![(color online) Plot of the effective dispersion of magnetic excitations, $E$ vs. $0.5\pm\delta$. Blue circles (green diamonds) correspond to $T=1.5$K (60 K). Blue squares represent the elastic magnetic peak positions at low temperature from a previous study \[21\]. Horizontal bars indicate peak widths. []{data-label="fg:disp"}](figure_s3.pdf){width="\columnwidth"}
[^1]: See Supplemental Material at \[URL to be inserted\] for line cuts of low-energy data and a plot of the effective dispersion at low energy.
|
---
abstract: 'Recently, the precise performance of the Generalized LASSO algorithm for recovering structured signals from compressed noisy measurements, obtained via i.i.d. Gaussian matrices, has been characterized. The analysis is based on a framework introduced by Stojnic and heavily relies on the use of Gordon’s Gaussian min-max theorem (GMT), a comparison principle on Gaussian processes. As a result, corresponding characterizations for other ensembles of measurement matrices have not been developed. In this work, we analyze the corresponding performance of the ensemble of isotropically random orthogonal (i.r.o.) measurements. We consider the constrained version of the Generalized LASSO and derive a sharp characterization of its normalized squared error in the large-system limit. When compared to its Gaussian counterpart, our result analytically confirms the superiority in performance of the i.r.o. ensemble. Our second result, derives an asymptotic lower bound on the minimum conic singular values of i.r.o. matrices. This bound is larger than the corresponding bound on Gaussian matrices. To prove our results we express i.r.o. matrices in terms of Gaussians and show that, with some modifications, the GMT framework is still applicable.'
author:
- |
Christos Thrampoulidis and Babak Hassibi\
Department of Electrical Engineering, Caltech, Pasadena
bibliography:
- 'compbib.bib'
---
|
---
abstract: 'We report a study of the ferromagnetism of ZrZn$_{2}$, the most promising material to exhibit ferromagnetic quantum criticality, at low temperatures $T$ as function of pressure $p$. We find that the ordered ferromagnetic moment disappears discontinuously at $p_c$=16.5 kbar. Thus a tricritical point separates a line of first order ferromagnetic transitions from second order (continuous) transitions at higher temperature. We also identify two lines of transitions of the magnetisation isotherms up to 12T in the $p-T$ plane where the derivative of the magnetization changes rapidly. These quantum phase transitions (QPT) establish a high sensitivity to local minima in the free energy in ZrZn$_{2}$, thus strongly suggesting that QPT in itinerant ferromagnets are always first order.'
author:
- 'M. Uhlarz'
- 'C. Pfleiderer'
- 'S. M. Hayden'
title: 'Quantum Phase Transitions in the Itinerant Ferromagnet ZrZn$_2$'
---
The transition of a ferromagnet to a paramagnet with increasing temperature is regarded as a canonical example of a continuous (second order) phase transition. This type of behavior has been well established in many materials ranging from nickel [@weiss26] to chromium tribromide [@ho69]. The detailed variation of the order parameter near the critical point, in this case the Curie temperature, has been analyzed in a wide variety of systems using *classical* statistical-mechanical models for the case when the Curie temperature is not too small. Classical statistics are appropriate when all fluctuating modes have energies much less than $k_BT_{c}$. It was pointed out by Hertz [@hertz76] that the system undergoes a *‘quantum’* phase transition (QPT) when the transition is driven by non-thermal fluctuations whose statistics are in the quantum limit. The search for a second order (critical) QPT in itinerant electron systems, which are believed to be responsible for enigmatic quantum phases like magnetically mediated superconductivity and non-Fermi liquid behavior, has become of particular interest in recent years. Experimental studies have thereby revealed notable differences from “standard” second order behaviour in *all* materials investigated to date. For example, in MnSi [@pfle97] and UGe$_2$ [@pfle02], itinerant-electron magnetism disappears at a first order transition as pressure is applied. The bilayer ruthenate Sr$_3$Ru$_2$O$_7$, undergoes a field induced QPT with multiple first-order metamagnetic transitions [@perry04] and associated non-Fermi liquid behavior in the resistivity [@grigera01]. However, these materials have complicating factors: the zero-field ground state of MnSi is a helical spin spiral; UGe$_2$ is a strongly uniaxial (Ising) system; Sr$_3$Ru$_2$O$_7$ is a strongly two-dimensional metal. In fact, theoretical studies suggest [@shimizu64; @belitz99; @vojta00; @belitz02] that ferromagnetic transitions in clean three-dimensional (3$D$) itinerant ferromagnets at $T=0$ are *always* first order.
In this Letter we address the nature of the ferromagnetic QPT experimentally. The system we have chosen is the itinerant ferromagnet ZrZn$_2$, which is a straight forward itinerant ferromagnet with a cubic (C15) structure and small magnetic anisotropy. [[ZrZn$_2$]{}]{} has a small ordered moment ($M = 0.17 \mu_B$ f.u.$^{-1}$) which is an order of magnitude smaller than the fluctuating Curie-Weiss moment $\mu_{\mathrm{eff}}=1.9\,\mu_{B} \mathrm{f.u.}^{-1}$ and the Curie temperature is low ($T_c=28.5$ K). The magnetisation is highly unsaturated as function of field up to 35T, the highest field studied. Neutron diffraction in ZrZn$_{2}$ is consistent with all the hallmarks of a three-dimensional itinerant ferromagnet [@neut]. Quantum oscillatory studies [@yate03] have shown that ZrZn$_2$ has a large quasiparticle mass enhancement [@yate03] as expected near quantum criticality.
Single crystals of ZrZn$_2$ have long been considered ideal in the search for quantum criticality (second order behaviour). Previous hydrostatic pressure studies suggested the existence of a second order QPT in ZrZn$_{2}$ [@smit71; @hube75; @gros95]. However, substantial differences of the critical pressure $p_{c}$ and the form of $T_{C}(p)$ were reported in different studies. We now believe that these differences can be traced to low sample quality. These studies underscore the need for high-quality single-crystals.
De Haas–van Alphen (dHvA) studies in high quality single crystals as function of pressure have recently [@kimura04] let to the suggestion that multiple first order QPT exist in [[ZrZn$_2$]{}]{}, notably a crossover between two ferromagnetic phases at ambient pressure and a first order suppression of ferromagnetism at high pressure. However, the evidence for a pressure induced QPT associated with the two ferromagnetic phases at ambient pressure was purely derived from a tiny pocket of the Fermi surface. Further, the first order QPT at $p_c$ was predicted theoretically but not experimental evidence has been reported until now.
Here we report an investigation of the question of multiple QPT in [[ZrZn$_2$]{}]{} using detailed measurements of the DC magnetisation, i.e., direct measurements of the order parameter. We establish for the first time that, while the ferromagnetic transition at ambient pressure is continuous, the ferromagnetism disappears in a first order fashion (discontinuously) as pressure is increased beyond $p_c=16.5$ kbar. The observation of a first order QPT at $p_c$ is strongly supported by the discovery of metamagnetic behavior, characterized by a sudden superlinear rise in the magnetisation as a function of applied field, for pressures above $p_c$. We also characterize the pressure dependence of a second transition *within* the ferromagnetic state for the first time using the DC magnetisation. These data suggest another first order QPT. Thus we establish experimentally the existence of multiple first order QPT based on measurements of the order parameter itself in the ideal candidate for the occurrence of ferromagnetic quantum criticality, [[ZrZn$_2$]{}]{}.
Two single crystals were studied, a short cylindrical piece and a half-cylinder, which produced identical results. Data from these samples are therefore not distinguished further. The samples studied here are the same in which superconductivity was originally discovered [@pfle01a]. The method of growth avoids the problems with the zinc vapor pressure and is similar to that described in [@schr89]. The residual resistivity ratio of our samples $\rho(293\,{\rm K})/\rho(T\to0) \approx 100$ is high and the residual resistivity $\rho_{0}\approx0.6\,\mu\Omega{\rm cm}$ low. Laue x-ray and neutron diffraction confirmed that the samples were single crystals. Extensive dHvA data [@yate03], the magnetic field dependence of the specific heat [@pfle01b], $T_{C}(p)$ in very low fields for $p<16$kbar [@uhla02] and the electrical resistivity up to 21kbar and magnetic field up to 12T [@uhla04] of the same samples are reported elsewhere.
![Typical magnetization cycles below and above the critical pressure $p_{c}=16.5$kbar at various temperatures. On the left hand side data correspond to the temperatures given in each panel, respectively. The arrow marks the metamagnetic transition field $B_{m2}$ that appears above $p_{c}$.[]{data-label="hysteresis"}](figure2.eps){width=".45\textwidth"}
The DC magnetization $M(B,T)$ was measured in an Oxford Instruments vibrating sample magnetometer (VSM) between room temperature and 1.5K at magnetic field in the range $\pm12$T. Additional measurements were carried out in a bespoke SQUID magnetometer in the range 4.2K to 60K at fields in the range 10 $\mu$T to 10mT. The sample was measured together with the nonmagnetic miniature clamp cell. The signal of the empty pressure cell was subtracted to obtain the contribution of the sample, which was typically between 50 and 80% of the total signal. Pressures were determined from the superconducting transition of Sn or Pb in the VSM and SQUID-magnetometer, respectively.
Fig.\[main-result\](a) shows the ferromagnetic ordered moment $M$ as a function at pressure and temperature (inset). The moment was obtained by extrapolating magnetization isotherms (Arrott plots) to zero field. When the pressure is varied at low temperature ($T = $2.3K), the magnetization drops discontinuously at a critical pressure $p_c=16.5$kbar. For comparison the inset shows the variation of the ferromagnetic moment at $p=0$ with increasing $T$ through the Curie temperature. At $p=0$ the transition is continuous (2nd order) presumably because we are in the classical (high temperature) limit. In contrast, when the transition is suppressed to zero through the application of hydrostatic pressure, the magnetization disappears discontinuously (first order). The Curie temperature $T_{C}$, shown in panel (b), qualitatively tracks $M$ and vanishes also discontinuously at $p_{c}$.
Fig. \[hysteresis\] shows magnetization curves near $p_{c}$. Below $p_{c}$ (Fig.\[hysteresis\](a)) and at the lowest temperatures, $M(B)$ initially raises rapidly with $B$ as a single domain is formed. Above $p_{c}$ a new feature appears at low fields where $M(B)$ initially increases approximately linearly. This is followed by a sudden superlinear increase with $B$ (‘a kink’), at the field $B_{m2}$ marked by the arrows in Fig.\[hysteresis\] (b) and (c). Above $B_{m2}$ the shape of the magnetization isotherms is reminiscent of those below $p_{c}$. A sudden rise in $M(B)$, such as that observed here, is usually called metamagnetism [@wohlfarth62]. It is a deep signature of a first order structure of the underlying free energy (local minimum) and proofs unambiguously that the QPT at $p_c$ is first order. The magnetization isotherms allow us to extract the pressure dependence of the cross-over or metamagnetic field $B_{m2}$ as shown in Fig.\[main-result\](c). Our data are consistent with $B_{m2}$ terminating near $p_{c}$ showing an intimate connection with the discontinuous drop at $p_c$.
![(Color online) (a) Magnetization $M$ as function of magnetic field $B$ at $T=2.3$K as function of pressure. Labels correspond to the following pressures in kbar: A=0, B=1.8, C=3.0, D=4.8, E=6.6, F=8.9, G=12.5, H=13.1, I=14.4, J=17.1, K=19.1, L=20.9. (b) Derivative $dM/dB$ for selected pressures (a). For pressures A-E the arrow marks the crossover field $B_{m1}$.[]{data-label="magnetization"}](figure3.eps){width=".35\textwidth"}
In addition to the low-field magnetization measurements described above, we also made measurements at high fields under hydrostatic pressure. Fig.\[magnetization\](a) shows the field dependence of the magnetisation for various pressures up to 20.9 kbar. As with our previous study at ambient pressure [@pfle01a] we observe a kink in the magnetization near $B \approx$ 5 Tesla (curve A in Fig. \[magnetization\](a)). The anomaly can be seen more clearly in the derivative $dM/dB$ shown in Fig.\[magnetization\](b). We are able to identify a cross-over field $B_{m1}$ from a low-field phase (FM1) to a high-field field phase (FM2), where the ordered moment of the high field phase is increased by $\sim$10% (see e.g. Ref.[@kimura04]). With increasing pressure $B_{m1}$ increases as plotted in Fig.\[main-result\] (c). The transition from FM1 to FM2 at $B_{m1}$ corresponds to a transition *within* the ferromagnetic state. By analogy with $B_{m2}(p)$ we extrapolate that $B_{m1}(p)$ terminates at a QPT at approximately -6 kbar.
![(Color online) The experimental variation of the magnetization $M$ with pressure and applied field in ZrZn$_{2}$. The Figure is based on the data shown in Figs. \[hysteresis\] and \[magnetization\]. The white lines show approximately the locations of ‘kinks’ in the magnetization reported in this paper.[]{data-label="M_color"}](figure4.eps){width=".45\textwidth"}
Fig. \[M\_color\] shows an overall representation of the magnetization based on the data in Figs. \[hysteresis\] and \[magnetization\]. The white lines denote the approximate positions of the transition fields $B_{m1}$ and $B_{m2}$. It is interesting to note that $B_{m1}$ occurs at approximately constant $M$. This strongly suggests that the transition is triggered by an exchange splitting that is insensitive to pressure and therefore indeed related to the electronic structure. As for the transition at $B_{m2}$ the anomaly at $B_{m1}$ hence is evidence of a further first order minimum in the free energy, establishing that the associated QPT must be first order. Fig. \[M\_color\] also shows that the high field magnetization varies very little with pressure above the critical pressure $p_c$.
We now discuss the interpretation of our results. Our observations show for the first time that the ordered magnetic moment in ZrZn$_2$ disappears discontinuously around $p_c \approx$ 16.5 kbar. This suggests that in the $p-T$ plane, a tricritical point separates a line of first order transitions from second order behavior at high temperature. Fig. \[f:tricritical\] shows the proposed schematic phase diagram for ZrZn$_2$ [@kimura04], which has not been verified experimentally until now. Crossing the shaded (blue) region corresponds to a first order phase transition. The related dotted line represents a crossover where there is a rapid change in $M(B)$. In our experiments we observe a discontinuous drop of $T_c(p)$, consistent with a tricritical point near $T_{t} \approx 5$ K and $p_{t} \approx 16.5$ kbar as qualitatively proposed in [@kimura04]. However, we have not observed a discontinuous change of $M$ with $T$ or $B$. Presumably this is because the region of first order transitions is small and none of the pressures chosen in our study sample this region. At high magnetic fields (second dotted line) we observe an unusual sudden change of the gradient in the magnetization isotherms $M(B)$ that translates into an increase of the ordered moment.
![(Color online) Systematic representation of a possible phase diagram of ZrZn$_2$ proposed in Ref. [@kimura04]. For $p \leq p_c$, a line of second order ferromagnetic ends at a tricrital point TP. For $p > p_c$, a first order jump in the magnetization occurs on crossing the shaded (blue) area which extends as a crossover to higher pressures (dotted line). At higher fields a crossover in $M(B)$ persists which emerges from a QPT at an extrapolated negative pressure.[]{data-label="f:tricritical"}](figure5.eps){width=".30\textwidth"}
The evidence for first order behaviour we observe bears on various theoretical descriptions of ferromagnetic phase transition in metals in the quantum limit. The first is a ‘Stoner’ picture where the effect of electrons are incorporated into a one-particle band structure and the exchange interaction is described by a molecular field $\lambda M$. In this case, the condition for ferromagnetism and field dependence of the magnetization $M(B)$ are determined by the structure of the electronic density of states near the Fermi energy. Within this model, Shimizu [@shimizu64] has shown that, if the Fermi energy lies near a peak in the density of states, ferromagnetism will disappear at a first order transition. An extension to this model [@wohlfarth62; @Sandeman03] suggests that the application of a magnetic field can lead to a metamagnetic QPT, if the criterium for ferromagnetism is not quite satisfied. The applicability of the ‘Stoner’ model to ZrZn$_2$ is supported by band structure calculations [@santi01; @singh02] and the experimental determination of the Fermi surface [@yate03], which suggest that the paramagnetic Fermi energy lies about 30 meV below a double-peak in the one-electron density of states. In a second description of ferromagnetic quantum criticality [@belitz99; @belitz02], the transition is found to be generically first order due to a coupling of long-wavelength magnetisation modes to soft particle-hole excitations. Near the ferromagnetic QPT, this leads to a nonanalytic term in the free energy that generates first order behaviour. This mechanism is independent of the band structure and *always* present.
In summary, we report for the first time that ferromagnetism in [[ZrZn$_2$]{}]{} disappears discontinuously at $p_c=$ 16.5 kbar and ‘crossover’ or ‘transition’ lines exist in the $p-B$ plane ($B_{m1}(p)$ and $B_{m2}(p)$). The first transition, $B_{m1}(p)$, occurs at ambient pressure and its pressure dependence shows the existence of a first order QPT at negative pressure. The second transition, $B_{m2}(p)$, is associated with the disappearance of ferromagnetism at the first order ferromagnetic QPT at $p_c$. Our experiments establish for the first time the existence of multiple first order QPT in [[ZrZn$_2$]{}]{}, as previously proposed [@kimura04]. The emergence of these multiple first order QPT with decreasing temperature in a material, [[ZrZn$_2$]{}]{}, that is in every respect considered to be *the* prime candidate for ferromagnetic quantum criticality supplies strong evidence that QPT in itinerant ferromagnets must be generically first order.
We wish to thank D. Belitz, S. Dugdale, M. Garst, J. Kübler, H. v. Löhneysen, G. Lonzarich, I. Mazin, K. Noriyaki, A. Rosch, G. Santi, T. Vojta and M. Vojta. Help by I. Walter, T. Wolf, V. Ziebat and P. Pfundstein are gratefully acknowledged. Financial support by the Deutsche Forschungsgemeinschaft (DFG), European Science Foundation under FERLIN and the UK EPSRC are gratefully acknowledged.
[99]{} P. Weiss and R. Forrer, Ann. Phys. (Paris) **5**, 153 (1926); J. S. Kouvel and M.E. Fisher, Phys. Rev. **136**, A1626 (1964).
J.T. Ho and J.D Lister, Phys. Rev. Lett. **22**,603 (1969).
J. Hertz, Phys. Rev. B **14**, 1165 (1976)
C. Pfleiderer et al., Phys. Rev. B [**55**]{}, 8330 (1997).
C. Pfleiderer, A. D. Huxley, Phys. Rev. Lett. [**89**]{}, 147005 (2002).
R.S. Perry *et al.*, Phys. Rev. Lett. **92**, 166602 (2004).
S.A. Grigera, et al., Science **294**, 329 (2001).
D. Belitz et al., Phys. Rev. Lett. [**82**]{}, 4707 (1999);
T. Vojta, Adv. Phys. **9**, 403 (2000).
D. Belitz and T.R. Kirkpatrick, Phys. Rev. Lett. [**89**]{}, 247202 (2002).
M. Shimizu, Proc. Phys. Soc. **84**, 397 (1964).
N. Kimura, et al., Phys. Rev. Lett. **92**, 197002 (2004).
P. J. Brown et al., J. Mag. Mag. Mater. **42** (1984) 12.
S. Yates et al., Phys. Rev. Lett. **90**, 057003 (2003).
T. F. Smith et al., Phys. Rev. Lett. [**27**]{}, 1732 (1971).
J. G. Huber et al., Solid State Commun. **16**, 211 (1975).
F. M. Grosche et al., Physica B **206-207**, 20 (1995).
C. Pfleiderer et al., Nature (London), [**412**]{}, 58 (2001); corrigendum ibid 660.
L. W. M. Schreurs et al., Mater. Res. Bull. **24**, 1141 (1989).
C. Pfleiderer et al., J. Mag. Mag. Mat. **226-230**, 258 (2001).
M. Uhlarz et al., Physica B **312-313**, 487 (2002).
M. Uhlarz, PhD thesis, Universität Karlsruhe (2004).
E. P. Wohlfarth and P. Rhodes, Philos. Mag. **7**, 1817 (1962).
K. G. Sandeman,1 G. G. Lonza r ich, 1 and A. J. Schofield, Phys. Rev. Lett. **90**, 167005 (2003).
G. Santi, S. B. Dugdale, and T. Jarlborg, Phys. Rev. Lett. **87**, 247004 (2001).
D. J. Singh and I. I. Mazin, Phys. Rev. Lett. **88**, 187004 (2002).
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---
abstract: 'We show how to map the states of an ergodic Markov chain to Euclidean space so that the squared distance between states is the expected commuting time. We find a minimax characterization of commuting times, and from this we get monotonicity of commuting times with respect to equilibrium transition rates. All of these results are familiar in the case of time-reversible chains, where techniques of classical electrical theory apply. In presenting these results, we take the opportunity to develop Markov chain theory in a ‘conformally correct’ way.'
author:
- 'Peter G. Doyle'
- Jean Steiner
bibliography:
- 'commute.bib'
date: |
Version 1A9 dated 5 March 2008\
[ No Copyright[^1]]{}
title: Commuting time geometry of ergodic Markov chains
---
Overview
========
In an eye-opening paper, Chandra, Raghavan, Ruzzo, Smolensky, and Tiwari [@crrst:commute] revealed the central importance of expected commuting times for the theory of time-reversible Markov chains. Here we extend the discussion to general, non-time-reversible chains.
We begin by showing how to embed the states in a Euclidean space so that the squared distance between states is the commuting time. In the time-reversible case, Leibon et al. have used Euclidean embeddings to great effect as a way to visualize a chain, and reveal natural clustering of states. Our embedding theorem shows that non-time-reversible chains should be amenable to the same treatment.
Looking beyond the Euclidean embedding, we find a natural minimax characterization of commuting times. From this we get the monotonicity law for commuting times: If all equilibrium interstate transition rates are increased, then all commuting times are diminished. For time-reversible chains, this monotonicity law is an ancient and powerful tool. It is questionable how useful it will prove to be in the general case.
In presenting these results, we will be taking a ‘conformally correct’ approach to Markov chains. Briefly, a conformal change to a Markov chain changes its equilibrium measure, but not its equilibrium transition rates. The opportunity to develop this conformally correct approach is at least as important to us as the particular results we’ll be discussing here.
The problem
===========
The *commuting time* $T_{ab}$ between two states $a,b$ of an ergodic Markov chain is the expected time, starting from $a$, to go to $b$ and then back to $a$. Evidently $T_{ab}=T_{ba}$ and $$T_{ac} \leq T_{ab}+T_{bc}
.$$ Thus it might seem natural to think of $T_{ab}$ as a measure of the distance between $a$ and $b$. But in fact it is most natural to think of $T_{ab}$ as the *squared distance* between $a$ and $b$. The reason is that, as we will see, there is a natural way to identify the states of the chain with points in a Euclidean space having quadratic form $||x||^2$ such that for any states $a,b$ we have $$T_{ab} = ||a-b||^2
.$$ Now that we are interpreting $T_{ab}$ as a squared distance, the inequality $T_{ac} \leq T_{ab}+T_{bc}$ tells us that $$||a-c||^2 \leq ||a-b||^2 +||b-c||^2
.$$ This means that all angles $\angle abc$ are acute (at least weakly: some might be right angles).
Realizing commuting times as squared distances is straight-forward for time-reversible chains. Here’s a sketch, meant only for orientation: We won’t rely on any of this below. Time-reversible chains correspond exactly to resistor networks, with $T_{ab}$ corresponding to the effective resistance between $a$ and $b$. This effective resistance is the energy of a unit current flow from $a$ to $b$. The energy of a flow is its squared distance with respect to the energy norm on flows. If we associate to state $i$ the unit current flow from $i$ to some arbitrary reference vertex (the ‘ground’), then the difference between the flows associated to $a$ and $b$ will be the unit current flow from $a$ to $b$, having square norm $T_{ab}$.
The trick will be to extend this result to non-time-reversible chains. Now, it may in fact be the case that to any chain there corresponds a time-reversible chain having the same $T$, up to multiplication by a positive constant. This would immediately take care of the extension beyond the time-reversible case. It is easy enough to compute what the transition rates of this time-reversible chain would have to be, but we don’t know that they are always positive. We leave this question for another day.
Before proceeding, we should observe that the triangle inequality for squared lengths is not in itself a sufficient condition for realizability of a Euclidean simplex. It *is* sufficient for tetrahedra (four vertices in 3-space), but for five vertices we have the following counterexample. Take $$T=
\left(
\begin{array}{ccccc}
0& 7& 7& 7& 13
\\
7& 0& 12& 12& 7
\\
7& 12& 0& 12& 7
\\
7& 12& 12& 0& 7
\\
13& 7& 7& 7& 0
\end{array}
\right)$$ This matrix is not realizable because the associated quadratic form with matrix $${\frac{1}{2}}\left(
\begin{array}{llll}
14 & 2 & 2 & 13 \\
2 & 14 & 2 & 13 \\
2 & 2 & 14 & 13 \\
13 & 13 & 13 & 26
\end{array}
\right)$$ is not positive definite: It has the eigenvalue ${\frac{1}{2}}(22-\sqrt{523}) \approx -0.434597$. Since we’re going to see that commuting time matrices are always realizable, this means in particular that this matrix $T$ cannot arise as the matrix of commuting times of a Markov chain.
The short answer
================
Below we will give the honest solution to this problem, developing in a thoroughgoing way what we will call the ‘conformally correct’ approach to Markov chains. Here we just extract the answer to our embedding question, and present it in a way that should be immediately accessible to those familiar with the standard theory of Markov chains, as developed for example in Grinstead and Snell [@grinsteadSnell:prob]. The only caveat is that we will be using tensor notation, i.e. writing some indices up rather than down. You can look at section \[sec:tensor\] below for remarks about this, but if you prefer you can just view this as an idiosyncracy, as long as you bear in mind that $\tensor{Z}{_i^j}$ represents a different array of numbers from $Z_{ij}$.
Consider a discrete-time Markov chain with transition probabilities $$\tensor{P}{_i^j} = {\mathrm{Prob}}(\mbox{next at $j$}|\mbox{start at $i$})
.$$ Assume the chain is ergodic so there is a unique equilibrium measure $w^i$ with $$\sum_i w^i \tensor{P}{_i^j} = w^j$$ and $$\sum_i w^i = 1
.$$ Define $${\Delta}^{ij} = w^i(\tensor{I}{_i^j}-\tensor{P}{_i^j})
,$$ and note that $$\sum_i {\Delta}^{ij} = \sum_j {\Delta}^{ij} = 0
.$$
Now define $$\tensor{Z}{_i^j}
=
(\tensor{I}{_i^j}-w^j)
+
(\tensor{P}{_i^j}-w^j)
+
(\tensor{{P^{(2)}}}{_i^j}-w^j)
+
\ldots
,$$ where $\tensor{{P^{(2)}}}{_i^j}
=
\sum_k \tensor{P}{_i^k} \tensor{P}{_k^j}$ represents the matrix square of $\tensor{P}{_i^j}$, and the elided terms involve higher matrix powers. Using conventional matrix notation if we define $\tensor{{{P^{(\infty)}}}}{_i^j} = w^j$ we can write $$\begin{aligned}
Z
&=&
(I-{{P^{(\infty)}}}) + (P-{{P^{(\infty)}}}) + (P^{(2)}-{{P^{(\infty)}}}) + \ldots
\\&=&
(I-P+{{P^{(\infty)}}})^{-1} - {{P^{(\infty)}}}.\end{aligned}$$ (Note that Grinstead and Snell [@grinsteadSnell:prob] use the alternate definition $Z=(I-P+{{P^{(\infty)}}})^{-1}$, which is less congenial but works just as well in this context.)
Set $$Z_{ij} = \frac{1}{w^j}\tensor{Z}{_i^j}
.$$ $Z_{ij}$ acts like an inverse to ${\Delta}^{ij}$ in the sense that for any $u^i$ with $\sum_i u^i=0$, we have $$\sum_{jk} u^j Z_{jk} {\Delta}^{kl} = u^l$$ and $$\sum_{jk} {\Delta}^{ij} Z_{jk} u^k = u^i
.$$
Standard Markov chain theory tells us that the expected time $M_{ab}$ to hit state $b$ starting from state $a$ is $$M_{ab} = Z_{bb} - Z_{ab}
.$$ So for the commuting time we have $$T_{ab}=M_{ab}+M_{ba}
=
Z_{aa}-Z_{ab}-Z_{ba}-Z_{bb}
.$$
For a vector $x=(x_i)_{i=1,\ldots,n}$ define $$||x||^2 =
\sum_{ij} x_i {\Delta}^{ij} x_j
.$$ Please note that this does not make ${\Delta}^{ij}$ the matrix of the quadratic form in the usual sense, because in general ${\Delta}^{ij} \neq {\Delta}^{ji}$. The matrix of the form in the usual sense is the symmetrized version ${\frac{1}{2}}({\Delta}^{ij}+{\Delta}^{ji})$.
Because $$\sum_i {\Delta}^{ij} = \sum_j {\Delta}^{ij} = 0$$ we have the key identity $$||x||^2 = - {\frac{1}{2}}\sum_{ij} {\Delta}^{ij} (x_i-x_j)^2
.$$ Recalling the definition of ${\Delta}^{ij}$ gives $$||x||^2 =
{\frac{1}{2}}\sum_{ij} w^i \tensor{P}{_i^j} (x_i-x_j)^2
.$$ Thus the quadratic form $||x||^2$ is weakly positive definite, but not strictly so, because it vanishes for constant vectors: $$||(c,\ldots,c)||^2 = 0
.$$ It becomes strictly positive definite if we identify vectors differing by a constant vector: $$(x_i)_{i=1,\ldots,n} \equiv (z_i+c)_{i=1,\ldots,n}
.$$ This Euclidean space (vectors mod constant vectors, with the pushed-down quadratic form) is where we will embed our chain.
To get the embedding, map state $a$ to the vector $$f(a) = (Z_{ai})_{i=1,\ldots,n}
.$$ For the difference between the images of $a$ and $b$ we have $$(f(a)-f(b))_i = Z_{ai}-Z_{bi}
=
\sum_k
({\tensor{\delta}}{_a^k}-{\tensor{\delta}}{_b^k}) Z_{ki}
,$$ with ${\tensor{\delta}}{_i^j}$ the Kronecker delta. We want to see that $f(a)-f(b)$ has square norm $T_{ab}$.
From the generalized inverse relationship between $Z_{ij}$ and ${\Delta}^{ij}$ and the fact that $$\sum_k
{\tensor{\delta}}{_a^k}-{\tensor{\delta}}{_b^k} = 0$$ we have $$\sum_{i} (Z_{ai}-Z_{bi}) {\Delta}^{ij}
=
\sum_{ki} ({\tensor{\delta}}{_a^k}-{\tensor{\delta}}{_b^k}) Z_{ki} {\Delta}^{ij}
=
{\tensor{\delta}}{_a^j}-{\tensor{\delta}}{_b^j}
.$$ So $$\begin{aligned}
||f(a)-f(b)||^2
&=&
\sum_{ij} (Z_{ai}-Z_{bi}) {\Delta}^{ij} (Z_{aj}-Z_{bj})
\\&=&
\sum_j ({\tensor{\delta}}{_a^j}-{\tensor{\delta}}{_b^j}) (Z_{aj}-Z_{bj})
\\&=&
Z_{aa}-Z_{ab}-Z_{ba}+Z_{bb}
\\&=&
T_{ab}
.\end{aligned}$$ There you have it.
What just happened
==================
We want to explain the proof we have just given in more conceptual terms.
Let $V$ be a finite-dimensional real vector space, and ${{V^\star}}$ the dual space, consisting of linear functionals $\phi:V \to {{\bf R}}$. For $u \in {{V^\star}}$, $x \in V$ write $${\langle}u,x {\rangle}_V = u(x)$$ for the natural pairing between $V$ and ${{V^\star}}$. Identify $V$ with ${{V^{\star \star}}}$ as usual: $${\langle}x,u {\rangle}_{{V^\star}}=
u(x)
=
{\langle}u,x {\rangle}_V
.$$ To a map $f:V \to W$ we associate the adjoint map $f^\star:W^\star \to V^\star$, such that for $u \in W^\star$, $x \in V$ $${\langle}f^\star(u),x {\rangle}_V = u(f(x))
.$$
A bilinear form on $V$ arises from a linear map $$\phi:V \to {{V^\star}}$$ via $$L_\phi(x,y) = {\langle}\phi(x),y {\rangle}_V
.$$ The adjoint map $${{\phi^\star}}: {{V^\star}}\to V$$ yields the transposed bilinear form $$L_{{\phi^\star}}(x,y)
=
{\langle}{{\phi^\star}}(x),y {\rangle}_V
=
{\langle}x,\phi(y) {\rangle}_{{V^\star}}=
{\langle}\phi(y),x {\rangle}_V
=
L_\phi(y,x)
.$$
If $\phi$ is invertible the inverse $${{\phi^{-1}}}: {{V^\star}}\to V$$ yields the form $L_{{\phi^{-1}}}$ on ${{V^\star}}$: $$L_{{\phi^{-1}}}(u,v)
=
{\langle}\phi^{-1}(u),v {\rangle}_{{V^\star}}=
{\langle}v, {{\phi^{-1}}}(u) {\rangle}_V
.$$
The forms $L_{{\phi^\star}}$ and $L_{{\phi^{-1}}}$ are conjugate, because $$L_{{\phi^{-1}}}(u,v)
=
{\langle}v,{{\phi^{-1}}}(u) {\rangle}_V
=
L_\phi({{\phi^{-1}}}(v),{{\phi^{-1}}}(u))
=
L_{{\phi^\star}}({{\phi^{-1}}}(u),{{\phi^{-1}}}(v))
.$$ Going back the other way, $$L_{{\phi^\star}}(x,y)
=
L_{{\phi^{-1}}}(\phi(x),\phi(y))
.$$
From these two equations, we get two distinct ways to conjugate $L_\phi$ to $L_{{\phi^{-1 \star}}}$. Plugging $\phi=({{\phi^{-1}}})^{-1}$ into the first and putting $(x,y)$ for $(u,v)$, we get $$L_\phi(x,y)=L_{{\phi^{-1 \star}}}(\phi(x),\phi(y))
.$$ Plugging $\phi=({{\phi^\star}})^\star$ into the second we get $$L_\phi(x,y)=L_{{\phi^{-1 \star}}}({{\phi^\star}}(x),{{\phi^\star}}(y))
.$$ Now putting ${{\phi^\star}}$ for $\phi$ we see that in fact there were two ways to conjugate $L_{{\phi^{-1}}}$ to $L_{{\phi^\star}}$: $$L_{{\phi^\star}}(x,y)
=
L_{{\phi^{-1}}}(\phi(x),\phi(y))
=
L_{{\phi^{-1}}}({{\phi^\star}}(x),{{\phi^\star}}(y))
.$$
Having two ways to conjugate $L_\phi$ to $L_{{\phi^{-1 \star}}}$ gives us an automorphism ${{\phi^{-1}}}{\circ}{{\phi^\star}}$ of $L_\phi$: $$L_\phi(x,y)=
L_\phi({{\phi^{-1}}}({{\phi^\star}}(x)),{{\phi^{-1}}}({{\phi^\star}}(y)))
.$$ Along with ${{\phi^{-1}}}{\circ}{{\phi^\star}}$ we also have the inverse automorphism ${{\phi^{-1 \star}}}{\circ}\phi$: $$L_\phi(x,y)=
L_\phi({{\phi^{-1 \star}}}(\phi(x)),{{\phi^{-1 \star}}}(\phi(y)))
.$$ We could also consider powers other than $-1$ of our automorphism, but we don’t need to, because the conjugacy between $L_\phi$ and $L_{{\phi^\star}}$ is canonical (in the sense of being equivariant with respect to taking duals and inverses) up to this factor of two. The difference between them, as measured by the automorphism ${{\phi^{-1}}}{\circ}{{\phi^\star}}$, measures the antisymmetry of $L_\phi$. It is destined to play an important role in our future.
Looking now at the level of quadratic forms $Q_\phi(x)=L_\phi(x,x)$, everything in sight is conjugate: $$Q_\phi(x)=Q_{{\phi^\star}}(x)
;$$ $$Q_{{\phi^{-1}}}(u) = Q_{{\phi^{-1 \star}}}(u)
= Q_\phi({{\phi^{-1}}}(u)) = Q_\phi({{\phi^{-1 \star}}}(u))
.$$
All this nonsense can be made much more concrete using matrices. Let $V = {{\bf R}}^n$ and represent $x \in V$, $u \in {{V^\star}}$ as column and row vectors respectively, so that the pairing is just multplying a row vector by a column vector: $${\langle}u, x {\rangle}_V = ux
.$$ Denote transposition of matrices by $\star$. Write $$L_\phi(x,y) = x^\star A y
,$$ so that $$\phi(x) = x^\star A = (A^\star x)^\star
.$$ Now $${{\phi^{-1}}}(u) = (u A^{-1})^\star = A^{{-1 \star}}u^\star
,$$ so $$L_{{\phi^{-1}}}(u,v) =
{\langle}v, {{\phi^{-1}}}(u) {\rangle}_V
= v A^{{-1 \star}}u^\star
= u A^{-1} v^\star
.$$ Good!
Now to see the two conjugacies of $L_{{\phi^\star}}$ with $L_{{\phi^{-1}}}$: $$A^\star A^{{-1}}A = A^\star
;$$ $$A A^{{-1}}A^\star = A^\star
.$$ These combine to give two automorphisms of $L_\phi$: $$(A^{{-1}}A^\star)^\star A (A^{{-1}}A^\star)
=
A A^{{-1 \star}}A A^{{-1}}A^\star
=
A
;$$ $$(A^{{-1 \star}}A)^\star A (A^{{-1 \star}}A)
=
A^\star A^{{-1}}A^{{-1 \star}}A
=
A
.$$ Hmm. Why didn’t we do it this way in the first place?
So, here’s what happened with our Markov chain. We started with the space $V={{\bf R}}^n {/}{\mathrm{1}}$ with quadratic form $L_\phi(x,y) = \sum_{ij} x_i {\Delta}^{ij} y_j$, embedded the states in ${{V^\star}}= {{\bf R}}^n \perp {\mathrm{1}}$ with quadratic form $L_{{\phi^{-1}}}(u,v) = \sum_{ij} u^i Z_{ij} v^j$, and proved that $L_{{\phi^{-1}}}$ is positive definite by showing that it is conjugate to $L_\phi$.
Tensor notation for Markov chains {#sec:tensor}
=================================
As you will already have noticed, we are using tensor notation, rather than trying to work within the confines of matrix notation, as is usual in the theory of Markov chains. For our purposes, a tensor may be viewed as an array where some of the indices are written as superscripts rather than subscripts. Thus, for example, we write the transition rates for a Markov chain as $\tensor{P}{_i^j}$, and the equilbrium measure as $w^i$.
Where the indices of a tensor are placed makes a difference: Thus $\tensor{Z}{_i^j}$ represents a different array from $Z_{ij}$. We may ‘raise’ and ‘lower’ these indices as is usual with tensors, though in this case the procedure is simpler than usual, because to raise or lower an index $i$ we just multiply or divide by the entries of $w^i$. Thus we get $Z_{ij}$ from $\tensor{Z}{_i^j}$ by lowering the index $j$: $$Z_{ij} = \frac{1}{w^i} \tensor{Z}{_i^j}
.$$ We get back to $\tensor{Z}{_i^j}$ from $Z_{ij}$ by raising the index $j$: $$\tensor{Z}{_i^j}
=
w^j Z_{ij}
.$$
We will still be able to use matrix notation to multiply matrices (two-index tensors) and vectors (one-index tensors). The beautiful thing is that when we do this, the indices take care of themselves, as long as the indices that get summed over when multiplying matrices are paired high with low. To show by example what this means, if we write $C=AB$, it will entail (among other things) that $$\tensor{C}{_i^j} =
\tensor{(AB)}{_i^j} =
\sum_k \tensor{A}{_i^k}\tensor{B}{_k^j}
=
\sum_k \tensor{A}{_i_k}\tensor{B}{^k^j}
,$$ and $$\tensor{C}{_i_j} =
\tensor{(AB)}{_i_j} =
\sum_k \tensor{A}{_i^k}\tensor{B}{_k_j}
=
\sum_k \tensor{A}{_i_k}\tensor{B}{^k_j}
=
\sum_k \tensor{A}{_i_k} w^k \tensor{B}{_k_j}
=
\sum_k \tensor{A}{_i_k}\tensor{B}{^k^j}\frac{1}{w^j}
.$$
[**Note.**]{} If you’re familiiar with the Einstein summation convention, be aware that we don’t use it here. It wouldn’t work well in this context, because we want to write $w^i Z_{ij}$ without automatically summing over $i$. Fortunately, for our purposes, using the notation of matrix multiplication turns out to be even more convenient than the summation convention.
What it means to be conformally correct
=======================================
We have said that we want our approach to be ‘conformally correct’. Before we go further, a word about what this means. (Skip this if you don’t care.)
Conformal equivalence of Markov chains is most natural for continuous time chains. In that context two chains with transition rates $\tensor{A}{_i^j}$ and $\tensor{B}{_i^j}$ are conformally equivalent if $$\tensor{B}{_i^j} = \frac{1}{a_i} \tensor{A}{_i^j}$$ where all $a_i>0$. Generally we will also want the additional condition that $\sum_i w^i a_i=1$ where $w^i$ is the equilibrium probability of being at $i$ for the $A$ chain. With this ‘volume condition’ the equilibrium probability of being at $i$ for the $B$ chain will be $w^i a_i$ and $$B^{ij} = w^i a_i \tensor{B}{_i^j} = w^i a_i \frac{1}{a_i} \tensor{A}{_i^j}
= A^{ij}
.$$ Thus while the raw transition rates $\tensor{A}{_i^j}$ are not conformal invariants, when we raise the index $i$ we get a new array $A^{ij} = w^i \tensor{A}{_i^j}$ whose entries are conformal invariants: They tell the rate at which transitions are made from $i$ to $j$ when the chain is in equilibrium.
It is possible to talk about conformal equivalence of discrete time chains, but it is not as pleasant as for continuous-time chains. This is true so often in the theory of Markov chains! And yet, for simplicity, we want to talk about discrete-time chains. So our approach will be to do everything in such a way that the discussion would be conformally invariant when translated from discrete to continuous time.
So that’s what it means for chains to be conformally equivalent. As for ‘conformal correctness’, we mean an approach that seeks to identify and emphasize quantities that are conformally invariant. And why should we do this? Because it will pay.
Visualizing commuting times
===========================
One way to determine the expected commuting time $T_{ab}$ between $a$ and $b$ is to run the chain for a long time $T$ (beware of confusion!), paying attention to when the chain is at $a$ or $b$ and ignoring other states. If $R$ is the number of runs of $a$’s (which is within $1$ of the number of runs of $b$’s), then $$T_{ab} \approx T/R
.$$ To keep track of $R$ we imagine painting our Markovian particle green when it reaches $a$ and red when it reaches $b$. Let $r_{ab}$ be the equilibrium rate at which red particles are being painted green. Ignoring end effects, over our long time interval $T$, $R$ above is the number of times a red particle gets painted green, thus roughly $T r_{ab}$, and it follows that $$T_{ab} = \frac{1}{r_{ab}}
.$$ This is an instance of the general principle from renewal theory that when events happen at rate $r$, the expected time between events is $1/r$.
[**Note.**]{} This painting business is very close to a model developed by Kingman [@kingman:paint] and Kelly [@kelly:paint]. (See exercise 1 in section 3.3 of Doyle and Snell [@doylesnell:walks].) However, I don’t know that Kingman and Kelley ever made the connection to commuting times, and it is possible that their discussion concerned only time-reversible chains. Somebody should check this.
It is high time to observe that if $\hat{T}_{ab}$ is the commuting time for the time-reversed chain (according to the general convention that time-reversed quantities wear hats), we have $$T_{ab}=T_{ba}=\hat{T}_{ab}=\hat{T}_{ba}
.$$ We claim to be able to see this from our way of approximating $T_{ab}$ by observing the chain over a long time. If we reverse a record of the chain moving forward for a long time, we see roughly a record of the time-reversed chain starting in equlibrium. In fact if we started the original chain in equilibrium we’re golden. If we started the chain not in equilibirum (e.g. by starting at $a$, as we might well be tempted to do), there will be problems toward the end of the time-reversed record, as the time-reversed chain gets drawn to end where the forward chain began. But this effect is negligible when $T$ is large.
The Laplacian and the cross-potential
=====================================
Consider a discrete-time Markov chain with transition probabilities $$\tensor{P}{_i^j} = {\mathrm{Prob}}(\mbox{next at $j$}|\mbox{start at $i$})
.$$ Assume the chain is ergodic, so that there is a unique equilibrium measure $w^i$ with $$\sum_i w^i \tensor{P}{_i^j} = w^j
,$$ $$\sum_i w^i = 1
.$$
Define the *Laplacian* $${\Delta}^{ij} = w^i(\tensor{I}{_i^j}-\tensor{P}{_i^j})
.$$ For $i \neq j$, $-{\Delta}^{ij}$ tells the equilibrium rate of transitions from $i$ to $j$; ${\Delta}^{ii}$ tells the total rate of transitions to and from states other than $i$. The time-reversed Markov chain has Laplacian $\hat{{\Delta}}^{ij} = {\Delta}^{ji}$. A time-reversible chain has ${\Delta}^{ij}={\Delta}^{ji}$.
We have $$\sum_i {\Delta}^{ij} = \sum_j {\Delta}^{ij} = 0
.$$ So considered as a matrix, ${\Delta}^{ij}$ is not invertible. However, it has a generalized inverse $Z_{ij}$ with the property that for any measure of total mass 0, which is to say for any $u^i$ with $\sum_i u^i=0$, we have $$\sum_{jk} u^j Z_{jk} {\Delta}^{kl} = u^l$$ and $$\sum_{jk} {\Delta}^{ij} Z_{jk} u^k = u^i
.$$ An equivalent way to write this is $$\sum_{jk} {\Delta}^{ij} Z_{jk} {\Delta}^{kl} = {\Delta}^{il}
,$$ because if we think of ${\Delta}^{ij}$ as a matrix, its rows and columns both span the space of measures with total mass 0.
A sensible choice for the generalized inverse $Z_{ij}$ is $$Z_{ij} = \frac{1}{w^j} \tensor{Z}{_i^j}$$ where $$\tensor{Z}{_i^j}
=
(\tensor{I}{_i^j}-w^j)
+
(\tensor{P}{_i^j}-w^j)
+
(\tensor{{P^{(2)}}}{_i^j}-w^j)
+
\ldots
,$$ where $\tensor{{P^{(2)}}}{_i^j}
=
\sum_k \tensor{P}{_i^k} \tensor{P}{_k^j}$ represents the matrix square of $\tensor{P}{_i^j}$, and the elided terms involve higher matrix powers. Define $\tensor{{{P^{(\infty)}}}}{_i^j} = w^j$, to suggest that the ‘infinitieth power’ of $\tensor{P}{_i^j}$ has all rows equal to the vector $w^i$. We can write $$\begin{aligned}
Z
&=&
(I-{{P^{(\infty)}}}) + (P-{{P^{(\infty)}}}) + (P^{(2)}-{{P^{(\infty)}}}) + \ldots
\\&=&
(I-P+{{P^{(\infty)}}})^{-1} - {{P^{(\infty)}}}.\end{aligned}$$ This naturally translates into the formula we’ve given for $\tensor{Z}{_i^j}$, and from there, by ‘lowering the index j’, we get $Z_{ij}$.
For this choice of $Z$ we have the natural interpretation that $\tensor{Z}{_i^j}$ is the expected excess number of visits to $j$ for a chain starting at $i$ compared to a chain starting in equilibrium. For the time-reversed chain we get $$\tensor{\hat{Z}}{_{ij}}=\tensor{Z}{_{ji}}
,$$ and so in particular if the chain is time-reversible we have $Z_{ij}=Z_{ji}$.
This is all very well, but we still do not want to prescribe this particular choice of $Z$ because it is not conformally invariant: It depends on the equilibrium measure $w^i$, and not just on the Laplacian ‘matrix’ ${\Delta}^{ij}$. This makes it insufficiently canonical for us.
What *is* canonical is the bilinear form $$B(u,v)= \sum_{ij} u^i Z_{ij} v^j$$ when $u$ and $v$ are restricted to the subspace $S$ of measures of total mass $0$: $$S = \{u^i: \sum_i u^i=0 \}$$ Fixing $a,b,c,d$ and setting $$u={\tensor{\delta}}{_a^i}-{\tensor{\delta}}{_b^i}
;\;
v={\tensor{\delta}}{_c^i}-{\tensor{\delta}}{_d^i}$$ gives us the *cross-potential* $$N_{abcd}
=
B({\tensor{\delta}}{_a^i}-{\tensor{\delta}}{_b^i},{\tensor{\delta}}{_c^i}-{\tensor{\delta}}{_d^i})
=
Z_{ac}-Z_{ad}-Z_{bc}+Z_{bd}
.$$ $N$ satisfies $$N_{bacd}=N_{abdc}=-N_{abcd}
.$$ For the time-reversed process $$\hat{N}_{abcd}=N_{cdab}
.$$
Clearly, knowing $N$ is the same as knowing $B$, or ${\Delta}$. If we know $w$ as well as $N$ we can recover our sensible-but-not-canonical $Z$: $$Z_{ij}=\sum_{kl} N_{ikjl}w^k w^l
.$$ Different choices of $w$ in this formula lead to different $Z$’s, but they all determine the same bilinear form $B$. From $Z$ and $w$ we can recover $P$.
In general, it is useful to think of an ergodic Markov chain as specified by the cross-potential $N$, which determines its conformally invariant properties, together with the equilibrium measure $w$. Expressing formulas in these terms allows us to see the extent to which quantities are conformally invariant (like $N$, $B$, and ${\Delta}$) or not (like $w$, $Z$, $P$).
[**Complaint.**]{} $N$ and $w$ together don’t quite determine the original transition rates for a continuous-time Markov chain, or rather, they wouldn’t do so if we had some way to distinguish between remaining at $i$ and moving from $i$ to $i$. Such a distinction is not possible for discrete-time chains represented by matrices, but we could handle it in the continuous case by allowing for non-zero transition rates on the diagonal. Better yet, we could reformulate Markov chain theory in the context of queuing networks based on $1$-complexes (graphs where loops and multiple edges are allowed). This would give us a way to distinguish different ways of stepping from $i$ to $j$. A further step would be to allow a general distribution for the time it takes to make a transition for $i$ to $j$. This would be very helpful when watching the chain only when it is in a subset of its states, as in the case above where we contemplated watching the chain only when it is at $a$ or $b$. We didn’t say just what we meant by this, because it doesn’t conveniently fit into the usual formulation of Markov chain theory.
Probabilistic and electrical interpretation
===========================================
We may interpret $N_{abcd}$ probabilistically as the equilibrium concentration difference between $c$ and $d$ due to a unit flow of particles entering at $a$ and leaving at $b$. Here’s what this means. Introduce Markovian particles at $a$ at a unit rate, and remove them when they reach $b$. Write the ‘dynamic equilibrium’ measure of particles at $i$ as $w^i \phi_i$, so that $\phi_i$ tells the concentration of particles relative to the ‘static equilibrium’ measure $w^i$. Conservation of particles implies that $$w^i \phi_i \sum_j \tensor{P}{_i^j}
- \sum_j w^j \phi_j \tensor{P}{_j^i}
= {\tensor{\delta}}{_a^i}-{\tensor{\delta}}{_b^i}
.$$ We hasten to rewrite this in the conformally correct form $$\sum_j \phi_j {\Delta}^{ji} = {\tensor{\delta}}{_a^i}-{\tensor{\delta}}{_b^i}
.$$ Since also $$\sum_j (Z_{aj}-Z_{bj}) {\Delta}^{ji} = {\tensor{\delta}}{_a^i}-{\tensor{\delta}}{_b^i}$$ and since the Laplacian ${\Delta}$ kills only constants, if follows that $$\phi_j = Z_{aj}-Z_{bj} + C
,$$ and thus $$\phi_c - \phi_d = Z_{ac}-Z_{bc}-Z_{ad}+Z_{bd} = N_{abcd}
.$$
From this probabilistic interpretation of $N$ we can see that $N_{abab} = C_{ab}$, the commuting time between $a$ and $b$. Indeed, in the particle-painting scenario introduced earlier, $C_{ab}$ is the reciprocal of the rate at which red particles are turning green at $a$. Paying attention only to green particles, we see green particles appearing at $a$ at rate $1/C_{ab}$, and disappearing at $b$. The equilibrium concentration of green particles at $i$ is the probability $p_i$ of hitting $a$ before $b$ for the time-reversed chain, and in particular $p_a=1$ and $p_b=0$, so the concentration difference between $a$ and $b$ is $1$. Multiplying the green flow by $C_{ab}$ normalizes it to a unit flow with concentration difference $C_{ab}$ between $a$ and $b$. So $$C_{ab}=N_{abab}
.$$
If we embellish this probabilistic scenario by imagining that our particles carry a positive charge, we may identify the net flow of particles with electrical current; the concentration of particles (relative to the equilibrium measure) with electrical potential; and differences of concentration with voltage drop. With this terminology, $N_{abcd}$ tells the voltage drop between $c$ and $d$ due to a unit current from $a$ to $b$. Traditionally this way of talking is reserved for time-reversible Markov chains, which are precisely those for which we have the ‘reciprocity law’ $N_{abcd}=N_{cdab}$. For such chains, if we build a resistor network where nodes $i \neq j$ are joined by a resistor of conductance (i.e., reciprocal resistance) $-{\Delta}^{ij}$, then $N_{abcd}$ will indeed be the voltage drop between $c$ and $d$ due to a unit current from $a$ to $b$. We propose to extend this way of talking to non-time-reversible chains.
In electrical terms, the voltage drop $N_{abab}$ between $a$ and $b$ due to a unit current between $a$ and $b$ is the *effective resistance*. This is the same as the reciprocal of the current that flows when a $1$-volt battery is connected up between $a$ and $b$—which is what we get in effect when we measure commuting times using green and red paint. So the commuting time $C_{ab}=N_{abab}$ is the same as the effective resistance between $a$ and $b$.
The connection of commuting time to effective resistance, and the general recognition that commuting times play a key role in understanding Markov chains, is due to Chandra et al. [@crrst:commute].
[**Note.**]{} Now we are in a position to understand the significance of the name ‘cross-potential’. This name is meant to indicate the connection of $N_{abcd}$ to the cross-ratio of complex function theory. If we extend our notions about Markov chains to cover Brownian motion on the Riemann sphere, we get $$\begin{aligned}
N_{abcd}
&=&
-\frac{1}{2\pi} (\log |a-c| - \log |a-d| - \log |b-c| + \log |b-d|)
\\&=&
-\frac{1}{2\pi} \log \left| \frac{a-c}{a-d} \frac{b-d}{b-c} \right|)
\\&=&
-\frac{1}{2\pi} \Re \log \frac{a-c}{a-d} \frac{b-d}{b-c}
.\end{aligned}$$ We don’t have to specify a metric on the sphere here, because the Laplacian is a conformal invariant in two dimensions. Thinking of the sphere as being an electrical conductor with constant conductivity (say, 1 mho ‘per square’), the electrical interpretation becomes exact. The advantage of having $N$ to take four ‘arguments’ now becomes apparent, because $N_{abcb} = \infty$. That’s why engineers using look for cracks in nuclear reactor cooling pipes with a emph[4-point probe]{}. To get a sensible generalization of $C_{ab}$ we will need to do some kind of renormalization, which will introduce a dependence on the metric. We should not be sorry about this, because it brings curvature into the picture—and you know that can’t be bad.
Realization
===========
Now, finally, to realize commuting times as squared distances. From the bilinear form $B$ we get the quadratic form $$Q(u)=||u||^2=B(u,u)
=
\sum_{ij} u^i Z_{ij} u^j
.$$ $$C_{ab}=N_{abab}=Q({\tensor{\delta}}{_a^i}-{\tensor{\delta}}{_b}) = ||{\tensor{\delta}}{_a}-{\tensor{\delta}}{_b}||^2
.$$ So if we map $i$ to ${\tensor{\delta}}{_i}$ then the commuting time $C_{ab}$ becomes the squared distance between the images in the $Q$-norm.
That is, if what we’re calling the $Q$-norm is indeed a norm. Is $Q$ really positive definite?
To understand better what is going on here, it is useful to look at the bilinear form $$L(\phi,\psi)=
\sum_{ij} \phi_i {\Delta}^{ij} \psi_j
,$$ where we think of $\phi$ and $\psi$ as being defined only modulo additive constants. If we think of $\phi_i$ as the potential of the measure $$\sum_i \phi_i {\Delta}^{ik}
,$$ then this is the same bilinear form as before, except that now instead of measures of total mass $0$ it takes as its arguments the corresponding potentials, the first with respect to the original chain, and the second with respect to the time-reversed chain: $$L(\phi,\psi)
=
B(\sum_i \phi_i {\Delta}^{ik},
\sum_i \psi_i {\Delta}^{ki}
)
=
B(\sum_i \phi_i {\Delta}^{ik},
\sum_i \hat{{\Delta}}^{ik} \psi_i
)
.$$ This follows from the formula ${\Delta}Z {\Delta}= {\Delta}$ above.
Now to get the equivalent of $Q$ in this context we restrict to the subspace $$V=\{
(\phi,\psi): \sum_i \phi_i {\Delta}^{ik} = \sum_j {\Delta}^{kj} \psi_j
\}$$ and take as our quadratic form $$R((\phi,\psi))=L(\phi,\psi)
.$$
In the case of a time-reversible chain, $V$ is just the diagonal $\phi = \psi$, and $$Q(\phi {\Delta})= R((\phi,\phi)) = L(\phi,\phi) =
\sum_{ij} \phi_i {\Delta}^{ij} \phi_j
= {\frac{1}{2}}\sum_{ij} (-{\Delta}^{ij}) (\phi_i-\phi_j)^2
.$$ This is evidently positive-definite. Indeed, if we associate to $(\phi,\phi)$ the vector with $n \choose 2$ coordinates $\sqrt{-{\Delta}^{ij}}(\phi_i-\phi_j)$, $i<j$, then we will have embedded the normed space $(V,R)$, and along with it our Markov chain, in Euclidean $n \choose 2$-space.
Electrically, what we have done here is to account for the energy being dissipated in the network by adding up the energy dissipated by individual resistors. And there should be some kind of probabilistic interpretation as well.
That’s how it works for time-reversible chains, for which ${\Delta}^{ij}={\Delta}^{ji}$. However, the argument extends to the general case by what amounts to a trick. The key is the observation that for $(\phi,\psi) \in V$ we have $$L(\phi,\psi)=L(\phi,\phi)=L(\psi,\psi)
.$$ (But please note that in general $L(\phi,\psi) \neq L(\psi,\phi)$!) So $$Q(\phi {\Delta})= R((\phi,\psi)) =
L(\phi,\psi)=
L(\phi,\phi)=
\sum_{ij} \phi_i {\Delta}^{ij} \phi_j
= {\frac{1}{2}}\sum_{ij} (-{\Delta}^{ij}) (\phi_i-\phi_j)^2
.$$ So there is the positive-definiteness we need.
Now, though, we don’t see any natural way to interpret the terms of the sum electrically or probabilistically. (Which is not to say that there isn’t one!) In putting $\phi$ in both slots of $L$ we leave the subspace $V$, and thereby commit what appears to be an unnatural act. But it seems to have paid off.
Minimax characterization of commuting times and hitting probabilities
=====================================================================
Fix states $a \neq b$, and let $$S_{a,b}= \{ \phi | \phi_a =1, \phi_b=0 \}$$ Here we really should be thinking of $\phi$ as being defined only up to an additive constant, which means we should write $\phi_a-\phi_b=1$, but we’re going to be sloppy about this, because we want to focus attention on two distinguished elements of $S_{a,b}$ which are naturally $1$ and $a$ and $0$ at $b$. These are $${\bar{\phi}}_i =
{\mathrm{Prob}}(\mbox{hit $a$ before $b$ starting at $i$ going backward in time})$$ and $${\bar{\psi}}_i =
{\mathrm{Prob}}(\mbox{hit $a$ before $b$ starting at $i$ going forward in time})
.$$
We’ve met ${\bar{\phi}}$ before: It’s proportional to the equilibrium concentration of green particles in our painting scenario. ${\bar{\psi}}$ is the analogous quantity for the reversed chain. The pair $({\bar{\phi}},{\bar{\psi}})$ belongs to our subset $V$, because $$({\bar{\phi}}{\Delta})^i = ({\Delta}{\bar{\psi}})^i =
r_{ab}({\tensor{\delta}}{_a^i}-{\tensor{\delta}}{_b^i})
.$$ Here we once again are writing $r_{ab} = \frac{1}{T_{ab}}$ for the equilibrium rate of commuting between $a$ and $b$. Observe that any $f$ we have $$L({\bar{\phi}},f) = L(f,{\bar{\psi}}) = r_{ab} (f_a-f_b)
.$$ So whenever $f$ is in $S_{a,b}$ we have $$L({\bar{\phi}},f) = L(f,{\bar{\psi}}) = r_{ab}
,$$ and in particular $$L({\bar{\phi}},{\bar{\psi}})=r_{ab}
.$$
[**Theorem.**]{} $$r_{ab} = \frac{1}{T_{ab}}
\min_{\alpha} \max_{\phi+\psi=2\alpha} L(\phi,\psi)
.$$ Here and below, $\alpha$, $\phi$, and $\psi$ are restricted to lie in $S_{a,b}$, i.e. to take value $1$ at $a$ and $0$ at $b$.
[**Proof.**]{} Whatever $\alpha$ is, we may take $\phi={\bar{\phi}}$ (and thus $\psi=2\alpha-{\bar{\phi}}$), and have $$L(\phi,\psi) = L({\bar{\phi}}, \psi) = r_{ab}$$ as above. So $$\min_{\alpha} \max_{\phi+\psi=2\alpha} L(\phi,\psi)
\geq r_{ab}
.$$
To prove the inequality in the other direction, and in the process identify where the minimax is achieved, take $$\alpha = {\frac{1}{2}}({\bar{\phi}}+{\bar{\psi}})
.$$ If $\phi+\psi=2\alpha$ then we can write $$\phi={\bar{\phi}}+f$$ and $$\psi={\bar{\psi}}-f
,$$ where $f_a=f_b=0$.
Now $$L({\bar{\phi}},f)=L(f,{\bar{\psi}})=r_{ab}(f_a-f_b)=0
,$$ so $$L(\phi,\psi)
=
L({\bar{\phi}}+f,{\bar{\psi}}-f)
=
L({\bar{\phi}},{\bar{\psi}})-L(f,f)
= r_{ab}-L(f,f)
.$$ And even though we claim it is a travesty to put the same $f$ into both slots of $L$, we still have $$L(f,f) \geq 0
:$$ That was the upshot of our embedding investigation. So $$L(\phi,\psi) \leq r_{ab}
,$$ still assuming $\alpha={\frac{1}{2}}({\bar{\phi}}+{\bar{\psi}})$ and $\phi+\psi=2\alpha$. Hence $$\min_{\alpha} \max_{\phi+\psi=2\alpha} L(\phi,\psi)
\geq r_{ab}
. \quad {\rule{2mm}{2.5mm}}$$
In the time-reversible case, where ${\Delta}^{ij}={\Delta}^{ji}$, this minimax can be reduced to a straight minimum. That’s because in this case for any $g,f$ we have $L(f,g)=L(g,f)$, and hence $$L(g+f,g-f)
=
L(g,g)-L(f,f)
.$$ So to maximize $L(\phi,\psi)$ while fixing the sum $\phi+\psi=2\alpha$ we take $\phi=\psi=\alpha$.
[**Corollary.**]{} When ${\Delta}^{ij}$ is symmetric $$r_{ab} =
\min_{\phi(a)=1,\phi(b)=0} L(\phi,\phi)
.
\quad {\rule{2mm}{2.5mm}}$$
This minimum principle for resistances was known already to 19th century physicists, specifically Thomson (a.k.a. Kelvin), Maxwell, and Rayleigh: For more about this, see Doyle and Snell [@doylesnell:walks].
Having a straight minimum is a lot better than having a minimax, because now we can plug in any $\phi$ with $\phi(a)=1,\phi(b)=0$ and get an upper bound for $r_{ab}$, corresponding to a lower bound for $T_{ab}$. This method is a staple of electrical theory—the part of electrical theory that doesn’t extend to non-time-reversible chains because it depends on the relation $L(f,g)=L(g,f)$.
For time-reversible chains there are also complementary methods for finding lower bounds for $r_{ab}$, and thus upper bounds for $T_{ab}$. These emerge from the minimum principle through the mystery of convex duality. In practice, though, it is generally conceptually simpler to work instead with the monotonicity law described in the next section. This monotonicity law extends to all chains, but sadly, for all we can tell thus far, its usefulness appears to get left behind.
Monotonicity
============
From the minimax characterization of commuting times we immediately get the following:
[**Monotonicity Law**]{} Commuting times decrease monotonically when equilibrium interstate transition increase: Using barred and unbarred quantities to refer to two different Markov chains, if ${\Delta}^{ij} \leq \bar{{\Delta}}^{ij}$ for all $i \neq j$ then $\bar{T_{ij}} \leq T_{ij}$ for all $i,j$. $\quad {\rule{2mm}{2.5mm}}$
Actually it would be better to think of ${\Delta}$ and $\bar{{\Delta}}$ here as referring to conformal classes of chains, rather than individual chains, because as we know ${\Delta}^{ij}$ and $T_{ij}$ are conformal invariants.
This law holds for all chains, time-reversible or not. As we said above, for time-reversible chains this law can be used to get upper and lower bounds for commuting times, and hence for hitting probabilities: This is discussed in great detail by Doyle and Snell [@doylesnell:walks].
Sadly, even though the law extends to the non-time-reversible case, its usefulness does not extend, at least not in any obvious way. How can this be? There seem to be a number of reasons.
First, for time-reversible chains, if we block transitions back and forth between states $c,d$, requiring the particle to remain where it is when it attempts to make such a transition, we get a new $\bar{{\Delta}}$ dominated by the original ${\Delta}$ in the sense that $\bar{{\Delta}}^{ij} \leq {\Delta}^{ij}$ for $i \neq j$. Electrically speaking, blocking transitions between $c$ and $d$ amounts to cutting the wire between them. In the non-time-reversible case, this will change the equilibrium measure $w^i$ and thereby destroy the relation $\bar{{\Delta}}^{ij} \leq {\Delta}^{ij}$ that we need for monotonicity.
Second, for time-reversible chains, it is simple and natural to introduce intermediate states. Electrically speaking, introducing a state between $c$ and $d$ amounts to dividing the ‘wire’ connecting $c$ and $d$ into two pieces, if only in our mind’s eye. By combining this with the putting or taking of wires, we can produce chains to bound $T_{ab}$ above or below as closely as we please. And we can do this in such a way that our approximating chains are easy to analyze. Here lies the third apparent shortcoming of the non-time-reversible case: A seeming paucity of chains whose commuting times are easy to compute.
So, of what use is this monotonicity law in the non-time-reversible case? That remains to be seem.
The obstruction to time-reversibility
=====================================
Let $M_{ij}$ be the expected time to reach $j$ starting from $i$. Coppersmith, Tetali, and Winkler showed that a Markov chain is time-reversible just if for all $a,b,c$ $$M_{ab}+M_{bc}+M_{ca} =
M_{ac}+M_{cb}+M_{ba}
.$$ And in this case the expected time to traverse a cycle of any length will be the same in either direction. Note that the $M_{ij}$s themselves are not conformally invariant, these cycle sums are. For a cycle of length $2$, the cycle sum is our best friend the commuting time.
We always have $$M_{ab}+M_{bc}+M_{ca} =
\hat{M}_{ac}+\hat{M}_{cb}+\hat{M}_{ba}$$ (look at a long record of the chain backwards), so an equivalent condition is that for all $a,b,c$ $$M_{ab}+M_{bc}+M_{ca} =
\hat{M}_{ab}+\hat{M}_{bc}+\hat{M}_{ca}
.$$ This is true despite the fact that in general $$\hat{M}{ab} \neq M_{ba}
.$$
So, why is this true? It comes down to the fact that a conformal class of chains is reversible just if our bilinear for $L(\phi,\psi)$ on $V=\{x^i | \sum_i x_i = 0 \}$ is symmetric. To any bilinear form $\sum_{ij} u^i Z_{ij} v^j$ on $V$ their corresponds a natural cohomology class $$Z_{ij} - Z_{ji}
,$$ which is to say, an antisymmetric matrix defined up to addition of a matrix of the form $B_{ij}=a_i-a_j$. This class represents the obstruction to symmetrizing the matrix of the form within its $ab$-equivalence class. This class vanishes just if it integrates to $0$ around any cycle, and cycles of length $3$ span the space of cycles. Indeed, they span it in a very redundant way. To verify reversibility, it would suffice to check any basis for the space of cycles, e.g. only cycles of length $3$ involving the fixed state $n$ (the ‘ground’).
More to be said
===============
The next step would be discuss how to use the knee-jerk mapping to make a chain time-reversible without changing its commuting times. The knee-jerk method will produce the desired time-reversible chain whenever such a thing exists, but we still don’t know if this is always the case. What we do know is that if it turns out that no suitable time-reversible chain exists, the knee-jerk method will delivers a time-reversible chain whose commuting times agree as well as possible with those of the original chain. (See Coppersmith et al. [@cdrs:cat], Doyle [@doyle:kneejerk] .)
Then we should discuss uniformization of Markov chains, whereby we prescribe a canonical representative chain within each conformal class (or in other words, we prescribe a canonical $w$ to accompany a given $N$). This canonical chain extremizes the Kemeny constant $K$, which is the expected time to hit a point chosen according to the equilibrium distribution. (As Kemeny observed, $K$ doesn’t depend on where you start.) The extremal chain is characterized by constancy of the expected time $K_i$ to hit $i$ starting from equilibrium (the so-called ‘preKemeny non-constant’). It’s easy to write down the transition probabilities for this extremal chain. But, are they necessarily positive?
Beyond this lies the extension of this whole business to diffusion on surfaces, where we must renormalize hitting times because Brownian motion in dimension $2$ never hits a given point. (Cf. Doyle and Steiner [@doyleSteiner:hideandseek].) Now to uniformize we extremize not Kemeny’s constant, but a variant with a correction term involving the Gaussian curvature. Again, it is easy to write down the extremizing metric, or rather the extremizing area measure, which is not a priori positive everywhere. For spheres, all the round metrics tie for the extremum. For tori, the flat metrics win. For higher genus surfaces, the winners are not hyperbolic surfaces, nor should they be, because having constant curvature is a local condition that doesn’t know thick from thin. The canonical measure is sensitive to thickness in a conformally correct way. But is it a positive measure? If it isn’t, could it still be good for something?
[^1]: The authors hereby waive all copyright and related or neighboring rights to this work, and dedicate it to the public domain. This applies worldwide.
|
---
abstract: 'In the quest for applicable quantum information technology miniaturised, compact and scalable sources are of paramount importance. Here, we present the concept for the generation of 2-photon N00N states without further post-processing in a single non-linear optical element. Based upon a periodically poled waveguide coupler, we present the principle of state generation via type-0 parametric down-conversion inside this type of devices. With the eigenmode description of the linear optical element, we utilise the delocalised photon pair generation to generate a N00N state in the measurement basis. We show, that we are able to eliminate the need for narrow-band spectral filtering, as well as for phase-stabilisation of the pump light, making this approach an elegant way to produce 2-photon N00N states.'
author:
- 'Regina Kruse,$^1$ Linda Sansoni,$^1$ Sebastian Brauner,$^1$ Raimund Ricken,$^1$ Craig S. Hamilton,$^2$ Igor Jex,$^2$ and Christine Silberhorn'
bibliography:
- '2-wg-arxiv.bib'
title: 'N00N states from a single non-linear directional coupler'
---
Introduction
============
During the last decades integrated optics has become a working horse for the photonic industry. Waveguide based lasers, combined with passive beam splitters as well as active modulators have enabled a lot of progress in the optics research field. In a more recent development, the integrated quantum optics community has strived to profit from the achievements in classical integrated optics for the miniaturisation of quantum circuits [@tanzilli_highly_2001; @politi_silica--silicon_2008; @peruzzo_quantum_2010; @owens_two-photon_2011; @crespi_integrated_2011; @krapick_efficient_2013; @corrielli_rotated_2014; @metcalf_quantum_2014], e.g. for compact quantum communication or quantum computation devices. However, it still remains a challenge to combine the generation of single photons with linear circuits on one chip. Recently integrated devices have been fabricated, combining the state generation with linear manipulation on a chip [@silverstone_-chip_2014; @jin_-chip_2014]. This approach eliminates the incoupling losses of single photons to the linear networks, which is a prerequisite for the combination of multiple sources, e.g. for boson sampling [@crespi_integrated_2013; @spring_boson_2013; @broome_photonic_2013; @tillmann_experimental_2013]. Still, the reliable preparation of sophisticated quantum states from multichannel devices requires generally a well defined and stabilised phase for the pump light. Furthermore, achieving indistinguishability of photons generated by different sources remains difficult and is usually realised by narrow-band filtering, introducing high loss in the experiment. The integration of state generation into the linear element additionally offers many new possibilities [@mista_nonclassical_1997; @solntsev_spontaneous_2012; @kruse_spatio-spectral_2013; @solntsev_generation_2014]. Firstly, it pushes the miniaturisation even further and secondly, the more important aspect, we gain access to new dynamics [@hamilton_driven_2014], which are not available in conventional single photon sources. Recently, Lugani et al. [@lugani_generation_2011] proposed a scheme for the integrated generation of 2-photon N00N states. However, due to its use of propagation constant matching, it requires careful parameter design and has low tolerance for fabrication imperfections.
In this paper, we integrate photon-pair generation into a directional coupler and generate genuine 2-photon N00N states [@dowling_quantum_2008; @afek_high-noon_2010; @israel_experimental_2012; @boyd_quantum_2012] without postprocessing. We eliminate the need for narrow-band spectral filtering to prepare indistinguishable photons and are able to fully forego phase-stabilised pumping of the process. By using the eigenmode description of the linear waveguide coupler, show that the photon pairs are generated into linear superpositions of the waveguide modes and that with a suitable choice of the pump frequency we are able to obtain a genuine N00N state at the output. First, we give a detailed description of the integrated system and discuss the linear properties of this device. Then, we analytically describe the concept of the state generation in this source type and investigate numerically the expected fidelity for realistic fabrication parameters.
Analytical Description
======================
The system which we are considering is sketched in figure \[fig:sys\_sketch\].
![Our source consists of a waveguide coupler, where the coupling region of length $L$ is periodically poled. We couple ultrafast pump light into one waveguide of the fan-in region, where it propagates along the $z$-axis. After the periodically poled area, we use the fan-out region to separate the two output ports for our spatially entangled 2-photon N00N state $\ket {\Psi_\mathrm{out}}$.[]{data-label="fig:sys_sketch"}](system_sketch.pdf){width=".4\textwidth"}
The underlying waveguide structure is that of a directional coupler. It consists of a fan-in region for easier access to the waveguide inputs and a coupling region in the middle, where the two waveguides run parallely at a short distance of a few $\mu$m. The strength of the coupling, described by the coupling parameter $C$ is directly given by the distance between the two waveguides and the operating wavelength $\lambda$ of the directional coupler. The coupling region is then followed by a fan-out region to separate the two waveguide outputs again. We define that the fields propagate along the $z$-axis, while coupling happens in $x$-direction.
So far this optical element is purely linear in nature. However, we add a periodic poling to the coupling part of length $L$ in the directional coupler. This allows for parametric down-conversion (PDC) [@louisell_quantum_1961; @burnham_observation_1970] to take place for a specific parameter combination. We assume, that only the generated quantum fields in the telecom regime are affected by the directional coupler, while the pump stays unmodified in the pumped waveguide channel.
Linear Analysis
---------------
First, we need to review the linear properties of a directional coupler, as they are the key to understanding the unique properties of the non-linear process in this structure.
From integrated optics it is long known, how to describe the linear properties of a directional coupler [@somekh_channel_1973; @marom_relation_1984]. Here, we use the coupled-mode approximation, where the description of the coupled system is given via a linear combination of the modes in the uncoupled system. Solving the linear differential equation of the electric fields in the coupled waveguide system yields two eigenmodes with two non-degenerate eigenvalues $$\begin{aligned}
i\beta_{A} &= i[\beta^{(0)}+C]\\
i\beta_{S} &= i[\beta^{(0)}-C]\, ,
\end{aligned}
\label{eq:eigenvalues}$$ where $\beta^{(0)}$ is the propagation constant of the uncoupled waveguide. We choose the labels $A$ and $S$ based on the shape (antisymmetric, symmetric) of the eigenmodes in the coupler structure. They are in this approximation given by a linear combination of the spatial modes of the electric field $E$ in the uncoupled waveguides 1 and 2 $$\begin{aligned}
E_A &= \frac{1}{\sqrt{2}} (E_1-E_2)\\
E_S &= \frac{1}{\sqrt{2}} (E_1+E_2)\, .
\end{aligned}
\label{eq:eigenmodes}$$ This result is not only a solution to the coupled differential equation for the coupler system, but it also gives instruction on how to switch between the waveguide basis, which is our measurement basis, and the eigenmode basis of the directional coupler.
Non-Linear Analysis
-------------------
Using the eigenmode description we calculate the generated PDC state in a directional coupler following the approach of [@christ_spatial_2009; @solntsev_spontaneous_2012; @kruse_spatio-spectral_2013]. To compute the generated bi-photon state in the directional coupler, we express the electric fields in the PDC-Hamiltonian $${\hat{H}}_{PDC} = \chi^{(2)}\int_V d^3r (\mathcal{E}_p^{(+)} {\hat{E}}^{(-)}{\hat{E}}^{(-)} + h.c.)$$ with the eigenmodes of the coupler, as they diagonalise the linear part of the full system Hamiltonian. Here, $\chi^{(2)}$ denotes the strength of the second order non-linear coefficient, $\mathcal{E}_p$ the classical pump field and ${\hat{E}}$ the generated quantum fields. As we will only consider type-I PDC, where the photons are fundamentally indistinguishable [@loudon_quantum_2000], the two generated quantum fields can be described by the same operator.
![The different eigenmode combinations for the photon-pair generation yield different phase-matching conditions, as their propagation constants are modified uniquely. The process generating two photons into the symmetric mode is modified by $-2C$, while two photons into the antisymmetric mode is modified by $+2C$. In the case that one photon is generated into the symmetric and antisymmetric mode (as highlighted by the yellow shading), the modifications cancel each other. $\omega_0$ denotes the degenerate phase-matching condition for signal and idler fields in an uncoupled waveguide.[]{data-label="fig:phasematching"}](phasematching.pdf){width=".3\textwidth"}
After a lengthy, but straightforward calculation, we arrive at the final PDC state in the eigenmode basis of the waveguide coupler $$\begin{aligned}
\ket{\Psi}^{\mathrm{Eig.}} &= \frac{1}{\mathcal{N}} \int d\omega_s \int d\omega_i \alpha(\omega_s+\omega_i)\\
&\left[\gamma\, {\mathrm{sinc}}\left(\Delta\beta_{S,S}\frac{L}{2}\right) e^{-i\Delta\beta_{S,S}\frac{L}{2}} {\hat{a}}_S^\dagger(\omega_s){\hat{a}}^\dagger_S(\omega_i)\right.\\
&+\delta\, {\mathrm{sinc}}\left(\Delta\beta_{S,A}\frac{L}{2}\right) e^{-i\Delta\beta_{S,A}\frac{L}{2}} {\hat{a}}_S^\dagger(\omega_s){\hat{a}}_A^\dagger(\omega_i)\\
&+ \delta\, {\mathrm{sinc}}\left(\Delta\beta_{A,S}\frac{L}{2}\right) e^{-i\Delta\beta_{A,S}\frac{L}{2}} {\hat{a}}_A^\dagger(\omega_s){\hat{a}}_S^\dagger(\omega_i)\\
&+\left. \gamma\, {\mathrm{sinc}}\left(\Delta\beta_{A,A}\frac{L}{2}\right) e^{-i\Delta\beta_{A,A}\frac{L}{2}} {\hat{a}}_A^\dagger(\omega_s){\hat{a}}_A^\dagger(\omega_i) \right] \ket{0}\, ,
\end{aligned}$$ where $\gamma$ and $\delta$ are the pump excitation amplitudes for the symmetric and the antisymmetric eigenmodes, $\alpha(\omega_s+\omega_i)$ is the spectral pump shape depending on the signal and idler photon frequencies $\omega_s$ and $\omega_i$, $\Delta\beta_{ij}$ is the phase-mismatch for the eigenmode combination $(i,j)$, ${\hat{a}}_k^\dagger(\omega)$ is the creation operator for a photon of frequency $\omega$ in mode $k$ and $\mathcal{N}$ the normalisation constant. In the coupled system, the PDC process generates the photons into superpositions of the eigenmodes [@kruse_spatio-spectral_2013]. As the coupler only features two eigenmodes, there are four combinations generate photons in different eigenmodes. It is only possible to generate either two photons into the symmetric, two photons into the antisymmetric or one photon in each of the two eigenmodes. So far, this is simple combination. However, remembering the linear description of the directional coupler, we find, that these different combinations of eigenmodes correspond to different phase-matching conditions, as the propagation constant is modified uniquely for different eigenmodes. This enables us to specifically excite spatial properties of the generated photon pairs via spectral selection of the corresponding phase-matching condition. In the following, we specifically select the yellow shaded phase-matching condition (corresponding to a selection of the pump wavelength in this region) of figure \[fig:phasematching\], which belongs to the generation of 1 photon in each eigenmode.
While the PDC state in the eigenmode basis of the directional coupler is a full description of the system, we still need to translate the properties of this state into the measurement basis of the laboratory. For this purpose, we choose the waveguide basis and switch back from the eigenmodes using equation . This yields for the yellow shaded phase-matching condition (i.e. only $\Delta \beta_{A,S}\: \&\: \Delta\beta_{S,A}$ contribute to the final state)
$$\begin{aligned}
\ket{\Psi}^{WG}&=\frac{\delta}{2\sqrt{\mathcal{N}}}\int d\omega_s d\omega_i \alpha(\omega_s+\omega_i)\\
&\left[ \left\{{\mathrm{sinc}}\left(\Delta\beta_{S,A}\frac{L}{2}\right) e^{-i\Delta\beta_{S,A}\frac{L}{2}}+ {\mathrm{sinc}}\left(\Delta\beta_{A,S}\frac{L}{2}\right) e^{-i\Delta\beta_{A,S}\frac{L}{2}}
\right\}{\hat{a}}^\dagger_1(\omega_s) {\hat{a}}^\dagger_1(\omega_i)\right. \\
&- \underbrace{\left\{ {\mathrm{sinc}}\left(\Delta\beta_{S,A}\frac{L}{2}\right) e^{-i\Delta\beta_{S,A}\frac{L}{2}}- {\mathrm{sinc}}\left(\Delta\beta_{A,S}\frac{L}{2}\right) e^{-i\Delta\beta_{A,S}\frac{L}{2}}
\right\}}_{=0}{\hat{a}}^\dagger_1(\omega_s) {\hat{a}}^\dagger_2(\omega_i) \\
&+\underbrace{\left\{ {\mathrm{sinc}}\left(\Delta\beta_{S,A}\frac{L}{2}\right) e^{-i\Delta\beta_{S,A}\frac{L}{2}}- {\mathrm{sinc}}\left(\Delta\beta_{A,S}\frac{L}{2}\right\} e^{-i\Delta\beta_{A,S}\frac{L}{2}}
\right)}_{=0}{\hat{a}}^\dagger_2(\omega_s) {\hat{a}}^\dagger_1(\omega_i) \\
&-\left\{\left.{\mathrm{sinc}}\left(\Delta\beta_{S,A}\frac{L}{2}\right) e^{-i\Delta\beta_{S,A}\frac{L}{2}}+ {\mathrm{sinc}}\left(\Delta\beta_{A,S}\frac{L}{2}\right) e^{-i\Delta\beta_{A,S}\frac{L}{2}}\right\}{\hat{a}}^\dagger_2(\omega_s) {\hat{a}}^\dagger_2(\omega_i) \right] \ket{0} \, ,
\end{aligned}$$
where ${\hat{a}}^\dagger_k(\omega)$ creates a photon of frequency $\omega$ in waveguide $k$. The key to the N00N state generation is embedded in the two middle terms of this state. We have already stated, that it is possible to simultaneously generate one photon (signal) in the symmetric and the other (idler) in the antisymmetric eigenmode. However, the interchanged combination (idler in symmetric, signal in antisymmetric) is also possible, but with a phase-flip. As these two possibilities are indistinguishable (type-I PDC), the two terms cancel out during the basis transformation. This is the main reason for the post-processing free generation of 2-photon N00N states in this device. To give more physical understanding to this state, we explicitly illustrate the generation principle in the following.
Figure \[fig:generation\_protocol\] shows the schematic generation process of photon pairs in the non-linear coupler.
![ At the beginning of the periodically poled coupler structure, the eigenmodes are unoccupied. During the PDC process, we generate photons with a certain probability only in the combination of symmetric and antisymmetric eigenmode. During the basis transformation a Hong-Ou-Mandel type interference takes place and cancels the coincidences between the two waveguide outputs, leading to the creation of post-processing free 2-photon N00N states.[]{data-label="fig:generation_protocol"}](generation-protocol.pdf){width=".45\textwidth"}
At the beginning of our waveguide based source, we find the unpopulated eigenmodes of the system, as we are not using a seed to stimulate the process. During the PDC process, we find, that we are populating the eigenmodes with one photon each from a generated pair. The probability of this generation is given via the strength of the non-linear interaction. After the poled region, we have to perform the basis transformation to the measurement basis in our lab. Incidentally, this basis transformation is mathematically fully equivalent to a 50/50 beam splitter with the eigenmodes as inputs and the waveguide modes as outputs. Note, however, that there is no *physical* beam splitter implemented on the chip. Here, only the basis transformation to the measurement basis gives rise to the beam splitter transformation working on our quantum state. Seeing, that we put one photon into each port of our basis beam splitter, we receive Hong-Ou-Mandel type interference [@hong_measurement_1987], cancelling the coincidence contributions between the two waveguide outputs. This results in the post-processing free 2-photon N00N state.
Numerical Results
=================
In this section we use the analytically calculated output state for the non-linear coupler PDC to calculate the expected fidelity for the 2-photon N00N state, generated by our device.
For a perfect N00N state, we would expect 2-photon events (coincidences) in either the pumped waveguide (WG 1) or the unpumped one (WG 2) with perfect suppression of coincidences between the waveguides. However, we need a careful selection of the pump wavelength to achieve this state, as shown in figure \[fig:coincidences\]. We tune the pump wavelength to scan the different phase-matching conditions and plot the coincidence rates for the different measurement combinations; coincidences in the pumped (unpumped) waveguide in green (blue) or between waveguides (red). The eigenmode combinations (S, S) and (A, A) show a clear peak for the coincidences between waveguides, while coincidences in a single waveguide show a less pronounced coincidence peak. Using the beam splitter interpretation of the basis transformation this behaviour can be explained easily. For these two eigenmode combinations, we are inserting two photons on the same side of our basis beam splitter and therefore generate coincidences between waveguides with a 50$\,$% probability. In the middle of the phase-matching roughly at $758\,$nm, exciting the (SA, AS) phase-matching condition, we observe a suppression of coincidence counts between waveguides, a clear sign for the generation of 2-photon N00N states. To characterise the fidelity of the generated state, we use the coincidence probabilities for the above mentioned measurement combinations $$\mathcal{F}=\frac{p_{\mathrm{Coinc,WG 1}}+p_{\mathrm{Coinc,WG 2}}-p_{\mathrm{Coinc,WG 1\&2}}}{p_{\mathrm{Coinc,WG 1}}+p_{\mathrm{Coinc,WG 2}}+p_{\mathrm{Coinc,WG 1\&2}}}\, .$$ For the used parameter combination we achieve a fidelity of $\mathcal{F}\approx 93\, \%$. It is restricted by the sinc-sidepeak contributions from the other phase-matching conditions. However, by careful parameter design we can eliminate this restriction on the state fidelity, by e.g. choosing a longer coupler stem length leading to a narrowing the peaks or a higher coupling parameter leading to a larger spectral separation.
![The phase-matching scan by tuning the pump wavelength shows a clear suppression of coincidences between the two waveguide ports (red) in the middle of the phase-matching, while coincidences in the single waveguides peak (green: pumped waveguide, blue: unpumped waveguide).[]{data-label="fig:coincidences"}](theory.pdf){width=".45\textwidth"}
Conclusion
==========
In this paper, we have introduced a single device producing post-processing free 2-photon N00N states. We have discussed the system in detail and have described the generation process of the PDC state. In future, an experimental implementation of this device is planned, showing high state fidelity, as well as the 2-photon N00N state double fringe frequency.
R. K., L. S. and C. S. acknowledge financial support from DFG TRR 142. C. S. H. and I. J. received financial support from Grants No. RVO 68407700 and No. GACR 13-33906 S.
Correspondence should be addressed to:\
regina.kruse@upb.de
|
---
abstract: 'We investigate the impact of dust-induced gas fragmentation on the formation of the first low-mass, metal-poor stars ($<1M_\odot$) in the early universe. Previous work has shown the existence of a critical dust-to-gas ratio, below which dust thermal cooling cannot cause gas fragmentation. Assuming the first dust is silicon-based, we compute critical dust-to-gas ratios and associated critical silicon abundances ($\mbox{[Si/H]}_{\text{crit}}$). At the density and temperature associated with protostellar disks, we find that a standard Milky Way grain size distribution gives $\mbox{[Si/H]}_{\text{crit}} = -4.5 \pm 0.1$, while smaller grain sizes created in a supernova reverse shock give $\mbox{[Si/H]}_{\text{crit}} = -5.3 \pm 0.1$. Other environments are not dense enough to be influenced by dust cooling. We test the silicate dust cooling theory by comparing to silicon abundances observed in the most iron-poor stars ($\mbox{[Fe/H]}<-4.0$). Several stars have silicon abundances low enough to rule out dust-induced gas fragmentation with a standard grain size distribution. Moreover, two of these stars have such low silicon abundances that even dust with a shocked grain size distribution cannot explain their formation. Adding small amounts of carbon dust does not significantly change these conclusions. Additionally, we find that these stars exhibit either high carbon with low silicon abundances or the reverse. A silicate dust scenario thus suggests that the earliest low-mass star formation in the most metal-poor regime may have proceeded through two distinct cooling pathways: fine structure line cooling and dust cooling. This naturally explains both the carbon-rich and carbon-normal stars at extremely low \[Fe/H\].'
author:
- 'Alexander P. Ji'
- Anna Frebel
- Volker Bromm
title: 'The Chemical Imprint of Silicate Dust on the Most Metal-Poor Stars'
---
Introduction {#section:introduction}
============
The formation of the first stars marks the beginnings of structure formation, cosmic reionization, and chemical enrichment (e.g., @Bromm09 and references within). These so-called Population III stars formed out of metal-free primordial gas at the centers of dark matter minihalos [@Couchman86; @Haiman96; @Tegmark97; @Yoshida03]. Due to relatively weak feedback and inefficient cooling, they had high characteristic masses of order at least tens of solar masses and therefore short life spans (e.g., @Abel02 [@Bromm02; @Stacy10; @Stacy12; @Greif11; @Hosokawa11]).
Although the short lives of Population III stars implies that they cannot be directly observed anymore, it is believed that the metals released in their supernovae trigger a transition from predominantly high mass star formation to a low mass mode [@Bromm01; @Schneider02]. The chemical abundances of low-mass, metal-poor Population II stars in the Milky Way stellar halo have been interpreted as traces of the Population III star era (e.g., @Beers05, @Frebel13a). If this is indeed the case, then an understanding of the formation process for Population II stars is one way to probe the epoch of the first stars [@Tumlinson06; @Karlsson13].
However, unlike the formation of Population III stars, whose gas properties and formation environments are relatively well-understood, the conditions for Population II star formation are quite uncertain (e.g., @Bromm13). Introducing even trace amounts of metals significantly affects the thermal behavior of collapsing gas clouds (e.g., @Omukai05). There are also many possible candidate environments that might be the formation sites of these stars, ranging from the atomic cooling halos of the first protogalaxies (e.g., @Wise07 [@Greif08; @Greif10]) to post-supernova shock regions (e.g., @Salvaterra04 [@Chiaki13b]). The two main theories for how metals cause low mass star formation are gas cooling through atomic fine structure lines [@Bromm03; @Santoro06] and gas fragmentation induced by dust continuum radiation (e.g., @Schneider06 [@Omukai10]). We will refer to these as “fine structure cooling” and “dust cooling”, respectively.
Fine structure cooling argues that in the absence of sufficient atomic metal line cooling, gas clouds cannot quickly collapse beyond a “loitering state” of $n \sim 10^4\,$cm$^{-3}$ and $T \sim 200\,$K [@Bromm02]. The presence of molecular hydrogen may smooth out this metallicity threshold (e.g., @Jappsen09a [@Jappsen09b]), but only if there is no soft UV Lyman-Werner (LW) background produced by the first stars, capable of destroying molecular hydrogen [@Bromm01; @SafShrad10]. Arguably, the presence of such a LW background is natural, as the same stars that produced the first heavy elements would also emit LW radiation; thus, the fine-structure threshold is clearly imprinted, without H$_2$ cooling smoothing it out. If the gas metallicity is above a critical metallicity of $Z/Z_\odot
\sim 10^{-3.5}$, the gas is unstable to vigorous fragmentation (e.g., @Santoro06 [@Smith09]). The most important atomic species are carbon and oxygen [@Bromm03], so the theory predicts enhancements in these elements. If correct, this is a natural explanation for the measured carbon enhancement in many metal-poor stars [@Frebel07]. However, the Jeans mass of gas fragments formed by just fine structure cooling is $\geq 10M_\odot$, which is too massive for a star formed early in the universe to survive until the present day [@Klessen12].
In contrast, dust cooling easily causes gas fragmentation at Jeans masses of $\sim 0.1-1\, M_\odot$ because it becomes efficient only at high gas densities and temperatures around $10^{12}$ cm$^{-3}$ and $1000\,$K. The critical metallicity required for dust cooling to cause fragmentation is also much lower at $Z/Z_\odot \sim 10^{-5}$ [@Omukai05; @Tsuribe06; @Schneider06; @Clark08; @Omukai10; @Schneider12; @Dopcke13]. This dust must have been formed in early supernovae [@Gall11]. Many dust models have been produced which turn supernova yields into dust masses (e.g., @Todini01 [@Nozawa03; @Schneider04; @Bianchi07]). Most of these models assume steady-state chemistry and use classical nucleation theory to calculate dust yields. These approximations may not applicable in a supernova outflow environment [@Donn85; @Cherchneff08; @Cherchneff09; @Cherchneff10], although see @Paquette11 and @Nozawa13. Furthermore, significant amounts of dust can also be destroyed in supernova reverse shocks [@Silvia10].
The large difference in the critical metallicity between these two cooling mechanisms has sparked some debate about which one is most relevant for the formation of low-mass metal-poor stars. This can be observationally tested, as the relevant cooling mechanisms should leave an imprint on the observed chemical abundances. @Frebel07 observationally tested the fine structure cooling theory by introducing the transition discriminant $D_{\text{trans}}$. They predicted that metal-poor stars forming through this mechanism must have $D_{\text{trans}} > -3.5 \pm 0.2$. Nearly all stars satisfy this criterion (see @Frebel13a for an updated $D_{\text{trans}}$ figure). The only star known to violate the $D_{\text{trans}}$ criterion is SDSS J1029151+1729 [@Caffau11]. @Schneider12b and @Klessen12 showed that dust cooling was instead able to explain the formation of this star. More generally, @Schneider12 calculate a critical dust-to-gas ratio ($\mathcal{D}_{\text{crit}}$) that could in principle place an observational restriction on dust cooling, similar to the $D_{\text{trans}}$ restriction on fine structure cooling. However, for metal-poor stars besides SDSS J1029151+1729, the impact of dust cooling has not been evaluated in detail.
Ideally, there would be general properties of supernova dust that could be tested with observations of abundances in metal-poor stars. Recently, @Cherchneff10 have shown that when accounting for non-equilibrium chemical kinetics in dust formation, dust yields are significantly lower and dominated by silicon-based grains, rather than the carbon grains that are typical results of most steady-state models. There is some debate about the extent to which carbon dust formation is suppressed (e.g., @Nozawa13). However, if indeed carbon dust formation is generally suppressed in the early universe, the silicon abundance of metal-poor stars could be used as an observational constraint on dust cooling processes.
In this paper, we investigate the impact that silicon-based dust could have had on the formation process of the first low-mass stars. Using the silicon-based dust compositions from @Cherchneff10, we compute critical silicon abundances and compare them to observations of chemical abundances in long-lived metal-poor stars. In Section \[section:dustmodelintro\], we describe the dust models used for this paper. In Section \[section:critical\], we calculate critical silicon abundances for our dust models, assessing how differences in chemistry, grain size distribution, and environment affect this critical threshold. Our main results are found in Section \[section:data\], where we compare our critical silicon abundances to measurements of metal-poor stars. Section \[section:dustvsFS\] considers evidence for two distinct formation pathways of low-mass metal-poor stars, and Section \[section:DLA\] discusses the potential for Damped Lyman-$\alpha$ systems to help constrain the star formation environments. After outlining important caveats in Section \[section:caveats\] (particularly related to the production of carbon dust), we conclude in Section \[section:conclusion\].
Dust Models {#section:dustmodelintro}
===========
We first present the dust models used in this paper in Section \[section:dustmodels\]. We then discuss some processes in these dust models which strongly inhibit carbon dust formation in Section \[section:whynocarbon\].
Dust Chemical Composition and Size Distributions {#section:dustmodels}
------------------------------------------------
We use the eight different silicon-based dust chemistries presented in @Cherchneff10. We assume these are representative of typical dust yields in the early universe. The dust masses are given in Table \[tbl:1\]. Although the eight different models represent different assumptions about the nature of the supernovae and the dust condensation process, we simply take them as plausible variations in the chemical composition of dust. The dominant dust species are SiO$_2$, Mg$_2$SiO$_4$, amorphous Si, and FeS.
For our calculations in Section \[section:dustcooling\], we require a dust grain size distribution. However, @Cherchneff10 do not compute grain size distributions for their dust models. We thus consider two simple but well-motivated grain size distributions. The first is a @Pollack94 “standard” size distribution. This was used in @Omukai10, and it is similar to the Milky Way grain size distribution used in @Dopcke13. For spherical dust grains of radius $a$: $$\label{eq:stdsizedistr}
\frac{dn_{\text{standard}}}{da} \propto \begin{cases}
1 & a < 0.005 \mu \text{m} \\
a^{-3.5} & 0.005 \mu \text{m} < a < 1 \mu \text{m} \\
a^{-5.5} & 1 \mu \text{m} < a < 5 \mu \text{m}
\end{cases}$$ We also consider a grain size distribution that approximates the effect of running a post-supernova reverse shock through newly created dust, based on the size distributions calculated in @Bianchi07: $$\label{eq:shocksizedistr}
\frac{dn_{\text{shock}}}{da} \propto \begin{cases}
1 & a < 0.005 \mu \text{m} \\
a^{-5.5} & a > 0.005 \mu \text{m}
\end{cases}$$ From now on, we will refer to these two grain size distributions as the “standard” and “shock” size distributions. For simplicity, we assume that each type of dust grain has the same grain size distribution, though it may also be possible to calculate a good approximation to the grain size distribution using classical nucleation theory [@Paquette11]. We normalize the size distributions to number of particles per unit dust mass (cm$^{-1}$ g$^{-1}$) by using the amount of dust mass formed and the solid-phase chemical density of each type of dust [@Semenov03; @HICC].
Silicate or Carbon Dust? {#section:whynocarbon}
------------------------
We use the @Cherchneff10 dust models to establish a critical silicon criterion (Section \[section:sicrit\]). Thus, our results crucially depend on the assumption that the dust composition is largely silicon-based. The most significant non-silicate dust is typically amorphous carbon. We thus briefly describe why carbon dust formation is almost completely inhibited in these models. We refer the reader to @Cherchneff09 [@Cherchneff10] for a more extensive discussion.
The chemical mechanisms that inhibit carbon dust formation depend on the C/O ratio in the supernova ejecta. When the C/O ratio is less than one, CO formation rapidly depletes the available carbon. Although there are processes that can destroy this supply of CO and form short carbon chains, subsequent oxidation of these chains inhibits dust formation. This effect is seen despite accounting for non-thermal processes such as the destruction of CO through high energy Compton electrons [@Cherchneff10]. When the C/O ratio is greater than one, small carbon clusters can form but are rapidly destroyed by the He$^+$ ions that accompany large amounts of carbon.
In radial distributions of supernova ejecta, carbon is always accompanied by large oxygen or helium abundances (e.g., @Nozawa03). However, if the supernovae ejecta is poorly mixed at a microscopic level, then carbon rich clouds may form significant amounts of carbon dust in addition to silicate dust [@Cherchneff10; @Nozawa13]. Thus, the suppression of carbon dust may heavily depend on the level of mixing, which itself depends on the details of the supernova explosion.
We will follow the assumption of microscopically-mixed supernova ejecta as in @Cherchneff10, which leads to silicate dust being the dominant form of dust in the early universe. A major motivation for investigating the consequences of silicon-based dust is that silicon abundances measured from the most metal-poor stars are comparable to the theoretical critical silicon abundances we derive in Section \[section:sicrit\], thus offering an empirical test of this important assumption. For completeness, in Section \[section:cdust\], we also explore the impact that the formation of carbon dust would have on our results.
Critical Silicon Abundance for Gas Fragmentation {#section:critical}
================================================
In this section, we present the method for calculating the critical silicon abundance ($\mbox{[Si/H]}_{\text{crit}}$) required for gas fragmentation. We use a simplified model that only considers dust thermal cooling and adiabatic compressional heating. Many previous papers have studied these in detail, using a more comprehensive set of cooling mechanisms that influence a large range of gas densities (e.g., @Omukai00 [@Omukai05; @Schneider06; @Omukai10; @Schneider12]). To derive the critical silicon abundance, we focus on the density regime where dust cooling dominates.
In Section \[section:dustcooling\] we show how we calculate the dust cooling rate for a given dust model. In Section \[section:dcrit\] we use the cooling rate to calculate a critical dust-to-gas ratio [@Schneider12], which we convert to a critical silicon abundance in Section \[section:sicrit\]. In Section \[section:environments\], we discuss uncertainties in the Population II star forming environment and the implications this may have for our critical silicon abundance.
Calculating the Dust Cooling Rate {#section:dustcooling}
---------------------------------
We describe how to calculate the gas cooling rate due to dust emission. This calculation closely follows the method in @Schneider06. For completeness and convenience of the reader, we here give a brief summary.
Dust grain emission is well approximated by thermal radiation [@Draine01], in which case the cooling rate can be written $$\label{eq:lambdad}
\Lambda_{\text{d}} = 4 \sigma_{\text{SB}} T_{\text{d}}^4 \kappa_{\text{P}} \rho_{\text{d}} \beta_{\text{esc}}$$ where $\sigma_{\text{SB}}$ is the Stefan-Boltzmann constant, $T_{\text{d}}$ is the dust temperature, $\kappa_{\text{P}}$ is the temperature-dependent Planck mean opacity of dust grains per unit *dust* mass, $\rho_{\text{d}}$ is the dust mass density, and $\beta_{\text{esc}}$ is the photon escape fraction. We define the dust-to-gas ratio as $$\label{eq:dusttogasratio}
\mathcal{D} \equiv \rho_{\text{d}}/\rho$$
For a given dust composition model, the Planck mean opacity is given by $$\label{eq:kappaplanck}
\kappa_{\text{P}}(T_{\text{d}}) = \frac{\int_0^\infty\kappa_\lambda B_\lambda(T_{\text{d}})
d\lambda}{\int_0^\infty B_\lambda(T_{\text{d}}) d\lambda}$$ where $\kappa_\lambda$ is the wavelength-dependent opacity in cm$^2$ g$^{-1}$ and $B_\lambda(T_{\text{d}})$ is the Planck specific intensity. $\kappa_\lambda$ can be calculated by $$\label{eq:kappalambda}
\kappa_\lambda = \sum_i f_i \kappa_\lambda^i \text{ with }
\kappa_\lambda^i = \int_0^\infty Q_\lambda^i(a) \pi a^2
\frac{dn^i}{da} da$$ where $i$ denotes different dust species, $f_i$ is the mass fraction, $Q_\lambda^i$ is the area-normalized absorption cross section, and $dn^i/da$ is the size distribution. We calculate $Q_\lambda^i$ using Mie theory, with optical constants taken from the sources listed in Table \[tbl:1\]. If required, we linearly extrapolate the optical constants on a log-log basis. We plot the Planck mean opacities for all our dust models in Figure \[fig:kplanck\], and for comparison we also include the Planck mean opacities for carbon-heavy dust models in @Schneider06 and @Schneider12.
![Planck mean opacity. Top panel shows the standard size distribution, bottom panel the shock size distribution. The vertical dashed line indicates the dust sublimation temperatures of $1500\,$K. For comparison, we also include Planck mean opacities for the core-collapse supernova model in @Schneider06 (dotted line) and the metal-free supernova from @Schneider12 (dash-dotted line), both of which contain significant amounts of carbon dust. These lines terminate when the dust has sublimated. \[fig:kplanck\]](paper_std_kPlanck.pdf "fig:"){width="9cm"}\
![Planck mean opacity. Top panel shows the standard size distribution, bottom panel the shock size distribution. The vertical dashed line indicates the dust sublimation temperatures of $1500\,$K. For comparison, we also include Planck mean opacities for the core-collapse supernova model in @Schneider06 (dotted line) and the metal-free supernova from @Schneider12 (dash-dotted line), both of which contain significant amounts of carbon dust. These lines terminate when the dust has sublimated. \[fig:kplanck\]](paper_x55_kPlanck.pdf "fig:"){width="9cm"}
To calculate the dust temperature, we set the dust cooling rate in Equation \[eq:lambdad\] equal to the gas-dust collisional heating rate [@Hollenbach79]: $$\label{eq:energybalance}
\Lambda_{\text{d}} = H_{\text{d}} = n n_{\text{d}} \sigma_{\text{d}}
v_{\text{th}} f (2k_{\text{B}}T - 2k_{\text{B}}T_{\text{d}})$$ where $n$ is the number density of atomic hydrogen, $n_{\text{d}}$ is the number density of dust, $\sigma_{\text{d}}$ is the dust geometrical cross section, $v_{\text{th}}$ is the thermal velocity of atomic hydrogen, $f$ is a correction factor for species other than atomic hydrogen, $T$ is the gas temperature, and $T_{\text{d}}$ is the dust temperature. Note that the kinetic energy per colliding gas particle is $2k_{\text{B}}T$ instead of $1.5k_{\text{B}}T$ because higher energy particles collide more frequently [@DraineBook]. We assume the gas has a Maxwellian velocity distribution so the average velocity of atomic hydrogen is $$v_{\text{th}} = \left(\frac{8k_{\text{B}}T}{\pi m_{\text{p}}} \right)^{1/2}$$ Since dust is most important at high gas densities, we assume the hydrogen in the gas is fully molecular. Then neglecting the effects of charge or sticking probabilities, we account for the differences in number density and thermal velocity by setting $f = 1/2\sqrt{2} +
y_{\text{He}}/2$, where $y_{\text{He}} = n_{\text{He}}/n = 1/12$ for primordial gas. We can also rewrite $$n_{\text{d}} \sigma_{\text{d}} = \rho_{\text{d}} S = \mathcal{D} \mu m_{\text{p}} n S$$ where $S$ is the total dust geometrical cross section per unit dust mass defined by $$S = \sum_i f_i S_i \text{ with } S_i = \int_0^\infty\pi a^2
\frac{dn^i}{da}da$$ and $\mu = 1 + 4 y_{\text{He}} = 4/3$.
In general, solving Equation \[eq:energybalance\] depends on the dust-to-gas ratio $\mathcal{D}$ because the amount of dust may influence $\beta_{\text{esc}}$. We assume $\beta_{\text{esc}} = \min
(1,\tau^{-2})$ which is suitable for radiative diffusion out of an optically thick gas [@Omukai00]. The optical depth $\tau$ is given by: $$\label{eq:gasopacity}
\tau = (\kappa_{\text{gas}}\rho + \kappa_{\text{d}}\rho_{\text{d}}) \lambda_{\text{J}}$$ $\kappa_{\text{gas}}$ is the continuum Planck mean opacity of primordial gas from @Mayer05, $\kappa_{\text{d}}$ is the Planck mean opacity of dust calculated in this paper, $\rho$ and $\rho_{\text{d}}$ are the densities of gas and dust respectively, and $\lambda_{\text{J}}$ is the Jeans length. The Jeans length is the typical size of a dense core of a uniformly collapsing spherical gas cloud (e.g., @Larson69). If the gas is optically thin ($\beta_{\text{esc}}=1$), it is possible to solve for the dust temperature independently of $\mathcal{D}$. However for the optically thick case, dust opacity will affect the solution and cause some nonlinear dependence on $\mathcal{D}$.
If the dust temperature becomes too high, the dust will sublimate. Different dust grains sublimate at different temperatures. We simplify this effect by assuming all grains sublimate at $T_{\text{d}} = 1500\,$K, a typical temperature for non-carbon grains [@Schneider06]. We set $\kappa_{\text{P}} = 0$ when the dust sublimates. Also, when there is negligible dust heating from gas collisions, the cosmic microwave background (CMB) provides a temperature floor. We include this effect by modifying the dust radiation rate to $\Lambda_{\text{d}}(T_{\text{d}}) -
\Lambda_{\text{d}}(T_{\text{CMB}})$ (e.g., @Schneider10). We assume that $T_{\text{CMB}} = 50\,$K, corresponding to $z \sim 15$.
In summary, the inputs into this model are the gas properties $n$ and $T$; and the dust properties $\kappa_{\text{P}}$, $S$, and $\mathcal{D}$. The output is a dust temperature $T_{\text{d}}$ with a corresponding cooling rate $\Lambda_{\text{d}}$. In Figure \[fig:opticalthickthin\], we show a representative calculation of $\Lambda_{\text{d}}$ using dust model 1 and $\mathcal{D}=10^{-7}$. Our simple thermal model is sufficient to capture many important features of a full thermal evolution calculation [@Omukai05; @Schneider06]. For example, we see that dust cooling becomes comparable to adiabatic heating at densities $\gtrsim~10^{10-12}\,$cm$^{-3}$; the smaller grains in the shock size distribution increase gas cooling; and opacity begins to shut off dust cooling at densities $\gtrsim~10^{14}\,$cm$^{-3}$. Note that the $T=2000\,$K lines terminate at $n=10^{13}\,$cm$^{-3}$ because the dust sublimates when it reaches $1500\,$K.
![Ratio between dust cooling rate and adiabatic heating rate as a function of gas density and temperature for dust model 1 and $\mathcal{D}=10^{-7}$. Top panel shows the standard size distribution, bottom panel the shock size distribution. Dotted lines correspond to dust cooling when $\beta_{\text{esc}}=1$ while solid lines correspond to dust cooling with dust and gas opacity included. For $T=2000\,$K, the cooling terminates around $n=10^{13}\,$cm$^{-3}$ because the dust sublimates. The solid black line indicates where dust cooling is equal to adiabatic heating, and the intersection with the colored lines indicate densities and temperatures where $\mathcal{D}_{\text{crit}}=10^{-7}$. \[fig:opticalthickthin\]](paper_optthickthin_v2a.pdf "fig:"){width="9cm"}\
![Ratio between dust cooling rate and adiabatic heating rate as a function of gas density and temperature for dust model 1 and $\mathcal{D}=10^{-7}$. Top panel shows the standard size distribution, bottom panel the shock size distribution. Dotted lines correspond to dust cooling when $\beta_{\text{esc}}=1$ while solid lines correspond to dust cooling with dust and gas opacity included. For $T=2000\,$K, the cooling terminates around $n=10^{13}\,$cm$^{-3}$ because the dust sublimates. The solid black line indicates where dust cooling is equal to adiabatic heating, and the intersection with the colored lines indicate densities and temperatures where $\mathcal{D}_{\text{crit}}=10^{-7}$. \[fig:opticalthickthin\]](paper_optthickthin_v2b.pdf "fig:"){width="9cm"}
Critical Dust-to-Gas Ratio {#section:dcrit}
--------------------------
Following @Schneider12, we define the critical dust-to-gas ratio $\mathcal{D}_{\text{crit}}$ as the minimum mass fraction of dust that causes gas fragmentation. We solve for this in a manner similar to @Bromm03, by finding the dust to gas ratio such that $$\label{eq:solveforcrit}
\Lambda_{\text{d}} = \Gamma_{\text{ad}}$$ where $\Lambda_{\text{d}}$ is given by Equation \[eq:energybalance\] and $\Gamma_{\text{ad}}$ is the adiabatic compressional heating rate, given by $$\label{eq:adiabaticheating}
\Gamma_{\text{ad}} \simeq 1.5 n \frac{k_{\text{B}}T}{t_{\text{ff}}}$$ where $t_{\text{ff}}$ is the free fall time. @Schneider12 show that this method of finding the dust-to-gas ratio gives a $\mathcal{D}_{\text{crit}}$ that is very close to a full calculation that accounts for other thermal effects in the gas. Also note that the value of $\mathcal{D}_{\text{crit}}$ depends on the gas density and temperature. Following @Schneider12, we use a gas density of $n=10^{12}$ cm$^{-3}$ and gas temperature of $T=1000\,$K as our fiducial values (but see Section \[section:environments\]).
Critical Silicon Abundance {#section:sicrit}
--------------------------
Given a dust composition with a corresponding $\mathcal{D}_{\text{crit}}$, we can calculate the minimum amount of silicon required for gas fragmentation. To do this, we write two expressions for the mass fraction of Si at the critical point.
The fraction of silicon in the dust is given by $$\label{eq:Dcritdust}
\frac{M_{\text{Si}}}{M_{\text{dust}}}\mathcal{D}_{\text{crit}}$$ where $M_{\text{Si}}$ is the mass of silicon in the dust, $M_{\text{dust}}$ is the total mass of dust, and $\mathcal{D_{\text{crit}}}$ is the critical dust-to-gas ratio. Note that $M_{\text{Si}}$ and $M_{\text{dust}}$ depend on the specific dust model used, and the ratios $M_{\text{Si}}/M_{\text{dust}}$ for our dust models are given in Table \[tbl:1\]. The fraction of silicon in the gas is given by $$\label{eq:Dcritgas}
\frac{\mu_{\text{Si}} \, n_{\text{Si,crit}}}{\mu \, n_{\text{H}}}$$ where $\mu_{\text{Si}}$ is the molecular weight of silicon ($28.1 m_{\text{p}}$), $\mu$ is the molecular weight of the gas, $n_{\text{Si,crit}}$ is the number density of silicon at the critical point, and $n_{\text{H}}$ is the hydrogen number density.
We now assume that these two fractions are equal. In other words, we assume that all silicon present in the gas cloud is locked up in dust. This maximizes the amount of dust and provides the most conservative way to calculate a critical silicon threshold. Setting Equations \[eq:Dcritdust\] and \[eq:Dcritgas\] equal and rewriting them in terms of an abundance, we obtain $$\label{eq:SiHcrit}
\log \frac{n_{\text{Si,crit}}}{n_{\text{H}}} =
\log \mathcal{D}_{\text{crit}} +
\log\left(\frac{\mu}{\mu_{\text{Si}}}\right) +
\log\left(\frac{M_{\text{Si}}}{M_{\text{dust}}}\right)$$ and we can find $\mbox{[Si/H]}_{\text{crit}}$ by subtracting the solar abundances from @Asplund09[^1]. A star whose measured $\mbox{[Si/H]}$ is less than $\mbox{[Si/H]}_{\text{crit}}$ thus has a sub-critical $\mathcal{D}$, too low to trigger dust-induced gas fragmentation.
In Figure \[fig:sizedistr\], we show the effect of varying the size distribution and the dust composition on the dust cooling solution at the fiducial gas density and temperature of $n=10^{12}$ cm$^{-3}$ and $T=1000\,$K. Table \[tbl:1\] shows the numerical values for $\mbox{[Si/H]}_{\rm crit}$. The differences between chemical compositions are quite small, but changing the size distribution makes a very large difference. In particular, $\mathcal{D}_{\text{crit}}$ and $\mbox{[Si/H]}_{\text{crit}}$ for the shocked size distribution are about an order of magnitude lower for all the different chemical models. This is a direct result of differences in the average cross section $S$, as a larger $S$ causes the grains to heat up more quickly [@Schneider06]. In contrast, changing the chemical composition mostly affects $\kappa_{\text{P}}$, but the steep temperature dependence of dust cooling (Equation \[eq:lambdad\]) implies that large changes in $\kappa_{\text{P}}$ can be compensated by relatively small changes in $T_{\text{d}}$. For comparison, in Figure \[fig:sizedistr\] we show the $\mathcal{D}_{\rm crit}$ calculated in @Schneider12, where the dotted line indicates $\mathcal{D}_{\text{crit}} = 4.4 \times 10^{-9}$ and the shaded box indicates $\mathcal{D}_{\rm crit} \in
[2.6,6.3]\times 10^{-9}$. The range in $\mathcal{D}_{\rm crit}$ corresponds to differences just in the grain size distribution/cross section. Most of the dust models in @Schneider12 are composed primarily of carbon dust, and the similarity in $\mathcal{D}_{\rm crit}$ between these models and our silicate dust models emphasizes that changing the dust composition produces only a small effect compared to changing the grain size distribution. Many previous authors have also noted the importance of the dust grain size distribution in determining the cooling properties of dust (e.g., @Omukai05 [@Hirashita09]).
![Differences in dust properties at the critical point for a gas density of $n=10^{12}$ cm$^{-3}$ and temperature $T=1000\,$K for the eight dust models in Table \[tbl:1\]. From top to bottom: equilibrium dust temperature, Planck mean opacity at the equilibrium dust temperature, dust geometric cross section, critical dust-to-gas ratio, and critical silicon abundance. Differences across chemical compositions are relatively small, but differences across different size distributions are very large. The $\mathcal{D}_{\text{crit}}$ from the shock size distribution is similar to the $\mathcal{D}_{\text{crit}}$ range from @Schneider12 (dotted line and shaded box in fourth panel). \[fig:sizedistr\] ](paper_compare_sizedistr.pdf){width="9cm"}
Population II Star Forming Environments {#section:environments}
---------------------------------------
The critical dust-to-gas ratio, $\mathcal{D}_{\text{crit}}$, is a function of the ambient gas density and temperature in the regions where second-generation, Population II, star formation takes place. Their physical conditions are still rather uncertain, as opposed to the well-defined initial conditions for Population III star formation [@Bromm13]. Thus far we have assumed a fiducial density and temperature of $n
\sim 10^{12}\,$cm$^{-3}$ and $T \sim 1000\,$K where dust cooling will certainly be important [@Omukai05; @Schneider12]. This naturally corresponds to the protostellar disks explored in simulations [@Clark08; @Stacy10; @Greif11; @Clark11; @Dopcke13]. However, other Population II star forming environments may also achieve high densities, with likely environments including the turbulent cores of atomic cooling halos [@Wise07; @Greif08; @SafShrad12] or in the post-shock region of a supernova [@Mackey03; @Salvaterra04; @Nagakura09; @Chiaki13b]. To provide a broader view, we consider how dust could impact these environments by estimating their maximum densities and temperatures.
We do not expect Population II stars to form in the first dark matter minihalos since Population III supernova evacuate much of the gas from the minihalo, preventing future star formation [@Whalen08]. However, a $\sim 10^8 M_\odot$ dark matter halo can cool efficiently through Lyman-$\alpha$ lines [@Wise07; @Greif08]. These atomic cooling halos are supersonically turbulent, which can cause densities as high as $10^6\,$cm$^{-3}$ [@SafShrad12]. The virial temperatures of these halos are quite high ($\sim 10^4\,$K) but H$_2$ cooling can reduce the temperature to $\sim 400\,$K [@Oh02; @SafShrad12]. These conditions will not be sufficient for dust fragmentation [@Omukai05; @Schneider12]. However, at the center of these halos gas can continue collapsing, eventually forming into protostellar disks.
An additional way to obtain a density enhancement is through a supernova shockwave. The supernova shell and post-shock region can achieve density enhancements of $10^4$ above the ambient ISM density [@Mackey03]. Thus the maximum density achievable in a shell may be around $10^6$ cm$^{-3}$, which will again be too low to immediately fragment through dust cooling. However, shell instabilities may still cause fragmentation, and subsequent collapse may cause dust-induced low-mass star formation [@Salvaterra04; @Nagakura09; @Chiaki13b].
It is clear that in these environments, dust cannot cause widespread fragmentation until the disk stage of collapse. However, our density estimates of these environments are rather crude, and future studies may find other Population II star forming environments with extremely high densities. Also, there will certainly be variations in the density and temperature in a protostellar disk. Thus, for completeness, we show how $\mathcal{D}_{\text{crit}}$ varies with density and temperature in Figure \[fig:ntplane\]. It is clear that $\mathcal{D}_{\text{crit}}$ (and thus $\mbox{[Si/H]}_{\text{crit}}$) is somewhat sensitive to the choice of density and temperature. We also show the analytic scaling of $\mathcal{D}_{\text{crit}}$ derived in @Schneider12 as dotted lines in Figure \[fig:ntplane\] (using $\log
\mathcal{D}_{\text{crit}}= -7.5$). This scaling matches our calculation well at higher gas temperatures, as expected based on the approximation $T_{\text{d}}=0$ used to derive the formula.
We note that we use a simple thermal model that only considers adiabatic heating and dust thermal cooling. Thus, the $\mathcal{D}_{\text{crit}}$ values in Figure \[fig:ntplane\] should be treated as guidelines that approximate what would be obtained from a more complete thermal model or from simulations.
![$\mathcal{D}_{\text{crit}}$ as a function of gas density and temperature for dust model 1. Other dust models are qualitatively similar. Left: standard distribution. Right: shock distribution. Horizontal dashed lines indicate the CMB and dust sublimation temperatures. Dotted line shows the analytical approximation for dust cooling at $\mathcal{D}_{\text{crit}}=10^{-7.5}$ from @Schneider12. Thick slanted dashed lines indicate Jeans masses of $10M_\odot$ and $1M_\odot$. \[fig:ntplane\]](paper_ntplane_Dcrit.pdf){width="9cm"}
Comparison with Metal-Poor Star Abundances {#section:data}
==========================================
We now compare the critical silicon abundances for star forming gas, as derived from all eight of our silicon-based dust models with both grain size distributions, to abundance measurements of metal-poor stars. We evaluate $\mbox{[Si/H]}_{\text{crit}}$ at $n=10^{12}\,$cm$^{-3}$ and $T=1000\,$K (Figure \[fig:sizedistr\], Table \[tbl:1\]).
We use metal poor halo stars and dwarf galaxy stars with $\mbox{[Fe/H]} < -3.5$ taken from the literature [@Suda08; @Frebel10; @Yong13]. References to individual abundances can be found for all but the most iron-poor stars in the SAGA database [@Suda08]. Figure \[fig:sih\] shows $\mbox{[Si/H]}$ as a function of $\mbox{[Fe/H]}$ for our stars. For consistency, we use abundances derived from 1D LTE stellar atmosphere models (but see further discussion below). The horizontal dashed lines indicate the critical silicon abundances from our dust models. The lines are colored by size distribution: green lines correspond to the standard size distribution, and red lines correspond to the shock size distribution. As previewed in Figure \[fig:sizedistr\], the critical silicon abundances are higher for the standard size distribution by almost an order of magnitude, but variation between different chemical compositions is relatively low and less than $0.3\,$dex. For reference, we also show the solar silicon-to-iron ratio as a thin black line. As can be seen, the stellar abundances cover a large range in the diagram. Stars with $\mbox{[Fe/H]} > -4.0$ have typical $\alpha$-abundance ratios of $\mbox{[Si/Fe]} \sim 0.4$ and higher, albeit with one exception. For this study, stars with $\mbox{[Fe/H]} \lesssim -4.5$ or $\mbox{[Si/H]} \lesssim -4.5$ are of particular interest. Indeed, there are several objects in this range which we use as test objects for our modeling of dust cooling in the earliest star forming environments. The higher metallicity stars are unfortunately not usable in this context as they likely formed at a later time from gas that already contained enough metals for cooling.
![Silicon and iron abundances for our sample compiled from the SAGA database [@Suda08; @Frebel10; @Yong13]. We include our new silicon abundance measurement for HE 0557$-$4840 and upper limits for HE 0107$-$5240 and HE 1327$-$2326. We show typical errors on the abundance measurements in the top left corner. We plot critical silicon abundances calculated for $n=10^{12}$ cm$^{-3}$ and $T=1000\,$K, indicated by the dashed horizontal lines. The green dashed lines are computed using the standard size distribution, and the red lines are computed with the shock size distribution. The black line indicates $\mbox{[Si/Fe]}=0$ as a reference. Five stars are emphasized by larger black symbols. The black squares are, from low to high $\mbox{[Fe/H]}$: HE 1327$-$2326 [@Frebel08], HE 0107$-$5240 [@Christlieb04], and HE 0557$-$4840 [@Norris07]. The black pentagon is HE 1424$-$0241 [@Cohen08]. The black diamond is SDSS J1029151+1729 [@Caffau11]. The blue hexagons show the three most iron poor DLAs from @Cooke11 and the upper limits from @Simcoe12 (see Section \[section:DLA\]). \[fig:sih\] ](paper_sih.pdf){width="9cm"}
We note that silicon abundance measurements can be challenging in the most metal-poor stars given the overall weakness of absorption lines. Moreover, the strongest Si line at $3905\,$[Å]{} is blended with a molecular CH line. As most of these stars are carbon-enhanced, Si abundances or upper limits are difficult to derive. As a result, HE 0557$-$4840 ($\mbox{[Fe/H]}=-4.7$, @Norris07 [@Norris12]), HE 1327$-$2326 ($\mbox{[Fe/H]}=-5.7$, @Frebel05 [@Frebel06; @Frebel08]), and HE 0107$-$5240 ($\mbox{[Fe/H]}=-5.4$, @Christlieb02 [@Christlieb04; @Bessell04]) do not have published silicon abundances or (tight) upper limits. From having available spectra of these objects, we used the spectrum synthesis technique (see e.g., @Frebel13a for further details) and published stellar parameters and carbon abundances [@Norris07; @Frebel05; @Christlieb02] to derive a silicon abundance for HE 0557$-$4840 and upper limits for HE 1327$-$2326 and HE 0107$-$5240. For HE 0557$-$4840, the silicon line is somewhat distorted in addition to the carbon blend, but two different spectra yield a consistent result of $\mbox{[Si/H]} = -4.85 \pm 0.2$. For HE 0107$-$5240 and HE 1327$-$2326, a visual examination of the spectra shows no apparent absorption at $3905\,$[Å]{}, although again there is a strong CH feature very close to the position of the Si line. Our newly determined upper limits are $\mbox{[Si/H]} < -5.5$ for HE 0107$-$5240 and $\mbox{[Si/H]} < -5.4$ for HE 1327$-$2326. In the case of HE 0107$-$5240, we thus found a much improved limit compared to an equivalent width-based upper limit @Christlieb04.
Before comparing our critical silicon abundances to those observed in the metal-poor stars, it is important to briefly consider effects on abundances derived from 1D LTE model atmospheres, which can yield different abundances compared to using more physical 3D LTE hydrodynamic models or carrying out additional NLTE corrections. Although the carbon and oxygen abundances derived from 3D models have abundance corrections of order $+0.5\,$dex, the available 3D iron and silicon abundances appear to be within $+0.2\,$dex of the 1D abundances [@Collet06; @Caffau11]. However, NLTE effects on 1D abundances can increase the silicon abundances in metal-poor stars by $0.2$ to $0.5\,$dex when $T_{\text{eff}} > 5500\,$K. This effect becomes larger as stars become hotter [@Shi09; @Zhang11]. Of the interesting stars, HE 1327$-$2326 ($T_{\text{eff}}=6180\,$K) and SDSS J1029151+1729 ($T_{\text{eff}}=5811\,$K) may be affected. But since only the most iron-poor stars have 3D LTE abundances available, we show the 1D LTE abundances of all stars in Figure \[fig:sih\]. We then assume that within the given error bars, these abundances are reasonably accurately describing the Si and Fe content of the stars, especially relative to each other.
In Figure \[fig:sih\], the three black squares are HE 1327$-$2326, HE 0107$-$5240, and HE 0557$-$4840. These stars all have silicon abundances that fall below the critical lines for the standard size distribution, showing that they could not have formed from gas cooled by silicon-based dust of this size distribution. Furthermore, HE 0107$-$5240 and HE 1327$-$2326 have silicon upper limits that are even slightly below the critical silicon abundances derived from the shock size distribution. This suggests that both of these stars did not form because of the agency of silicon-based dust cooling at all, but instead relied on some other mechanism to enable low-mass star formation.
The star HE 1424$-$0241 is also interesting because of an abnormally low silicon abundance, $\mbox{[Si/Fe]}=-1.00$, despite its somewhat higher iron abundance ($\mbox{[Fe/H]}=-3.96$; @Cohen08) compared to the stars described above. It also has only an upper limit on the carbon abundance and anomalously low $\mbox{[Ca/Fe]}$ and $\mbox{[Ti/Fe]}$ abundance, but significant enhancements in $\mbox{[Mn/Fe]}$ and $\mbox{[Co/Fe]}$. This star is shown as the black pentagon in Figure \[fig:sih\]. It also falls beneath the critical silicon abundances derived from the standard size distribution. While this is certainly interesting in the context of testing for cooling mechanisms, it may be possible that this star’s abundance pattern does not reflect nucleosynthesis products of typical supernovae, as such a low Si abundance has never before been found in similar metal-poor stars [@Cohen08].
There is another interesting star with low iron, SDSS J1029151+1729 (@Caffau11, $\mbox{[Fe/H]}=-4.73$, black diamond in Figure \[fig:sih\]). It has $\mbox{[Si/H]} = -4.3$ which places it above the critical silicon values for all of our models. This star is also not carbon-enhanced (see further discussion in Section \[section:dustvsFS\]) and it has previously been suggested that this star formed from dust-cooled gas [@Schneider12b; @Klessen12]. Our results agree with this finding.
Overall, from Figure \[fig:sih\], it is apparent that within our framework, the four stars falling beneath the standard size distribution’s critical silicon abundances are unable to have formed in a cloud cooled by silicate dust with a Milky Way grain size distribution. Thus, the fragmentation seen in simulations using metallicity-scaled Milky Way dust (e.g., @Omukai10 [@Dopcke13]) cannot explain the formation of these four stars[^2]. It follows that either this type of dust is not an accurate model of dust in the early universe, or that the presence of such dust in early gas clouds was subject to stochastic events (e.g., individual supernovae), only rarely leading to the cooling required for star formation to occur.
Two pathways for early low-mass star formation? {#section:dustvsFS}
===============================================
In Section \[section:data\], and assuming the suppression of carbon-based dust, we found that some stars apparently cannot form from gas cooled by only silicon-based dust. In a broader context, it is then interesting to consider the relative importance of dust thermal cooling and carbon/oxygen fine structure line cooling. We can derive new constraints on a star’s formation process by considering its silicon abundance in conjunction with $D_{\text{trans}}$ from @Frebel07. Hence, we calculate $D_{\text{trans}}$ for our star sample with the updated formula from @Frebel13a: $$\label{eq:dtrans}
D_{\text{trans}} = \log(10^{\mbox{[C/H]}}+0.9 \times 10^{\mbox{[O/H]}})$$ To emphasize our notation, note the difference between $\mathcal{D}$ which represents a dust-to-gas ratio, and $D_{\text{trans}}$ which is the transition discriminant of @Frebel07.
In Figure \[fig:dtranssih\], we show $D_{\text{trans}}$ as a function of the silicon abundance. Stars that have both carbon and oxygen abundances available are plotted in black. Following @Frebel13a, stars missing either carbon or oxygen are plotted in red, with a vertical bar denoting the $D_{\text{trans}}$ range corresponding to $-0.7<$\[C/O\]$<+0.2$.
The four stars with the lowest silicon abundances appear to all have large carbon abundances, placing them above the critical $D_{\text{trans}}$ value of $-3.5$. SDSS J1029151+1729 however has a relatively high silicon abundance (at $\mbox{[Si/H]} = -4.3$) and a low carbon abundance placing it below the critical $D_{\text{trans}}
=-3.5$ level. This combination of low Si/high C and high Si/low C abundances is an interesting finding which warrants further exploration in future work. However, if dust in the early universe is silicon-based, then the currently available data suggest a bifurcation in the dominant cooling mechanisms of the gas clouds that produced these low-mass stars. This lends support to the arguments made by @Norris13 who suggest different paths of star formation for carbon-enhanced metal-poor stars ($\mbox{[C/Fe]} >0.7$) and those that do not show such a significant overabundance of carbon relative to iron. After all, nearly a quarter of extremely metal-poor stars with $\mbox{[Fe/H]} < -2.5$ are carbon-enhanced (e.g., @Beers05), with the carbon-rich fraction increasing with decreasing \[Fe/H\].
![$D_{\text{trans}}$ vs $\mbox{[Si/H]}$ for our sample of stars using 1D LTE abundances. The shaded red and green bars are the range of critical silicon abundances shown in Figure \[fig:sih\]. The critical $D_{\text{trans}}$ value and errors are shown as dashed and dotted horizontal black lines. The four stars that fall to the left of the green bar likely cannot form through dust cooling, while the star that falls below the dashed line likely cannot form through fine structure cooling. This is evidence that both dust cooling and fine structure cooling can be relevant for low mass star formation. It is also tentative evidence that fine structure cooling and dust cooling are mutually exclusive. \[fig:dtranssih\]](paper_dtranssih.pdf){width="9cm"}
We thus further examine the physics driving these two potential pathways, which may guide future work towards clarifying emerging bimodal picture of first low-mass star formation. We start by considering gas collapse within an atomic cooling halo [@Wise07; @Greif08]. Here, a bifurcation occurs into two different pathways depending on the fragmentation properties of the gas. A schematic view of these two pathways is shown in Figure \[fig:cartoon\].
In the first pathway, a large collapsing gas cloud undergoes vigorous fragmentation into many medium-mass clumps [@Bromm01; @SafShrad13]. The presence of a LW background from the first stars (e.g., @Ciardi00) prevents molecular hydrogen from dominating the cooling rate, and thus fine structure cooling is required to enable fragmentation at these intermediate densities [@SafShrad13]. The result is a strongly clustered star formation mode, where typical stars may grow to masses $\gtrsim 10M_\odot$, but also leading to a retinue of lower-mass cluster members. Specifically, many-body gravitational interactions may eject some of these protostars from their parent clouds, thus shutting-off further accretion, so that they remain at low masses. We call this mode the “dynamic pathway”, which could be reflected in the carbon-enhanced metal-poor stars. Alternatively, the atomic carbon may condense into dust grains at high densities, inducing gas fragmentation [@Chiaki13a].
The second pathway involves monolithic collapse of a Jeans-unstable gas cloud. In the absence of significant fine-structure cooling, the gas just continues to collapse until a protostellar disk forms at the center of the cloud (e.g., @Clark08). The LW background prevents fragmentation at intermediate densities from molecular hydrogen cooling [@SafShrad13]. In the disk, the density is high enough for dust cooling to be significant, and the disk fragments into low-mass clumps [@Dopcke13]. We call this the “thermal pathway” and note that rotation support is critical for providing an environment that is stable for longer than the gravitational free fall time [@Tohline80; @Clark08]. Although the LW background inhibits fragmentation from molecular cooling, it is possible that other processes could cause additional fragmentation away from the center of the halo. For example, a shell instability in a supernova shock may cause fragmentation, creating additional star clusters in the atomic cooling halo (e.g., @Salvaterra04 [@Nagakura09; @Chiaki13b]).
While our two-pathway interpretation is still largely qualitative at this stage, the current body of metal-poor stellar abundance data can only be satisfactorily explained with such different star formation processes occuring in the early universe. Future modeling of gas cooling and metal mixing processes will shed more light on the matter. Additional discoveries of metal-poor stars with iron, carbon, oxygen, and silicon abundance measurements and upper limits will greatly help to confirm or refute this two-pathway theory by populating the parameter space presented in Figure \[fig:dtranssih\].
![ Two potential pathways for low-mass metal-poor star formation. We start with a collapsing gas cloud on the left. In the dynamic pathway, fine structure cooling induces vigorous fragmentation into many sub-clumps (e.g., @Bromm01). Many-body dynamics can then cause the ejection of a protostar from its parent cloud, creating a low-mass star without dust cooling. In the thermal pathway, the absence of fine structure cooling causes the entire cloud to collapse without experiencing subfragmentation. The center of the cloud forms a protostellar disk with high density, and dust cooling causes low-mass fragmentation in the disk (e.g., @Dopcke13). \[fig:cartoon\]](cartoon.pdf){width="9cm"}
Damped Lyman-$\alpha$ Hosts {#section:DLA}
===========================
Chemical abundances of Damped Lyman-$\alpha$ (DLA) systems have potential to help us understand the star formation environment that may have hosted these early metal-poor stars. DLA’s have indeed been hypothesized to be observational probes of the environment where metal-poor Population II stars may form (e.g., @Cooke11 and references within). They may also be able to constrain the Population III initial mass function [@Kulkarni13]. Most DLAs observed to date have $\mbox{[Fe/H]} > -3.5$ [@Cooke11], but recently a high-redshift DLA candidate has been discovered with only upper limits on metal abundances [@Simcoe12]. There has also been evidence that gas may remain very pristine at lower redshifts as well [@Fumagalli11].
We show the chemical abundances of the three most iron-poor DLAs from @Cooke11 and the upper limits from @Simcoe12 as blue hexagons in Figures \[fig:sih\] and \[fig:dtranssih\]. The three DLAs from @Cooke11 have abundances that fall within the scatter of the more metal-rich stars of our sample, which is consistent with the interpretation that these DLAs could be the formation sites of the metal-poor stars in our halo. The DLA candidate from @Simcoe12 has abundance limits at the critical values of both the fine structure and dust cooling criteria. This could be interpreted such that neither dust nor fine-structure line cooling have operated in this system, leading to no low-mass star formation. However, the nature of this DLA remains somewhat ambiguous [@Simcoe12], so this interpretation may need to be revised. Future observations of metal-poor DLAs will show whether additional systems can be found with such low abundances, and whether any will be below the critical silicon and carbon/oxygen abundances as presented in this paper. In fact, more metal-poor DLA’s would greatly help to further constrain the formation environment of the most metal-poor stars in the Milky Way halo.
Caveats {#section:caveats}
=======
Impact of Carbon Dust {#section:cdust}
---------------------
The most important assumption in our work is that dust in the early universe is largely silicon-based. If large amounts of non-silicate dust is produced, then the critical silicon abundance may not be suitable for testing dust cooling with the most metal-poor stars. In particular, as mentioned in Section \[section:whynocarbon\] and discussed in @Cherchneff10, significant amounts of carbon dust may form if carbon-rich regions are not microscopically mixed with helium ions in the supernova ejecta. @Cherchneff10 calculate an upper limit on carbon dust produced in this situation by assuming no mixing between the carbon and helium layers. 95% of the carbon-rich/oxygen-poor layer is depleted for a total of $0.0145 M_\odot$ of carbon dust, or about 10% of the final dust mass in dust models 1 and 2.
The level of mixing, and thus how much carbon dust is produced, depends on many variables including the nature of the supernova. Thus, we recompute $\mbox{[Si/H]}_{\rm crit}$ for our dust models after adding different amounts of carbon dust directly to the dust models in Table \[tbl:1\]. The results are shown in Figure \[fig:carbfrac\]. The general shape is logarithmic, corresponding to the silicon mass term in Equation \[eq:SiHcrit\]. Changes in $\mathcal{D}_{\rm crit}$ affect $\mbox{[Si/H]}_{\rm crit}$ mostly at low carbon fractions.
When $\sim$20% of the dust mass is in carbon, there is a $\sim$0.2 dex shift down in $\mbox{[Si/H]}_{\rm crit}$ (see Figure \[fig:carbfrac\]). This does not significantly affect our conclusions from Section \[section:data\]. However, it complicates our interpretation in Section \[section:dustvsFS\] (see Figure \[fig:dtranssih\]) as the carbon cannot be directly associated with fine structure cooling. Above $\sim$50% dust mass in carbon, $\mbox{[Si/H]}_{\rm crit}$ shifts down by $\gtrsim$0.5 dex. As a consequence, silicon is no longer a useful element for empirically evaluating the role of dust.
In principle, the methodology described in Section \[section:sicrit\] could be applied to derive critical abundances for carbon dust. An upper limit on the critical carbon abundance can be found by assuming pure carbon dust. We find $\mbox{[C/H]}_{\rm crit, max} \sim-4.9$ for the standard size distribution and $\sim-5.8$ for the shock size distribution. These thresholds are so low that falsifying a carbon dust theory is observationally intractable at the present time. We estimate that $\mbox{[C/H]} \lesssim -5$ could be measured for a suitable bright, cool ($T \sim 4600\,$K) giant if the signal-to-noise is over 300. This is at the edge of current telescope capabilities, but spectrographs on the next generation of extremely large telescopes (e.g., GCLEF on GMT) should enable observations of extremely low carbon abundances. Thus, although testing carbon dust with a critical carbon criterion is currently impractical, it may be accessible in the future.
{width="18cm"}
Other Considerations
--------------------
Unlike @Schneider12b, we do not fit separate supernovae yields to individual stellar abundance patterns. However, we have verified that the supernovae abundances fall within the abundance range of known metal-poor stars [@Frebel10]. Different supernova yields would affect the dust compositions computed by the @Cherchneff10 models. In particular, if the ejecta were to be very abundant in carbon, regions of unmixed carbon are more probable and thus larger amounts of carbon dust would form.
We chose two simple grain size distributions to derive our critical silicon abundances instead of calculating them specifically for our dust models. In doing so, we made the simplifying assumption that all types of dust follow the same size distribution. This assumption is likely not accurate as different chemical species condense to different initial sizes and undergo different amounts of destruction in a supernova reverse shock (e.g., @Todini01 [@Bianchi07; @Nozawa07; @Silvia10]). Since different grain chemical species may not be in thermal equilibrium with each other due to their low density, it may be important to treat grain types separately instead of lumping them together into a single dust model as is typically done in the literature as well as in this paper. We also note that although smaller dust grains lead to more efficient gas cooling, the supernova reverse shocks generally responsible for breaking up dust grains also completely destroy a significant fraction of the dust [@Bianchi07; @Silvia10].
We did not consider grain growth, which can create significantly more dust [@Chiaki13a]. However, at low metallicities, this is not important for our fiducial density of $10^{12}\,$cm$^{-3}$ [@Hirashita09]. We also neglected the effect of increased H$_2$ formation on the surfaces of dust grains. However, the increased H$_2$ cooling should be roughly balanced by the heat released in forming H$_2$ (e.g., @Omukai10 [@Glover13; @Dopcke13]). We do not expect that including this effect would significantly change the critical silicon abundances, but it may be relevant for causing additional fragmentation in the thermal pathway [@SafShrad13].
Conclusion {#section:conclusion}
==========
We have computed critical silicon abundances using the silicon-based dust models in @Cherchneff10. We found that different dust chemical compositions introduce only small variations ($\sim 0.2$ dex) in the critical silicon abundance, but assumptions about the size distribution can produce an order of magnitude difference, with smaller grains being much more effective at cooling the gas (Figure \[fig:sizedistr\]). At the densities and temperatures associated with protostellar disks, the critical silicon abundance is $\mbox{[Si/H]}=-4.5 \pm 0.1$ for a standard Milky Way grain size distribution and $\mbox{[Si/H]}= -5.3 \pm 0.1$ for a shocked grain size distribution. Other Population II star forming environments are not likely to be influenced by dust because their densities are too low.
We then compare our critical silicon abundances to chemical abundances of metal-poor stars. For the standard Milky Way grain size distribution, four of the nine stars with $\mbox{[Fe/H]} < -4.0$ and three of the four stars with $\mbox{[Fe/H]} \lesssim -4.5$ have silicon abundances too low to be explained by silicon-based dust cooling. All stars that cannot form through silicon-based dust cooling satisfy the $D_{\text{trans}}$ criterion, with the possible exception of HE 1424$-$0241. (Figures \[fig:sih\] and \[fig:dtranssih\]).
In fact, two stars have silicon abundances below even the critical silicon abundances for the shocked size distribution, suggesting that silicon-based dust may not have played a dominant role in their formation. With the caution required in interpreting a small sample of stars, we thus see hints of two distinct pathways for the formation of low-mass metal-poor stars in the early universe. One pathway depends on fine structure cooling, and the other depends on dust cooling (Figure \[fig:cartoon\]).
The most important uncertainty in this analysis is the production of carbon dust, which can occur if carbon-rich regions of supernova ejecta are not microscopically mixed with helium ions. If significant amounts of carbon dust can form, the critical silicon abundance will decrease (Figure \[fig:carbfrac\]). If carbon dust is less than 20% of the total dust mass, the critical silicon abundances shift by less than 0.2 dex and our comparison with data is not significantly affected. However, if more of the dust is in carbon, the critical silicon abundance may not be a good criterion to evaluate dust cooling and our interpretation of Figures \[fig:sih\] and \[fig:dtranssih\] may need revisiting. A more complete understanding of microscopic mixing and dust formation in Population III supernova ejecta may allow us to better determine a carbon dust fraction.
Given these results, we note that many potentially interesting metal-poor stars in the literature do not have silicon abundances measured. We encourage observers to consider measurements of silicon abundances or upper limits, both in future data and in currently available spectra. Additional discoveries of metal-poor DLAs may furthermore help to understand the birth clouds of metal-poor stars in the early universe. Only with more data can we observationally evaluate this and other potential models for the formation of the most metal-poor stars.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank an anonymous referee for improving the manuscript, especially regarding our treatment of the dust models. We thank Ralf Klessen and Naoki Yoshida for conversations regarding dust properties, Chalence Safranek-Shrader and Gen Chiaki for clarifying many points regarding their work, and John Norris and Norbert Christlieb for providing us with spectra of HE 0557$-$4840 and HE 0107$-$5240. A.J and A.F. are supported by NSF grant AST-1255160. V.B. is supported by NSF grant AST-1009928 and by NASA ATFP grant NNX09-AJ33G.
[104]{} natexlab\#1[\#1]{}
, T., [Bryan]{}, G. L., & [Norman]{}, M. L. 2002, Science, 295, 93
, M., [Grevesse]{}, N., [Sauval]{}, A. J., & [Scott]{}, P. 2009, , 47, 481
, T. C., & [Christlieb]{}, N. 2005, , 43, 531
, M. S., [Christlieb]{}, N., & [Gustafsson]{}, B. 2004, , 612, L61
, S., & [Schneider]{}, R. 2007, , 378, 973
, V. 2013, Reports on Progress in Physics, 76, 112901
, V., [Coppi]{}, P. S., & [Larson]{}, R. B. 2002, , 564, 23
, V., [Ferrara]{}, A., [Coppi]{}, P. S., & [Larson]{}, R. B. 2001, , 328, 969
, V., & [Loeb]{}, A. 2003, , 425, 812
, V., [Yoshida]{}, N., [Hernquist]{}, L., & [McKee]{}, C. F. 2009, , 459, 49
, E., [Bonifacio]{}, P., [Fran[ç]{}ois]{}, P., [et al.]{} 2011, , 477, 67
, I., & [Dwek]{}, E. 2009, , 703, 642
—. 2010, , 713, 1
, I., & [Lilly]{}, S. 2008, , 683, L123
, G., [Nozawa]{}, T., & [Yoshida]{}, N. 2013, , 765, L3
, G., [Yoshida]{}, N., & [Kitayama]{}, T. 2013, , 762, 50
, N., [Bessell]{}, M. S., [Beers]{}, T. C., [et al.]{} 2002, , 419, 904
, N., [Gustafsson]{}, B., [Korn]{}, A. J., [et al.]{} 2004, , 603, 708
, B., [Ferrara]{}, A., & [Abel]{}, T. 2000, , 533, 594
, P. C., [Glover]{}, S. C. O., & [Klessen]{}, R. S. 2008, , 672, 757
, P. C., [Glover]{}, S. C. O., [Smith]{}, R. J., [et al.]{} 2011, Science, 331, 1040
, J. G., [Christlieb]{}, N., [McWilliam]{}, A., [et al.]{} 2008, , 672, 320
, R., [Asplund]{}, M., & [Trampedach]{}, R. 2006, , 644, L121
, R., [Pettini]{}, M., [Steidel]{}, C. C., [Rudie]{}, G. C., & [Nissen]{}, P. E. 2011, , 417, 1534
, H. M. P., & [Rees]{}, M. J. 1986, , 221, 53
, B., & [Nuth]{}, J. A. 1985, , 288, 187
, G., [Glover]{}, S. C. O., [Clark]{}, P. C., & [Klessen]{}, R. S. 2013, , 766, 103
, B. T. 2011, [Physics of the Interstellar and Intergalactic Medium]{} (Princeton University Press)
, B. T., & [Li]{}, A. 2001, , 551, 807
, D., [Henning]{}, T., [J[ä]{}ger]{}, C., [et al.]{} 2001, , 378, 228
, A. 2010, Astronomische Nachrichten, 331, 474
, A., [Aoki]{}, W., [Christlieb]{}, N., [et al.]{} 2005, , 434, 871
, A., [Christlieb]{}, N., [Norris]{}, J. E., [Aoki]{}, W., & [Asplund]{}, M. 2006, , 638, L17
, A., [Collet]{}, R., [Eriksson]{}, K., [Christlieb]{}, N., & [Aoki]{}, W. 2008, , 684, 588
, A., [Johnson]{}, J. L., & [Bromm]{}, V. 2007, , 380, L40
, A., & [Norris]{}, J. E. 2013, [Metal-Poor Stars and the Chemical Enrichment of the Universe]{}, ed. T. D. [Oswalt]{} & G. [Gilmore]{}, 55
, M., [O’Meara]{}, J. M., & [Prochaska]{}, J. X. 2011, Science, 334, 1245
, C., [Hjorth]{}, J., & [Andersen]{}, A. C. 2011, , 19, 43
, S. 2013, in Astrophysics and Space Science Library, Vol. 396, Astrophysics and Space Science Library, ed. T. [Wiklind]{}, B. [Mobasher]{}, & V. [Bromm]{}, 103
, T. H., [Glover]{}, S. C. O., [Bromm]{}, V., & [Klessen]{}, R. S. 2010, , 716, 510
, T. H., [Johnson]{}, J. L., [Klessen]{}, R. S., & [Bromm]{}, V. 2008, , 387, 1021
, T. H., [Springel]{}, V., [White]{}, S. D. M., [et al.]{} 2011, , 737, 75
, Z., [Thoul]{}, A. A., & [Loeb]{}, A. 1996, , 464, 523
, H., & [Omukai]{}, K. 2009, , 399, 1795
, D., & [McKee]{}, C. F. 1979, , 41, 555
, T., [Omukai]{}, K., [Yoshida]{}, N., & [Yorke]{}, H. W. 2011, Science, 334, 1250
, A.-K., [Klessen]{}, R. S., [Glover]{}, S. C. O., & [Mac Low]{}, M.-M. 2009, , 696, 1065
, A.-K., [Mac Low]{}, M.-M., [Glover]{}, S. C. O., [Klessen]{}, R. S., & [Kitsionas]{}, S. 2009, , 694, 1161
, T., [Bromm]{}, V., & [Bland-Hawthorn]{}, J. 2013, Reviews of Modern Physics, 85, 809
, R. S., [Glover]{}, S. C. O., & [Clark]{}, P. C. 2012, , 421, 3217
, G., [Rollinde]{}, E., [Hennawi]{}, J. F., & [Vangioni]{}, E. 2013, arXiv:1301.4201
, R. B. 1969, , 145, 271
, D. W., & [Hunter]{}, W. R. 1998, Handbook of Optical Constants of Solids III, ed. E. D. [Palik]{} (Academic Press, New York), 233
, J., [Bromm]{}, V., & [Hernquist]{}, L. 2003, , 586, 1
, M., & [Duschl]{}, W. J. 2005, , 358, 614
, T., [Hosokawa]{}, T., & [Omukai]{}, K. 2009, , 399, 2183
, J. E., [Christlieb]{}, N., [Bessell]{}, M. S., [et al.]{} 2012, , 753, 150
, J. E., [Christlieb]{}, N., [Korn]{}, A. J., [et al.]{} 2007, , 670, 774
, J. E., [Yong]{}, D., [Bessell]{}, M. S., [et al.]{} 2013, , 762, 28
, T., & [Kozasa]{}, T. 2013, , 776, 24
, T., [Kozasa]{}, T., [Habe]{}, A., [et al.]{} 2007, , 666, 955
, T., [Kozasa]{}, T., [Umeda]{}, H., [Maeda]{}, K., & [Nomoto]{}, K. 2003, , 598, 785
, S. P., & [Haiman]{}, Z. 2002, , 569, 558
, K. 2000, , 534, 809
, K., [Hosokawa]{}, T., & [Yoshida]{}, N. 2010, , 722, 1793
, K., [Tsuribe]{}, T., [Schneider]{}, R., & [Ferrara]{}, A. 2005, , 626, 627
, J. A., & [Nuth]{}, III, J. A. 2011, , 737, L6
Patnaik, P. 2003, Handbook of Inorganic Chemical Compounds, McGraw-Hill Handbooks (McGraw-Hill Professional Publishing)
, H. R. 1985, Handbook of Optical Constants of Solids, ed. E. D. [Palik]{} (Academic Press, New York), 719
, H. 1985, Handbook of Optical Constants of Solids, ed. E. D. [Palik]{} (Academic Press, New York), 571
, J. B., [Hollenbach]{}, D., [Beckwith]{}, S., [et al.]{} 1994, , 421, 615
, D. M., & [Hunter]{}, D. R. 1991, Handbook of Optical Constants of Solids II, ed. E. D. [Palik]{} (Academic Press, New York), 919
, C., [Agarwal]{}, M., [Federrath]{}, C., [et al.]{} 2012, , 426, 1159
, C., [Bromm]{}, V., & [Milosavljevi[ć]{}]{}, M. 2010, , 723, 1568
, C., [Milosavljevic]{}, M., & [Bromm]{}, V. 2013, arXiv:1307.1982
Salvaterra, R., Ferrara, A., & Schneider, R. 2004, New Astronomy, 10, 113
, F., & [Shull]{}, J. M. 2006, , 643, 26
, R., [Ferrara]{}, A., [Natarajan]{}, P., & [Omukai]{}, K. 2002, , 571, 30
, R., [Ferrara]{}, A., & [Salvaterra]{}, R. 2004, , 351, 1379
, R., & [Omukai]{}, K. 2010, , 402, 429
, R., [Omukai]{}, K., [Bianchi]{}, S., & [Valiante]{}, R. 2012, , 419, 1566
, R., [Omukai]{}, K., [Inoue]{}, A. K., & [Ferrara]{}, A. 2006, , 369, 1437
, R., [Omukai]{}, K., [Limongi]{}, M., [et al.]{} 2012, , 423, L60
, D., [Henning]{}, T., [Helling]{}, C., [Ilgner]{}, M., & [Sedlmayr]{}, E. 2003, , 410, 611
, J. R., [Gehren]{}, T., [Mashonkina]{}, L., & [Zhao]{}, G. 2009, , 503, 533
, D. W., [Smith]{}, B. D., & [Shull]{}, J. M. 2010, , 715, 1575
, R. A., [Sullivan]{}, P. W., [Cooksey]{}, K. L., [et al.]{} 2012, , 492, 79
, B. D., [Turk]{}, M. J., [Sigurdsson]{}, S., [O’Shea]{}, B. W., & [Norman]{}, M. L. 2009, , 691, 441
, A., [Greif]{}, T. H., & [Bromm]{}, V. 2010, , 403, 45
—. 2012, , 422, 290
, T., [Katsuta]{}, Y., [Yamada]{}, S., [et al.]{} 2008, , 60, 1159
, M., [Silk]{}, J., [Rees]{}, M. J., [et al.]{} 1997, , 474, 1
, P., & [Ferrara]{}, A. 2001, , 325, 726
, J. E. 1980, , 239, 417
, O. B., [Pollack]{}, J. B., & [Khare]{}, B. N. 1976, , 81, 5733
, T., & [Omukai]{}, K. 2006, , 642, L61
, J. 2006, , 641, 1
, D., [van Veelen]{}, B., [O’Shea]{}, B. W., & [Norman]{}, M. L. 2008, , 682, 49
, J. H., & [Abel]{}, T. 2007, , 665, 899
, D., [Norris]{}, J. E., [Bessell]{}, M. S., [et al.]{} 2013, , 762, 26
, N., [Abel]{}, T., [Hernquist]{}, L., & [Sugiyama]{}, N. 2003, , 592, 645
, S., [Posch]{}, T., [Mutschke]{}, H., [Richter]{}, H., & [Wehrhan]{}, O. 2011, , 526, A68
, L., [Karlsson]{}, T., [Christlieb]{}, N., [et al.]{} 2011, , 528, A92
, V. G., [Mennella]{}, V., [Colangeli]{}, L., & [Bussoletti]{}, E. 1996, , 282, 1321
[ccrrrrrrrrrrrr]{} 1 & UM ND 20 & 0.039 & 0 & 0 & 0.030 & 4.6 $\times 10^{-5}$ & 0.033 & 3.9 $\times 10^{-4}$ & 3.9 $\times 10^{-4}$ & 8.6 $\times 10^{-5}$ & 0.469 & -4.51 & -5.27\
2 & UM D 20 & 0 & 0 & 0.089 & 0.030 & 4.6 $\times 10^{-5}$ & 0.033 & 3.9 $\times 10^{-4}$ & 3.9 $\times 10^{-4}$ & 8.6 $\times 10^{-5}$ & 0.312 & -4.62 & -5.30\
3 & M ND 20 & 0.105 & 0 & 0 & 0.049 & 4.3 $\times 10^{-4}$ & 0 & 1.4 $\times 10^{-3}$ & 0 & 8.8 $\times 10^{-4}$ & 0.625 & -4.45 & -5.26\
4 & M D 20 & 0 & 0.125 & 0.160 & 0.049 & 0 & 0 & 0 & 0 & 8.8 $\times 10^{-4}$ & 0.293 & -4.63 & -5.37\
5 & UM ND 170 & 3.638 & 0 & 0 & 1.963 & 6.7 $\times 10^{-5}$ & 0.011 & 8.02 $\times 10^{-3}$ & 2.5 $\times 10^{-6}$ & 0.0297 & 0.648 & -4.39 & -5.17\
6 & UM D 170 & 2.577 & 0 & 2.474 & 1.963 & 6.7 $\times 10^{-5}$ & 0.011 & 2.5 $\times 10^{-4}$ & 2.5 $\times 10^{-6}$ & 0.0296 & 0.519 & -4.53 & -5.33\
7 & M ND 170 & 17.3 & 0 & 0 & 8.1 & 0.004 & 0 & 0 & 0 & 0.003 & 0.637 & -4.44 & -5.25\
8 & M D 170 & 12.9 & 6.6 & 5.7 & 8.1 & 0.004 & 0 & 0 & 0 & 0.003 & 0.486 & -4.50 & -5.29\
[^1]: $\mbox{[X/Y]} =
\log_{10}(N_X/N_Y)_* - \log_{10}(N_X/N_Y)_\odot$ for element X,Y
[^2]: As a consistency check: we calculate $\mathcal{D}_{\text{crit}}$ of $\sim 10^{-7.5}$ for the standard size distribution. The Milky Way dust-to-gas ratio is $\sim
10^{-2}$ [@DraineBook]. Thus, when scaling by $Z/Z_\odot =
10^{-5}$ this is above $\mathcal{D}_{\text{crit}}$, but when scaling by $Z/Z_\odot = 10^{-6}$ this is below $\mathcal{D}_{\text{crit}}$. This matches the simulation results of @Omukai10 and @Dopcke13.
|
---
abstract: 'We investigate non-linear elastic deformations in the phase field crystal model and derived amplitude equations formulations. Two sources of non-linearity are found, one of them based on geometric non-linearity expressed through a finite strain tensor. It reflects the Eulerian structure of the continuum models and correctly describes the strain dependence of the stiffness. In general, the relevant strain tensor is related to the left Cauchy-Green deformation tensor. In isotropic one- and two-dimensional situations the elastic energy can be expressed equivalently through the right deformation tensor. The predicted isotropic low temperature non-linear elastic effects are directly related to the Birch-Murnaghan equation of state with bulk modulus derivative $K''=4$ for bcc. A two-dimensional generalization suggests $K''_{2D}=5$. These predictions are in agreement with ab initio results for large strain bulk deformations of various bcc elements and graphene. Physical non-linearity arises if the strain dependence of the density wave amplitudes is taken into account and leads to elastic weakening. For anisotropic deformations the magnitudes of the amplitudes depend on their relative orientation to the applied strain.'
author:
- 'C. Hüter'
- 'M. Friák'
- 'M. Weikamp'
- 'J. Neugebauer'
- 'N. Goldenfeld'
- 'B. Svendsen'
- 'R. Spatschek'
bibliography:
- 'references.bib'
title: 'Non-linear elastic effects in phase field crystal and amplitude equations: Comparison to ab initio simulations of bcc metals and graphene'
---
Introduction
============
For the understanding and development of new materials with specific mechanical properties, a good knowledge of the elastic response is mandatory. A complete parametrisation of elastic properties either experimentally or via [*ab initio*]{} techniques is however challenging, especially if information beyond the linear elastic regime is required, which is important for high-strength materials. Whereas in the linear elastic regime the number of elastic constants is limited, it is obvious that a complete characterization of the mechanical response in the non-linear regime increases the number of required parameters tremendously. A reduction of this parameter set, together with an increased understanding for the non-linear behaviour, would therefore be beneficial. Therefore, the present paper aims at a reduction of this complexity by exploiting the intrinsic description of non-linear elasticity provided by the phase field crystal (PFC) model, in combination with [*ab initio*]{} calculations as well as analytical energy-volume relations.
The PFC method[@Elder:2004ys; @PhysRevLett.88.245701] has become a popular method for simulating microstructure evolution on diffusive timescales and with atomic resolution. In contrast to conventional phase field models, this approach allows to describe e.g. the detailed structure of grain boundaries, as the atomic density distribution is maintained. The PFC community has extended the scope of the model tremendously over the years, and we just mention few of the recent remarkable developments here. Hydrodynamics have been included [@HeinonenV:2016prl], as well as growth from vapor phases [@KocherG:2015prl], dislocation dynamics [@TarpJM:2014prl], and glass formation [@BerryJ:2011prl], and recently also polycrystalline 2D materials, in particular graphene [@MSeymour:2016aa]. General structural transformations became accessible by constructing free energy functionals from generic two-particle correlation functions [@GreenwoodM:2010prl]. One of the advantages of the PFC model is that it automatically contains elasticity, as a deformation of the lattice, expressed through a change of the lattice constant, raises the energy. For small deformations this energy change is quadratic, hence linear elasticity is captured, and for larger deformations non-linear effects appear[@Chan:2009aa; @Huter:2015aa]. Whereas the original PFC model is fully phenomenological, later extensions have shown that it can be linked to the classical density functional theory of freezing [@Singh1991351; @ISI:A1987K583500046; @Spatschek:2010fk], which allows to determine the model parameters from fundamental physical quantities, which can for example be determined from molecular dynamics simulations[@Wu:2006uq; @Wu:2007kx; @Adland:2013ys; @kar13]. The obtained elastic constants can then be obtained from properties of the liquid structure factor and give reasonable estimates of the high temperature values near the melting point. Conceptually, this means that the theory is applied in the high temperature regime, formally at the coexistence between solid and melt phases. Here, in contrast, we aim at an understanding of the ability of the model to capture elastic effects also in the low temperature regime. The predictions will be compared to [*ab initio*]{} results using electronic structure density functional theory (DFT). We note that for the elastic constants and non-linearity the reliability of DFT calculations is excellent, as it is supported by experimental benchmarks[@LejaeghereK:2013crit].
The article is organized as follows: In Section \[ModelingApproachSection\] we revisit the ingredients for the work in the present article. It starts with general concepts concerning the description of non-linear elasticity and then discusses them in the context of the PFC model. The section is concluded with details on the [*ab initio*]{} simulations, which are used to benchmark the continuum descriptions. Section \[1DPFCsection\] analyses the non-linear elastic response of the PFC model in one dimension, where the analysis is particularly simple, emphasizing the Eulerian character of the elastic description and the different roles of geometric and physical non-linearity. Section \[2DPFCsection\] continues with the two dimensional situation of crystals with triangular symmetry. Section \[sectionBCC\] investigates the same behavior for the three-dimensional case of bcc crystal structures. The results are compared to classical descriptions of non-linear elasticity in Section \[BirchMurnaghanSection\] and also to [*ab initio*]{} simulations of bcc materials and graphene. The article concludes with a summary and discussion in Section \[SummaryConclusion\].
Modeling approach {#ModelingApproachSection}
=================
It is one of the primary goals of the present article to link expressions for the elastic response under large deformations using modeling approaches on different scales. For small deformations, the energy increases quadratically with the strain, as known from linear theory of elasticity. For larger deformations, deviations appear, which require a careful distinction between the undeformed reference and the present state of the material. We first investigate these non-linear effects from a mechanical perspective, which we then apply to the phase field crystal model. Here, a primary goal is to see which of the different non-linear strain tensors is most suitable to describe the elastic response in these models. The results are compared to [*ab initio*]{} simulations of large bulk deformations for various bcc metals and graphene.
Finite strain description
-------------------------
We start the investigations with a brief reminder and definition of the different finite strain tensors and quantities relevant for mechanical applications and modelling as described e.g. in Refs. .
We begin with the deformation gradient tensor ${\mathbf{F}}$ which describes how a medium at a reference point ${\mathbf{X}}$ is deformed when it changes to a new coodinate ${\mathbf{X}}\mapsto {\mathbf{x}}$. The local geometry is represented by a segment $d{\mathbf{X}}$ in the reference system which is mapped to $d{\mathbf{x}}$ in the deformed system. We define $dx_j = F_{jk} dX_k$ in coordinate representation, using the Einstein convention for summation over repeated indices. We note that intuitively the deformation gradient tensor can be constructed as product of a rotation tensor and and a stretch tensor, or, more generally a non-rotational tensor. For the description in terms of finite displacements, we introduce the relative displacement vector $d{\mathbf{u}}$ which describes the difference between the two segments $d{\mathbf{X}},d{\mathbf{x}}$ as $d{\mathbf{u}}= d{\mathbf{x}}-d{\mathbf{X}}$ in addition to the displacement of the reference point from ${\mathbf{X}}$ as ${\mathbf{u}}= {\mathbf{x}}-{\mathbf{X}}$. Then the right Cauchy-Green deformation tensor ${\mathbf{C}}= {\mathbf{F}}^\dagger {\mathbf{F}}$ (with $\dagger$ for the transposition) describes the square of local change of distances by deformation as $d{\mathbf{x}}^2 = d{\mathbf{X}}{\mathbf{C}}d{\mathbf{X}}$. In coordinate representation, it is connected to the Green-Lagrange finite strain tensor $\epsilon_{ij}$ components as $$\label{LagrangianStrain}
\epsilon_{ij} =\frac{1}{2} \left( C_{ij} - I_{ij} \right) = \frac{1}{2} \left( \frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i} + \frac{\partial u_m}{\partial X_i}\frac{\partial u_m}{\partial X_j} \right).$$ The corresponding tensor in the reference frame of the deformed medium is the Piola tensor ${\mathbf{c}}= {\mathbf{F}}^{-1\dagger}{\mathbf{F}}^{-1}$, which is related to the Euler-Almansi finite strain tensor as $$\label{eulerStrain}
e_{ij} =\frac{1}{2} \left( I_{ij} - c_{ij} \right) = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} - \frac{\partial u_m}{\partial x_i}\frac{\partial u_m}{\partial x_j} \right).$$ Finally, we introduce the tensor $$\label{strangeEulerStrain}
\bar{e}_{ij} = \frac{1}{2}\left( I_{ij} - C^{-1}_{ij}\right) =\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} - \frac{\partial u_i}{\partial x_k}\frac{\partial u_j}{\partial x_k} \right),$$ corresponding to the $D$ strain of Clayton [@Clayton:2014aa]. It involves the inverse of the right Cauchy-Green deformation tensor $C^{-1}_{ij}=(\delta_{ik}-\partial u_i/\partial x_k)(\delta_{jk}-\partial u_j/\partial x_k)$. Here, the differences between the tensors (\[eulerStrain\]) and (\[strangeEulerStrain\]) appear as contraction either in the numerator or denominator of the non-linear part. The above finite strain tensors all agree up to the level of leading terms (as used in linear elasticity), and deviations due to the geometric effects show up at the quadratic level. For a recent discussion of the three different strain tensors from a continuum mechanics perspective of large deformations we refer to Ref. .
In the following, we use the term [*geometric non-linearity*]{} to express the fact the elastic energy depends on finite deformation measures (e.g., stretch or strain). These will be identified below. Due to the non-linear terms in the strain tensors the energy is therefore not a quadratic function in terms of displacement gradients. Besides geometric non-linearity, also [*physical non-linearity*]{} contributes to deviations from linear elasticity. Physical non-linearity pertains when the terms in the elastic energy of cubic or higher-order in the (geometrically linear or non-linear) strain become non-negligible. A classic example of this is anharmonic elastic behavior. It is obvious that such effects should show up at sufficiently large strains. Under tension, complete dissociation of the material leads to independent atoms or molecules with vanishing interaction and stress. Under strong compression, the Pauli repulsion leads to stress increases due to “hard core contributions”, and these effects are not captured by geometric non-linearity alone.
Phase field crystal modeling {#pfcintrosec}
----------------------------
The phase field crystal model uses an order parameter $\psi$ to describe a material state. In contrast to conventional phase field models this order parameter is not spatially constant, but exhibits periodic modulations in a crystalline phase.
For simplicity, we use here only the original and very basic phase field crystal model, which is described by the energy functional $$\label{pfcfunctional}
F = \int_V d\mathbf{r} \left\{ \psi \left[(q_0^2+\nabla^2)^2 -{\varepsilon}\right]\frac{\psi}{2} +\frac{\psi^4}{4} \right\}. $$ The atom density is denoted by $\psi$, which is periodic in a crystalline state, and $V$ is the system volume. All quantities are assumed to be dimensionless, and ${\varepsilon}$ is a control parameter, which corresponds to a dimensionless temperature. In the following we set $q_0=1$. We focus here on crystalline phases and ignore the melt phase, in agreement with the concept of a low temperature limit. The average density $\bar{\psi}$ is a second control parameter. It is defined as $$\bar{\psi}=\frac{1}{V} \int_V \psi(\mathbf{r}) d\mathbf{r}.$$ The free energy density $f$ in the expression (\[pfcfunctional\]), i.e. the expression in curly brackets $\{\cdots\}$, averaged over a unit cell, will later allow the comparison to [*ab initio*]{} calculated energies.
Equilibrium is obtained via the evolution equation for a conserved order parameter $$\label{PFCevolution}
\frac{\partial\psi}{\partial t} = \nabla^2\left( \frac{\delta F}{\delta\psi} \right).$$ Here we focus on equilibrium elastic properties only, therefore the precise (conserved) dynamics is not important.
A sketch of a density profile in one dimension is shown in Fig. \[figsketch\].
With the basic energy functional given by Eq. (\[pfcfunctional\]) different crystal structures like smectic or triangular (in 2D) or bcc (in 3D) are found. Therefore, the interpretation of the order parameter of an atom density is suggested. Before starting with the actual analysis it is worthwhile to discuss the interpretation of elasticity within the phase field crystal model, and to become aware of limitations. Typically, we perform simulations in a fixed volume, as indicated by the grey box in Fig. \[figsketch\]. When the material is deformed, this would physically lead to a change of the system size, which we do not consider in the simulations. Instead, the domain, in which the equations are solved, is still the grey shaded area. This already hints at the understanding of elasticity in the PFC model in a Eulerian spirit. As a result of this fixed system size the number of “atoms” – the peaks in the density profiles – is not conserved (but see also the discussion about physical atomic density in PFC and vacancies in Ref. ). Additionally, the conservation of the particle number can be violated by the creation or annihilation of atoms under large strain (Eckhaus instability[@EckhausW:1965b]), and will not be further considered here.
Physically, one would expect from such an interpretation that the density is related to the volume change during deformation. With the original system size of the undeformed reference state being $V_0$ and the deformed system having volume $V$, one may suggest the relation $\bar{\psi}_0 V_0 = \bar{\psi} V$ as conservation of the average density, which would then change from $\bar{\psi}_0$ to $\bar{\psi}$. However, such an interpretation is misleading, as instead $\bar{\psi}$ is considered as constant control parameter in the simulations using a fixed volume, and there is no direct connection between the average density $\bar{\psi}$ and the number of atoms. This is most striking in the one-mode expansion, see Eqs. (\[1Donemode\]) and (\[onemode\]) below, where average density and atom spacing can be varied independently, noting that this approximation gives an excellent description of the true density $\psi$ in particular in the regime of small values of ${\varepsilon}$. Most interpretations of elasticity in the PFC model use the picture of following the atoms’ positions during the deformation, which allows to define the elastic response. An exception is the analysis in Ref. , which defines the bulk modulus via the density dependence of the free energy. According to the above discussion it is not surprising that this interpretation leads to different bulk moduli in comparison to the first approach. Many of the conceptual questions related to the interpretation of elasticity in the PFC model become prominent only when non-linear effects are considered. In the framework of (geometrically) linear elasticity the difference between Eulerian and Lagrangian strains vanishes, as obvious from the expressions (\[LagrangianStrain\])-(\[strangeEulerStrain\]) above, and also the distinction between undeformed and deformed configurations is ignored. In an atomistic description one considers the energy per unit cell, and by the inspection of this [*integrated energy*]{} as a function of the strain one can determine the linear and non-linear elastic behavior. As we will see in the following, one arrives at a physically useful interpretation of elasticity in the PFC model, if one considers the [*energy density*]{} in a Eulerian sense as measure. This has implicitly been used in many investigations in the literature [@Elder:2004ys; @PhysRevLett.88.245701; @Spatschek:2010fk] for small deformations, where is it appropriate. However, a thorough investigation in the non-linear regime is still missing, apart from investigations in Ref. . In particular we find that this interpretation leads to a description analogous to the Birch-Murnaghan equation of state[@MurnaghanFD:1944pnas; @BirchF:1947pr], which is frequently used in [*ab initio*]{} simulations to fit the elastic energy, and also sheds light on the strength of non-linearity for bcc elements and graphene. This will become more transparent in Section \[BirchMurnaghanSection\]. On the practical level we use an analytical description which is based on a one-mode expansion of the density field, as used also to derive amplitude equations descriptions. This means that we write the density as a superposition of plane waves. As mentioned before, for small values of $\bar{\psi}$ and ${\varepsilon}$ such a sine wave approximation is very good and allows to treat the problem of non-linear elastic deformation analytically. This will be shown explicitly in one, two and three dimensions in the following sections, taking care of the important role of geometric non-linearity. The analysis builds up on the work by Chan and Goldenfeld [@Chan:2009aa], rectifying an improper interpretation of the non-linear strain tensor.
Ab initio modeling
------------------
The quantum mechanical calculations within the framework of density functional theory [@Hohenberg1964; @Kohn1965] are performed using the Vienna Ab Initio Simulation Package (VASP) [@Kresse1993; @Kresse1996]. The exchange and correlation energy is treated in the generalized gradient approximation as parametrized by Perdew, Burke, and Ernzerhof [@Perdew1996] and implemented in projector augmented wave pseudopotentials [@Bloechl1994]. We use a plane-wave cutoff of 450 eV with a 18 $\times$ 18 $\times$ 18 Monkhorst-Pack k-point mesh for the 2-atom elementary body-centered cubic (bcc) supercells, yielding total-energy accuracy better than 1 meV per atom. All calculations are performed at $T=0$. The corresponding VASP calculations for graphene are performed similarly, with a plane-wave cutoff of 350 eV and a 48 $\times$ 48 $\times$ 3 Monkhorst-Pack k-point mesh for the 2-atom hexagonal cells. The computational cells are designed to be highly anisotropic in their shapes so as to separate individual graphene sheets from their periodic images by 32 Ångstrom of vacuum.
The one-dimensional PFC model {#1DPFCsection}
=============================
We use a one-dimensional situation first to illustrate the conceptual approach. It briefly summarizes results from the literature[@Chan:2009aa] and extends them, elucidating the role of non-linear elasticity.
Geometric non-linearity
-----------------------
For an analysis of the elastic energy we use a one-mode approximation of the order parameter. $$\label{1Donemode}
\psi(x) = A\sin(q x) +\bar{\psi}$$ With this one gets the averaged free energy density[@Elder:2004ys] $$\begin{aligned}
\label{1DPFCelasticEnergy}
f &=& \frac{\bar{\psi}^2}{2} \left[ -{\varepsilon}+1 +\frac{3A^2}{2} + \frac{\bar{\psi}^2}{2} \right] \nonumber \\
&& + \frac{A^2}{4} \left[ -{\varepsilon}+ (1-q^2)^2 + \frac{3A^2}{8} \right]. \label{eq3}\end{aligned}$$ Averaging is done over multiples of the “unit cell”, i.e. periods with “lattice unit” $a=2\pi/q$. Obviously, $q=1$ minimises the energy density for fixed amplitude $A$ and $\bar{\psi}$ (for $q_0=1$). For the moment, we keep the amplitude constant. Then a variation of $q$ leads to an elastic energy change proportional to $(1-q^2)^2$ for small deviations from the ground state. A value $q\neq 1$ expresses a homogeneous strain in the system, and therefore the displacement field has the form $$\label{eq2}
u(x) = (1-q) x,$$ which turns out to be defined in a Eulerian frame, which will become more obvious below in Section \[1DEulerlagrange\]. The displacement gradient is $\partial u/\partial x = 1-q$. In our present one-dimensional setup we get from the strain definitions (\[eulerStrain\]) and (\[strangeEulerStrain\]) $$\label{nonumbersofar}
e_{xx} = \bar{e}_{xx} = \frac{1}{2} (1-q^2),$$ noting that for a one-dimensional situation the tensors (\[eulerStrain\]) and (\[strangeEulerStrain\]) coincide. Only later, in three dimensional situations we will see that in fact $\bar{e}_{ij}$ is the most suitable tensor in the context of PFC modeling. Here we see that the elastic energy density can be written in terms of $\bar{e}_{ij}$, as it is proportional to $e_{xx}^2=\bar{e}_{xx}^2$, i.e. $f_{el}=[f(e_{xx})-f(0)]= A^2\bar{e}_{xx}^2$. An important result is that the non-linear elasticity on this level can be completely attributed to the [*geometric*]{} non-linearity. The constitutive law, which connects stress and strain, is still purely linear, since the elastic energy is quadratic in $\bar{e}_{ij}$.
We can plot the elastic energy density (for fixed amplitude) as a function of the lattice constant $a=2\pi/q$, as shown as solid curve in Fig. \[fig2\].
![ Elastic energy per unit cell as a function of the lattice constant $a$ in the one-dimensional phase field crystal model for $\bar{\psi}=0$ and ${\varepsilon}=0.6$. The solid curve uses a constant amplitude $A=A(q_0)$, see Eq. (\[eq4\]), whereas the dashed curve is based on a strain dependent amplitude $A=A(q)$ according to the same equation. []{data-label="fig2"}](fig2.pdf){width="8.5cm"}
One has to keep in mind that in this representation the energy changes asymmetrically around $a_0=2\pi/q_0$, in contrast to the dependence as a function of the strain $\bar{e}_{xx}$. It is important to mention that in the non-linear elastic regime the stiffness is higher under compression ($a<a_0$) then under tension, as one would expect physically.
Physical non-linearity
----------------------
So far we have assumed that the amplitude $A$ is constant and does not depend on the strain, which leads to geometric non-linearity only. We follow here the analysis by Chan and Goldenfeld [@Chan:2009aa] to account for physical non-linearity.
In equilibrium, the value of $A$ is optimised via the condition $$\frac{\partial f(A, q, {\varepsilon}, \bar{\psi})}{\partial A} = 0.$$ From Eq. (\[eq3\]) we get $$\label{eq4}
A = \pm 2\left(\frac{{\varepsilon}}{3} - \bar{\psi}^2 - \frac{1}{3} (1-q^2)^2 \right)^{1/2}.$$ Close to $q=1$, i.e. in the linear elastic regime, the amplitude is unaffected by the strain. For larger deformations, the amplitudes are reduced as a precursor of a strain induced melting process. Inserting this amplitude back into the energy expression leads to the dashed curve in Fig. \[fig2\]. For larger strains, the physically non-linear effects overcompensate the geometric non-linearity, as discussed above. This is more pronounced for smaller values of ${\varepsilon}$, and then we get agreement of the solid and dashed curve essentially only in the linear elastic regime. This is in line with the interpretation of ${\varepsilon}$ as an undercooling with respect to the solid-liquid coexistence, and therefore for lower values of ${\varepsilon}$ a strain induced melting is more favorable.
We mention that for higher dimensional situations the strain dependence of the amplitudes will be orientation dependent. Hence, different amplitudes will then depend differently on an anisotropic strain. This effect has not been considered in Ref. .
Eulerian vs. Lagrangian description {#1DEulerlagrange}
-----------------------------------
Here we demonstrate that a precise distinction of the strain tensors and the reference states is essential for a correct description. As mentioned before beyond linear elasticity a careful use of deformed and reference configurations is mandatory, and this will be investigated here.
To illustrate this one could naively use the definition (\[LagrangianStrain\]) to calculate the Lagrangian strain from the displacement (\[eq2\]), identifying $x$ as the reference coordinates $X$. This would lead to $$\label{inconvinientStrain}
\epsilon_{xx}=(1-q)(3-q)/2,$$ and the elastic energy contained in (\[eq3\]) cannot easily be represented through this strain expression. However, this expression for the strain tensor would be based on an erroneous mixing of reference frames.
In the Lagrangian perspective, one consistently has to work in the undeformed reference state. For the non-deformed case, we have there a density profile $\psi_0(x) \sim \cos(q_0 x)$ (using $q_0=1$) and for the deformed one $\psi(x) \sim \cos(q x)$, where $x$ is a Eulerian coordinate. The “atom” which is originally located at the (Lagrangian) density peak position $X=2\pi/q_0$ is displaced to $X+u_L(X)=2\pi/q$. Hence the displacement at the position $X$ is given by $$\label{exampledispl}
u_L(2\pi) = 2\pi/q - 2\pi.$$ With such a homogeneously strained solid the (Lagrangian) deformation gradient is $$\frac{\partial u_L(X)}{\partial X} = \frac{u_L(2\pi)}{2\pi} = \frac{1}{q}-1,$$ since the reference length in the undeformed crystal is $2\pi/q_0$ (notice that the displacement at $x=0$ is zero). With this, the Lagrangian strain (\[LagrangianStrain\]) becomes $$\epsilon_{xx} = \frac{1}{2} \left( \frac{1}{q^2} -1 \right),$$ which clearly differs from the (incorrect) expression in Eq. (\[inconvinientStrain\]). Furthermore, the strain is the relative length change of a material, as expressed through $dx^2=dX^2 + 2 \epsilon_{ij}dX_i dX_j$ in Lagrangian formulation. Here this leads consistently to $dx^2=dX^2/q^2$, in agreement with the wavelength change.
Let us contrast this to the Eulerian description. Here the displacement is the same as above in Eq. (\[exampledispl\]), but read as a function of the deformed coordinate $x$, $$u_E(2\pi/q) = 2\pi/q - 2\pi.$$ For the inverse deformation gradient the reference is now the deformed system, hence $$\frac{\partial u_E(x)}{\partial x} = \frac{u_E(2\pi/q)}{2\pi/q} = 1-q.$$ Consequently, the Eulerian strain reads according to Eqs. (\[eulerStrain\]) and (\[strangeEulerStrain\]) $$e_{xx}=\bar{e}_{xx} = \frac{1}{2}(1-q^2),$$ which coincides with Eq. (\[nonumbersofar\]) and shows that we are indeed operating in a Eulerian description in the PFC model. The length change is expressed through the relation $dx^2=dX^2 + 2 e_{ij}dx_idx_j$, which reads here again consistently $dX^2=q^2 dx^2$.
The 2D triangular model {#2DPFCsection}
=======================
As in the one-dimensional case we use the amplitude equation formulation to extract the non-linear elastic response of a two-dimensional stable or metastable triangular phase. The density field is expressed as $$\label{onemode}
\psi = \bar{\psi} + \sum_{j=1}^N \left[ A_j\exp(i\mathbf{k}^{(j)}\cdot\mathbf{r}) + A_j^*\exp(-i\mathbf{k}^{(j)}\cdot\mathbf{r}) \right]$$ with $N=3$ here. The normalised reciprocal lattice vectors (RLVs) are $$\label{2DRLVs}
\mathbf{k}^{(1)} = \left(
\begin{array}{c}
0\\
1
\end{array}
\right),
\quad
\mathbf{k}^{(2)} = \left(
\begin{array}{c}
\sqrt{3}/2\\
-1/2
\end{array}
\right),
\quad
\mathbf{k}^{(3)} = \left(
\begin{array}{c}
-\sqrt{3}/2\\
-1/2
\end{array}
\right).$$ In the following we work in the parameter regime $\bar{\psi}>0$. With the above definition of the RLVs for an undeformed state the amplitudes are equal in magnitude, but not in sign, $$\label{2Damplitudesigns}
-A_1 = A_2 = A_3= A.$$ Chan and Goldenfeld [@Chan:2009aa] derived the free energy functional, which follows from insertion of the amplitude expansion into the free energy functional (\[pfcfunctional\]) and assuming that the amplitudes vary on a scale which is large in comparison to the atomic spacing. Then, only terms which correspond to closed polygons of reciprocal lattice vectors contribute, and one arrives at the functional $$\begin{aligned}
F &=& \int d\mathbf{r} \Bigg\{ -\sum_{j=1}^3 A_j^* (\Gamma-L_j^2)A_j + 3 \sum_{j,\ell=1}^3 |A_j|^2 |A_\ell|^2 \nonumber \\
&& - \frac{3}{2} \sum_{j=1}^3 |A_j|^4 + 6 \bar{\psi} (A_1A_2A_3 +A_1^*A_2^*A_3^*) \Bigg\} \nonumber \\
&=& \int d\mathbf{r} (f_{local} + f_{nonlocal}), \label{eq22}\end{aligned}$$ where an offset, which is independent of the amplitudes, is skipped, see Appendix \[appendix::aederivation\] for details. This free energy functional contains the operator $$L_j = \nabla^2 + 2 i \mathbf{k}^{(j)}\cdot\nabla $$ and $\Gamma = {\varepsilon}- 3\bar{\psi}^2$. The nonlocal term is $$f_{nonlocal} = A_j^* L_j^2 A_j.$$ After an integration by part we can represent it more conveniently as $$f_{nonlocal} = |L_j A_j|^2,$$ where we have skipped boundary terms.
For a deformed state the amplitudes are $$A_j = A_{j, 0} \exp[-i\mathbf{k}^{(j)}\cdot \mathbf{u}(\mathbf{r})],$$ which uses the proper sign in the exponential compared to Ref. and . This is in line with the above discussion in Section \[1DEulerlagrange\] and Ref. . For each mode we obtain $$L_j A_j = A_{j, 0} \exp[-i\mathbf{k}^{(j)}\cdot \mathbf{u}(\mathbf{r})] \left\{\cdots\right\}$$ with $$\begin{aligned}
\left\{\cdots\right\} &=& -k_\beta^{(j)} k_\gamma^{(j)} (\partial_\alpha u_\beta) (\partial_\alpha u_\gamma) + 2 k_\alpha^{(j)} k_\beta^{(j)} \partial_\alpha u_\beta \nonumber \\
&& - i k_\beta^{(j)} \partial_\alpha^2 u_\beta.\end{aligned}$$ In terms of the tensor (\[strangeEulerStrain\]) we can rewrite this as $$\left\{\cdots\right\} = 2 k_\alpha^{(j)} k_\beta^{(j)} \bar{e}_{\alpha\beta} - i k_\beta^{(j)} \partial_\alpha^2 u_\beta.$$ Therefore the elastic energy density for fixed amplitudes is for each mode $j$ $$\begin{aligned}
f_{nonlocal}^{(j)} &=& |A_{j,0}|^2 |\left\{\cdots\right\}|^2 \nonumber\\
&=& 4 |A_{j,0}|^2 \left(k_\alpha^{(j)} k_\beta^{(j)} \bar{e}_{\alpha\beta}\right)^2 \nonumber \\
&& + |A_{j,0}|^2 \left( k_\beta^{(j)} \partial_\alpha^2 u_\beta \right)^2.\end{aligned}$$ The first term corresponds to an elastic term, the second to a strain gradient contribution. For long wave distortions the second term is negligible and will not be considered here. Whereas the energy expression for the individual modes contains the strain tensor $\bar{e}_{ij}$, which may be somewhat unexpected from point of view of elasticity, the situation changes if we sum over the three modes. We then get for the elastic term $$\label{intermediateresult}
f_{nonlocal} = 3 |A_0|^2 \bar{\Delta},$$ with $$\label{eq33}
\bar{\Delta} = \frac{3}{2} \bar{e}_{xx}^2 + \frac{3}{2} \bar{e}_{yy}^2 + 2 \bar{e}_{xy}^2 + \bar{e}_{xx}\bar{e}_{yy}.$$ Here we skipped the strain gradient term and assumed that all amplitudes have the same magnitude $|A_{j, 0}| = |A_0|$. This expression is analogous to the one in Refs. , which contain an incorrect sign in the definition of the displacement. Therefore, here the Eulerian variant of the strain tensor appears. This correction is important as it reflects that materials get stiffer (softer) under compression (tension), and not vice versa. For the present case of triangular systems, this expression coincides with the one defined through the Euler-Almansi strain, $$\label{eq34}
{\Delta} = \frac{3}{2} {e}_{xx}^2 + \frac{3}{2} {e}_{yy}^2 + 2 {e}_{xy}^2 + {e}_{xx}{e}_{yy},$$ hence $\Delta=\bar{\Delta}$ and we explain this coincidence in section \[sectionBCC\] in detail. This implies again that for constant amplitudes the non-linear elasticity is described entirely through geometric non-linearity. In Fig. \[fig3\] we show the elastic energy for the particular case of isotropic straining as a function of the lattice constant, $a/a_0$, where the geometric non-linearity results in the material indeed becoming stiffer (softer) under compression (tension).
![ Elastic energy density in the two-dimensional triangular PFC model, as a function of the relative lattice constant $a/a_0$ for isotropic straining. The parameters $\bar{\psi}=0.3$ and ${\varepsilon}=0.6$ are used. The solid curve is for fixed amplitude $A=A(\Delta=0)$, whereas the dashed curve includes the physical non-linearity due to the elastic weakening $A=A(\Delta)$. []{data-label="fig3"}](fig3.pdf){width="8.5cm"}
The Eulerian strains are $$e_{xx}=e_{yy}=\frac{a-a_0}{a} - \frac{1}{2}\left( \frac{a-a_0}{a} \right)^2,\qquad e_{xy}=0.$$
If we now take the situation of isotropic deformations and also minimise the energy with respect to the amplitudes, all of them change their magnitude equally due to symmetry. Hence we have $A_j = A_{j,0}\exp(-i\mathbf{k}^{(j)}\cdot\mathbf{u})$ with the same real and positive value $A_0=-A_{1,0}=A_{2,0}=A_{3,0}$. Evaluation of the free energy density $f=f_{local}+f_{nonlocal}$ and minimisation with respect to $A_0$ gives $$\label{isotropicweakening2D}
A_0(\Delta) = \frac{1}{5} \left( \bar{\psi} + \frac{1}{3} \sqrt{9\bar{\psi}^2 + 15(\Gamma-\Delta)}\right),$$ which is the same as in Ref. , written here for the case $\bar{\psi}>0$. We note that for large values of ${\varepsilon}\gg \Delta$ the amplitudes hardly change with the strain, and then geometric non-linearity is essentially the only source for deviations from linear elasticity, as before in the one-dimensional case. The energy density is (again for general values of ${\varepsilon}$ and $\Delta$) $$f(\Delta) = \frac{45}{2}A_0^4(\Delta) - 12\bar{\psi}A_0^3(\Delta) - 3(\Gamma-\Delta)A_0^2(\Delta),$$ which is valid for isotropic deformations. Again the strain dependent amplitudes lead to physical non-linearity. Due to the amplitude as additional degree of freedom, which is used here for minimization, the energy is lower than for fixed amplitude, see Fig. \[fig3\].
The assumption of all amplitudes being the same in magnitude is valid for isotropic deformations only. Although the expression involving $\Delta$ may suggest that it holds also for other cases, this is not the case. For anisotropic deformations the amplitudes will in general change differently as a function of the applied strain. For general amplitudes $A_j = A_{j,0}\exp(-i\mathbf{k}^{(j)}\cdot\mathbf{u})$ the nonlocal energy contribution becomes $$\begin{aligned}
f_{nonlocal} &=& 4 |A_{1,0}|^2 \bar{e}_{yy}^2 \nonumber \\
&+& 4 |A_{2,0}|^2 \left ( \frac{3}{4} \bar{e}_{xx} - \frac{\sqrt{3}}{2} \bar{e}_{xy} + \frac{1}{4} \bar{e}_{yy} \right)^2 \nonumber \\
&+& 4 |A_{3,0}|^2 \left ( \frac{3}{4} \bar{e}_{xx} + \frac{\sqrt{3}}{2} \bar{e}_{xy} + \frac{1}{4} \bar{e}_{yy} \right)^2.\end{aligned}$$ From now on we assume that all prefactors $A_{j,0}$ are real. Then the local energy density reads $$\begin{aligned}
f_{local} &=& -\Gamma (A_{1,0}^2 + A_{2,0}^2 + A_{3,0}^2) \nonumber \\
&+& \frac{3}{2} \Big( A_{1,0}^4 + A_{2,0}^4 + A_{3,0}^4 + 4A_{1,0}^2A_{2,0}^2+ 4A_{1,0}^2A_{3,0}^2 \nonumber \\
&+& 4A_{2,0}^2A_{3,0}^2 \Big) + 12\bar{\psi} A_{1,0} A_{2,0}A_{3,0}.\end{aligned}$$ We have to minimise (for given strain) the energy with respect to all amplitudes $A_{j,0}$. To simplify the situation, we consider the case of uniaxial stretching in $x$ direction, i.e. $\bar{e}_{xy}=\bar{e}_{yy}=0$. Then by symmetry two amplitudes are equal, and we write $A_{1,0}=A<0$ and $A_{2,0}=A_{3,0}=B>0$. With this the energy densities become $$\begin{aligned}
f_{local} &=& -\Gamma(A^2+2B^2) + \frac{3}{2} (A^4+6B^4+8A^2B^2) \nonumber \\
&&+ 12AB^2\bar{\psi}\end{aligned}$$ and $$f_{nonlocal} = \frac{9}{2} B^2 \bar{e}_{xx}^2.$$ Minimization of $f$ has to be performed with respect to $A$ and $B$. From the minimisation with respect to $A$ we get $$B = \sqrt{\frac{(\Gamma-3A^2)A}{6(2A+\bar{\psi)}}},$$ where we have chosen the branch $B>0$. From the minimisation with respect to $B$ we get the condition $$-4\Gamma+ 36 B^2 +24A^2 + 24 A\bar{\psi} + 9\bar{e}_{xx}^2=0.$$ Fig. \[fig5\] shows the amplitudes as a function of the applied strain.
![Absolute values of the amplitudes as a function of the uniaxial strain $e_{xx} = \bar{e}_{xx}$ for $e_{xy}=e_{yy}=0$ in the two-dimensional PFC model. The solid line is the case of strain independent amplitudes, where only geometric non-linearity arises. The long dashed curve uses the approximation of equal strain dependence of the amplitudes according to Eq. (\[isotropicweakening2D\]), where all amplitudes are subject to the same elastic weakening. For that, the expression (\[eq33\]) is used, with $\bar{e}_{xx}$ being the only nonvanishing component. The remaining two curves show the unequal response of the amplitudes as a result of the uniaxial strain. Parameters are ${\varepsilon}=0.6$, $\bar{\psi}=0.3$.[]{data-label="fig5"}](fig4.pdf){width="8.5cm"}
As one can see the amplitudes indeed depend differently on the strain. The mode related to $\mathbf{k}^{(1)}$, which has a RLV perpendicular to the applied load, increases in magnitude as a function of strain; this mode does not carry elastic energy, therefore its increase in magnitude is not penalised. In contrast, the other two modes decrease in magnitude, and this more strongly than in the isotropic approximation.
The energy density can then be written as $$f = f - \frac{1}{2} \frac{\partial f}{\partial B} B = -\Gamma A^2 + \frac{3}{2} A^4 - 9B^4,$$ where the partial derivative is zero by the minimisation condition and leads to the first identity. The total energy as a function of the uniaxial change of the lattice constant is shown in Fig. \[fig5a\].
![Elastic energy density of the two-dimensional PFC model for a uniaxial strain $e_{xx} = \bar{e}_{xx}$ and $e_{xy}=e_{yy}=0$. The elastic energy is highest in the non-linear regime if the amplitudes are considered as constant (solid curve). The equal dependence on the strain, $-A_{1,0}=A_{2,0}=A_{3,0}$ reduces the energy and leads to the dashed curve. The dotted curve correctly considers unequal weakening of the amplitudes due to strain and leads to the lowest elastic energy. Parameters are ${\varepsilon}=0.6$, $\bar{\psi}=0.3$. []{data-label="fig5a"}](fig5.pdf){width="8.5cm"}
As expected, the energy is lower in the full anisotropic description compared to the isotropic approximation. The reason is that we allow for additional degrees of freedom, $A\neq-B$, which allow to further reduce the energy. By this, the contribution of physical non-linearity to the elastic response becomes more important.
Body-centered cubic materials {#sectionBCC}
=============================
The body-centred cubic (bcc) phase exists in equilibrium in some parameter regions of the three-dimensional phase field crystal model. Again we use a one-mode approximation according to Eq. (\[onemode\]), this time summing over $N=6$ normalised reciprocal lattice vectors, $$\begin{aligned}
&&\mathbf{k}_{110}= \left(
\begin{array}{c}
1/\sqrt{2}\\
1/\sqrt{2}\\
0
\end{array}
\right),
\qquad
\mathbf{k}_{101} = \left(
\begin{array}{c}
1/\sqrt{2}\\
0\\
1/\sqrt{2}
\end{array}
\right),
\nonumber \\
&&
\mathbf{k}_{011} = \left(
\begin{array}{c}
0\\
1/\sqrt{2}\\
1/\sqrt{2}
\end{array}
\right),
\qquad
\mathbf{k}_{1\bar{1}0} = \left(
\begin{array}{c}
1/\sqrt{2}\\
-1/\sqrt{2}\\
0
\end{array}
\right),
\nonumber \\
&&\mathbf{k}_{10\bar{1}} = \left(
\begin{array}{c}
1/\sqrt{2}\\
0\\
-1/\sqrt{2}
\end{array}
\right),
\qquad
\mathbf{k}_{01\bar{1}} = \left(
\begin{array}{c}
0\\
1/\sqrt{2}\\
-1/\sqrt{2}
\end{array}
\right). \label{3DRLVs}\end{aligned}$$
Similarly to above we get by insertion into the free energy and orthogonality (see Appendix \[appendix::aederivation\] and Refs. ) $$\begin{aligned}
F &=& \int d\mathbf{r} \Bigg[ 4 \sum_{j=1}^6 \left|\Box_j A_j\right|^2 + (3\bar{\psi}^2-{\varepsilon}) \sum_{j=1}^{6} A_j A_j^* \nonumber \\
&&+ 3 \Bigg\{ \left( \sum_{j=1}^{6} A_j A_j^* \right)^2 - \frac{1}{2}\sum_{j=1}^{6} |A_j|^4 \nonumber \\
&&+ 2A_{110}^* A_{1\bar{1}0}^* A_{101} A_{10\bar{1}} + 2A_{110} A_{1\bar{1}0} A_{101}^* A_{10\bar{1}}^*
\nonumber \\
&& + 2A_{1\bar{1}0} A_{011} A_{01\bar{1}} A_{110}^*
+ 2A_{1\bar{1}0}^* A_{011}^* A_{01\bar{1}}^* A_{110}
\nonumber \\
&& + 2A_{01\bar{1}} A_{10\bar{1}}^* A_{101} A_{011}^*
+ 2A_{01\bar{1}}^* A_{10\bar{1}} A_{101}^* A_{011} \Bigg\}
\nonumber \\
&&
+6\bar{\psi} \Big( A_{011}^* A_{101} A_{1\bar{1}0}^* + A_{011} A_{101}^* A_{1\bar{1}0} + A_{011}^* A_{110} A_{10\bar{1}}^* \nonumber \\
&&
+ A_{011} A_{110}^* A_{10\bar{1}}
+ A_{01\bar{1}}^* A_{110} A_{101}^* + A_{01\bar{1}} A_{110}^* A_{101}
\nonumber \\
&&
+ A_{01\bar{1}}^* A_{10\bar{1}} A_{1\bar{1}0}^* + A_{01\bar{1}} A_{10\bar{1}}^* A_{1\bar{1}0} \Big) \nonumber \\
&&
+ \frac{1}{2} \bar{\psi}^2(1-{\varepsilon}) + \frac{1}{4} \bar{\psi}^4 \Bigg], \label{nonlinAE}\end{aligned}$$ expressed here through the box operator $$\Box_j=\mathbf{k}_j\cdot\nabla - \frac{i}{2q_0}\nabla^2 =-\frac{i}{2q_0} L_j$$ with $q_0=|\mathbf{k}_j|=1$ and $\bar{\psi}<0$.
The nonlocal contribution from the box operator can be evaluated as before, and we get $$F_{nonlocal} = 4 \int d{\mathbf r} \sum_{j=1}^6 \left|\Box_j A_j\right|^2 = 4 \int d\mathbf{r} \bar{\Delta} |A_0|^2$$ with $$\begin{aligned}
\bar{\Delta} &=& \bar{e}_{xx}^2 + \bar{e}_{yy}^2 + \bar{e}_{zz}^2
+ 2 (\bar{e}_{xy}^2 + \bar{e}_{yz}^2 + \bar{e}_{xz}^2) \nonumber \\
&& + \bar{e}_{xx}\bar{e}_{yy}+ \bar{e}_{yy} \bar{e}_{zz} + \bar{e}_{xx} \bar{e}_{zz}, \label{eq49}\end{aligned}$$ where strain gradient terms are suppressed. Here we have assumed that all amplitudes have the same magnitude, $A_{j, 0}=A_0$. This is the case for an isotropic deformation $\bar{e}_{ij} = \bar{e}\delta_{ij}$, which leads to $$\begin{aligned}
F &=& \int d\mathbf{r} \Big\{ \left[4\bar{\Delta}+6(3\bar{\psi}^2-{\varepsilon})\right] A_0^2 + 48 \bar{\psi} A_0^3 \nonumber \\
&&+ 135 A_0^4 + \frac{1}{2} \bar{\psi}^2(1-{\varepsilon}) + \frac{1}{4} \bar{\psi}^4 \Big\}. \label{eq50}\end{aligned}$$ For fixed amplitudes $A_0$ we therefore see that — as before for the one- and two-dimensional case — the elastic part of the energy is linear in $\bar{\Delta}$ and therefore quadratic in the (Eulerian) strains $\bar{e}_{ij}$. As before, this gives rise to the geometric non-linearity, see Fig. \[figaa\].
![ Elastic energy density in the three-dimensional bcc PFC model, as a function of the relative lattice constant $a/a_0$ for isotropic straining. The parameters $\bar{\psi}=-0.18$ and ${\varepsilon}=0.1$ are used. The solid curve is for fixed amplitude $A_0=A(\Delta=0)$, whereas the dashed curve includes the physical non-linearity due to the elastic weakening $A_0=A_0(\Delta)$, see Eq. (\[isotropicamplitudebcc\]). []{data-label="figaa"}](fig6.pdf){width="8.5cm"}
We point out that here only the strain tensor $\bar{e}_{ij}$ as defined in Eq. (\[strangeEulerStrain\]) allows the compact notation of the elastic energy through $\bar{\Delta}$, similar to the two-dimensional case. In contrast, it is here not possible to represent the elastic energy in terms of $\Delta$ directly, which is defined via the Euler strains $e_{ij}$. The reason is that in the two dimensional triangular case (indicated here through a superscript ‘tri’) both $\Delta^{tri}$ and $\bar{\Delta}^{tri}$ can be expressed through the (identical) traces of the tensors $e$ and $\bar{e}$ or powers of them. Explicitly, one gets from Eqs. (\[eq33\]) and (\[eq34\]) $$\begin{aligned}
\Delta^{tri} &=& \mathrm{tr}(e^2) + \frac{1}{2} \mathrm{tr}(e)^2, \\
\bar{\Delta}^{tri} &=& \mathrm{tr}(\bar{e}^2) + \frac{1}{2} \mathrm{tr}(\bar{e})^2, \end{aligned}$$ indicating elastic isotropy [@LandauLifshitz:7]. In contrast, for the bcc system we cannot expect the equality of $\bar{\Delta}$ and $\Delta$ due to the cubic symmetry. Indeed, such a representation is not possible for the three-dimensional bcc expression (\[eq49\]) and an analogous term $\Delta$ involving $e_{ij}$. One can readily check that the equivalence of $\bar{\Delta}$ and $\Delta$ fails for specific situations with nonvanishing shear.
For an isotropic deformation $e_{ij}=e_{xx}\delta_{ij}$ we can identify in the small strain regime (where all strain tensors coincide and the deformed and reference volume are the same) the bulk modulus $K$ by the comparison with the elastic part of the energy $F_{nonlocal}=9 K V e_{xx}^2/2$ as $K=16 |A_0|^2/3$. In the following we relax the amplitudes to obtain physical non-linearity, first again for the isotropic and then an anisotropic situation.
If we minimise the energy (\[eq50\]) with respect to $A_0$ in the isotropic case, it becomes a function of $\bar{\Delta}$. Explicitly, we get $$\label{isotropicamplitudebcc}
A_0(\bar{\Delta}) = \frac{1}{45} \left(-6\bar{\psi} + \sqrt{3} \sqrt{15{\varepsilon}-33\bar{\psi}^2-10\bar{\Delta}}\right).$$ In this case the free energy becomes a non-linear function of $\bar{\Delta}$, see Fig. \[figaa\]. This is important, as through the quadratic nature of $\bar{\Delta}$ the elastic energy is symmetric if plotted versus the isotropic strain $\bar{e}_{xx}=\bar{e}_{yy}$, even with strain dependent amplitudes. Already at this point we mention that this outcome suggests to inspect the elastic energy of real materials as a function of $\bar{e}_{ij}$ instead of the Lagrangian variant $\epsilon_{ij}$. This will be pursued in the following section.
We conclude the analysis similar to the previous two-dimensional case with the situation of an anisotropic strain, where the amplitudes depend differently on the mechanical load. We use a homogeneous uniaxial strain $e_{xx}=\bar{e}_{xx}$, assuming that all other strain components $\bar{e}_{ij}$ vanish. In this case the amplitudes group into two sets, having the same magnitude in each of these groups. The first set contains amplitudes with RLVs perpendicular to the strain, i.e. $\mathbf{k}_j\cdot\hat{\mathbf{x}}=0$, namely $A:=A_{011, 0}=A_{01\bar{1}, 0}$. The remaining amplitudes with $\mathbf{k}_j\cdot\hat{\mathbf{x}}\neq 0$ have the same magnitude, denoted as $B$, i.e. $B:=A_{110, 0}=A_{1\bar{1}0, 0}=A_{101, 0}=A_{10\bar{1}, 0}$. With $\bar{\Delta} = \bar{e}_{xx}^2$ we obtain the free energy density $$\begin{aligned}
&&f = 9 A^4 + 48 AB^2 \bar{\psi} + A^2 (72 B^2 - 2 {\varepsilon}+ 6 \bar{\psi}^2) \\ \nonumber
&& + \frac{1}{4} \left[ 216 B^4 + \bar{\psi}^2 (2-2{\varepsilon}+ \bar{\psi}^2) + 16 B^2 (\bar{\Delta} - {\varepsilon}+ 3 \bar{\psi}^2) \right]. \end{aligned}$$ By minimization with respect to $A$ and $B$ we obtain the strain dependent amplitudes. The isotropic and anisotropic amplitude relaxation is shown in Fig. \[fig5d1\] and its effect on the free energy in Fig. \[fig5d2\].
![Values of the amplitudes for the bcc model as a function of the uniaxial strain $e_{xx} = \bar{e}_{xx}$, and all other strain components vanish, $\bar{e}_{ij}=0$. The solid line is the case of strain independent amplitudes. The dashed curve is based on the isotropic approximation, where all amplitudes equally depend on the strain according to Eqs. (\[eq49\]) and (\[isotropicamplitudebcc\]). The two dotted curves are based on the true minimization with two independent amplitudes $A$ and $B$. Parameters are ${\varepsilon}=0.1$, $\bar{\psi}=-0.18$. []{data-label="fig5d1"}](fig7.pdf){width="8.5cm"}
![ Elastic energy density of the three-dimensional bcc model for a uniaxial strain $e_{xx} = \bar{e}_{xx}$, and all other strain components vanish. The elastic energy is highest in the non-linear regime if the amplitudes are considered as constant (solid curve). The equal dependence of the amplitudes on the strain according to Eqs. (\[eq49\]) and (\[isotropicamplitudebcc\]) reduces the energy and leads to the dashed curve. Consideration of anisotropic weakening leads to the lowest energy (dotted curve). Parameters are ${\varepsilon}=0.1$, $\bar{\psi}=-0.18$. []{data-label="fig5d2"}](fig8.pdf){width="8.5cm"}
For the used parameters the influence of the amplitude relaxation (physical non-linearity) is weaker than for the previous one- and two-dimensional cases, where we used a lower undercooling ${\varepsilon}$. For smaller values of ${\varepsilon}$ we find here a rather small range of strains before the phase becomes unstable. This is related to the narrow single phase region of bcc in the phase diagram. Also, the influence of anisotropic versus isotropic amplitude relaxation is lower than for the two-dimensional triangular model. However, we recall that already a decrease of few percent in the free energy can significantly influence phase coexistence regimes. In the low temperature limit the non-linear elastic energy stems from geometric non-linearity alone, and then the elastic energy becomes a quadratic function of the strain components $\bar{e}_{ij}$. This prediction will be compared to [*ab initio*]{} results in the following section.
The Birch-Murnaghan equation: Comparison with ab initio simulations {#BirchMurnaghanSection}
===================================================================
The Murnaghan[@MurnaghanFD:1944pnas] and Birch-Murnaghan equations of state[@BirchF:1947pr] are used to describe the non-linear elastic response under isotropic stretching or compression. They are frequently used to fit [*ab initio*]{} data for energy-volume curves. Birch[@BirchF:1947pr] emphasises the importance of the distinction between Lagrangian and Eulerian descriptions, and notes that the representation is simpler in the Eulerian frame. Our findings support this concept from a PFC perspective. It is therefore the goal of this section to compare the PFC model with the classical equations for energy-volume curves and to further link it to [*ab initio*]{} calculated energy-strain curves of elementary bcc systems and graphene.
The Birch-Murnaghan model describes the energy as a function of volume as $$\begin{aligned}
\label{BM3D}
E_\mathrm{BM}(V) &=& E_0 + \frac{9 V_0 K}{16} \Bigg\{ \left[ \left( \frac{V_0}{V} \right)^{2/3} -1 \right]^3 K' \nonumber \\
&+& \left[ \left( \frac{V_0}{V} \right)^{2/3} -1 \right]^2 \left[ 6-4\left( \frac{V_0}{V}\right)^{2/3} \right] \Bigg\}.\end{aligned}$$ Here $V_0$ is the equilibrium volume, $V$ the actual volume of the isotropically deformed system, $K$ the zero pressure bulk modulus and $K'=(dK/dP)_{P=0}$ the derivative of the bulk modulus. The latter quantity is (usually) positive, as materials get stiffer under compression. The above equation is applicable for three dimensions.
One can derive the pressure for the present isotropic case from the standard thermodynamic relation $$\label{thermointen1}
P = -\left( \frac{\partial E_\mathrm{BM}}{\partial V} \right)_{N, T}$$ and from this get the bulk modulus as $$\label{thermointen2}
K(V) = - V \left( \frac{\partial P}{\partial V} \right)_{N, T}.$$ In the limit of vanishing pressure, i.e. for $V\to V_0$, one gets the leading contribution, which is denoted above as a constant $K$. The derivative of the bulk modulus for zero pressure then follows from $$\label{thermointen3}
K'= \lim_{V\to V_0} \frac{\displaystyle \left( \frac{\partial K}{\partial V}\right)_{N, T}}{\displaystyle \left( \frac{\partial P}{\partial V}\right)_{N, T}}.$$
In order to link this equation to the phase field crystal model we rewrite the Birch-Murnaghan expression (\[BM3D\]) in terms of the Eulerian strain, $$e_{xx} = e_{yy} = e_{zz} = \frac{a-a_0}{a} - \frac{1}{2} \left( \frac{a-a_0}{a} \right)^2 = \frac{a^2-a_0^2}{2a^2}.$$ Here we particularly have $\bar{e}_{xx}=e_{xx}$, $\bar{e}_{yy}=e_{yy}$, $\bar{e}_{zz}=e_{zz}$. With $V=a^3$ and $V_0=a_0^3$ we therefore get in three dimensions $$\label{Mur3}
\left( \frac{V_0}{V} \right)^{2/3} = 1-2e_{xx}$$ and arrive at the compact representation $$\label{Mur1}
E_\mathrm{BM}(e_{xx}) = \frac{9}{2} K V_0 e_{xx}^2 \left[ 1+ (4-K') e_{xx} \right].$$ For small strains $|e_{xx}|\ll 1$ it reduces to the usual linear elastic energy $E_\mathrm{BM}(V) \approx 9 KV_0 \epsilon_{xx}^2/2$. For many materials the bulk modulus derivative turns out to be close to $K'=4$, and in this case we obtain the simple expression $$\label{Mur2}
E_{\mathrm{BM}, K'=4}(e_{xx}) = \frac{9}{2} K V_0 e_{xx}^2,$$ which also holds in the non-linear regime. Notice that the reference volume $V_0$ instead of the actual volume $V$ appears here. As will be shown below this formula fits very well the [*ab initio*]{} data for various elemental metals. From Eq. (\[Mur1\]) we see that deviations from $K'=4$ break the symmetry between compression and expansion $e_{xx}\to -e_{xx}$.
The older Murnaghan model[@MurnaghanFD:1944pnas] is given by $$\begin{aligned}
E_\mathrm{M}(V) &=& E_0 + K V_0 \Bigg[ \frac{1}{K'(K'-1)} \left( \frac{V}{V_0} \right)^{1-K'} \nonumber \\
&+& \frac{1}{K'} \frac{V}{V_0} - \frac{1}{K'-1} \Bigg].\end{aligned}$$ Expanding it in terms of the Euler strain gives $$\begin{aligned}
E_\mathrm{M}(e_{xx}) &=& \frac{9}{2} K V_0 e_{xx}^2 \Bigg[ 1+ (4-K') e_{xx} \nonumber \\
&+& \frac{1}{12}(143-63K'+9{K'}^2) e_{xx}^2 + {\cal O}(e_{xx}^3) \Bigg].\end{aligned}$$ It agrees with the Birch-Murnaghan model up to third order in $e_{xx}$.
In comparison the three-dimensional bcc phase field crystal model with constant amplitudes delivers the comparable expression for the averaged elastic free energy density $$\label{eq65}
f_\mathrm{PFC}(e_{xx}) = \frac{9}{2} K e_{xx}^2,$$ with the identification $K = 16|A_0|^2/3$. In the spirit of the discussion in Section \[pfcintrosec\] we can therefore conclude that the phase field crystal is analogous to the Birch-Murnaghan model for $K'=4$ in the low temperature limit. This includes in particular that the elastic energy is symmetric with respect to the Eulerian strain for bulk deformations. As discussed before, the effect of strain dependent amplitudes for isotropic deformations can lower the elastic energy, which can lead to deviations from the Birch-Murnaghan curve. Still, the PFC energy will remain quadratic in the strains, and therefore in particular symmetric under the exchange $e_{xx}\to-e_{xx}$. Hence it affects only higher order corrections of the elastic energy starting at ${\cal O}(e_{xx}^4)$. This effect is most pronounced for small values of $\epsilon$, which corresponds to high temperatures.
To shed light on the specific value $K'=4$, which is suggested by the three-dimensional PFC model, we performed nonmagnetic [*ab initio*]{} simulations for various bcc elements. $T=0 K$ results for the total energy as a function of the lattice constant are shown in Fig. \[figabinitio1\].
![(Color online) Elastic energy as a function of the lattice constant for various nonmagnetic bcc metals. Notice that in this representation the energy is not symmetric around the minimum position $a_0$.[]{data-label="figabinitio1"}](fig9.pdf){width="8.5cm"}
If we present this data as a function of the Eulerian strain, we find that it becomes symmetric for many metals apart from lithium. This symmetry corresponds to $K'=4$ in the Murnaghan models, in agreement with the PFC prediction. The exception Li has a slight asymmetry and a value of $K'\approx 3.5$. We note that the fitted values of $K'$ have an uncertainty, as can be seen from the (small) difference between Li and the other elements. The main conclusion is that the shown bcc elements essentially lead to a parabolic curve if represented in terms of the Euler strain, which means that $K'$ is at least not too far from $K'=4$. Fitting the bulk modulus from the curvature near the minimum therefore allows to reduce all data (apart from Li) to one curve, see Fig. \[figabinitio2\].
![(Color online) Elastic energy as a function of the isotropic Eulerian strain for various elemental metals, normalised to the bulk modulus. All data collapses to a simple Master curve $E=9 K V_0 e_{xx}^2/2$, which is equivalent to the Birch-Murnaghan model with $K'=4$ (continuous black curve). Lithium slightly deviates from this curve and has a value of $K'\approx 3.5$.[]{data-label="figabinitio2"}](fig10.pdf){width="8.5cm"}
In essence, we can therefore conclude that the PFC and amplitude equations models, which predict $K'=4$ for bcc, are able to capture well the low temperature non-linear elasticity for various elements. For these low temperature applications, a large value of $\epsilon\gg\bar{\Delta}$ has to be chosen, such that the amplitudes are essentially strain independent. By adjusting the value of $q_0$ according to the equilibrium lattice constant, $q_0=2\pi/a_0$, and multiplying the phase field crystal energy with a dimensional energy prefactor, one matches the bulk modulus of each element.
We can exploit the comparison between the phase field crystal and the [*ab initio*]{} calculations even further. For uniaxial stretching in \[100\] direction we can again predict the elastic energy and compare it to the $T=0$ [*ab initio*]{} results. The continuum theory predicts that the energy should be linear in $\Delta=e_{xx}^2=\bar{\Delta}=\bar{e}_{xx}^2$ in the low temperature regime. The [*ab initio*]{} data for tungsten fully confirms this expectation, see Fig. \[fig11\].
![Uniaxial and isotropic straining of tungsten. The normalized elastic energy is shown versus the parameter $\Delta=\bar{\Delta}$ for these loadings. The [*ab initio*]{} data falls on a straight line for compression and tension for isotropic loading; the theoretical prediction is the solid line. For uniaxial loading the data also collapses onto a straight (dashed) line both for tension and compression, but exhibits a slightly different slope due to violations of the Cauchy relation. []{data-label="fig11"}](fig11.pdf){width="8.5cm"}
Also, the behavior is symmetric under tension and compression in this representation, as the data (open and filled squares) falls onto a single straight line. Moreover, from the definition of $\bar{\Delta}$ in Eq. (\[eq49\]), we expect that the elastic energy for the isotropic stretching should be six times larger than for the uniaxial stretching for the same value of $e_{xx}$, as then $\bar{\Delta}=6\bar{e}_{xx}^2$. Fig. \[fig11\] therefore also contains the previous data for tungsten for isotropic straining, both in the compressive and tensile regime (open and filled circles, respectively). The describing straight line has a similar, but slightly different slope compared to the uniaxial case. We attribute this to slight deviations from the Cauchy relation $C_{12}=C_{44}$ for tungsten; this relation is exactly fulfilled in the PFC model[@Spatschek:2010fk]. Still, the PFC model gives an excellent description also for this type of mechanical loading.
Analogous to the three-dimensional expression (\[Mur1\]) we can propose a similar expression for two dimensions. For that we start with the ansatz $$\label{Mur2Dguessed}
E_{2D} = 2 K_{2D} V_0 e_{xx}^2 [1+\alpha(\beta-K'_{2D})e_{xx}],$$ where the undeformed two-dimensional volume is $V_0=a_0^2$, in comparison to the deformed volume $V=a^2$. We note that the choice of the global prefactor $2$ is here a matter of choice and only rescales the two-dimensional bulk modulus $K_{2D}$, which is not in the focus of the present investigations. The non-linear Euler strain is $$e_{xx}=\bar{e}_{xx}=e_{yy}=\bar{e}_{yy}= \frac{1}{2} \left(1-\frac{V_0}{V} \right), $$ and all other strain components vanish for an isotropic deformation. The coefficients $\alpha$ and $\beta$ in Eq. (\[Mur2Dguessed\]) are determined by the requirement that the zero strain bulk modulus derivative (\[thermointen3\]) is recovered. From this we get $\alpha=2/3$ and $\beta=5$. This can be compared with the low temperature limit ${\varepsilon}\gg\Delta$ of the 2D PFC result (\[intermediateresult\]), $$\label{Mur2DPFC}
f_{\mathrm{PFC}, 2D} = \frac{1}{2} K_{2D} e_{xx}^2,$$ where a term, which is cubic in the strain $e_{xx}$, does not appear for constant amplitudes. Here we have identified $K_{2D}=24 |A_0|^2$. These results therefore suggest $K'_{2D}=5$.
We have performed [*ab initio*]{} simulations of graphene for isotropic deformations, $e_{xx}=e_{yy}$. The data, plotted versus the Euler strain $e_{xx}$ is shown in Fig. \[figgraphene\].
![Elastic energy of graphene, plotted versus the Euler strain, as obtained from the [*ab initio*]{} simulations. The functional form slightly deviates from a pure parabola and is well described by Eq. (\[Mur2Dguessed\]) with $K'_{2D}\approx 4.3$. The filled symbols are the DFT data with the closed curve according to Eq. (\[Mur2Dguessed\]). The open symbols are the DFT data shown as a function of negative strain $e_{xx}$ to visualize the slight asymmetry and hence the deviation from a purely parabolic function. []{data-label="figgraphene"}](fig12.pdf){width="8cm"}
The functional form is again very close to the parabolic (and symmetric) form (\[Mur2DPFC\]), but shows a small asymmetry, which can be fitted by $K'_{2D}\approx 4.3$ in Eq. (\[Mur2Dguessed\]). The deviation may be due to the effect that the graphene structure deviates from the triangular structure of the 2D PFC model. The extension of the analysis to the recent graphene model [@MSeymour:2016aa] may shed light on this issue. Still, we can conclude from Fig. \[figgraphene\] that the (standard) PFC model gives a good description of the elastic response in a wide strain regime also for graphene.
Summary and conclusions {#SummaryConclusion}
=======================
We have analysed the non-linear elastic response of the phase field crystal in one, two and three dimensions for different crystal structures. First, we have elaborated that the proper interpretation of (non-linear) elasticity is via the energy [*density*]{} in the PFC model, which has to be compared to the energy [*per unit cell*]{} for discrete atomistic descriptions.
A natural outcome of the differential operators in the amplitude equation formulation of the PFC model is the appearance of geometric non-linearity. For elevated temperatures, additional physical non-linearity appears, which shows up via strain dependent amplitudes and can be understood as precursors of stress induced melting.
Both with and without physical non-linearity the response to deformation can be described through the non-linear strain tensor $\bar{e}_{ij}$ as given by Eq. (\[strangeEulerStrain\]), which is based on the right Cauchy-Green deformation tensor. For the one- and two-dimensional case and isotropic loading the elastic response can equivalently be described through the Euler-Almansi tensor $e_{ij}$, but this is not the case for bcc ordering. In general, the phase field crystal model has to be interpreted as a Eulerian description of elasticity. A particular outcome is that the non-linear elastic energy depends on the strain tensor components $\bar{e}_{ij}$ in a symmetric way under compression and tension, as expressed through the dimensionless quantity $\bar{\Delta}$.
In the low temperature limit the PFC predictions for energy-volume curves coincide with the Birch-Murnaghan expression in three dimensions with bulk modulus derivative $K'=4$, and $K'_{2D}=5$ in two dimensions. These suggested values are in good agreement with [*ab initio*]{} calculated energy-volume curves for various nonmagnetic bcc elements and graphene. Also, other deformations like large uniaxial strains are well described by the PFC model. We can therefore conclude that the phenomenological PFC model is well suitable to describe non-linear elastic deformation.
It is quite remarkable that the heuristic PFC model in its simplest form can capture a wide range of non-linear elastic response so well, as compared to electronic structure calculations. It suggests that the effect of elastic deformations is already well described by the effective atom densities, which are the basis for the classical density functional theory and therefore the PFC model. Moreover, the results indicate that the one-mode approximation is particularly good for the bcc elements. It is known that the representation of e.g. fcc requires to include more modes and reciprocal lattice vectors[@Wu:2010vn]. The investigation of the non-linear elastic response for these cases will be subject of future research activities.
This work has been supported by the DFG priority program SPP 1713. The authors gratefully acknowledge the computing time granted on the supercomputer JURECA at the Jülich Supercomputing Centre (JSC). Furthermore, this research was supported by the Academy of Sciences of the Czech Republic through the Fellowship of Jan Evangelista Purkyně (M.F.). The access to the computational resources provided by the MetaCentrum under the program LM2010005 and the CERIT-SC under the program Center CERIT Scientific Cloud, part of the Operational Program Research and Development for Innovations, Reg. No. CZ.1.05/3.2.00/08.0144, is highly appreciated. This work was also supported by the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070), funded by the European Regional Development Fund and the national budget of the Czech Republic via the Research and Development for Innovations Operational Program, as well as Czech Ministry of Education, Youth and Sports via the project Large Research, Development and Innovations Infrastructures (LM2011033).
Derivation of the amplitude equations {#appendix::aederivation}
=====================================
We use here a derivation of the amplitude equations which starts directly from the energy functional instead from the evolution equations, following the approach in Ref. . This offers a direct way to obtain the proper equations without the need to create a generating functional a posteriori [@Wu:2007kx]. We also point out that the rotationally invariant different operator, which is frequently found in amplitude equations comes out automatically, without the need to justify it from more complex renormalization group theory approaches [@PhysRevLett.80.3888; @PhysRevLett.76.2185]. We briefly illustrate this for the two-dimensional triangular model and then give the result for the three-dimensional bcc case, which can be obtained similarly.
We start from the PFC energy functional (\[pfcfunctional\]) for $q_0=1$. The amplitude expansion is written according to Eq. (\[onemode\]) with the reciprocal lattice vectors given in (\[2DRLVs\]). To get the amplitude equation energy functional we insert the one-mode approximation into the PFC energy and assume that the amplitudes are varying on a scale much larger than $1/q_0$. Then we get under the assumption that the system volume $V$ is a multiple of the lattice unit $$\int_V f(\{A_j\}) \exp(i\mathbf{k}\cdot\mathbf{r}) d\mathbf{r} = \int_V f(\{A_j\}) \delta_{\mathbf{k}, 0} d\mathbf{r}$$ for any slow function $f(\{A_j\})$ of the amplitudes. In particular, this step includes the averaging over unit cells. For the quartic term $$F_4 := \frac{1}{4} \int d\mathbf{r} \psi^4,$$ only terms, which belong to a closed polygon of RLVs, contribute. In the following we decompose $F_4=F_{44}+F_{43}+F_{42}+F_{40}$, where the second subscript denotes the order of the amplitudes.
For the contribution, which is quartic in the amplitudes $A_j$, there are two types of RLV configurations, namely $\mathbf{k}^{(i)}-\mathbf{k}^{(i)}+\mathbf{k}^{(i)}-\mathbf{k}^{(i)}=0$ and $\mathbf{k}^{(i)}-\mathbf{k}^{(i)}+\mathbf{k}^{(j)}-\mathbf{k}^{(j)}=0$ with $i\neq j$. Forming all possible combinations we get for this $$\begin{aligned}
F_{44}/V &=& \frac{1}{4} \Big( 6 |A_1|^4 + 6 |A_2|^4 + 6 |A_3|^4 + \\
&=& 24|A_1|^2|A_2|^2 + 24|A_1|^2|A_3|^2 + 24|A_2|^2|A_3|^2 \Big). \nonumber\end{aligned}$$ For each product of amplitudes, which leads to a closed polygon of reciprocal lattice vectors, we have to count the number of combinations with which it appears in the term proportional to $\psi^4$. For the products, which consist of $A_j^2 {A_j^*}^2$ we therefore have to arrange all possible combinations of these four factors. The coefficient 6 appears, because there are $4\cdot 3/2$ possibilities for placing the amplitude $A_j$ twice, and the remaining two places in a product are occupied by $A_j^*$. For the mixed terms we get $4\cdot 3 \cdot 2 \cdot 1=24$ possibilities to arrange $A_1$, $A_1^*$, $A_2$ and $A_2^*$. This completes the calculation of the quartic terms.
The cubic terms are also generated by $F_4$. Here the only closed polygon is $\mathbf{k}^{(1)}+\mathbf{k}^{(2)}+\mathbf{k}^{(3)}=0$, hence we get terms containing $A_1A_2A_3$ and $A_1^*A_2^*A_3^*$. Notice that one of the amplitudes ($A_1$) is negative, hence these products are negative for positive $\bar{\psi}$, see Eq. (\[2Damplitudesigns\]). This is necessary to stabilise the solid phase. We get $$F_{43}/V = \frac{1}{4} \bar{\psi} \left( 24 A_1A_2A_3 + 24 A_1^*A_2^*A_3^* \right).$$ We have $4 \cdot 3 \cdot 2 \cdot 1$ possibilities to arrange the four factors $A_1$, $A_2$, $A_3$, $\bar{\psi}$ or $A_1^*$, $A_2^*$, $A_3^*$, $\bar{\psi}$.
The quadratic terms stem from the $\psi^4$, $\psi^2$ and the gradient term. We start with $F_{42}$. Only combinations of antiparallel reciprocal lattice vectors contribute here. There we get $$F_{42}/V = \frac{1}{4}\bar{\psi}^2 \left( 12|A_1|^2 + 12|A_2|^2 + 12|A_3|^2\right),$$ with $12=4\cdot3\cdot2/2$ for choosing the positions of the individual factors in a product of $2\times\bar{\psi}$, $A$, $A^*$.
There is no contribution of the type $F_{41}$, as the oscillating factors cancel. However, there is a term $F_{40}$, which only involves the constant contributions, $$F_{40}/V = \frac{1}{4} \bar{\psi}^4.$$
The local quadratic energy is defined as $$F_2 := \frac{1}{2} (1-{\varepsilon})\int d\mathbf{r} \psi^2.$$ It gives $$F_2/V = \frac{1}{2} (1-{\varepsilon}) \left( 2 |A_1|^2+2 |A_2|^2+ 2|A_3|^2 + \bar{\psi}^2\right).$$
The most difficult term is the one which contains gradients of $\psi$. It leads both to terms which are local and nonlocal in the amplitudes. It is defined as $$\begin{aligned}
F_{grad} &=& \int f_{grad} d\mathbf{r} = \int \left( \frac{\psi}{2} (\nabla^4\psi + 2\nabla^2\psi) \right) d\mathbf{r} \nonumber \\
&=& F_{grad, A} + F_{grad, \nabla A}.\end{aligned}$$ The term $F_{grad, A}$ contains only the terms which are local in $A$, because the differentiation acts on the exponential term in the product rule. Then, each differentiation simply brings down a factor $\pm i\mathbf{k}$, and we get with $|\mathbf{k}|=1$ $$\begin{aligned}
F_{grad, A} &=& \int \bigg[ 2\times\frac{1}{2} \left( |A_1|^2 + |A_2|^2 + |A_3|^2 \right) \nonumber \\
&& - 2\times \left( |A_1|^2 + |A_2|^2 + |A_3|^2 \right) \bigg] d\mathbf{r} \nonumber \\
&=& -\int \left( |A_1|^2 + |A_2|^2 + |A_3|^2 \right) d\mathbf{r},\end{aligned}$$ where again we retained only the term which do not contain fast oscillating factors. The factor $2\times$ comes from the fact that each combination $A_iA_i^*$ can be obtained with either $A_i^*$ or $A_i$ being in front in a product $\psi^2$.
Altogether, the local terms therefore form the “double well potential”, in analogy to classical phase field models. It is given by $$\begin{aligned}
F_{dw} &=& F_{44} + F_{43} + F_{42} + F_{40} + F_2 + F_{grad, A} , \nonumber \\
&=& \int d\mathbf{r} \Bigg\{ \frac{3}{2} |A_1|^4 + \frac{3}{2} |A_2|^4 + \frac{3}{2} |A_3|^4 + 6|A_1|^2|A_2|^2 \nonumber \\
&& + 6|A_1|^2|A_3|^2 + 6|A_2|^2|A_3|^2 \nonumber \\
&&+ \bar{\psi} \left( 6 A_1A_2A_3 + 6 A_1^*A_2^*A_3^* \right) \nonumber \\
&&+ (3\bar{\psi}^2-{\varepsilon}) \left( |A_1|^2 + |A_2|^2 + |A_3|^2 \right) \nonumber \\
&& + \frac{1}{2}(1-{\varepsilon})\bar{\psi}^2 + \frac{1}{4} \bar{\psi}^4 \Bigg\},\end{aligned}$$ which coincides with Chan’s and Goldenfeld’s result[@Chan:2009aa], apart from the terms independent of the amplitudes. We point out that they are not relevant for the amplitude dynamics, as they vanish during the variational derivative. However, these terms still influence the energy and are therefore required to compare the PFC energy (\[pfcfunctional\]) with the one expressed through the amplitudes.
For evaluating the nonlocal terms in $A$ we perform an integration by part of $F_{grad}$ and retain afterwards only the derivatives acting on $A$, not on the exponential factor, which are already covered by $F_{grad, A}$ (the previous local contribution is the same whether we use the integration by part or not). Hence $$F_{grad} = \int \left( \frac{1}{2} (\nabla^2\psi)^2 - (\nabla\psi)^2 \right) d\mathbf{r}.$$ From this we get $$F_{grad, \nabla A} = \int \sum_{j=1}^3 \left| (\nabla^2 + 2i \mathbf{k}^{(j)}\cdot\nabla)A_j \right|^2 d\mathbf{r},$$ where again we retain only the terms which contain gradients of the amplitudes. With ${L}_j=\nabla^2+2i\mathbf{k}^{(j)}\cdot\nabla$ therefore altogether $$\begin{aligned}
F &=& \int d\mathbf{r} \Bigg\{ \sum_{j=1}^3 \left| {L}_j A_j \right|^2 + \frac{3}{2} |A_1|^4 + \frac{3}{2} |A_2|^4 + \frac{3}{2} |A_3|^4 \nonumber \\
&&+ 6|A_1|^2|A_2|^2 + 6|A_1|^2|A_3|^2 + 6|A_2|^2|A_3|^2 \nonumber \\
&&+ \bar{\psi} \left( 6 A_1A_2A_3 + 6 A_1^*A_2^*A_3^* \right) \nonumber \\
&&+ (3\bar{\psi}^2-{\varepsilon}) \left( |A_1|^2 + |A_2|^2 + |A_3|^2 \right) \nonumber \\
&& + \frac{1}{2}(1-{\varepsilon})\bar{\psi}^2 + \frac{1}{4} \bar{\psi}^4 \Bigg\}.\end{aligned}$$ Apart from the last two amplitude independent terms this expression is the same as in Eq. (\[eq22\]). Alternatively, we express the differential operator as ${L}_j = 2i\,\Box_j$ using $$\Box_j=\mathbf{k}^{(j)}\cdot\nabla - \frac{i}{2q_0}\nabla^2$$ with $q_0=|\mathbf{k}^{(j)}|$.
For the three-dimensional bcc model we can proceed in the same way. We start from the same phase field crystal model, but this time with the reciprocal lattice vectors given in (\[3DRLVs\]). Notice that $\bar{\psi}$ is assumed to be negative there, in agreement with the calculation by Wu and Karma[@Wu:2006uq; @Wu:2007kx]. By inserting the amplitude expansion into the functional and integrating over multiples of the unit cells we obtain similarly to above the functional (\[nonlinAE\]).
|
---
abstract: 'A measurement of neutrinoless double beta decay in one isotope does not allow to determine the underlying physics mechanism. We discuss the discrimination of mechanisms for neutrinoless double beta decay by comparing ratios of half life measurements for different isotopes. Six prominent examples for specific new physics contributions to neutrinoless double beta decay are analyzed. We find that the change in corresponding ratios of half lives varies from 60% for supersymmetric models up to a factor of 5-20 for extra-dimensional and left-right-symmetric mechanisms.'
author:
- Frank Deppisch
- Heinrich Päs
title: Pinning down the mechanism of neutrinoless double beta decay with measurements in different nuclei
---
An uncontroversial detection of neutrinoless double beta ($0\nu\beta\beta$) decay [@Doi:1982dn; @revs; @vogel; @paesrev] will be a discovery of uttermost significance. Most importantly, it will prove lepton number to be broken in Nature, and neutrinos to be Majorana particles [@petcov]. On the other hand, it will immediately generate another puzzle: what is the mechanism that triggers the decay? The most prominently discussed mechanism for neutrinoless double beta decay is the exchange of light Majorana neutrinos. But other mechanisms, like the exchange of SUSY superpartners with R-parity violating or conserving couplings, leptoquarks, right-handed W-bosons or Kaluza-Klein excitations, among others, have been discussed in the literature as well. Possibilities to disentangle at least some of the possible mechanisms include the analysis of angular correlations between the emitted electrons [@Doi:1982dn; @Ali:2006iu] or a comparative study of $0\nu\beta\beta$ and $0\nu\beta^+$ with electron capture ($EC$) decay [@Hirsch:1994es]. Another possibility seems to be the study of double beta decay to excited $0^+$ states [@0+]. Unfortunately, the search for $0\nu\beta^+/EC$ decay is complicated due to small rates and the experimental challenge to observe the produced X-rays or Auger electrons, and most double beta experiments of the next generation are not sensitive to electron tracks or transitions to excited states.
Without identification of the underlying mechanism, an experimental evidence for neutrinoless double beta decay will only provide ambiguous information about the concrete physics underlying the decay. For example, no information about the neutrino mass can be obtained from a measurement of the neutrinoless double beta decay half life.
In general, contributions to neutrinoless double beta decay can be categorized as either long-range or short-range interactions. In the first case, the diagram involves two vertices which are pointlike at the Fermi scale, and the exchange of a light neutrino in between, and is described by an effective Lagrangian of the type [@Pas:1999fc] $${\cal L} = \frac{G_F}{\sqrt{2}}\left( j_{V-A}^{\mu}J_{V-A,\mu}+ \sum
\epsilon_{NP} j_{NP} J_{NP}\right), \label{lrlag}$$ where the sum runs over all Lorentz invariant combinations of hadronic and leptonic Lorentz currents of defined helicity, $J_{NP,V-A}=\bar{u} {\cal O}_J d$ and $j_{NP,V-A} = \bar{e} {\cal
O}_j \nu$, respectively. Here ${\cal O}_{J,j} $ denotes the corresponding transition operator. The effective coupling strengths in new physics contributions are denoted as $\epsilon_{NP}$ throughout. For short-ranged contributions, on the other hand, the interactions are described by a single vertex being pointlike at the Fermi scale. The decay rate therefore results from first order perturbation theory, and is described by the Lagrangian [@Pas:2000vn] $$\begin{aligned}
\label{lagsr}
{\cal L} &=& \frac{G^2_F}{2} m_p^{-1} \sum \epsilon_{NP} J_{NP}
J_{NP} j'_{NP}.\end{aligned}$$ Here $m_p$ denotes the proton mass and the sum runs over all Lorentz invariant combinations of hadronic, $J_{NP}=\overline{u}{\cal O}_J d $, and leptonic, $j'_{NP}=\overline{e}{\cal O}_{j} e^C$, currents of defined chirality.
The combination involving two vertices of the first term in (\[lrlag\]) leads to the usual neutrinoless double beta decay half life formula for the mass mechanism, $$[T_{1/2}^{m_\nu}]^{-1}=
(\langle m_\nu \rangle/m_e)^2 G_{01}|{\cal M}^{m_\nu}|^2,$$ where $\langle m_\nu \rangle$ is the effective neutrino mass in which the contributions of individual neutrino mass eigenstates are weighted by mixing matrix elements squared, $\langle m_\nu
\rangle = |\sum U_{ei}^2 m_i|$. The combination of the first term in (\[lrlag\]) with any of the latter terms as well as the short-range Lagrangian (\[lagsr\]) leads to the expression $$\label{t12np}
[T_{1/2}^{NP}]^{-1}=\epsilon_{NP}^2 G_{NP} |{\cal M} ^{NP}|^2.$$ Here, ${\cal M}^{m_\nu}$ and ${\cal M}^{NP}$ are the nuclear matrix elements for the mass mechanism and alternative new physics contributions, and $G_{01}$ and $G_{NP}$ denote the corresponding phase space integrals from the list given in [@Doi:1982dn]. We have assumed, that one mechanism dominates the double beta decay rate, and we do not consider interference between different mechanisms. Calculational details and results for the relevant matrix elements involved have been given elsewhere [@Pas:1999fc; @Pas:2000vn], and numerical results for all common double beta emitter isotopes will be published soon [@paes06].
In the present context, we will concentrate on the observation that the combinations of leptonic and hadronic currents specific to different mechanisms result in different nuclear matrix elements. This fact taken alone is not of much help in order to disentangle the different mechanisms, since e.g. a smaller nuclear matrix element for the mass mechanism as compared to any alternative new physics mechanism can be compensated by a larger value for the neutrino mass, at least within the constraints implied by other observations such as Tritium beta decay and cosmology. However, under the assumption that one mechanism dominates in triggering the decay, the new physics parameter $\langle m_\nu \rangle$ or $\epsilon_{NP}$ drops out in the ratio of experimentally determined half lives for two different emitter isotopes, $$\frac{ T_{1/2}(^AX)}{T_{1/2}(^{76}{\rm Ge})} =
\frac{|{\cal M}(^{76}{\rm Ge})|^2 G(^{76}{\rm Ge})}
{|{\cal M}(^AX)|^2 G(^AX)}.$$ Consequently, half life ratios depend on the mechanism of double beta decay, but not on the new physics parameter, and thus can be compared with the theoretical prediction for different mechanisms. Moreover, the error in the isotope nuclear matrix element ratio can be reduced compared to the theoretical error in one matrix element, due to cancellations of systematic effects.
In the following we study several prominent examples of specific alternative new physics contributions by calculating the corresponding ratios of half lives $${\cal R}^{NP}(^AX)=\frac {T_{1/2}^{NP}(^AX)}
{T_{1/2}^{NP}(^{76}{\rm {\rm Ge}})}, \label{ratiosdef}$$ where we concentrate on a comparison with $^{76}$Ge as it constitutes the best tested isotope to date. We choose the following mechanisms for a detailed discussion:
- [**SUSY-accompanied neutrinoless double beta decay: ${\cal R}^{\rm SUSYacc}$**]{}\
This mechanism has been first discussed in [@susyacc]. The effective Lagrangian for the dominant contribution assumes the form $$\begin{aligned}
{\cal L}&\supset&
\frac{G_F U^*_{e i}}{4 \sqrt{2}}\epsilon^{\rm SUSYacc}
\Big[ \left( \overline{\nu}_i (1+\gamma_5) e^c\right)
\left( \overline{u} (1+\gamma_5) d \right)
+ \frac{1}{2} \left( \overline{\nu}_i \sigma^{\mu\nu}
(1+\gamma_5) e^c\right)
\left( \overline{u} \sigma^{\mu \nu}(1+\gamma_5) d \right)
\Big],\end{aligned}$$ and results from integrating out a heavy $d$-squark of the $k$-th generation with $R$-parity violating couplings $\lambda'_{11k}$ and $\lambda'_{1k1}$, and exchanging a light neutrino of the $i$-th generation between the nucleons. The new physics parameter is given by $$\epsilon^{\rm SUSYacc} =
\sum_k\frac{\lambda'_{11k}\lambda'_{1k1}}{2 \sqrt{2} G_F} \sin 2
\theta_k \left(\frac{1}{m^2_{\tilde{d}_1}}- \frac{1}
{m^2_{\tilde{d}_2}} \right),$$ where $\theta_k$ parametrizes the left-right sfermion mixing of the mass eigenstates $\tilde{d}_1$ and $\tilde{d}_2$.
- [**Gluino exchange mechanism in R-parity violating SUSY**]{}: ${\cal R}^{{\rm SUSY}-\tilde{g}}$\
In this short-range contribution discussed in [@Mohapatra; @Hirsch:1995ek], integrating out $u$- and $d$-squarks and a gluino leads to the effective Lagrangian $$\begin{aligned}
{\cal L}&\supset&
\frac{G_F^2}{2} m_p^{-1} \epsilon^{\tilde{g}}
\left( (\overline{u} (1+\gamma_5) d) (\overline{u} (1+\gamma_5) d)
- \frac{1}{4} (\overline{u} \sigma^{\mu \nu} (1+\gamma_5) d)
(\overline{u} \sigma^{\mu \nu} (1+\gamma_5) d) \right)
(\overline{e}(1+\gamma_5) e^c),\end{aligned}$$ with $$\epsilon^{\tilde{g}}=
\frac{2 \pi \alpha_s}{9}
\frac{\lambda'^2_{111}}{G_F^2 m^4_{\tilde{d}_R}}
\frac{m_p}{m_{\tilde{g}}}
\left[1+\left(\frac{m_{\tilde{d}_R}}{m_{\tilde{u}_L}}\right)^4
\right].$$
- [**Right-handed currents**]{}: ${\cal R}^{LR-\eta\eta}$ and ${\cal R}^{LR-\lambda\lambda}$\
Integrating out right-handed $W$-bosons occurring in left-right symmetric models can lead to two types of new contributions with right-handed leptonic currents [@Doi:1982dn], $${\cal L} \supset \frac{G_F}{\sqrt{2}}
\left(\overline{\nu}_i\gamma_\mu(1+\gamma_5) e^c\right)
\Big(\eta (\overline{u}\gamma^\mu (1-\gamma_5) d)
+ \lambda (\overline{u} \gamma^\mu(1+\gamma_5) d) \Big),$$ where the new physics parameters are given by $\eta$ and $\lambda$.
- [**Kaluza-Klein neutrino exchange in extra-dimensional models**]{}: ${\cal R}^{KK}$\
In extra-dimensional theories, the double beta observable is given by a sum over contributions from all Kaluza-Klein excitations with masses $m_{(n)}$, weighted with the mass dependent matrix element ${\cal M}^{m_{\nu}}(m_{(n)})$ [@Bhattacharyya:2002vf]: $$\epsilon^{KK} =
\frac{1}{{\cal M}^{m_{\nu}}}\sum_{-\infty}^{\infty} U^2_{en} m_{(n)}
\left({\cal M}^{m_{\nu}}(m_{(n)}) -{\cal M}^{m_{\nu}}\right).$$ In this case the effective coupling constant $\epsilon^{KK}$ depends on the nuclear matrix element ${\cal
M}^{m_{\nu}}(m_{(n)})$, and therefore the particle physics does not decouple from the nuclear physics. This is because the masses of the Kaluza-Klein excitations vary from values much smaller than the nuclear Fermi momentum $p_F$ to values much larger than $p_F$, while the $m_{(n)}$-dependence of ${\cal M}^{m_{\nu}}(m_{(n)})$ changes around $p_F$. Therefore the Kaluza-Klein spectrum has to be fixed by choosing specific values for the brane shift parameter $a$ and the radius of the extra dimension $R$. In the limit of $a\rightarrow 0$ or $R\rightarrow 0$, ${\cal R}^{KK}$ approaches ${\cal R}^{m_\nu}$.
The matrix elements for the mass mechanism and for the SUSY accompanied neutrino exchange have been calculated in the pn-QRPA approach of [@Staudt:1990qi; @Hirsch:1994es], in the latter case for the first time. For the other mechanisms, existing numerical values obtained with the same nuclear structure model have been adopted from the literature. The values for the phase space integral factors $G_{01}$, $G_{NP}$ have been calculated in [@Doi:1982dn]. Numerical values for ${\cal R}^{NP}(^AX)$ are given in Table 1, and Fig. 1 displays the relative change expected from various new physics contributions, compared to the mass mechanism. An application of the procedure to any other alternative new physics contribution by using the matrix elements listed in [@paes06] is straightforward.
All isotope ratios have been normalized to the half life of the most extensively studied nucleus $^{76}$Ge. Moreover, while at present no experiment using a $^{128}$Te source has been proposed, we included this isotope since it provides a particularly powerful discriminator and thus may encourage future experimental efforts to study this nucleus.
$^{82}$Se $^{100}$Mo $^{128}$Te $^{130}$Te $^{136}$Xe $^{150}$Nd Ref.
----------------------------------- ----------- ------------ ------------ ------------ ------------ ------------ -------------------------
${\cal R}^{m_\nu}$ 0.26 0.11 3.26 0.18 0.77 0.02 this paper
${\cal R}^{\rm SUSYacc}$ 0.28 0.11 3.22 0.17 0.53 0.02 this paper
${\cal R}^{{\rm SUSY}-\tilde{g}}$ 0.28 0.10 3.16 0.17 0.53 0.01 [@Hirsch:1995ek]
${\cal R}^{LR-\eta\eta}$ 0.29 0.13 2.96 0.20 0.54 0.02 [@Muto:1989cd]
${\cal R}^{LR-\lambda\lambda}$ 0.14 0.13 18.40 0.13 0.67 0.01 [@Muto:1989cd]
${\cal R}^{KK}$ (10 GeV$^{-1}$) 0.24 0.08 3.26 0.19 3.31 0.08 [@Bhattacharyya:2002vf]
${\cal R}^{KK}$ (0.1 GeV$^{-1}$) 0.26 0.11 3.26 0.18 0.78 0.02 [@Bhattacharyya:2002vf]
: Ratios ${\cal R}(^AX)$ of half lives for various important double beta decay emitter isotopes, normalized to the half-life of $^{76}$Ge. For the exchange of Kaluza-Klein excitations in extra dimensional theories the brane shift parameter and bulk radius do not factorize, and are chosen to be $a=10~ {\rm GeV}^{-1},~~ 0.1~ {\rm GeV}^{-1}$ and $R=(1/300)$ eV$^{-1}$.
![Relative deviations of half life ratios ${\cal
R}^{NP}(^AX)$, normalized to the half-life of $^{76}$Ge, compared to the ratio in the mass mechanism ${\cal R}^{m_\nu}(^AX)$. []{data-label="bulkpath"}](reldiff.eps){width="95.00000%"}
The two supersymmetric contributions show similar deviations, which are rather small for all isotopes. It is obvious that these mechanisms are most effectively discriminated from the mass mechanism by comparing the half life ratios between $^{82}$Se and $^{136}$Xe which vary by 60%. In left-right symmetric models, strong deviations can be found for the $\lambda\lambda$ combinations, while deviations for the $\eta\eta$ combination are rather small. A comparison of half life ratios between $^{100}$Mo and $^{136}$Xe yields a variation of 70 % for the $\eta\eta$ contribution with right-handed hadronic currents, while a comparison of measurements in $^{128}$Te and $^{150}$Nd will provide a powerful discriminator with a variation of more than a factor of 20 for the $\lambda\lambda$ contribution with left-handed hadronic currents. Similarly in extra-dimensional neutrino models with a large brane shift parameter, large deviations can be found for $^{136}$Xe and $^{150}$Nd, and the half life ratios for $^{150}$Nd and $^{100}$Mo vary by more than a factor of 5. Some caution is necessary when referring to the half life ratio of the heavily deformed $^{150}$Nd, which is ignored in most QRPA calculations (compare the discussion in [@def]). Finally it should be stressed that not necessarily two positive results are needed - already the comparison of one half life measurement and one upper bound in another isotope could provide non-trivial information on the double beta mechanism.
Since the theoretical errors of the nuclear matrix element calculation dominate the experimental errors, it is difficult to determine the confidence level with which either mechanism can be excluded to generate the observed double beta evidence. If, for example, a statistical distribution of matrix element values is assumed, a relative variation of 60% in ${\cal R}^{NP}(^AX)$ with respect to ${\cal R}^{m_\nu}(^AX)$ is significant only if the corresponding nuclear matrix elements would be known with an accuracy of 15%, which seems to be unrealistic, if only one pair of isotopes is being analyzed. Indeed, estimates of errors in nuclear matrix elements vary from a factor 3-5, when the spread of published values is used as a measure, to only 30%, according to an assessment of uncertainties inherent in QRPA [@unc].
However, the significance of the comparison of two isotopes will increase if a whole set of measurements in different isotopes resembles the expected pattern. Moreover, one would expect that systematical effects, like an overestimation of the nuclear matrix elements due to a too small value for the particle-particle interaction $g_{pp}$ in the pn-QRPA approach, a different value for the axial-vector coupling $g_A$, the inclusion of higher-order terms or a different model-space would influence calculations for the different isotopes in a similar way, and thereby cancel in the half life ratios discussed. This expectation is confirmed by the comparison of the results of different QRPA codes in [@unc], and of QRPA and shell model codes in [@vogel]. Finally it has been pointed out in [@bilenky] that the half life ratios (\[ratiosdef\]) can also be used to single out the correct nuclear structure model. In this case the correct combination of mechanism and nuclear structure code can be determined by the best fit of the theoretical half life ratios to half life measurements in various nuclei. Thus the results presented in this letter should be complemented and checked with alternative codes for the nuclear matrix element calculation. Moreover, other mechanisms, including pion exchange [@pion], may be dominating in some of the models discussed, and should be discussed as well.
In summary, we discussed how different mechanisms of neutrinoless double beta decay would manifest themselves in half life ratios involving different isotopes. We thus conclude that complementary measurements in different isotopes would be strongly encouraged. At present, next-generation experiment proposals exist for $^{76}$Ge (GERDA, MAJORANA, GEM, GeH$_4$), $^{82}$Se (Super-NEMO, DCBA, SeF$_6$), $^{100}$Mo (MOON), $^{130}$Te (CUORE), $^{136}$Xe (EXO, XMASS, Xe), as well as for the isotopes $^{48}$Ca, $^{116}$Cd and $^{160}$Gd not discussed in this letter (CANDLES, COBRA and GSO) (for recent overviews of the experimental status see [@exps]). An experimental study of this kind should be complemented by neutrino mass searches in Tritium beta decay experiments and cosmology, as well as searches for effects of the alternative new physics source of lepton number violation in other processes, such as lepton flavor violating decays [@Cirigliano:2004tc].
After this paper had been submitted for publication, the paper [@Gehman:2007qg] appeared, which comes to similar conclusions and estimates the number of required measurements and their precision needed.
Acknowledgements {#acknowledgements .unnumbered}
================
HP thanks B. Allanach, K.S. Babu and S. Pascoli for discussions and the University of Hawaii and DESY for kind hospitality.
[99]{} M. Doi, T. Kotani, H. Nishiura and E. Takasugi, Prog. Theor. Phys. [**69**]{}, 602 (1983). C. Aalseth [*et al.*]{}, arXiv:hep-ph/0412300. P. Vogel, AIP Conf. Proc. [**870**]{}, 124 (2006). H. V. Klapdor-Kleingrothaus and H. Päs, arXiv:hep-ph/9808350. J. Schechter and J. W. F. Valle, Phys. Rev. D [**25**]{}, 2951 (1982). A. Ali, A. V. Borisov and D. V. Zhuridov, arXiv:hep-ph/0606072. M. Hirsch, K. Muto, T. Oda and H. V. Klapdor-Kleingrothaus, Z. Phys. A [**347**]{}, 151 (1994). F. Simkovic, M. Nowak, W. A. Kaminski, A. A. Raduta and A. Faessler, Phys. Rev. C [**64**]{}, 035501 (2001). H. Päs, M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Kovalenko, Phys. Lett. B [**453**]{}, 194 (1999). H. Päs, M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Kovalenko, Phys. Lett. B [**498**]{}, 35 (2001). M. Hirsch, S.G. Kovalenko, H. Päs, in preparation.
K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. [**75**]{}, 2276 (1995); M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Kovalenko, Phys. Lett. B [**372**]{}, 181 (1996) \[Erratum-ibid. B [**381**]{}, 488 (1996)\]; H. Päs, M. Hirsch and H. V. Klapdor-Kleingrothaus, Phys. Lett. B [**459**]{}, 450 (1999). R. N. Mohapatra, Phys. Rev. D [**34**]{}, 3457 (1986); J. D. Vergados, Phys. Lett. B [**184**]{}, 55 (1987). M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Kovalenko, Phys. Rev. D [**53**]{}, 1329 (1996). G. Bhattacharyya, H. V. Klapdor-Kleingrothaus, H. Päs and A. Pilaftsis, Phys. Rev. D [**67**]{}, 113001 (2003). A. Staudt, K. Muto and H. V. Klapdor- Kleingrothaus, Europhys. Lett. [**13**]{}, 31 (1990). K. Muto, E. Bender and H. V. Klapdor, Z. Phys. A [**334**]{}, 187 (1989). F. Simkovic, L. Pacearescu and A. Faessler, Nucl. Phys. A [**733**]{}, 321 (2004). V. A. Rodin, A. Faessler, F. Simkovic and P. Vogel, Nucl. Phys. A [**766**]{}, 107 (2006). S. M. Bilenky and J. A. Grifols, Phys. Lett. B [**550**]{}, 154 (2002). A. Faessler, S. Kovalenko, F. Simkovic and J. Schwieger, Phys. Rev. Lett. [**78**]{}, 183 (1997). S. R. Elliott, arXiv:nucl-ex/0609024; K. Zuber, Acta Phys. Polon. B [**37**]{}, 1905 (2006); A. Barabash, arXiv:hep-ex/0608054; A. Piepke, Nucl. Phys. A [**752**]{}, 42 (2005). V. Cirigliano, A. Kurylov, M. J. Ramsey-Musolf and P. Vogel, Phys. Rev. Lett. [**93**]{}, 231802 (2004). V. M. Gehman and S. R. Elliott, J. Phys. G [**34**]{}, 667 (2007) \[arXiv:hep-ph/0701099\].
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abstract: 'Deep neural networks (DNNs) have shown very promising results for various image restoration (IR) tasks. However, the design of network architectures remains a major challenging for achieving further improvements. While most existing DNN-based methods solve the IR problems by directly mapping low quality images to desirable high-quality images, the observation models characterizing the image degradation processes have been largely ignored. In this paper, we first propose a denoising-based IR algorithm, whose iterative steps can be computed efficiently. Then, the iterative process is unfolded into a deep neural network, which is composed of multiple denoisers modules interleaved with back-projection (BP) modules that ensure the observation consistencies. A convolutional neural network (CNN) based denoiser that can exploit the multi-scale redundancies of natural images is proposed. As such, the proposed network not only exploits the powerful denoising ability of DNNs, but also leverages the prior of the observation model. Through end-to-end training, both the denoisers and the BP modules can be jointly optimized. Experimental results on several IR tasks, e.g., image denoisig, super-resolution and deblurring show that the proposed method can lead to very competitive and often state-of-the-art results on several IR tasks, including image denoising, deblurring and super-resolution.'
author:
- 'Weisheng Dong, , Peiyao Wang, Wotao Yin, , Guangming Shi, , Fangfang Wu, and Xiaotong Lu [^1] [^2]'
title: Denoising Prior Driven Deep Neural Network for Image Restoration
---
denoising-based image restoration, deep neural network, denoising prior, image restoration.
Introduction
============
Image restoration (IR) aiming to reconstruct a high quality image from its low quality observation has many important applications, such as low-level image processing, medical imaging, remote sensing, surveillance, etc. Mathematically, IR problem can be expressed as ${\bm{y}}={\textbf{A}}{\bm{x}}+{\bm{n}}$, where ${\bm{y}}$ and ${\bm{x}}$ denote the degraded image and the original image, respectively, ${\textbf{A}}$ denotes the degradation matrix relating to an imaging/degradation system, and ${\bm{n}}$ denotes the additive noise. Note that for different settings of ${\textbf{A}}$, different IR problems can be expressed. For example, the IR problem is a denoising problem [@KSVD; @BM3D; @CSR; @LASSC; @WNNM] when ${\textbf{A}}$ is an identical matrix and becomes a deblurring problem [@Dong:TIP11; @IDDBM3D; @NCSR; @Dong:IJCV15] when ${\textbf{A}}$ is a blurring matrix/operator, or a super-resolution problem [@TVSR; @Yang:SR08; @NCSR; @Gao:TIP12] when ${\textbf{A}}$ is a subsampling matrix/operator. Essentially, restoring ${\bm{x}}$ from ${\bm{y}}$ is a challenging ill-posed inverse problem. In the past a few decades, the IR problems have been extensively studied. However, they still remain as an active research area.
Generally, existing IR methods can be classified into two main categories, i.e., model-based methods [@Osher:TV05; @KSVD; @ISTA; @Mairal:TIP08; @NCSR; @Zoran:ICCV11; @Dong:IJCV15; @Roth:IJCV09; @Yu:TIP12] and learning-based methods [@Freeman:02; @Aplus; @Schmidt:CVPR14; @SRCNN; @Wang:CVPR15; @Zhang:TIP17]. The model-based methods attack this problem by solving an optimization problem, which is often constructed from a Bayesian perspective. In the Bayesian setting, the solution is obtained by maximizing the posterior $P({\bm{x}}|{\bm{y}})$, which can be formulated as $${\bm{x}}= \operatorname*{argmax}_{{\bm{x}}} \log P({\bm{x}}|{\bm{y}}) = \operatorname*{argmax}_{{\bm{x}}} \log P({\bm{y}}|{\bm{x}}) + \log P({\bm{x}}), \label{MAP}$$ where $\log P({\bm{y}}|{\bm{x}})$ and $\log P({\bm{x}})$ denote the data likelihood and the prior terms, respectively. For additive Gaussian noise, $P({\bm{y}}|{\bm{x}})$ corresponds to the $\ell_2$-norm data fidelity term, and the prior term $P({\bm{x}})$ characterizes the prior knowledge of ${\bm{x}}$ in a probability setting. Formally, Eq. (\[MAP\]) can be rewritten as $${\bm{x}}= \operatorname*{argmin}_{{\bm{x}}} ||{\bm{y}}-{\textbf{A}}{\bm{x}}||_2^2 + \lambda J({\bm{x}}), \label{Obj_fun}$$ where $J({\bm{x}})$ denotes the regularizer associated with the prior term $P({\bm{x}})$. Then, the desirable solution is the one that minimizes both the $\ell_2$-norm data fidelity term and the regularization term weighted by parameter $\lambda$. Clearly, the regularization term plays a critical role in searching for high-quality solutions. Numerous regularizers have been developed, ranging from the well-known total variation (TV) regularizer [@Osher:TV05], the sparsity-base regularizers with off-the-shelf transforms or learned dictionaries [@KSVD; @ISTA; @CSR; @Mairal:TIP08], to the nonlocal self-similarity (NLSS) inspired regularizers [@NLM; @BM3D; @NCSR]. The TV regularizer is good at characterizing the piecewise constant signals but unable to model more complex image edges and textures. The sparsity-based techniques are more effective in representing local image structures with a few elemental structures (called atoms) from an off-the-shelf transformation matrix (e.g., DCT and Wavelets) or a learned dictionary. Indeed, the IR community has witnessed a flurry of sparsity-based IR methods [@KSVD; @CSR; @Mairal:TIP08; @Yang:SR08] in the past decade. Motivated by the fact that natural images often contain rich repetitive structures, nonolocal regularization techniques [@BM3D; @NCSR; @LASSC; @WNNM] combining the NLSS with the sparse representation and low-rank approximation, have shown significant improvements over their local counterparts. Using those carefully designed prior, significant progresses of IR have been achieved. In addition to these explicitly regularized IR methods, denoising-based IR methods have also been proposed [@PPP:13; @Brifman:ICIP16; @Teodoro:ICIP16; @RED:17; @Chan:17]. In these methods, the original optimization problem is decoupled into two separated subproblems - one for dealing with the data fidelity term and the other for the regularization term, yielding simpler optimization problems. Specifically, the subproblem related to the regularization is a pure denoising problem, and thus other more complex denoising methods that cannot be expressed as regularization terms can also be adopted, e.g., BM3D [@BM3D], NCSR [@NCSR] and GMM [@Zoran:ICCV11] methods.
Different from the model-based methods that rely on a carefully designed prior, the learning-based IR methods learn mapping functions to infer the missing high-frequency details or desirable high-quality images from the observed image. In the past decade, many learning-based image super-resolution methods [@Freeman:02; @Aplus; @SRCNN; @Zhang:TIP17] have been proposed, where mapping functions from the low-resolution (LR) patches to high-resolution (HR) patches are learned. Inspired by the great successes of the deep convolution neural network (DCNN) for image classification [@Alexnet; @Resnet], the DCNN models have also been successfully applied to image IR tasks, e.g., SRCNN [@SRCNN], FSRCNN [@FSRCNN] and VDSR [@VDSR] for image super-resolution, and TNRD [@TNRD] and DnCNN [@Zhang:TIP17] for image denoising. In these methods, a DCNN is used to learn the mapping function from the degraded images to the original images. Due to its powerful representation ability, the DCNN based methods have shown better IR performances than conventional optimization-based IR methods in various IR tasks [@SRCNN; @VDSR; @TNRD; @Zhang:TIP17]. Though training of DCNN is very expensive, testing the DCNN is much more efficient than previous optimization-based IR methods. Though the DCNN models have shown promising results, the DCNN methods lack flexibilities in adapting to different image recovery tasks, as the data likelihood term has not been explicitly exploited. To address this issue, hybrid IR methods that combine the optimization-based methods and DCNN denoisers have been proposed. In [@Zhang:CVPR17], a set of DCNN models are pre-trained for image denoising task and are integrated into the optimization-based IR framework for different IR tasks. Compared with other optimization-based methods, the integration of the DCNN models has advantages in exploiting the large training dataset and thus leads to superior IR performance. Similar idea has also been exploited in the autoencoder-based IR method [@Bigdeli:arXiv], where denoising autoencoders are pre-trained as a natural image prior and a regularzer based on the pre-trained autoencoder is proposed. The resulting optimization problem is then iteratively solved by gradient descent. Despite the effectiveness of the methods [@Zhang:CVPR17; @Bigdeli:arXiv], they have to iteratively solve optimization problems, and thus their computational complexities are high. Moreover, the CNN and autoencoder models adopted in [@Zhang:CVPR17; @Bigdeli:arXiv] are pre-trained and cannot be jointly optimized with other algorithm parameters.
In this paper, we propose a denoising prior driven deep network to take advantages of both the optimization- and discriminative learning-based IR methods. First, we propose a denoising-based IR method, whose iterative process can be efficiently carried out. Then, we unfold the iterative process into a feed-forward neural network, whose layers mimic the process flow of the proposed denoising-based IR algorithm. Moreover, an effective DCNN denoiser that can exploit the multi-scale redundancies is proposed and plugged into the deep network. Through end-to-end training, both the DCNN denoisers and other network parameters can be jointly optimized. Experimental results show that the proposed method can achieve very competitive and often state-of-the-art results on several IR tasks, including image denoising, deblurring and super-resolution.
Related Work
============
We briefly review the IR methods, i.e., the denoising-based IR methods and the discriminative learning-based IR methods, which are related to the proposed method.
Denoising-based IR methods
--------------------------
Instead of using an explicitly expressed regularizer, denoising-based IR methods [@PPP:13] allow the use of a more complex image prior by decoupling the optimization problem of Eq. (\[Obj\_fun\]) into two subproblems, one for the data likelihood term and the other for the prior term. By introducing an auxiliary variable ${\bm{v}}$, Eq. (\[Obj\_fun\]) can be rewritten as $$({\bm{x}},{\bm{v}}) = \operatorname*{argmin}_{{\bm{x}}, {\bm{v}}} \frac{1}{2}||{\bm{y}}-{\textbf{A}}{\bm{x}}||_2^2 + \lambda J({\bm{v}}), s.t.~{\bm{x}}= {\bm{v}}. \label{den-based}$$ In [@PPP:13; @Chan:17], the ADMM technique is used to convert the above equally constrained optimization problem into two subproblems $$\begin{split}
&{\bm{x}}^{(t+1)} = \operatorname*{argmin}_{{\bm{x}}} \frac{1}{2}||{\bm{y}}-{\textbf{A}}{\bm{x}}||_2^2 + \frac{\mu}{2}||{\bm{x}}-{\bm{v}}^{(t)}+{\bm{u}}^{(t)}||_2^2, \\
&{\bm{v}}^{(t+1)} = \operatorname*{argmin}_{{\bm{v}}} \frac{\mu}{2}||{\bm{x}}^{(t+1)}-{\bm{v}}+{\bm{u}}^{(t)}||_2^2 + \lambda J({\bm{v}}),
\end{split}$$ where ${\bm{u}}$ denotes the augmented Lagrange multiplier updated as ${\bm{u}}^{(t+1)} = {\bm{u}}^{(t)}+\rho({\bm{x}}^{(t+1)}-{\bm{v}}^{(t+1)})$. The ${\bm{x}}$-subproblem is a simple quadratic optimization that admits a closed-form solution as $${\bm{x}}^{(t+1)} = ({\textbf{A}}^{\top}{\textbf{A}}+\lambda{\textbf{I}})^{-1}({\textbf{A}}^{\top}{\bm{y}}+\lambda({\bm{v}}^{(t)}-{\bm{u}}^{(t)})).$$ The intermediately reconstructed image ${\bm{x}}^{(t+1)}$ depends on both the observation model and a fixed estimate of ${\bm{v}}$. The ${\bm{v}}$-subproblem is also called the proximity operator of $J({\bm{v}})$ computed at point ${\bm{x}}^{(t+1)}+{\bm{u}}^{(t)}$, whose solution can be obtained by a denoising algorithm. By alternatively updating ${\bm{x}}$ and ${\bm{v}}$ until convergence, the original optimization problem of Eq. (\[Obj\_fun\]) is then solved. The advantage of this framework is that other state-of-the-art denoising algorithms, which cannot be explicitly expressed in $J({\bm{x}})$, can also be used to update ${\bm{v}}$, leading to better IR performance. For example, the well-known BM3D [@BM3D], Gaussian mixture model [@Zoran:ICCV11], NCSR [@NCSR] have been used for various IR applications [@PPP:13; @Brifman:ICIP16; @Teodoro:ICIP16]. In [@Zhang:CVPR17], the sate-of-the-art CNN denoiser has also been plugged as an image prior for general IR. Due to the excellent denoising ability, state-of-the-art IR results for different IR tasks have been obtained. Similar to [@Bigdeli:arXiv], an autoencoder denoiser is plugged into the objective function of Eq. (\[Obj\_fun\]). However, different from the variable splitting method described above, the objective function of [@Bigdeli:arXiv] is minimized by gradient descent. Though the denoising-based IR methods are very flexible and effective in exploiting sate-of-the-art image prior, they require a lot of iterations for convergence and the whole components cannot be jointly optimized.
Deep network based IR methods
-----------------------------
Inspired by the great success of DCNNs for image classification [@Alexnet; @Resnet], object detection [@FRCNN:PAMI17], semantical segmentation [@FCN:CVPR15], etc., DCNNs have also been applied for low-level image processing tasks [@SRCNN; @VDSR; @TNRD; @Zhang:TIP17]. Similar to the coupled sparse coding [@Yang:SR08], DCNNs have been used to learn nonlinear mapping from the LR patch space to the HR patch space [@SRCNN]. By designing very deep CNNs, state-of-the-art image super-resolution results have been achieved [@VDSR]. Similar network structures have also been applied for image denoising [@Zhang:TIP17] and also achieved state-of-the-art image denoising performance. For non-blind image deblurring, multiplayer perceptron network [@MLP:CVPR12] has been developed to remove the deconvolution artifacts. In [@Xu:NIPS14], Xu et al. propose to use DCNN for non-blind image deblurring. Though excellent IR performances have been obtained, these DCNN methods generally treat the IR problems as denoising problems, i.e., removing the noise or artifacts of the initially recovered images, and ignore the observation models.
There has been some attempts to leverage the domain knowledge and the observation model for IR. In [@Wang:CVPR15], based on the learned iterative shrinkage/thresholding algorithm (LISTA) [@LISTA], Wang et al. developed a deep network whose layers correspond to the steps of the sparse coding based image SR. In [@TNRD], the classic iterative nonlinear reaction diffusion method is also implemented as a deep network, whose parameters are jointly trained. The DNN inspired from the ADMM-based sparse coding algorithm has also been developed for compressive sensing based MRI reconstruction [@ADMM-Net]. In [@Xin:NIPS16], DNNs constructed from truncated iterative hard thresholding algorithm has also been developed for solving $\ell_0$-norm sparse recovery problem. These model-based DNNs have shown significant improvements in terms of both efficiency and effectiveness over original iterative algorithms. However, the strict implementations of the conventional sparse coding based methods result in a limited receipt field of the convolutional filters and thus cannot exploit the spatial correlations of the feature maps effectively, leading to limited IR performance.
Proposed Denoising-based Image Restoration Algorithm
====================================================
In this section, we develop an efficient iterative algorithm for solving the denoising-based IR methods, based on which a feed-forward DNN will be proposed in the next section. Considering the denoising-based IR problem of Eq. (\[den-based\]), we adopt the half-quadratic splitting method, by which the equally constrained optimization problem can be converted into a non-constrained optimization problem, as $$({\bm{x}},{\bm{v}}) = \operatorname*{argmin}_{{\bm{x}}, {\bm{v}}} \frac{1}{2}||{\bm{y}}-{\textbf{A}}{\bm{x}}||_2^2 + \eta||{\bm{x}}-{\bm{v}}^{(t)}||_2^2 + \lambda J({\bm{v}}). \\$$ The above optimization problem can be solved by alternatively solving two sub-problems, $$\begin{split}
&{\bm{x}}^{(t+1)} = \operatorname*{argmin}_{{\bm{x}}} ||{\bm{y}}-{\textbf{A}}{\bm{x}}||_2^2 + \eta||{\bm{x}}-{\bm{v}}^{(t)}||_2^2, \\
&{\bm{v}}^{(t+1)} = \operatorname*{argmin}_{{\bm{v}}} \eta||{\bm{x}}^{(t+1)}-{\bm{v}}||_2^2 + \lambda J({\bm{v}}).
\end{split}$$ The ${\bm{x}}$-subproblem is a quadratic optimization problem that can be solved in closed-form, as ${\bm{x}}^{(t+1)} = {\textbf{W}}^{-1}{\bm{b}}$, where ${\textbf{W}}$ is a matrix related to the degradation matrix ${\textbf{A}}$. Generally, ${\textbf{W}}$ is very large, so it is impossible to compute its inverse matrix. Instead, the iterative classic conjugate gradient (CG) algorithm can be used to compute ${\bm{x}}^{(t+1)}$, which requires many iterations for computing ${\bm{x}}^{(t+1)}$. In this paper, instead of solving for an exact solution of the ${\bm{x}}$-subproblem, we propose to compute ${\bm{x}}^{(t+1)}$ with a single step of gradient descent for an inexact solution, as $$\begin{split}
{\bm{x}}^{(t+1)} &= {\bm{x}}^{t} - \delta[ {\textbf{A}}^{\top}({\textbf{A}}{\bm{x}}^{(t)}-{\bm{y}}) + \eta({\bm{x}}^{(t)}-{\bm{v}}^{(t)}) ] \\
&= \bar{{\textbf{A}}}{\bm{x}}^{(t)} + \delta{\textbf{A}}^{\top}{\bm{y}}+ \delta{\bm{v}}^{(t)},
\end{split}$$ where $\bar{{\textbf{A}}}=[(1-\delta\eta){\textbf{I}}-\delta{\textbf{A}}^{\top}{\textbf{A}}]$ and $\delta$ is the parameter controlling the step size. By pre-computing $\bar{{\textbf{A}}}$, the update of ${\bm{x}}^{(t)}$ can be computed very efficiently. As will be shown later, we do not have to solve the ${\bm{x}}$-subproblem exactly. Updating ${\bm{x}}^{(t+1)}$ once is sufficient for ${\bm{x}}^{(t)}$ to converge to a local optimal solution. The ${\bm{v}}$-subproblem is a proximity operator of $J({\bm{v}})$ computed at point ${\bm{x}}^{(t+1)}$, whose solution can be obtained by a denoiser, i.e., ${\bm{v}}^{(t+1)}=f({\bm{x}}^{(t+1)})$, where $f(\cdot)$ denotes a denoiser. Various denoising algorithms can be used, including that cannot be explicitly expressed by the MAP estimator with $J({\bm{x}})$. In this paper, inspired by the success of DCNN for image denoising, we choose a DCNN-based denoiser to exploit the large training dataset. However, different from existing DCNN models for IR, we consider the network that can exploit the multi-scale redundancies of natural images, as will be described in the next section. In summary, the proposed iterative algorithm for solving the denoising-based IR problems is summarized in **Algorithm 1**. We now discuss the convergence property of **Algorithm 1**.
$\bullet$ **Initialization**:
\(1) Set observation matrix ${\textbf{A}}$, $\bar{{\textbf{A}}}$, $\delta>0$, $\eta>0$, $t=0$;
\(2) Initialize ${\bm{x}}$ as ${\bm{x}}^{(0)} = {\textbf{A}}^{\top}{\bm{y}}$, ${\bm{v}}^{(0)}=\bm{0}$;
$\bullet$ **While** not converge **do**
\(1) Compute ${\bm{x}}^{(t+1)} = \bar{{\textbf{A}}}{\bm{x}}^{(t)} + \delta{\textbf{A}}^{\top}{\bm{y}}+ \delta{\bm{v}}^{(t)}$
\(2) Compute ${\bm{v}}^{(t+1)} = f({\bm{x}}^{(t+1)})$
**End while**
Output: ${\bm{x}}^{(t)}$
Consider the energy function $$\xi({\bm{x}},{\bm{v}}) := \frac{1}{2}\|{\bm{y}}-{\textbf{A}}{\bm{x}}\|_2^2 + \frac{\eta}{2}\|{\bm{x}}-{\bm{v}}\|_2^2 + \lambda J({\bm{v}}).$$ Assume that $\xi$ is lower bounded and coercive[^3]. For Algorithm 1, $({\bm{x}}^{(t)}, {\bm{v}}^{(t)})$ has a subsequence that converges to a stationary point of the the energy function provided that the denoiser $f(\cdot)$ satisfies the sufficient descent condition: $$\begin{aligned}
\label{fcond}
&\frac{\eta}{2}||{\bm{x}}-{\bm{v}}||_2^2 + \lambda J({\bm{v}})-\frac{\eta}{2}||{\bm{x}}-f({\bm{x}})||_2^2 - \lambda J(f({\bm{x}}))\nonumber\\
&\ge c_2\|\tilde\nabla_{{\bm{v}}}\xi({\bm{x}},{\bm{v}})\|_2^2,$$ where $c_2>0$ and $\tilde\nabla_{{\bm{v}}}\xi({\bm{x}},\cdot)$ is a continuous limiting subgradient of $\xi$.
*Proof* See the Appendix.
Let us discuss the condition . We list some combinations of the function $J$ and mapping $f$ that satisfy :
1. $J$ is $L$-Lipschitz differentiable, and $f:({\bm{x}},{\bm{v}})\mapsto {\bm{v}}-\alpha \nabla_{{\bm{v}}} \xi({\bm{x}},{\bm{v}})$ is a gradient descent map, where $\alpha \in (0,\frac{2}{\eta+L})$ if $\xi({\bm{x}},{\bm{v}})$ is convex in ${\bm{v}}$ or $\alpha \in (0,\frac{1}{\eta+L})$ otherwise. Then, follows from standard gradient analysis.
2. $J$ is proper and lower semi-continuous, the function $\xi'({\bm{u}};{\bm{x}},{\bm{v}}):=\frac{\mu}{2}\|{\bm{x}}-{\bm{u}}\|_2^2 + \lambda J({\bm{u}})+\frac{\beta}{2}\|{\bm{v}}-{\bm{u}}\|_2^2$ is at least $\beta$-strongly convex in ${\bm{u}}$, and $f: ({\bm{x}},{\bm{v}}) \mapsto {\bm{v}}^+:=\operatorname*{argmin}_{{\bm{u}}}\xi'({\bm{u}};{\bm{x}},{\bm{v}})$. This $f$ is known as the proximal mapping of $\frac{\mu}{2}\|{\bm{x}}-\cdot\|_2+J(\cdot)$. The properties of $J$ ensures ${\bm{v}}^+$ to be well defined. Then, by convexity and optimality condition of the “$\operatorname*{argmin}$” subproblem, $$\begin{aligned}
&\frac{\mu}{2}\|{\bm{x}}-{\bm{v}}\|_2^2 + \lambda J({\bm{v}})-\frac{\mu}{2}\|{\bm{x}}-{\bm{v}}^+\|_2^2 - \lambda J({\bm{v}}^+)\nonumber\\
&\ge \beta\|{\bm{v}}-{\bm{v}}^+\|_2^2 = \frac{1}{\beta}\|\mu({\bm{x}}-{\bm{v}}^+)+\lambda \tilde\nabla J({\bm{v}}^+)\|_2^2\nonumber\\
&=\frac{1}{\beta}\|\tilde\nabla_{{\bm{v}}}\xi({\bm{x}},{\bm{v}}^+)\|_2^2.\label{xvp}\end{aligned}$$ This is different from since the right-hand side uses ${\bm{v}}^+$ rather than ${\bm{v}}$. However, applying the right-hand side term $\|{\bm{v}}-{\bm{v}}^+\|_2^2$ in the proof yields $\lim_{t}\|{\bm{v}}^{(t)}-{\bm{v}}^{(t+1)} \|_2=0$ and thus is satisfied asymptotically and the proof results still apply.
3. Let $\mathcal{M}$ denote a manifold of (noiseless) images and $J({\bm{v}}) := \mathrm{dist}({\bm{v}},\mathcal{M})^2$ be a function that measures a certain kind of squared distance between ${\bm{v}}$ and $\mathcal{M}$. In particular, consider the squared Euclidean distance $J({\bm{v}})=\frac{1}{2}\|{\bm{v}}- \Pi_{\mathcal{M}}({\bm{v}})\|_2^2$, where $\Pi_{\mathcal{M}}({\bm{v}})$ denotes orthogonal projection of ${\bm{v}}$ to $\mathcal{M}$. Then, for $f({\bm{x}}) := \operatorname*{argmin}_{{\bm{u}}}\{\frac{\mu}{2}\|{\bm{x}}-{\bm{u}}\|_2^2 + \frac{\lambda}{2}\|{\bm{u}}- \Pi_{\mathcal{M}}({\bm{u}})\|_2^2+\frac{\beta}{2}\|{\bm{v}}-{\bm{u}}\|_2^2\}$, we have $f({\bm{x}})=\frac{1}{\lambda+\mu+\beta}(\mu{\bm{x}}+\beta{\bm{v}}+\lambda\Pi_{\mathcal{M}}(\mu{\bm{x}}+\beta{\bm{v}})).$ Similar to the last point, we have and thus asymptotically.
4. For the same $\mathcal{M}$ in the last part, define $J({\bm{x}})=\delta_{\mathcal{M}}({\bm{x}})$, which returns 0 if ${\bm{x}}\in\mathcal{M}$ and $\infty$ if ${\bm{x}}\not\in\mathcal{M}$. If the manifold $\mathcal{M}$ is bounded and differentiable, then $J({\bm{x}})$ is known as *restricted prox-regular*. For $f({\bm{x}}) := \operatorname*{argmin}_{{\bm{u}}}\{\frac{\mu}{2}\|{\bm{x}}-{\bm{u}}\|_2^2 + \delta_{\mathcal{M}}({\bm{x}})+\frac{\beta}{2}\|{\bm{v}}-{\bm{u}}\|_2^2\}$, It is discussed in [@Yin:18] that holds and thus holds in the asymptotic sense.
In parts 2–4 above, we can remove the proximity term $\frac{\beta}{2}\|{\bm{v}}-{\bm{u}}\|_2^2$, which is used in defining the mapping $f$, and still ensure the same result, i.e., subsequence convergence to a stationary point. However, the proof must be adapted to each $J({\bm{v}})$ separately. We leave this to our future work.
It has been shown in [@KL:07] that if $\xi$ has the Kurdyka-Lojasiewicz property, the subsequence convergence can be upgraded to the convergence of full sequence, which has been a standard argument in recent convergence analysis. As shown in [@Yin:13], functions satisfying the KL property include, but not limited to, real analytic functions, semi-algebraic functions, and locally strongly convex functions. Therefore, $({\bm{x}}^{(t)}, {\bm{v}}^{(t)})$ converges to a stationary point. It is possible that the stationary point $({\bm{x}}^{*},{\bm{v}}^{*})$ is a saddle point rather than a local minimizer. However, it is known that first-order methods almost always avoid saddle points assuming the initial solution is randomly selected [@Lee:arXiv]. Therefore, converging to a saddle point is extremely unlikely.
It has been shown in [@Alain:14] that the denoiser autoencoder can be regarded as a approximately orthogonal projection of the noisy input ${\bm{y}}$ to the manifold of noiseless images. Therefore, as shown in the above parts $2$ and $3$, **Algorithm 1** with the mapping function $f(\cdot)$ defined by the DCNN denoiser in a loose sense converges to a local minimizer, based on the above analysis.
Denoising Prior Driven Deep Neural Network
==========================================
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In general, **Algorithm 1** requires many iterations to converge and is computationally expensive. Moreover, the parameters and the denoiser cannot be jointly optimized in an end-to-end training manner. To address these issue, here we propose to unfold the **Algorithm 1** into a deep network of the architecture shown in Fig. \[fig:network\] (a). The network exactly executes $K$ iterations of **Algorithm 1**. The input degraded image ${\bm{y}}\in \mathbb{R}^{n_y}$ first goes through a linear layer parameterized by the degradation matrix ${\textbf{A}}\in \mathbb{R}^{n_y\times m_x}$ for an initial estimate ${\bm{x}}^{(0)}$. ${\bm{x}}^{(0)}$ is then fed into the linear layer parameterized by matrix $\bar{{\textbf{A}}}\in \mathbb{R}^{m_x\times m_x}$, whose output is added with ${\bm{x}}^{(0)}$ weighted by $\delta_1$ via a shortcut connection. The updated ${\bm{x}}^{(1)}$ is fed into the denoiser module, whose structure is shown in Fig. \[fig:network\](b). The denoised signal ${\bm{v}}^{(1)}$ is fed into the linear layer parameterized by $\bar{{\textbf{A}}}$, whose output is further added with ${\bm{x}}^{(0)}$ and ${\bm{v}}^{(1)}$ via two shortcut connections for the updated ${\bm{x}}^{(2)}$. Such a process is repeated $K$ times. In our implementation, $K=6$ was always used. Instead of using fixed weights, all the weights ($\delta_1$, $\delta_{k,1}$, $\delta_{k,2}$, $k=1,2,\cdots, K$) involved in the $K$ recurrent stages can be discriminatively learned through end-to-end training. Regarding the denoising module, as we are using a DCNN-based denoiser that contains a large number of parameters, we enforce all the denoising modules to share the same parameters to avoid over-fitting.
The linear layers ${\textbf{A}}^{\top}$ and $\bar{{\textbf{A}}}$ are also trainable for a typical degradation matrix ${\textbf{A}}$. For image denoising, ${\textbf{A}}={\textbf{A}}^{\top}={\textbf{I}}$, and $\bar{{\textbf{A}}}$ also reduces to a weighted identity matrix $\bar{{\textbf{A}}}=\lambda{\textbf{I}}$, where $\lambda=1-\delta(1+\eta)$. For image deblurring, the layer ${\textbf{A}}^{\top}$ can be simply implemented with a convolutional layer. The layer $\bar{{\textbf{A}}}=a{\textbf{I}}-\delta{\textbf{A}}^{\top}{\textbf{A}}$ can also be computed efficiently by convolutional operations. The weight $a$ and filters correspond to ${\textbf{A}}^{\top}$ and ${\textbf{A}}$ can also be discriminatively learned. For image super-resolution, two types of degradation operators are considered: the Gaussian downsampling and the bicubic downsampling. For Gaussian downsampling, ${\textbf{A}}={\textbf{D}}{\textbf{H}}$, where ${\textbf{H}}$ and ${\textbf{D}}$ denote the Gaussian blur matrix and the downsampling matrix, respectively. In this case, the layer ${\textbf{A}}^{\top}={\textbf{H}}^{\top}{\textbf{D}}^{\top}$ corresponds to first upsample the input LR image by zero-padding and then convolute the upsampled image with a filter. Layer $\bar{{\textbf{A}}}$ can also be efficiently computed with convolution, downsampling and upsampling operations. All convolutional filters involved in these operations can be discriminatively learned. For bicubic downsampling, we simply use the bicubic interpolator function with scaling factor $s$ and $1/s$ ($s=2,3,4$) to implement the matrix-vector multiplications ${\textbf{A}}^{\top}{\bm{y}}$ and ${\textbf{A}}{\bm{x}}$, respectively.
The DCNN denoiser
-----------------
Inspired by the recent advances on semantical segmentation [@FCN:CVPR15] and object segmentation [@Sharpmask], the architecture of the denoising network is illustrated in Fig. \[fig:network\](b). Similar to the U-net [@UNet] and the sharpMask net [@Sharpmask], the proposed network contains two parts: the feature encoding and the decoding parts. In the feature encoding part, there are a series of convolutional layers followed by pooling layers to reduce the spatial resolution of the feature maps. The pooling layer helps increase the receipt field of the neurons. In the feature encoding stage, all the convolutional layers are grouped into $L$ feature extraction blocks ($L=4$ in our implementation), as shown by the blue blocks in Fig. \[fig:network\](b). Each block contains four convolutional layers with ReLU nonlinearity and $3\times 3$ kernels. The first three layers generate $64$-channel feature maps, while the last layer doubles the number of channels followed by a pooling layer to reduce the spatial resolution of the feature maps with scaling factor $0.5$. In the pooling layers, the feature maps are first convoluted with $2\times 2$ kernels and then subsampled by a scaling factor of $2$ along both axes.
The feature decoding part also contains a series of convolutional layers, which are also grouped into four blocks followed by an upsampling layer to increase the spatial resolution of the feature maps. As the finally extracted feature maps lose a lot of spatial information, directly reconstructing images from the finally extracted features cannot recover fine image details. To address this issue, the feature maps of the same spatial resolution generated in the encoding stage are fused with the upsampled feature maps generated in the decoding stage, for obtaining newly upsampled feature maps. Each reconstruction block also consists of four convolutional layers with ReLU nonlinearity and $3\times 3$ kernels. In each reconstruction block, the first three layers produce $128$-channels feature maps and the fourth layer generate $512$-channels feature maps, whose spatial resolutions are upsampled with a scaling factor of $2$ by a deconvolution layer. The upsampled feature maps are then fused with the feature maps of the same spatial resolution from the encoding part. Specifically, the fusion is conducted by concatenating the feature maps. The last feature decoding block reconstructed the output image. A skip connection from the input image to the reconstructed image is added to enforce the denoising network to predict the residuals, which has be verified to be more robust [@Zhang:TIP17].
Overall network training
------------------------
Note that the DCNN denoisers do not have to be pre-trained. Instead, the overall deep network shown in Fig. \[fig:network\] (a) is trained by end-to-end training. To reduce the number of parameters and thus avoid over-fitting, we enforce each DCNN denoiser to share the same parameters. Mean square error (MSE) based loss function is adopt to train the proposed deep network, which can be expressed as $${\mathbf{\Theta}}= \operatorname*{argmin}_{{\mathbf{\Theta}}} \sum_{i=1}^N ||\mathcal{F}({\bm{y}}_i; {\mathbf{\Theta}}) - {\bm{x}}_i||_2^2,$$ where ${\bm{y}}_i$ and ${\bm{x}}_i$ denote the $i$-th pair of degraded and original image patches, respectively, and $\mathcal{F}({\bm{y}}_i; {\mathbf{\Theta}})$ denotes the reconstructed image patch by the network with parameter set ${\mathbf{\Theta}}$. It is also possible to train the network with other the perceptual based loss functions, which may lead to better visual quality. We remain this as future work. The ADAM optimizer [@ADAM] is used to train the network with setting $\beta_1=0.9$, $\beta_2=0.999$ and $\epsilon=10^{-8}$. The learning rate is initialize as $10^{-4}$ and halved at every $2\times 10^5$ minibatch updates. The proposed network is implemented with framework and trained using $4$ Nvidia Titan X GPUs, taking about one day to converge.
Experimental Results
====================
In this section, we perform several IR tasks to verify the performance of the proposed network, including image denoising, deblurring, and super-resolution. We trained each model for different IR tasks. We empirically found that implementing $K=5$ iterations of **Algorithm 1** in the network generally lead to satisfied IR results for image denoising, deblurring and super-resolution tasks. Thus, we fixed $K=5$ for all IR tasks. To train the networks, we constructed a large training image set, consisting of $1000$ images of size $256\times 256$ used in [@Dong:TIP11].
Image denoising
---------------
For image denoising, ${\textbf{A}}={\textbf{I}}$ and **Algorithm 1** reduce to the iterative denoising process, i.e., the weighted noise image is added back to the denoised image for the next denoising process. Such iterative denoising has shown improvements over conventional denoising methods that only denoise once [@CSR]. Here, we also found that implementing multiple denoising iterations in the proposed network improves the denoising results. To train the network, we extracted image patches of size $40\times 40$ from the training images and added additive Gaussian noise to the extracted patches to generate the noisy patches. Totally $N=450,000$ patches were extracted for training. Note that none of the test images was included into the training image set. The training patches were also augmented by flip and rotations. We compared the proposed network with several leading denoising methods, including three model-based denoising methods, i.e., BM3D method [@BM3D], the EPLL method [@Zoran:ICCV11], and the low-rank based method WNNM method [@WNNM], and two deep learning based methods, i.e., the TNRD method [@TNRD] and the DnCNN-S method [@Zhang:TIP17].
Table \[table:denoising\] shows the PSNR results of the competing methods on a set of commonly used test images shown in Fig. \[fig:den0\]. It can be seen that both the DnCNN-S and the proposed network outperform other methods. For most of the test images and noise levels, the proposed network outperforms the DnCNN-S method, which is the current state-of-the-art denoising method. On average, the PSNR gain over DnCNN-S can be up to $0.32$ dB. To further verify the effectiveness of the proposed method, we also employ the Berkeley segmentation dataset (BSD68) that contains 68 natural images for comparison study. Table \[table:denoising2\] shows the average PSNR and SSIM results of the test methods on BSD68. One can seen that the PSNR gains over the other test methods become even larger for higher noise levels. The proposed method outperforms the DnCNN-S method by up to $0.78$ dB on average on the BSD68, demonstrating the effectiveness of the proposed method. Parts of the denoised images by the test methods are shown in Figs. \[fig:den1\]-\[fig:den2\]. One can see that the image edges and textures recovered by model-based methods, i.e., BM3D, WNNM and EPLL are over-smoothed. The deep learning based methods, TNRD, DnCNN-S and the proposed method produce much more visually pleasant image structures. Moreover, the proposed method generates even better results in recovering more details than TNRD and DnCNN-S.
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\[table:denoising\]
[![width1.2pt]{}l![width1.2pt]{}l ![width1.2pt]{}l![width1.2pt]{}l![width1.2pt]{}l ![width1.2pt]{}l![width1.2pt]{}l![width1.2pt]{}l ![width1.2pt]{}l![width1.2pt]{}l![width1.2pt]{}l ![width1.2pt]{}l![width1.2pt]{}l![width1.2pt]{}l![width1.2pt]{}]{}
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\[table:denoising2\]
[![width1.2pt]{}c![width1.2pt]{}c![width1.2pt]{}c |c|c|c|c|c|c|c|c|c![width1.2pt]{}]{} & & & & & &\
& & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM\
& 15 & 31.08 & 0.872 & 31.19 & 0.883 & 31.42 & 0.883 & 31.74 & **0.891** & **32.29** & 0.888\
& 25 & 28.57 & 0.802 & 28.68 & 0.812 & 28.91 & 0.816 & 29.23 & **0.828** & **29.88** & 0.827\
& 30 & 25.62 & 0.687 & 25.68 & 0.688 & 25.96 & 0.702 & 26.24 & 0.719 & **27.02** & **0.726**\
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Image deblurring
----------------
To train the proposed network for image deblurring, we first convoluted the training images with a blur kernel to generate the blurred images and then extracted the training image patches of size $120\times 120$ from the blurred images. The additive Gaussian noise of standard deviation $\sigma_n$ was also added to the blurred images. Patch augmentation with flips and rotations were adopted, generating total $450,000$ patches for training. Two types of blur kernels were considered, i.e., the $25\times 25$ Gaussian blur kernel of standard deviation $1.6$ and two motion blur kernels adopted in [@Levin:CVPR09] of sizes $19\times 19$ and $17\times 17$. We trained each model for different blur settings. We compared the proposed method with several leading deblurring methods, i.e., three leading model-based deblurring methods (EPLL [@Zoran:ICCV11], IDDBM3D [@IDDBM3D] and NCSR [@NCSR]) and the current state-of-the-art denoising-based deblurring methods with CNN denoisers [@Zhang:CVPR17] (denoted as DD-CNN). The test images involved in this comparison study are shown in Fig. \[fig:deblur0\]. In this experiment, we only conduct deconvolution for grayscale images. However, the proposed method can be easily extended for color image deblurring. The PSNR results of the test deblurring methods are reported in Table \[tab:deblur1\]. For fair comparisons, all the PSNRs of the other methods are generated by the codes released by the authors or directly written according to their papers. From table \[tab:deblur1\], we can see that the DD-CNN method performs much better than conventional model-based EPLL, IDDBM3D and NCSR methods. For Gaussian blur, the proposed method outperforms DD-CNN by 0.27 dB on average. For other motion blur kernels with higher noise levels, the proposed method is slightly worse than DD-CNN method. Parts of the deblurred images by the competing methods are shown in Figs. \[fig:deblur1\]-\[fig:deblur4\]. From Figs. \[fig:deblur1\]-\[fig:deblur4\], one can see that the proposed method not only produces more sharper edges but also recovers more details than the other methods.
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\[tab:deblur1\]
[![width1.2pt]{}c![width1.2pt]{}c![width1.2pt]{}c|c|c|c|c|c|c|c|c|c|c![width1.2pt]{}]{} Methods & $\sigma_n$ & Butterfly & Peppers & Parrot & starfish & Barbara & Boats & C.Man & House & Leaves & Lena & Average\
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IDD-BM3D & & 29.79 & 29.64 & 31.90 & 30.57 & 25.99 & 31.17 & 27.68 & 33.56 & 30.13 & 30.91 & 30.13\
EPLL & & 25.78 & 26.73 & 31.32 & 28.52 & 24.22 & 28.84 & 26.57 & 31.76 & 25.29 & 29.46 & 27.85\
NCSR & & 29.72 & 30.04 & 32.07 & 30.83 & **26.54** & 31.22 & 27.99 & 33.38 & 30.13 & 30.99 & 30.29\
DD-CNN & & 30.44 & **30.69** & 31.83 & 30.78 & 26.15 & 31.41 & 28.05 & 33.80 & **30.44** & 31.05 & 30.48\
**Ours** & & **30.67** & 30.18 & **32.40** & **32.00** & 26.47 & **31.54** & **28.24** & **34.25** & 30.23 & **31.48** & **30.75**\
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EPLL & & 26.23 & 27.40 & 33.78 & 29.79 & 29.78 & 30.15 & 30.24 & 31.73 & 25.84 & 31.37 & 29.63\
DD-CNN & & 32.23 & 32.00 & 34.48 & 32.26 & 32.38 & 33.05 & 31.50 & 34.89 & 33.29 & 33.54 & 32.96\
**Ours** & & **32.58** & **32.05** & **34.98** & **32.71** & **32.39** & **33.39** & **31.70** & **35.34** & **32.99** & **33.80** & **33.19**\
EPLL & & 24.27 & 26.15 & 30.01 & 26.81 & 26.95 & 27.72 & 27.37 & 29.89 & 23.81 & 28.69 & 27.17\
DD-CNN & & **28.51** & **28.88** & **31.07** & 27.86 & **28.18** & 29.13 & **28.11** & 32.03 & **28.42** & **29.52** & **29.17**\
**Ours** & & 28.24 & 28.42 & 31.03 & **28.00** & 28.01 & **29.19** & 27.77 & **32.06** & 27.98 & 29.42 & 29.01\
\
EPLL & & 26.48 & 27.37 & 33.88 & 29.56 & 28.29 & 29.61 & 29.66 & 32.97 & 25.69 & 30.67 & 29.42\
DD-CNN & & **31.97** & **31.89** & 34.46 & 32.18 & **32.00** & **33.06** & **31.29** & 34.82 & **32.96** & **33.35** & **32.80**\
**Ours** & & 31.86 & 31.38 & **34.72** & **32.28** & 31.36 & 32.86 & 31.21 & **35.09** & 32.29 & **33.35** & 32.64\
EPLL & & 23.85 & 26.04 & 29.99 & 26.78 & 25.47 & 27.46 & 26.58 & 30.49 & 23.42 & 28.20 & 26.83\
DD-CNN & & **28.21** & **28.71** & **30.68** & 27.67 & **27.37** & **28.95** & **27.70** & **31.95** & **27.92** & **29.27** & **28.84**\
**Ours** & & 27.47 & 28.02 & 30.46 & **27.82** & 26.86 & 28.84 & 27.48 & 31.91 & 27.28 & 29.23 & 28.54\
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Image super-resolution
----------------------
For image super-resolution, we consider two image subsampling operators, i.e., the bicubic downsampling and the Gaussian downsampling. For the former case, the HR images are downsampled by applying the bicubic interpolation function with scaling factor $1/s$ ($s=2, 3, 4$) to simulate the LR images. For the latter case, the LR images are generated by applying the Gaussian blur kernel to the original images followed by subsampling. The $7\times 7$ Gaussian blur kernel of standard deviation of 1.6 is used in this case. The LR/HR patch pairs are extracted from the LR/HR training image pairs and augmented by flip and rotations, generating $450,000$ patch pairs. The LR patch size is $32\times 32$, while the HR patch size is $32*s\times 32*s$. We train each network for the two downsampling cases. The image data sets commonly used in the image super-resolution (SR) literature are adopted for performance verification, including the set5, set14, the Berkeley segmentation dataset containing 100 images (denoted as BSD100), and the Urban 100 dataset [@VDSR] containing 100 high-quality images. We compared the proposed method with several leading image SR methods, including two DCNN based SR methods (SRCNN [@SRCNN] and VDSR [@VDSR]) and two denoising methods (TNRD [@TNRD] and DnCNN [@Zhang:TIP17]), which produce the HR images by first upsampling the LR images with the bicubic interpolator and then denoising the upsampled images to recovery the high-frequency details. For fair comparisons, the results of the others are directly borrowed from their papers or generated by the codes released by the authors.
The PSNR results of the test methods for bicubic downsampling are reported in Tables \[tab:SR1\]-\[tab:SR2\], from which we can see that the proposed method outperforms other competing methods. We observed that the PSNR gains over other methods becomes larger for large scaling factors, verifying the importance of observation consistencies for IR. The PSNR results of the test methods for Gaussian downsampling with scaling factor 3 are reported in Table \[tab:SR3\]. For this case, we compare the proposed method with DD-CNN [@Zhang:CVPR17], which has much better results than their earlier DnCNN [@Zhang:TIP17]. Since VDSR and SRCNN are trained for bicubic downsampling, it is unfair to directly apply these methods to the LR images generated by Gaussian downsampling and thus we didn’t include their results into this table. Parts of the reconstructed HR images by the test methods are shown in Fig. \[fig:SR1\]-\[fig:SR3\], from which we can see that the proposed method can produce sharper edges than other methods.
[![width1.2pt]{}c|c|c|c|c|c|c![width1.2pt]{}]{} Images & Scaling factor & TNRD & SRCNN & VDSR & DnCNN & Ours\
Baby & & 38.53 & 38.54 & 38.75 & 38.62 &**38.88**\
Bird & & 41.31 & 40.91 & 42.42 & 42.20 & **42.89**\
Butterfly & & 33.17 & 32.75 & 34.49 & 34.51 & **34.72**\
Head & & 35.75 & 35.72 & 35.93 & 35.84 & **36.00**\
Woman & & 35.50 & 35.37 & 36.05 & 35.99 & **36.24**\
**Average** & & 36.85 & 36.66 & 37.53 & 37.43 & **37.75**\
Baby & & 35.28 & 35.25 & 35.38 & 35.42 & **35.57**\
Bird & & 36.09 & 35.48 & 36.66 & 36.61 & **37.51**\
Butterfly & & 28.92 & 27.95 & **29.96** & 29.95 & 29.81\
Head & & 33.75 & 33.71 & 33.96 & 33.92 & **34.03**\
Woman & & 31.79 & 31.37 & 32.36 & 32.31 & **32.71**\
**Average** & & 33.17 & 32.75 & 33.66 & 33.64 & **33.93**\
Baby & & 31.30 & 33.13 & 33.41 & 33.23 & **33.64**\
Bird & & 32.99 & 32.52 & 33.54 & 33.06 & **34.09**\
Butterfly & & 26.22 & 25.46 & 27.28 & 26.94 & **27.68**\
Head & & 32.51 & 32.44 & 32.70 & 32.36 & **32.88**\
Woman & & 29.20 & 28.89 & 29.81 & 29.46 & **30.34**\
**Average** & & 30.85 & 30.48 & 31.35 & 31.01 & **31.72**\
[![width1.2pt]{}c![width1.2pt]{}c|c|c|c|c|c|c|c|c|c|c![width1.2pt]{}]{} & & & & & &\
& & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM\
& 2 & 32.54 & 0.907 & 32.42 & 0.906 & 33.03 & 0.912 & 33.03 & 0.911 & **33.30** & **0.915**\
& 3 & 29.46 & 0.823 & 29.28 & 0.821 & 29.77 & 0.831 & 29.82 & 0.830 & **30.02** & **0.836**\
& 4 & 27.68 & 0.756 & 27.49 & 0.750 & 28.01 & 0.767 & 27.83 & 0.755 & **28.28** & **0.773**\
& 2 & 31.40 & 0.888 & 31.36 & 0.888 & 31.90 & 0.896 & 31.84 & 0.894 & **32.04** & **0.898**\
& 3 & 28.50 & 0.788 & 28.41 & 0.786 & 28.82 & 0.798 & 28.80 & 0.795 & **28.91** & **0.801**\
& 4 & 27.00 & 0.714 & 26.90 & 0.710 & 27.29 & 0.725 & 27.08 & 0.709 & **27.39** & **0.729**\
& 2 & 29.70 & 0.899 & 29.50 & 0.895 & 30.76 & 0.914 & 30.63 & 0.911 & **31.50** & **0.922**\
& 3 & 26.44 & 0.807 & 26.24 & 0.799 & 27.14 & 0.828 & 27.08 & 0.824 & **27.61** & **0.842**\
& 4 & 24.62 & 0.729 & 24.52 & 0.722 & 25.18 & 0.752 & 24.94 & 0.735 & **25.53** & **0.768**\
[![width1.2pt]{}c![width1.2pt]{}c|c|c|c|c![width1.2pt]{}]{} Dataset & NCSR & SRCNN & VDSR & DD-CNN & Ours\
Set5 & 32.07 & - & - & 33.88 & **34.22**\
Set14 & 29.30 & - & - & 29.63 & **29.88**\
\[tab:SR3\]
\
\
\
Conclusion
==========
In this paper, we have proposed a novel deep neural network for general image restoration (IR) tasks. Different from current deep network based IR methods, where the observation models are generally ignored, we construct the deep network based on a denoising-based IR framework. To this end, we first developed an efficient algorithm for solving the denoising-based IR method and then unfolded the algorithm into a deep network, which is composed of multiple denoising modules interleaved with back-projection modules for data consistencies. A DCNN-based denoiser exploiting multi-scale redundancies of natural images was developed. Therefore, the proposed deep network can exploit not only the effective DCNN denoising prior but also the prior of the observation model. Experimental results show that the proposed method can achieve very competitive and often state-of-the-art results on several IR tasks, including image denoising, deblurring and super-resolution.
Convergence {#convergence .unnumbered}
===========
Consider the energy function $$\xi({\bm{x}},{\bm{v}}) := \frac{1}{2}\|{\bm{y}}-{\textbf{A}}{\bm{x}}\|_2^2 + \frac{\eta}{2}\|{\bm{x}}-{\bm{v}}\|_2^2 + \lambda J({\bm{v}}).$$ Assume that $\xi$ is lower bounded and coercive[^4]. For Algorithm 1, $({\bm{x}}^{(t)}, {\bm{v}}^{(t)})$ has a subsequence that converges to a stationary point of the the energy function provided that the denoiser $f(\cdot)$ satisfies the sufficient descent condition: $$\begin{aligned}
\label{fcond1}
&\frac{\eta}{2}||{\bm{x}}-{\bm{v}}||_2^2 + \lambda J({\bm{v}})-\frac{\eta}{2}||{\bm{x}}-f({\bm{x}})||_2^2 - \lambda J(f({\bm{x}}))\nonumber\\
&\ge c_2\|\tilde\nabla_{{\bm{v}}}\xi({\bm{x}},{\bm{v}})\|_2^2,$$ where $c_2>0$ and $\tilde\nabla_{{\bm{v}}}\xi({\bm{x}},\cdot)$ is a continuous limiting subgradient of $\xi$.
Since $\nabla_{{\bm{x}}}\xi({\bm{x}},{\bm{v}})$ is Lipschitz continuous with constant $\|{\textbf{A}}^{\top}{\textbf{A}}\|+\eta$, it is well known that the gradient step on ${\bm{x}}$ with step size $\delta\in(0,\frac{2}{\|{\textbf{A}}^{\top}{\textbf{A}}\|+\eta})$ satisfies the descent property $$\begin{aligned}
\label{desx}
\xi({\bm{x}}^{(t)},{\bm{v}}^{(t)})-\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t)}) \ge c_1 \|{\bm{x}}^{(t)}-{\bm{x}}^{(t+1)} \|_2^2, $$ where $c_1:=\frac{1}{\delta}-\frac{\|{\textbf{A}}^{\top}{\textbf{A}}\|+\eta}{2}>0$. By assumption, the ${\bm{v}}$-step satisfies $$\begin{aligned}
\label{desv}
\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t)})-\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t+1)}) \ge c_2\|\tilde\nabla_{{\bm{v}}}\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t)})\|_2^2.\end{aligned}$$ Since $\xi({\bm{x}},{\bm{v}})$ is coercive and, by and , $\xi({\bm{x}}^{(t)},{\bm{v}}^{(t)})$ is monotonically nonincreasing, the sequence $({\bm{x}}^{(t)},{\bm{v}}^{(t)})_{t=0,1,2,\ldots}$ is bounded (otherwise, it would cause the contradiction $\xi({\bm{x}}^{(t)},{\bm{v}}^{(t)})\to \infty$), so it has a convergent subsequence $({\bm{x}}^{(t_k)},{\bm{v}}^{(t_k)})_{k=0,1,\ldots}\overset{k}{\to}({\bm{x}}^{*},{\bm{v}}^{*})$. Since $\xi({\bm{x}},{\bm{v}})$ is lower bounded, adding (9) and (10) yields $$\begin{aligned}
&\xi({\bm{x}}^{(t)},{\bm{v}}^{(t)})-\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t+1)})\nonumber\\
&\ge c_1 \|{\bm{x}}^{(t)}-{\bm{x}}^{(t+1)} \|_2^2+c_2\|\tilde\nabla_{{\bm{v}}}\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t)})\|_2^2.\end{aligned}$$ and, by telescopic sum over $t=0,1,\ldots$ and by monotonicity and boundedness of $\xi({\bm{x}}^{(t)},{\bm{v}}^{(t)})$, we have the summability properties $\sum_t\|{\bm{x}}^{(t)}-{\bm{x}}^{(t+1)} \|_2^2 <\infty$ and $\sum_t\|\tilde\nabla_{{\bm{v}}}\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t)})\|_2^2 <\infty$, from which we conclude $$\begin{aligned}
&\lim_{t\to\infty}\|{\bm{x}}^{(t)}-{\bm{x}}^{(t+1)} \|_2=0,\label{txcvg}\\
&\lim_{t\to\infty}\|\tilde\nabla_{{\bm{v}}}\xi({\bm{x}}^{(t+1)},{\bm{v}}^{(t)})\|_2 = 0.\end{aligned}$$ Based on ${\bm{x}}^{(t+1)}-{\bm{x}}^{(t)}=\delta \nabla_{{\bm{x}}} \xi({\bm{x}}^{(t)},{\bm{v}}^{(t)})$, we get $\nabla_{{\bm{x}}} \xi({\bm{x}}^{*},{\bm{v}}^{*}) =\lim_{k\to\infty}\nabla_{{\bm{x}}} \xi({\bm{x}}^{(t_k)},{\bm{v}}^{(t_k)}) =0$, where we have used the continuity of $\nabla_{{\bm{x}}} \xi({\bm{x}},{\bm{v}})$ in ${\bm{x}}$. Also, $\lim_{k\to\infty}\tilde\nabla_{{\bm{v}}} \xi({\bm{x}}^{(t_k)},{\bm{v}}^{(t_k)}) =\lim_{k\to\infty}\tilde\nabla_{{\bm{v}}} \xi({\bm{x}}^{(t_k+1)},{\bm{v}}^{(t_k)})=0$, where the first “$=$” follows from the continuity of $\nabla_{{\bm{v}}} \xi({\bm{x}},{\bm{v}})=2\mu({\bm{v}}-{\bm{x}})+\lambda\tilde \nabla J({\bm{v}})$ in ${\bm{x}}$ and . Therefore, $({\bm{x}}^{*},{\bm{v}}^{*})$ is a stationary point of $\xi$.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
M. Elad and M. Aharon, “Image denoising via sparse and redundant representation over learned dictionaries,” *IEEE Transactions on Image Processing*, vol. 15, no. 12, pp. 3736–3745, 2006.
K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” *IEEE Transactions on image processing*, vol. 16, no. 8, pp. 2080–2095, 2007.
W. Dong, X. Li, L. Zhang, and G. Shi, “Sparsity-based image denoising via dictionary learning and structural clustering,” in *Proc. of the IEEE CVPR*, 2011, pp. 457–464.
W. Dong, G. Shi, and X. Li, “Nonlocal image restoration with bilateral variance estimation: a low-rank approach,” *IEEE Transactions on image processing*, vol. 22, no. 2, pp. 700–711, 2013.
W. Dong, X. Li, L. Zhang, and G. Shi, “Weighted nuclear norm minimization with application to image denoising,” in *Proc. of the IEEE CVPR*, 2014, pp. 2862–2869.
W. Dong, L. Zhang, G. Shi, and X. Wu, “Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization,” *IEEE Transactions on image processing*, vol. 20, no. 7, pp. 1838–1857, 2011.
A. Danielyan, V. Katkovnik, and K. Egiazarian, “Bm3d frames and variational image deblurring,” *IEEE Transactions on image processing*, vol. 21, no. 4, pp. 1715–1728, 2012.
W. Dong, L. Zhang, G. Shi, and X. Li, “Nonlocally centralized sparse representation for image restoration.” *IEEE Transactions on Image Processing*, vol. 22, no. 4, pp. 1620–1630, 2013.
W. Dong, G. Shi, Y. Ma, and X. Li, “Image restoration via simultaneous sparse coding: Where structured sparsity meets gaussian scale mixture,” *International Journal of Computer Vision*, pp. 1–16, 2015.
A. Marquina and S. J. Osher, “Image super-resolution by tv-regularization and bregman iteration,” *Journal of Scientific Computing*, vol. 37, no. 3, pp. 367–382, 2008.
J. Yang, J. Wright, T. Huang, and Y. Ma, “Image super-resolution as sparse representation of raw image patches,” in *Proc. of the IEEE CVPR*, 2008, pp. 1–8.
X. Gao, K. Zhang, D. Tao, and X. Li, “Image super-resolution with sparse neighbor embedding,” *Image Processing, IEEE Transactions on*, vol. 21, no. 7, pp. 3194–3205, 2012.
S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” *Multiscale Modeling and Simulation*, vol. 4, no. 2, pp. 460–489, 2005.
J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,” *IEEE Transactions on Image Processing*, vol. 16, no. 12, pp. 2992–3004, 2007.
J. Mairal, M. Elad, and G. Sapiro, “Sparse representation for color image restoration,” *IEEE Transactions on image processing*, vol. 17, no. 1, pp. 53–69, 2008.
D. Zoran and Y. Weiss, “From learning models of natural image patches to whole image restoration,” in *Proc. of the IEEE ICCV*, 2011, pp. 479–486.
S. Roth and M. J. Black, “Fields of experts,” *International Journal of Computer Vision*, vol. 82, no. 2, pp. 205–229, 2009.
G. Yu, G. Sapiro, and S. Mallat, “Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity,” *IEEE Transactions on Image Processing*, vol. 21, no. 5, pp. 2481–2499, 2012.
W. T. Freeman, T. R. Jones, and E. C. Pasztor, “Examplebased super-resolution,” *Computer Graphics and Applications*, vol. 22, no. 2, pp. 56–65, 2002.
R. Timofte, V. De Smet, and L. Van Gool, “A+: Adjusted anchored neighborhood regression for fast super-resolution,” in *Asian Conference on Computer Vision*.1em plus 0.5em minus 0.4emSpringer, 2014, pp. 111–126.
U. Schmidt and S. Roth, “Shrinkage fields for effective image restoration,” in *Proc. of the IEEE CVPR*, 2014, pp. 2774–2781.
C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” in *European Conference on Computer Vision*.1em plus 0.5em minus 0.4emSpringer, 2014, pp. 184–199.
Z. Wang, D. Liu, J. Yang, W. Han, and T. Huang, “Deep networks for image super-resolution with sparse prior,” in *Proc. of the IEEE ICCV*, 2015, pp. 370–378.
K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a gaussian denoiser: Residual learning of deep cnn for image denoising,” *IEEE Transactions on image processing*, vol. 26, no. 7, pp. 3142–3155, 2017.
A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in *Proc. of the IEEE CVPR*, 2005, pp. 60–65.
S. Venkatakrishnan, C. Bouman, E. Chu, and B. Wohlberg, “Plug-and-play priors for model based reconstruction,” in *Proc. of IEEE Global Conference on Signal and Information Processing*, 2013, pp. 945–948.
A. Brifman, Y. Romano, and M. Elad, “Turning a denoiser into a super-resolver using plug and play priors,” in *Proc. of the IEEE ICIP*, 2016, pp. 1404–1408.
A. M. Teodoro, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Image restoration and reconstruction using variable splitting and class-adapted image priors,” in *Proc. of IEEE ICIP*, 2016, pp. 3518–3522.
M. E. Y. Romano and P. Milanfar, “The little engine that could: Regularization by denoising (red),” *SIAM Journal on Imaging Sciences*, vol. 10, no. 4, pp. 1804–1844, 2017.
S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play admm for image restoration: Fixed-point convergence and applications,” *IEEE Transactions on Computational Imaging*, vol. 3, no. 1, pp. 84–98, 2017.
A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in *In Advances in Neural Information Processing Systems*, 2012, pp. 1097–1105.
K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in *Proc. of the IEEE CVPR*, 2016, pp. 770–778.
C. Dong, C. C. Loy, and X. Tang, “Accelerating the super-resolution convolutional neural network,” in *European Conference on Computer Vision*.1em plus 0.5em minus 0.4emSpringer, 2016, pp. 391–407.
J. Kim, J. K. Lee, and K. M. Lee, “Accurate image super-resolution using very deep convolutional networks,” in *IEEE Conference on Computer Vision and Pattern Recognition*, 2016, pp. 1646–1654.
Y. Chen and T. Pock, “Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration,” *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 39, no. 6, pp. 1256–1272, 2017.
K. Zhang, W. Zuo, S. Gu, and L. Zhang, “Learning deep cnn denoiser prior for image restoration,” in *Proc. of the IEEE CVPR*, 2017, pp. 2808–2817.
S. A. Bigdeli and M. Zwicker, “Image restoration using autoencoding priors,” in *arXiv:1703.09964v1*, 2017.
S. Ren, K. He, R. Girshick, and J. Sun, “Faster r-cnn: towards real-time object detection with region proposal networks,” *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 39, no. 6, pp. 1137–1149, 2017.
J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in *Proc. of IEEE CVPR*, 2015, pp. 3431–3440.
H. C. Burger, C. J. Schuler, and S. Harmeling, “Image denoising: can plain neural networks compete with bm3d?” in *Proc. of IEEE CVPR*, 2012, pp. 2392–2399.
L. Xu, J. S. Ren, C. Liu, and J. Jia, “Deep convolutional neural network for image deconvolution,” in *In Advances in Neural Information Processing Systems*, 2014.
K. Gregor and Y. LeCun, “Learning fast approximations of sparse coding,” in *Proc. of the IEEE ICML*, 2010.
Y. Yang, J. Sun, H. Li, and Z. Xu, “Deep admm-net for compressive sensing mri,” in *In Advances in Neural Information Processing Systems*, 2016.
B. Xin, Y. Wang, W. Gao, and D. Wipf, “Maximal sparsity with deep networks?” in *In Advances in Neural Information Processing Systems*, 2016.
Y. Wang, W. Yin, and J. Zeng, “Global convergence of admm in nonconvex nonsmooth optimization,” *Journal of Scientific Computing*, 2018.
J. Bolte, A. Daniilidis, and A. Lewis, “The lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems,” *SIAM Journal on Optimization*, vol. 17, no. 4, pp. 1205–1223, 2007.
Y. Xu and W. Yin, “A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion,” *SIAM Journal on Imaging Sciences*, vol. 6, no. 3, pp. 1758–1789, 2013.
J. Lee, I. Panageas, G. Piliouras, M. Simchowitz, M. Jordan, and B. Recht, “First-order methods almost always avoid saddle points,” *arXiv:1710.07406*.
G. Alain and Y. Bengio, “What regularized auto-encoders learn from the data-generating distribution,” *Journal of Machine Learning Research*, vol. 15, pp. 3743–3773, 2014.
P. O. Pinheiro, T.-Y. Lin, R. Collobert, and P. Dollar, “Learning to refine object segments,” in *Proc. of ECCV*, 2016.
O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in *In International Conference on Medical Image Computing and Computer-Assisted Intervention*, 2015, pp. 234–241.
D. Kingma and J. Ba, “Adam: a method for stochastic optimization,” in *Proc. of ICLR*, 2014.
A. Levin, Y. Weiss, F. Durand, and W. T. Freeman, “Understanding and evaluating blind deconvolution algorithms,” in *Proc. of CVPR*, 2009, pp. 1964–1971.
[^1]: W. Dong, P. Wang, G. Shi, Fangfang Wu, and X. Lu are with School of Electronic Engineering, Xidian University, Xi’an, 710071, China (e-mail: wsdong@mail.xidian.edu.cn)
[^2]: W. Yin is with Department of Mathematics, University of California, Los Angeles, CA 90095.
[^3]: $\xi({\bm{x}},{\bm{v}})\to\infty$ whenever $\|({\bm{x}},{\bm{v}})\|\to\infty$.
[^4]: $\xi({\bm{x}},{\bm{v}})\to\infty$ whenever $\|({\bm{x}},{\bm{v}})\|\to\infty$.
|
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abstract: 'In recent years, deep hashing methods have been proved to be efficient since it employs convolutional neural network to learn features and hashing codes simultaneously. However, these methods are mostly supervised. In real-world application, it is a time-consuming and overloaded task for annotating a large number of images. In this paper, we propose a novel unsupervised deep hashing method for large-scale image retrieval. Our method, namely unsupervised semantic deep hashing (**USDH**), uses semantic information preserved in the CNN feature layer to guide the training of network. We enforce four criteria on hashing codes learning based on VGG-19 model: 1) preserving relevant information of feature space in hashing space; 2) minimizing quantization loss between binary-like codes and hashing codes; 3) improving the usage of each bit in hashing codes by using maximum information entropy, and 4) invariant to image rotation. Extensive experiments on CIFAR-10, NUSWIDE have demonstrated that **USDH** outperforms several state-of-the-art unsupervised hashing methods for image retrieval. We also conduct experiments on Oxford 17 datasets for fine-grained classification to verify its efficiency for other computer vision tasks.'
bibliography:
- 'icme2018template.bib'
title: Unsupervised Semantic Deep Hashing
---
Ł[[L]{}]{}
Deep Learning, Unsupervised Hashing, Semantic Loss
Introduction
============
With the explosive increase of data, searching for content relevant image, or other media data remains a challenge because of large amount of computational cost and the accuracy requirement. In the early stage, researchers focus on data-independent methods. Locality-Sensitive Hashing [@andoni2006Near] and its variants are proposed. But it has a lower accuracy since the semantic information of data is not considered during coding process. In recent years, the data-dependent hashing methods [@lin2014fast] attract more attention since its compact representation and superior accuracy performance. Compared with data-independent hashing method, data-dependent hashing methods improve retrieval performance via training on the dataset.
Data-dependent methods mainly include supervised hashing methods [@liu2012supervised], unsupervised hashing methods [@weiss2009spectral] and semi-supervised hashing methods [@wang2010semi]. These supervised methods make use of the class information provided in the manual labels, where the supervised information is used in three forms: point-wise labels, pair-wise labels, and ranking labels. Some representative works have been proposed, e.g. Semantic Hashing [@salakhutdinov2009semantic], Binary Reconstruction Embedding [@kulis2009learning], Minimal Loss Hashing[@norouzi2011minimal], Kernel-based Supervised Hashing [@liu2012supervised], Hamming Distance Metric Learning [@norouzi2012hamming], and Column Generation Hashing [@li2013learning]. Although the supervised hashing methods and semi-supervised hashing methods have been proved to gain better accuracy with compacter hashing codes, it is a time-consuming and heavy workload task in practical application. In the past years, some classical unsupervised hashing methods also have been developed, e.g. Isotropic Hashing [@kong2012isotropic], Spherical Hashing [@heo2012spherical], Discrete Graph Hashing [@liu2014discrete], Locally Linear Hashing [@irie2014locally], Asymmetric Inner-product Binary Coding [@shen2015learning] and Scalable Graph Hashing [@jiang2015scalable].
In these traditional hashing methods, each image is initially represented by a hand-crafted feature. However, these features may not preserve accurate semantic information. And they also may not be suitable for generating binary codes. Due to these facts, the accuracy of image retrieval could not meet our requirement. Over the last five years, deep learning has been proved to be effective in computer vision because it could automatically extract high-level semantic feature to represent image that is robust to the variances of object. Hinton [@salakhutdinov2009semantic] et al. firstly proposed hashing method based on deep neural network. However, in [@salakhutdinov2009semantic], the input of the network is still hand-crafted features, which is the most crucial limitation.
Very recently, convolutional Neural Network Hashing [@xia2014supervised] introduces an end-to-end network for learning better hashing codes. However, this method has limitations since it cannot perform feature learning and hashing codes learning simultaneously. Followed [@xia2014supervised], new variants of deep hashing have been proposed, e.g, Deep Neural Network Hashing [@lai2015simultaneous], Deep Semantic Ranking Hashing [@zhao2015deep], deep supervised hashing [@liu2016deep] and DeepBit [@lin2016learning], which extract features and learn hashing codes simultaneously. These methods are more effective and perform more efficiently in image retrieval task. However, most of these deep hashing methods, except DeepBit [@lin2016learning] and DBD-MQ [@Duan2017learning], are pure supervised. DBD-MQ [@Duan2017learning] propose a quantization method for hashing learning. This method does not utilize the rigid sign function for binarization and considers the binarization as a multi-quantization task. DeepBit [@lin2016learning] tries to make hashing codes invariant to rotation by minimizing the difference between the hashing codes that describe the reference image and that of rotated one. However, this method only considers rotation invariance of images, and the invariance among different images with same class label can not be guaranteed.
In this paper, motivated by the success of DeepBit [@lin2016learning], we propose a novel unsupervised deep hashing method, called unsupervised semantic deep hashing (**USDH**).
{height="1.75in" width="5.38in"}
The main contributions of **USDH** are outlined as follows:
**USDH** is an unsupervised end-to-end deep hashing framework. Compared with DeepBit method, **USDH** not only considers rotation invariance in a single image, but also preserves the semantic information of image pairs.
**USDH** proposes a novel deep unsupervised hashing method to preserve the similarity information in the feature space. It regards the output of full-connected layer as representation descriptor of image. The loss function requires hashing codes learned by deep network approximating the similarity computed by representation descriptors of image.
Experiments on general datasets show that **USDH** can outperform other unsupervised methods to achieve the state-of-the-art performance in image retrieval applications. And it is also quite effective for fine-grained classification.
Unsupervised Semantic Deep Hashing
==================================
In this paper, we introduce a novel unsupervised deep hashing method. Compared with existing methods, our method utilizes relevant information preserved in the feature space to guide the learning process of hashing codes. Based on this motivation, the cost function of unsupervised semantic deep hashing contains four components: 1) preserving relevant information of feature space in hashing codes via a semantic loss 2) minimizing quantization loss between binary-like codes and hashing codes 3) improving the usage of each bit in hashing codes by maximizing information entropy 4) keeping the learned hashing codes invariant to rotation by pulling hashing codes of reference image and that of the rotated one together. The whole deep model is shown in Figure \[Fig2:\]. The cost function is written as below:
$$J=J_{1}+J_{2}+J_{3}+J_{4},$$
$J_{1}$ represents semantic loss, $J_{2}$ represents quantization loss, $J_{3}$ represents information loss, $J_{4}$ represents rotation loss.
Semantic Loss
-------------
To preserve semantic information in the feature space, firstly, we should adopt an optimal feature to represent images and use a proper formula to measure the similarity of images in the feature space, then we let similarity computed by the hashing codes of image pairs approximate the similarity measured in the original feature space.
Firstly, we adopt VGG-19 model to process the images and use the output of the second full-connected layer as our image feature. Many researches have proved that high-level feature of convolutional neural network has sufficient semantic information and these mid-features are robust to inner-class including rotation, shape and color variance. There also exist different metrics to measure similarity in the feature space. We adopt a widely-used metric that is defined as:
$$S_{i,j}=e^{\frac {-{\Vert{x_{i}-x_{j}\Vert}}_{2}}{\rho d}},$$
Where $d$ denotes the dimension of the second full-connected layer and $\rho$ is a positive constant parameter. $S_{i,j}\in(0,1]$ can represent a similarity degree of the images $i$ and $j$. The hashing codes of image $i$ is denoted as $b_i$.
We require hashing codes preserving relevant semantic information. More specifically, if $S_{i,j}$ is near to 1, we assume hashing codes $b_{i}$ and $b_{j}$ has smaller distance. But if $S_{i,j}$ is near to 0, then $b_{i}$ and $b_{j}$ has larger distance. For each training batch, we can obtain a similarity matrix. We try to use the similarity degree in the feature space to guide the learning of hashing codes. To do so, in hamming space, we also define a similarity measure, and then the similarity measure defined in hashing space is required to be as similar as possible to the similarity matrix defined in the original feature space.
According to this constraint, the neighbor points in the feature space are still neighbors in the hashing space. Specifically, $b_{i}\in\{0,1\}$ is relaxed to $(0,1)$, then the hashing codes is linearly transformed to $(-1,1)$: $$\widetilde b_{i}=2b_{i}-1,$$ where $\widetilde b_{i}\in(-1,1)$. The inner product of $\widetilde b_{i}$and $\widetilde b_{j}$ is in the range of $(-k,k)$, where $k$ is the length of hashing codes. Then the inner product is linear transformed to $(0,1)$ via $\frac{\widetilde b_{i}^T*\widetilde b_{j}+k}{2k}$. The result of linear transformation is also regarded as a similarity degree. And it fits in with the assumption on information loss that each bit of hashing codes plays the same role.The function of semantic loss is written as:
$$J_{1}=\sum_{i,j} {{\left \vert S_{i,j}-\frac{\widetilde b_{i}^T*\widetilde b_{j}+k}{2k}\right \vert}}_{1}.$$
With this loss function, the deep model is trained by back-propagation algorithm with batch gradient descent method. To solve this, the gradient of semantic loss function need to be computed. Since $l_{1}$ norm is non-differentiable at some certain point , we employ sub-gradient to overcome the problem and we define the sub-gradient at this point to be equal to right-hand derivative. The gradient of semantic loss is defined as:
$$%\left.\frac{\partial J_{1}}{\partial b_{i}}\right =\sum_{j} sgn(S_{i,j}-\frac{\widetilde b_{i}^T*\widetilde b_{j}+k}{2k})*{\frac{\widetilde b_{j}}{k}}
\frac{\partial J_{1}}{\partial b_{i}} =\sum_{j} sgn(S_{i,j}-\frac{\widetilde b_{i}^T*\widetilde b_{j}+k}{2k})*{\frac{\widetilde b_{j}}{2k}}$$
where
$$sgn(x)=\left\{\begin{array}{ll}
1& x \geq 0 ,\\
-1& x<0 .
\end{array}\right.$$
Quantization Loss
-----------------
Since it is difficult to directly optimize discrete loss function, we should relax the objective function to transform the discrete problem into a continuous optimization problem. As discussed in [@liu2016deep], some widely-used relaxation scheme working with non-linear functions, such as sigmoid and tanh function, would inevitably slow down or even restrain the convergence of the network [@krizhevsky2012imagenet]. To overcome such limitation, we still use the relu function as activation function of the second full-connected layer. Then the output of network is quantized to the binary codes. The quantization function is written as: $$f(b_i)=\left\{\begin{array}{ll}
1& b_i \geq 0.5 ,\\
0& b_i<0.5 .
\end{array}\right.$$ where $f(x)$ denotes the binarization function.
To decrease this loss, we let the value of network’s output near to 1 or 0. First, the hashing codes $b_i$ is linearly transformed to $(-1,1)$ in the same way. Then the result of linearly transformation is changed into an absolute value. The absolute value of the hashing codes $\vert \widetilde b_i \vert$ should be near to 1. Finally the quantization loss is defined as:
$$J_2=\alpha\sum_{i} {\left \vert |\widetilde b_i|-1 \right \vert}_1,$$
where$\left \vert . \right \vert $ denotes element-wise absolute value, and ${\left \Vert . \right \Vert}_1 $ denotes $l_1$ norm. $\alpha$ is a weighting parameter.
To train the model, the gradient of $J_2$ need to be computed. The sub-gradient is taken to replace the gradient of $J_2$ because of the non-differentiate point in the absolute operation and $l_1$ norm. The gradient is written as: $$\frac{\partial J_2}{\partial b_i}=\left\{\begin{array}{ll}
2\alpha & b_i \geq 1\quad or\quad 0<b_i<0.5,\\
-2\alpha & \quad otherwise .
\end{array}\right.$$
Information Loss
----------------
As the main assumption of semantic loss, each bit of hashing codes should play an equivalent impact, which means each bit should have the same mean value. Inspired by the efficiency of DeepBit [@lin2016learning] method, we also maximize capability of each bit in hashing codes to express information. So we further enhance the hashing codes by assuming that each bit has half chance to be one. Based on this constraint, the balanced distribution criterion can be written as below: $$\mu_i=\frac{1}{m}\sum_{i=1}^{m}{b_i(m)},$$ where $\mu_i$ denotes the mean value of $i$-th bite of hashing codes, ${\Vert . \Vert}_2 $ denotes $l_2$ norm and $m$ denotes the size of training batch.
Rotation Loss
-------------
Existing widely-used hand-crafted features should be invariant to rotation and scale. Inspired by this motivation, we also rotate the images and pull hashing codes that represent the reference image and that of the rotated one together. The proposed rotation-invariance criterion can be written as:
$$J_4=\sum_{i=1}^{m}\sum_{\theta=0}^{2R} {\left \Vert b_{\theta,i}-b_i\right \Vert},$$
Where $b_{\theta,i}$ denotes hashing codes of image $i$ with rotation $\theta$.
Experiment
==========
In order to test the performance of our proposed method, we conduct experiments on four datasets, including three widely used image retrieval datasets: CIFAR-10 and NUSWIDE dataset, as well as one recognition dataset: Oxford flower17. Similar to other image retrieval task, our method is also evaluated based on mean accuracy precision at top 1000. Compared with some representative unsupervised hashing methods, such as KMH [@He2013K], SphH [@heo2012spherical], SpeH [@weiss2009spectral], PCAH [@wang2010semi], LSH [@andoni2006Near], PCA-ITQ [@gong2013iterative], DH [@lin2015deep], DeepBit [@lin2016learning] and DBD-MQ [@Duan2017learning], experimental results verify that our proposed method outperforms these existing unsupervised hashing method. In order to prove our method is flexible for other computer vision applications, we also conduct experiments for fined-grained recognition on Oxford flower17 dataset.
Dataset
-------
[**CIFAR-10 dataset**]{} consists of 60000 32$\times$32 images in 10 classes. Each image in dataset belongs to one class.( 6000 images per class) The dataset is divided in two parts: train set(5000 images per class) and test set(1000 images per class).
[**NUSWIDE dataset**]{} is a multi-label dataset. NUSWIDE contains nearly 270k images associated with 81 semantic concepts. Followed [@xia2014supervised], We select the 21 most frequent concept. Each of concepts is associated with at least 5000 images. The dataset is splitted into training set and test set. We sample 100 images from each concepts to form a test set and the remaining images are treated as a training set.
[**Oxford 17 flower**]{} dataset consists of 1360 images belonging to 17 mutually classes. Each class contains 80 images. The dataset is divided into three parts, including train set, test set and validation set, with 40 images, 20 images and 20 images respectively. In our experiment, we ignore validation set.
Implementation Details
----------------------
The **USDH** method is implemented based on Caffe and the deep model is trained by batch gradient descend. As shown in Figure \[Fig2:\], We use VGG-19 as the base model, and the model is firstly trained on Imagenet dataset. Then the output layer of VGG-19 is replaced by hashing layer. In the training stage, image is regarded as input in the form of batch and every two images in same batch construct an image pair. The parameters of deep model are updated by minimizing objective function, including semantic loss, quantization loss, information loss and rotation loss. We conduct experiments for learning 16-bit, 32-bit, 48-bit hashing codes, respectively on cifar-10 dataset and NUSWIDE dataset. In this paper, we propose multiple loss function. So we further evaluate these loss functions. The semantic loss is proved more important and our quantization loss also improve performance. Since the efficiency of semantic loss, robustness analysis is discussed. We conduct experiments by different parameters $\rho$ in semantic loss. The constant parameters $\rho$ are respectively set as $d$, $\frac{d}{2}$, $\frac{d}{4}$. Where $d$ denotes as the dimension of output of second full-connected layer. To prove the efficiency of hashing codes learned by **USDH**, we also conduct experiments for other computer vision field, such as fined grained classification.
Results on image retrieval
--------------------------
Similar to DeepBit [@lin2016learning] method, the dataset is splitted into two parts. More specially, 10000 images is selected randomly as query image and then we conduct retrieval task on the remaining images for both CIFAR-10. We define similarity label based on semantic-level labels and images from the same class are considered similar. For NUSWIDE dataset, we follow the setting in [@xia2014supervised], and if two images share at least one same label, they are considered same. The Mean Average Precision (MAP,%) at top 1000 of different unsupervised hashing methods on CIFAR-10 dataset was shown in table1. The experiment results on Table \[tab1:\] show that **USDH** outperforms existing best retrieval performance by $4.6\%$, $10.1\%$, $7.3\%$ and improves DeepBit method by $6.7\%$, $11.7\%$, $11.5\%$, correspond to different hash bits, respectively 16 bits, 32 bits and 64 bits. we also conduct experiments for large-scale image retrieval. As shown in Table \[tab2:\], our method absolute increases of $25.77\%$, $25.58\%$, $24.67\%$ in average MAP for different bits on NUSWIDE dataset. Based on results of experiment, **USDH** is proved to be effective for image retrieval and the semantic information among different images in feature space improves significantly performance.
Method 16-bit 32-bit 64-bit
-------------- --------------- --------------- ---------------
KMH 13.59 13.93 14.46
SphH 13.98 14.58 15.38
SpeH 12.55 12.42 12.56
PCAH 12.91 12.60 12.10
LSH 12.55 13.76 15.07
PCA-ITQ 15.67 16.20 16.64
DH 16.17 16.62 16.96
DeepBit 19.43 24.86 27.73
DBD-MQ 21.53 26.50 31.85
[**USDH**]{} [**26.13**]{} [**36.56**]{} [**39.27**]{}
: Mean Average Precision (MAP) results for different number of bits CIFAR-10[]{data-label="tab1:"}
Method 16-bit 32-bit 48-bit
-------------- --------------- --------------- ---------------
SphH 41.30 42.40 43.10
SpeH 43.30 42.60 42.30
PCAH 42.90 43.70 41.40
LSH 40.30 42.60 42.30
PCA-ITQ 45.28 46.82 47.70
DH 42.20 44.80 48.00
DeepBit 38.30 40.10 41.20
[**USDH**]{} [**64.07**]{} [**65.68**]{} [**65.87**]{}
: MAP results for different number of bits NUSWIDE[]{data-label="tab2:"}
[**Component analysis of loss function:**]{} Our loss function consists of four components. In this section, we evaluate the effectiveness of two major components: semantic loss and quantization loss. The results on CIFAR-10 are shown in Table \[tab6:\] . It is worth mentioning that the semantic loss has improved the performance by 7.62% compared to DeepBit method. And the quantization loss proposed in our paper has further improved the performance by 4.07%.
Method MAP
----------------------- ---------------
DeepBit 24.86
DeepBit+semantic loss 32.48
[**our method**]{} [**36.55**]{}
: Effectiveness (MAP 32 bits) of different loss function[]{data-label="tab6:"}
[**Robustness analysis of semantic loss:**]{} Since all these experiment results have shown the effectiveness of semantic loss, the next experiment would focus on the influences of different parameter settings. We set the parameter $\rho$ in different value, including $1$, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, to conduct experiments on CIFAR10 to learn the 64-bits hashing codes, where $d$ denotes the dimension of second full-connected layer. Table \[tab3:\] reveals that semantic loss is robust to the value of parameter $\rho$. The experimental results suggest that the hashing codes learned by **USDH** focus on relative relationship of image features, instead of their exact similarity value.
$\rho$ $1$ $1/2$ $1/4$ $1/8$
-------- ------- ------- ------- -------
MAP 39.27 39.02 39.20 39.11
: Comparison of image retrieval MAP of our **USDH** with respect to different values of parameters $\rho$[]{data-label="tab3:"}
Results on fined grained classification
---------------------------------------
Different from supervised hashing method, **USDH** learns hashing codes without label information. Thus, it has more practical potential which benefits not only image retrieval but also other computer vision tasks such as fine-grained classification. To verify it, we conduct experiments on fine-grained classification compared with some traditional features, such as, SIFT, HOG, HSV and so on. It is worth mentioning that DeepBit is also a deep unsupervised hashing method. However, DeepBit method [@lin2016learning] only requires hashing codes invariant to rotation and not considers the within-class variance among different images. Fine grained classification is a classic computer vision task and refers to discriminating categories of same sub-class belong to different super class. This task requires image descriptors invariant to within-class variance. More specially, for flower classification, within-class variances include color difference, shape deformation and pose. We select multi-svm as classifier and conduct experiments with different features. Table \[tab4:\] and Figure \[fig4:\] shows classification accuracy of these experiment shows the experiment results of **USDH**. Since within-class variance limits the efficiency of traditional color descriptor and hand-crafted shape descriptor, hashing codes learned by deep network has a superior performance, improved $10.7\%$ than SIFT-Internal feature. Compared with DeepBit, our method still improvs $6.2\%$. Additionally, our method is same fast as DeepBit method and more faster than traditional descriptor since it has low dimension. From the above experiment, the proposed **USDH** method has been proved efficiency for classification task.
Conclusions
===========
In this paper, we propose a novel unsupervised deep hashing method, named unsupervised semantic deep hashing method. The parameters of deep neural network is fine-tuned according to four loss function: 1) semantic loss; 2) quantization loss; 3) information loss; and 4) rotation loss. Compare with previous unsupervised deep hashing methods, **USDH** requires hashing codes to preserve the relevant semantic information in the feature space. Extensive experiments on CIFAR-10 dataset and NUSWIDE dataset demonstrate that our proposed method outperforms existing unsupervised hashing method for image retrieval task. And the experimental results on Oxford17 dataset also prove that the hashing code learned by **USDH** is also effective on other computer vision tasks, such as fine-grained classification.
Feature Accuracy Training time(sec)
--------------- -------------------------- --------------------
Colour 60.9 $\pm$ 2.1% 3
Texture 70.2 $\pm$ 1.3% 4
HOG 63.7 $\pm$ 2.7% 3
HSV 58.5 $\pm$ 4.5% 4
SIFT-Boundary 59.4 $\pm$ 3.3% 4
SIFT-Internal 70.6 $\pm$ 1.6% 4
DeepBit 75.1 $\pm$ 2.5% 0.07
[**USDH**]{} [**81.3 $\pm$ 2.1%** ]{} [**0.07**]{}
: The recognition accuracy for fine grained classification on Oxford17 dataset compared with different features[]{data-label="tab4:"}
![ Confusion matrix of Oxford 17 flower classification using the proposed **USDH**.[]{data-label="fig4:"}](figure6.pdf){height="0.62\linewidth" width="0.9\linewidth"}
|
---
abstract: 'We use particle-in-cell (PIC) simulations to study the effects of variations of the incoming 400 GeV proton bunch parameters on the amplitude and phase of the wakefields resulting from a seeded self-modulation (SSM) process. We find that these effects are largest during the growth of the SSM, i.e. over the first five to six meters of plasma with an electron density of $7 \times10^{14}$ cm$^{-3}$. However, for variations of any single parameter by $\pm$5%, effects after the SSM saturation point are small. In particular, the phase variations correspond to much less than a quarter wakefield period, making deterministic injection of electrons (or positrons) into the accelerating and focusing phase of the wakefields in principle possible. We use the wakefields from the simulations and a simple test electron model to estimate the same effects on the maximum final energies of electrons injected along the plasma, which are found to be below the initial variations of $\pm$5%. This analysis includes the dephasing of the electrons with respect to the wakefields that is expected during the growth of the SSM. Based on a PIC simulation, we also determine the injection position along the bunch and along the plasma leading to the largest energy gain. For the parameters taken here (ratio of peak beam density to plasma density $n_{b0}/n_0 \approx 0.003$), we find that the optimum position along the proton bunch is at $\xi \approx -1.5 \; \sigma_{zb}$, and that the optimal range for injection along the plasma (for a highest final energy of $\sim$1.6 GeV after 10 m) is 5–6 m.'
author:
- Mariana Moreira
- Jorge Vieira
- Patric Muggli
bibliography:
- 'refs.bib'
title: 'Influence of proton bunch parameters on a proton-driven plasma wakefield acceleration experiment'
---
Introduction
============
The AWAKE experiment intends to demonstrate the concept of proton-driven plasma wakefield acceleration using 400GeV proton bunches supplied by the Super Proton Synchrotron (SPS) at CERN to accelerate externally injected electrons [@awake]. The concept underlying AWAKE is one of several that have been proposed for plasma-based acceleration, which could pave the way towards higher collision energies than what conventional accelerator technology can provide. An estimate for the maximum acceleration gradient supported by plasma is given by the cold non-relativistic wavebreaking field [@akhiezer; @dawson] $$E_0 = \frac{m_e c \, \omega_{pe}}{e} \, \approx \, 96 \sqrt{n_0 \left[ \mathrm{cm}^{-3} \right]} \, \left[ \mathrm{V/m} \right] \; ,$$ where $c$ is the speed of light, $m_e$ is the electron mass, $e$ is the elementary charge, $n_0$ is the plasma electron density, and $\omega_{pe} = \sqrt{e^2 n_0 / \varepsilon_0 \, m_e}$ is the electron plasma frequency and $\varepsilon_0$ is the vacuum permittivity. The plasma density used in AWAKE, for example, of the order of $10^{14} \; \mathrm{cm}^{-3}$, yields $E_0 \approx 1 \; \mathrm{GV/m}$, which is approximately ten times larger than what is feasible with RF cavities at the moment [@rfcavities]. For higher plasma densities ($10^{18} \; \mathrm{cm}^{-3}$), however, acceleration gradients of the order of $100 \; \mathrm{GV/m}$ could be reached.
Plasma-based acceleration can be accomplished using either a laser pulse or a particle bunch as a driver. AWAKE is an instance of the latter case, which is also known as plasma wakefield acceleration [@pwfa] (PWFA). As a particle bunch propagates in plasma, the fields caused by its space charge disturb the light plasma electrons, while the more massive plasma ions can be assumed to remain immobile (at the 1/$\omega_{pe}$ time scale) as long as the ion to electron mass ratio is sufficiently high [@vieiraiona; @vieiraionb].
The displaced plasma electrons in the wake of the particle driver oscillate at the plasma frequency $\omega_{pe}$, and this density oscillation is in turn associated with transverse and longitudinal fields, the wakefields. The wavelength of the resulting plasma wave (or wake) is thus related to $\omega_{pe}$ and is called the plasma wavelength: $\lambda_{pe} = 2 \pi v_b / \omega_{pe}$, where $v_b\simeq c$ is the proton bunch velocity.
When the drive bunch is short, i.e. with a typical length $L \apprle \lambda_{pe}$, the wake travels with the speed of the driver. A charged particle can then be trapped and accelerated if it is injected with roughly the same speed as the plasma wake in a region of the wakefields that is longitudinally accelerating and transversely focusing. In the linear regime, where the beam density of the drive bunch $n_b$ is much smaller than the plasma density ($n_b \ll n_0$), the transverse and longitudinal components of the wakefields are harmonic and phase-shifted by a fourth of a period with respect to each other, as expressed by a unique relationship between both components known as the Panofsky-Wenzel theorem [@PWtheo]. This means that each ideal region for acceleration, where the fields are both accelerating and focusing, is $\lambda_{pe}/4$ long.
In order to drive the wakefields effectively, the length of the driver should be of the order of $\lambda_{pe}$. This is not the case in AWAKE, where the bunches delivered by the SPS are considerably longer (6–12 cm) than the plasma wavelengths in the adjustable density range ($\sim$1–3 mm for (1–10) $\times 10^{14} \; \mathrm{cm}^{-3}$). This causes the long proton bunch to undergo the self-modulation instability (SMI) [@kumarsmi], whereby the bunch is progressively modulated into a train of shorter bunches, with lengths and separation distances of the order of $\lambda_{pe}$, due to periodic transversely focusing and defocusing fields. This instability eventually saturates and the initial proton bunch is self-consistently transformed into a bunch train, a format that can resonantly excite the wakefields.
The onset of an instability can either be due to noise or to a seed, i.e. a signal of higher amplitude than the noise level. When the SMI starts from noise, both the phase of the wakefields along the bunch as well as their amplitude vary randomly from event to event and thus prevent reliable acceleration of injected particles. In principle, seeding the instability is a means to fix the final phase and amplitude of the wakefields once the process has saturated. The process is then called seeded self-modulation (SSM) [@awake]. Seeded self-modulation was recently demonstrated experimentally using a sharp ionisation front created by an optical laser within the long proton bunch [@adliprl; @turnerprl].
It has been shown both theoretically [@schroeder] and through numerical simulations [@pukhov] that the phase velocity of the wakefields is smaller than that of the drive bunch during the growth of the SMI. This limits the maximum energy gain since electrons can easily find themselves in the defocusing and decelerating phase of the wakefields and be lost. External injection must therefore occur near or after saturation, when the wakefield phase velocity is very close to the driver velocity [@pukhov]. In addition, for this injection to succeed reliably, as is required for the application as a particle accelerator, the injected bunch must be deterministically placed in the accelerating and focusing phase of the wakefields, or within a range of $\lambda_{pe}/4$. The wakefield phase at the point of injection along the proton bunch and along the plasma must therefore be reproducible to within a fraction of that range. This must be true even in the presence of natural fluctuations of the drive bunch and of the accelerating structure, in this case the plasma. It is therefore essential to study the effect of parameter variations on the wakefield characteristics. Here we will assume that the plasma density and thus the frequency of the wakefields does not vary. This is an assumption that is addressed in experiments by carefully controlling the plasma parameters [@vaporsource].
In this work we focus on the effects of bunch parameter and plasma radius fluctuations on the amplitude and phase of the wakefields after saturation of the SSM, where acceleration over a long distance can in principle start [@savard]. We then use test electron calculations to infer the same effects on the energy of the accelerated electrons, and to study the optimal injection conditions that lead to the most acceleration.
The effects of initial bunch parameter variations are studied through numerical particle-in-cell (PIC) simulations in two-dimensional, axisymmetric cylindrical coordinates, performed with the code OSIRIS [@osiris; @osiris2]. The values of a set of proton bunch parameters are varied independently and the respective simulations compared to a baseline simulation with parameters similar to those of AWAKE. We note here that the hose instability, which can possibly compete with the SMI [@hose], is not described in 2D axisymmetric geometry. We therefore assume in this work that the seed for the self-modulation process is large enough to prevent the growth of the hose instability [@kumarsmi; @vieirahose].
Simulation and parameters
=========================
In the simulations used for this work, a moving window approximately 33 cm long and 1.6 mm high moves at $c$ with a proton bunch (moving at $\sim c$) as the latter propagates through 10 m of plasma. The simulation box consists of a grid of 20063 cells in the longitudinal and 425 cells in the transverse direction, which corresponds to a resolution of roughly $17 \; \mathrm{\mu m}$ and $4 \; \mathrm{\mu m}$ (or 74 and 333 cells per $\lambda_{pe}$), respectively. There are four particles per cell for each particle species (plasma electrons and beam protons) in the simulation.
The proton beam propagates with a Lorentz factor $\gamma_b \approx 480$ (corresponding to 450 GeV) with an energy spread of 0.035% and a normalized emittance of 2.5 mm mrad. The profile of the proton bunch is implemented with a sharp cut, which represents the plasma creation by the co-propagating laser pulse, i.e. the relativistic ionization front, that seeds the SSM process in the experiment. In these simulations the seeding of the self-modulation process is thus modeled by the sharp rising edge of the proton bunch. The bunch density profile is given by $$\begin{gathered}
\label{eq:nb}
n_b(\xi,r) = \\ \frac{ n_{b0} }{2} \left[ 1 + \cos \left( \sqrt{ \frac{\pi}{ 2 \sigma_{zb}^2 } } \left( \xi - \xi_s \right) \right) \right] e^{-r^2/\left( 2 \sigma_{rb}^2 \right) } \; ,\end{gathered}$$ for $\xi_0 \le \xi \le \xi_s$, where $\xi$ is the beam co-moving coordinate defined as $\xi = z - c t$, $n_{b0}$ is the peak bunch density, $\sigma_{zb}$ and $\sigma_{rb}$ are the RMS bunch length and width, respectively, $\xi_0$ is the position where the function crosses the $\xi$ axis (end of the bunch at $\xi_0=-\frac{\pi\sigma_{rb}}{2}$ for $\xi_s=0$), and $\xi_s$ is the seed position along the bunch. The plasma fills the simulation window up to the ionization radius $r_p = 1.5\;\mathrm{mm}$.
The following parameters were used in the simulations: $n_0 = 7 \times 10^{14} \; \mathrm{cm^{-3}}$, $\sigma_{zb} = 12.6 \; \mathrm{cm}$, and $\sigma_{rb} = 200 \; \mathrm{\mu m}$. The peak density in Eq. \[eq:nb\] is calculated according to $$\label{eq:nb0}
n_{b0} = \frac{N_b}{(2 \pi)^{3/2}\, \sigma_{rb}^2 \, \sigma_{zb} } \; ,$$ giving $n_{b0} \approx 1.89 \times 10^{12} \; \mathrm{cm^{-3}} $ for the proton bunch population $N_b = 1.5 \times 10^{11}$ in the full bunch.
The following parameters were independently varied by $\pm 5 \%$: $\sigma_{zb}$, $\sigma_{rb}$, $N_b$ and $r_p$. The RMS timing jitter of the proton bunch with respect to the ionizing laser pulse $\Delta t$ was also varied by $\pm 15 \; \mathrm{ps}$. Note that $\Delta t$ is in practice a phase shift of the cosine in Eq. with respect to the center of the profile $\xi_s$, thus encompassing either more or less charge depending on whether the maximum of the cosine is moved to the right or left of $\xi_s$.
These parameters are taken as representative for the AWAKE experiment. However, we expect the conclusions presented here to be quite general. In fact, we have confirmed that our conclusions will hold, by performing additional simulations with a new set of initial conditions (e.g. doubling the bunch charge).
Properties of the wakefields
============================
A reliable plasma accelerator necessarily requires both amplitude and phase stability of the wakefields in the face of natural drive bunch parameter fluctuations. Phase stability is especially critical since the accelerated electrons may otherwise slip into defocusing and decelerating regions of the wakefields and be lost before gaining a significant amount of energy.
Both the wakefield amplitude and the SSM growth rate depend on the bunch density. Wakefields driven by each self-modulated microbunch can reach an amplitude of the order of $E_z=\frac{n_{b0}}{n_0}E_{0}$ (in the linear regime). Therefore, at a given plasma density, variations of the wakefields with respect to bunch parameters are expected to follow dependencies similar to that of $n_{b0}\propto\frac{N_b}{\sigma_{zb}\sigma_{rb}^2}$ (see Eq.\[eq:nb0\]).
The effects of the bunch parameter variations on the wakefield amplitude were characterized by comparing the average absolute value of the oscillating field $E_z$ along the propagation distance $z$ ($\left< |E_z| \right>$) for each parameter. The average $\left< \cdot \right>$ is computed from $E_z$ values in the simulation window at radii smaller than the plasma skin depth $k_{pe}^{-1} = c/\omega_{pe}$ ($k_{pe}^{-1} \approx 201 \; \mathrm{\mu m}$ for $n_0 = 7 \times 10^{14} \; \mathrm{cm^{-3}}$). This limit corresponds to the radial extent beyond which the proton-driven plasma wakefields become negligible.
For example, the evolution of $\left< |E_z| \right>$ is shown in Fig. \[fig:e1ave\] for the baseline parameters and for variations in the bunch population $N_b$. In the three cases the average fields grow rapidly until around $z = 4\; \mathrm{m}$, signifying the growth of the SSM, after which the SSM process saturates and the overall amplitudes of the wakefields gradually decrease. We note here that this amplitude decay can in principle be avoided by using a small step in the plasma density early along the bunch propagation [@lotovpop]. As expected, more (less) bunch charge leads to a higher (lower) field amplitude. These profiles are typical of all the simulations in this study.
The relative difference in $\left< |E_z| \right>$ with respect to the baseline simulation is shown in Fig. \[fig:e1averel\] for all the parameter variations. In general, the effects of the parameter variations are maximum during the growth of the SSM ($z < 4 \; \mathrm{m}$), reaching a relative difference with respect to the baseline of approximately 26% at $z \approx 2.8\; \mathrm{m}$ for $0.95 \; \sigma_{rb}$. However, if electrons are injected only after the SSM process has saturated [@awake], at $z > 4$–$5 \; \mathrm{m}$, the potential for variations at $z < 4 \; \mathrm{m}$ to affect the final energy of the accelerated electrons is not critical. More relevantly, after $z \approx 6\;\mathrm{m}$ all field values converge to that of the baseline case, within $\pm 2\%$. This shows that the wakefield amplitude in these simulations is weakly dependent on the initial proton bunch parameters after 6 m along the plasma.
Before SSM saturation, i.e. where linear wakefield theory is still valid (before 4 m), the trends in Fig. \[fig:e1averel\] are consistent with $E_z\propto\frac{N_b}{\sigma_{zb}\sigma_{rb}^2}$: an increase of $N_b$ by +5% produces higher values for $\left< |E_z| \right>$, for example, and the variations in $\sigma_{zb}$ and $\sigma_{rb}$ cause inversely proportional effects, with the $\sigma_{rb}$ parameter variations causing the largest effects. There is also a clear effect on the growth rate, as evinced by the different slopes up to $z = 3 \; \mathrm{m}$ in Fig. \[fig:e1ave\].
[e1vd.pdf]{} (12.5,13)[(a)]{} (49.5,13)[(b)]{}
Since the timing jitter $\Delta t$ is small when compared to the bunch duration we expect its main effect to be associated with an increase or decrease in total charge driving the wakefields (corresponding to $N_b$ variations by $\pm 2.85\%$). With our choice of plasma radius ($r_p$), a $\pm$5% variation seems to have no significant effect on the wakefield amplitude. It has been shown that a smaller plasma radius can enhance the wakefield’s focusing force and hence the SMI’s growth rate by hindering the plasma’s shielding response to the charge in the drive bunch [@rp]. However, this effect only becomes prominent when $r_p$ approaches $\sigma_{rb}$, which, despite the variations of $\pm 5 \%$, is not the case here.
We now turn our attention to the behavior of the wakefield phase. Assuming that an electron is moving with a constant velocity $v_e$ in a region of the wakefields that is accelerating and focusing, when the phase velocity of the wakefields $v_\phi$ is below or above $v_e$, the electron will eventually slip out of this region and into an undesirable one (decelerating or defocusing). This happens at the latest when the electron and the wakefields dephase by $\lambda_{pe}/4$ with respect to each other (in linear wakefield theory).
Numerical simulation results show that during the SSM growth the phase velocity of the wakefields varies along the plasma and along the bunch, eventually converging towards that of the driver after the SSM has saturated [@schroeder; @pukhov]. This is also shown in Fig. \[fig:e1vd\], where the evolution of the wakefield phase velocity is visualized by plotting the on-axis longitudinal field component $E_z$ in a waterfall plot along the plasma. Since the simulation window moves at $c$, a negative slope in this type of graph means that the phase velocity of the wakefields is subluminal, while a positive slope indicates that it is superluminal. The relativistic proton bunch moves at nearly the speed of light, so its velocity essentially corresponds to a vertical line in Fig. \[fig:e1vd\] (the slope $\frac{\Delta z}{\Delta\xi}\approx -2 \: \gamma^2$ for bunch particles).
We use the longitudinal component $E_z$ to characterize the evolution of the phase of the wakefields, as we did for the amplitude. We make this choice because, though the transverse wakefields drive the SSM, they must be evaluated at the proper radius (e.g. at the bunch RMS transverse size for a Gaussian profile). Since both transverse radius and shape of the bunch change as the SSM evolves, the evaluation becomes ambiguous. In contrast, the longitudinal wakefield $E_z$ is well defined and maximum on the beam axis. Moreover, the transverse and longitudinal wakefields share a fixed phase relationship due to the Panofsky-Wenzel theorem [@PWtheo], which means that the phase behavior can be measured through either component.
[wrwz\_vd.pdf]{} (12.5,13)[(a)]{} (49.5,13)[(b)]{}
To illustrate this last point, we produce a similar plot to the one in Fig. \[fig:e1vd\], but for the product of the transverse and longitudinal force components $W_r$ and $W_z$, which, in cylindrical coordinates, are defined as $W_r = q \; (E_r - c \, B_\theta)$ and $W_z = q \; E_z$, respectively. Here, $q$ is the charge of the test particle, $E_r$ is the radial component of the electric field and $B_\theta$ is the azimuthal component of the magnetic field. The product $W_r \times W_z$ is evaluated at $r = 0.7 \; k_{pe}^{-1}$ in Fig. \[fig:wrwz\], since the transverse components of the wakefields are zero on the axis, and we only consider the accelerating regions, i.e. where $W_z > 0$.
Figures \[fig:e1vd\] and \[fig:wrwz\] show that, while the wakefields are growing ($z < 4 \; \mathrm{m}$), they are slower than the drive beam velocity (negative slope). In the region around $1.5\;\sigma_{zb}$ or 18.9 cm behind the seed \[Figs. \[fig:e1vd\](b) and \[fig:wrwz\](b)\] the phase velocity of the wakefields becomes essentially equal to the driver velocity after $z = 5 \; \mathrm{m}$ (vertical slope), which makes it a suitable position for the external injection of electrons. Further behind the seed \[around $-2.5 \; \sigma_{zb}$, Fig. \[fig:e1vd\](a)\] the phase velocity is superluminal for $z > 5 \; \mathrm{m}$, while earlier (for example around $\xi \approx - \sigma_{zb}$) it is subluminal (not shown). Experimentally, the injection position along the bunch can be scanned so as to find the optimal $\xi$ position for maximum electron acceleration.
The effects of the parameter variations on the phase of the wakefields are studied quantitatively by fitting the function $f(\xi) = ~A\; \sin \left[ k_{pe} \left( \xi - \xi_s \right) + \phi \right]$ (expected for linear wakefields) to 2.5-$\lambda_{pe}$-long segments (starting at $\xi_s$) of the waterfall plots discussed in Fig. \[fig:e1vd\], where $A$ and $\phi$ are the fitting parameters. The value of $\phi$ is always relative to the seed position $\xi_s$.
As an example, the fit to a segment located around $\xi \approx - 1.5 \; \sigma_{zb}$ is shown in Fig. \[fig:fit\] for three different propagation distances. The fit is worst around the saturation point of the SSM (see curves for $z = 5 \; \mathrm{m}$), where the fields show signs of nonlinearity (the presence of high harmonics which lead to wave steepening). However, the purpose of the fit is to define a local phase shift with respect to $\xi_s$, which is accomplished if the phases of both curves match, as is the case.
The result of this analysis for $\phi$ is shown in Fig. \[fig:phasepos\] for three different positions along the bunch. Note that the burgundy and black curves correspond to the cases in Fig. \[fig:e1vd\]. This figure again indicates that injection closer to $1.5 \; \sigma_{zb}$ rather than $1.0 \; \sigma_{zb}$ behind the seed would be more beneficial, since a slower wakefield phase velocity leads to early dephasing.
The position $\xi \approx - 1.5 \; \sigma_{zb}$ was chosen for the comparison of the effects from the parameter scans, shown in Fig. \[fig:phasecomp\]. In the linear and strongly-coupled regime, i.e. before saturation and for $k_{pe} |\xi| \gg \frac{k_b}{\sqrt{2 \gamma_b}} z$ where $k_b = \sqrt{e^2 n_{b0} / \varepsilon_0 \, M_b} $ and $M_b$ is the mass of the drive bunch particles, the longitudinal wakefield component behaves approximately as $E_z \propto \cos \left[ k_{pe} \; \xi - \frac{\pi}{4} + \varphi(\xi,z) \right]$, with the phase shift $\varphi(\xi,z) \propto n_{b0}^{1/3}$ [@schroeder]. The condition for the strongly-coupled regime is fulfilled for $\xi \approx - 1.5 \: \sigma_{zb}$ and $z \sim 10 \; \mathrm{m}$, with $k_{pe} |\xi| \approx 940.4$ and $\frac{k_b}{\sqrt{2 \gamma_b}} z \approx 1.9$. Nevertheless, the phase shift in Fig. \[fig:phasecomp\] only displays a relationship of the form $\phi \propto \left( \frac{N_b}{\sigma_{zb}\sigma_{rb}^2 } \right)^{1/3}$ (after substituting Eq. \[eq:nb0\]) roughly between $z = 3.5$–$5\;\mathrm{m}$.
The largest effects on the wakefield phase are again observed before the saturation of the SSM, at $z = 2$–$3 \; \mathrm{m}$ (see Fig. \[fig:phasecomp\]). Here, the largest difference is of roughly $2 \pi/20$ for $0.95 \; \sigma_{rb}$ at $z \approx 2.5\;\mathrm{m}$. After this point, phase variations are limited to $\pm$0.4 rad (corresponding to approximately $\lambda_{pe}/16$), an estimate constrained by simulation noise. Moreover, the phase stops changing after $z \approx 6\;\mathrm{m}$ in all cases, which is also the point after which the wakefield amplitude becomes essentially independent of the proton bunch parameter variations (see Fig. \[fig:e1averel\]).
This suggests that, at this plasma density and for the chosen proton bunch parameters, electrons injected at $z \approx 6 \;\mathrm{m}$ or further remain in phase with the wakefields for a long distance and can therefore be accelerated to high energies in wakefields with a constant phase.
Behavior of accelerated electrons
=================================
AWAKE aims to demonstrate the acceleration of an electron bunch, and therefore it is important to study the effects of initial parameter fluctuations on the properties of these electrons and not only on the wakefields, as was done so far. The characteristics of the accelerated electron bunch are the most important experimental output, and they are non-trivially dependent on several factors besides the wakefields themselves, such as the electrons’ initial velocity or the injection point along the plasma. Consequently, the wakefield variations reported above are not sufficient to infer possible effects on the accelerated bunches.
A simple diagnostic was devised to determine the energy gain acquired by an electron as a function of its injection point along the plasma ($z_\mathrm{inj})$ and its initial position along the bunch ($\xi_0$). This algorithm is in practice a one-dimensional (1D) particle pusher: for each possible $z_\mathrm{inj}$ along the plasma, a test particle is placed at $\xi_0$ along the on-axis wakefield (i.e. the data presented in Fig. \[fig:e1vd\]) and propagated forwards in the wakefields. All test electrons have an initial energy corresponding to $\gamma_0 = 39.1$, or approximately 20 MeV (the maximum range of the electron injector commissioned for AWAKE [@awake]).
[gammaf\_xi.pdf]{} (39,41)[(a)]{} (76,38)[(b)]{}
The spatial resolution of these results is limited to the resolution of the simulation box in the $\xi$ direction (which in this case means that at most 38 evenly-spaced test electrons can be tracked for every $\lambda_{pe}/2$), while the temporal resolution is limited to the number of simulation file dumps (in this case 300 over 10 m, giving a maximum resolution for $z_\mathrm{inj}$ of 3.55 cm). In this diagnostic, the electrons are assumed to remain on the axis at all times and no transverse forces are considered. Tracking particles in axisymmetric two-dimensional space (including transverse fields) would in effect entail full-fledged PIC simulations.
Since $E_z$ peaks on the axis and decays radially, an electron performing any transverse motion about the axis is subject to weaker longitudinal forces than if it is propagating exclusively along it (the most effective trajectory in terms of energy gain). This approach thus provides a best case scenario for the energy gained by accelerated electrons. It nonetheless includes their dephasing with respect to the wakefields, while the simplicity of the approach means that results can be obtained quickly for many different cases, e.g. for different injection points and for all the parameter scans performed in this work.
The result of this diagnostic is shown in Fig. \[fig:energy\](a) for the baseline simulation as a scatter plot of electrons that reach the end of the plasma, with their energy (color-coded) as a function of their injection position ($\xi_0$,$z_\mathrm{inj}$). The rest of the test electrons lose enough energy at some point along $z$ so as to slip out of the 33-centimeter-long simulation window, and hence not reach the end of the plasma.
The general features of the accelerating field \[see Fig. \[fig:e1vd\](b)\] are visible in the point density of Fig. \[fig:energy\](a). Regions with few test electrons correspond to decelerating regions. In regions where the field is accelerating ($E_z < 0$, for example $-19.00 < \xi_0 \: [\mathrm{cm}] < -18.95$), all the test electrons reach the end of the plasma. As expected, the final energies decrease as electrons are injected at later $z$ positions (shorter acceleration distances), though this is also because the wakefield amplitude decreases after $z \approx 5\; \mathrm{m}$ (see Fig. \[fig:e1ave\]). Figure \[fig:energy\](a) also implies that some electrons injected in the decelerating phase of the wakefields survive energy loss and dephasing to ultimately reach large energies (scattered red dots). The same is true for electrons injected before the saturation of the SSM ($z < 4 \; \mathrm{m}$).
The diagnostic was applied to the bunch parameter scans to evaluate possible effects on the final energy of injected electrons. We compared the maximum final energy attained by test electrons injected in the same wakefield period for each parameter variation, choosing the range $ -18.990 \le \xi_0 \: [\mathrm{cm}] \le -18.956 $ (approximately $\lambda_{pe}/4$-long), where $\gamma_f$ is maximal.
Figure \[fig:emaxpars\] shows the scatter plot of $\gamma_{f,\mathrm{max}}$ for the variations $\delta$ of $\sigma_{zb}$, $\sigma_{rb}$ and $N_b$, the parameters that caused the largest effects. We find trends of the form $\gamma_{f,\mathrm{max}} \propto\frac{N_b}{\sigma_{zb}\sigma_{rb}^2}$, which is consistent with the behavior observed above for the average wakefield amplitude $\left< |E_z| \right>$ and with the fact that the energy gain by trailing particles is directly linked to the amplitude of the axial field component $E_z$. The resulting maximum final energies vary at most between roughly $-3\%$ and $+5\%$ (the corresponding injection points lie between 4.15 and 4.52 m along the plasma).
To validate the diagnostic described above, we performed a full simulation with the baseline parameters, in which test electrons were injected at 41 equally-spaced injection points between 3.5 and 7.6 m. The electrons used in the simulation have zero emittance and are initially uniformly distributed in space (both longitudinally and transversely). The electron data was processed so as to obtain the same type of graph as Fig. \[fig:energy\](a). This data is shown in Fig. \[fig:energy\](b) for electrons injected close to the axis ($r_0 < 0.5 \; k_{pe}^{-1}$) that reached the end of the plasma.
We would expect to observe the influence of the transverse wakefields in the final energy distribution on the $(\xi_0,z_\mathrm{inj})$ plane of Fig. \[fig:energy\](b), which is indeed the case. The regions of electron loss in Fig. \[fig:energy\](b) (due to transverse forces) are much clearer than those on Fig. \[fig:energy\](a) (which are only due to longitudinal dephasing). The periodic regions with the most electrons in both plots \[i.e. accelerating phases in (a) and focusing phases in (b)\] also appear to be shifted by around $\lambda_{pe}/4$ with respect to each other \[note the shape of the scatter plot in (b) superimposed on (a)\], as would be expected from the Panofsky-Wenzel theorem [@PWtheo]. Other than this, the overall distribution matches well with that of Fig. \[fig:energy\](a).
A more quantitative comparison of the 1D pusher with direct simulation results can be seen in Fig. \[fig:gfmaxave\], which shows the average along each row of both graphs in Fig. \[fig:energy\] (pink) as well as each row’s maximum energy (blue) plotted against the injection point $z_\mathrm{inj}$.
The peak energies in the 2D simulation results are generally lower than the 1D results (compare dashed and solid blue curves), which is expected since the 1D diagnostic represents a best-case scenario. Furthermore, their trends do not agree before $z_\mathrm{inj} = 5.5\;\mathrm{m}$. This is the region where we expect the variation of the wakefield phase and associated defocusing to be the largest. For $z_\mathrm{inj} > 5\;\mathrm{m}$, however, where we expect these effects to be negligible, the trend in both curves is very similar. The average energies in turn show very good agreement (pink curves).
We can therefore conclude that the 1D diagnostic was an appropriate tool for a comparative analysis of the effects of the parameter variations on the final energies of electrons that are initially close to the axis.
[gammaf.pdf]{} (10,44)[(a)]{} (10,23.5)[(b)]{} (49,40.5)[(c)]{} (49,19.5)[(d)]{}
The peak energy curve obtained from the simulation in Fig. \[fig:gfmaxave\] (dashed blue line) suggests that the optimum injection point lies between 5–6 m. Although this graph only represents electrons initially close to the axis ($r_0 < 0.5 \; k_{pe}^{-1}$), the optimal injection range is confirmed when the final energies of all electrons at all possible radii (up to the plasma boundary $r_p$) are considered, as shown in Figs. \[fig:gfsim\](a) and (b). Each data point in Fig. \[fig:gfsim\](a) consists of an average of all the simulation particles that began at a given $\xi_0$ over the entire plasma radius, while Fig. \[fig:gfsim\](b) shows the peak energy out of all electrons with any $r_0$ for each $\xi_0$. Both scatter plots display the highest Lorentz factors for $z_\mathrm{inj} = 5$–$6\;\mathrm{m}$.
Figure \[fig:gfsim\](b) furthermore indicates that some electrons reach high energies when injected before saturation of the SSM ($z_\mathrm{inj} < 5\;\mathrm{m}$), which is also suggested by the 1D diagnostic results \[note red points for $z_\mathrm{inj} = 0$–$5\;\mathrm{m}$ in Fig. \[fig:energy\](a)\]. When we decompose the data in Fig. \[fig:gfsim\](b) into electrons originating above and below a radius of $1.5\;k_{pe}^{-1}$ (approximately 0.3 mm), we find that, for injections before 5 m, the electrons far from the axis attain the highest energies \[Fig. \[fig:gfsim\](d)\], while for $z_\mathrm{inj} = 5$–$6\;\mathrm{m}$ it is the electrons close to the axis that gain the most energy \[Fig. \[fig:gfsim\](c)\].
[e2-b3.pdf]{} (7,38.5)[(a)]{} (54,38.5)[(b)]{}
This difference is only observable for injections that take place before saturation of the SSM and could thus be explained by its development. In fact, the PIC simulations show that the phase velocity of the wakefields varies along the plasma radius as well. This is demonstrated by the waterfall plots of the transverse wakefield component $E_r - c \, B_\theta$ (which is responsible for focusing and defocusing) in Fig. \[fig:e2b3\]. For $z < 5 \; \mathrm{m}$, for example, the phase velocity closer to the axis \[Fig. \[fig:e2b3\](a), at $r = k_{pe}^{-1}$\] behaves as expected during the growth of the SSM and as previously discussed in Fig. \[fig:e1vd\]. At a larger radius, however, the phase is approximately stable between 4 and 5.5 m \[Fig. \[fig:e2b3\](b), at $r=3\;k_{pe}^{-1}$\]. This would explain why electrons starting before $z = 5\;\mathrm{m}$ at smaller radii would tend to be lost (due to the rapidly changing phase and their subsequent slippage into defocusing half-periods), while electrons further away from the axis would find a stable wakefield phase and thus gain energy over a larger distance.
Summary
=======
Using PIC simulations, we varied the bunch parameters $\sigma_{zb}$, $\sigma_{rb}$ and $N_b$, the plasma radius $r_p$, and the seed point timing $\Delta t$, and studied their effect on the wakefield amplitude and phase during the development of a seeded instability (SSM), and on the maximum energy gain as determined by test electrons.
We found that the parameter variations we considered ($\pm$5% and $\pm$15 ps) essentially lead to differences in wakefield amplitude and phase only in the growth region of the SSM along the plasma ($z<4\;\mathrm{m}$ in this case). The wakefield parameters all converge to similar values after saturation of the SSM, within a few percents for the amplitude and the equivalent of less than $\lambda_{pe}$/8 for the phase. While the results presented here were obtained for only one set of baseline parameters, the same analysis with different parameters showed similar trends. Furthermore, it is clear that in practice all initial parameters vary for each event. However, as variations may have counteracting effects, we assume that the conclusions reached through single parameter variation studies are still representative of experimental situations.
Based on the simulations, we also found that the optimal injection coordinates for our parameters ($n_0 = 7 \times 10^{14} \; \mathrm{cm^{-3}}$ and $N_b = 1.5 \times 10^{11}$) are 5–6 m into the plasma and around $1.5 \; \sigma_{zb}$ behind the wakefield seed. For an injection in this range, electrons close to the axis can reach energies of the order of 1.6 GeV over the last 4–5 m of plasma. Comparable final energies are also attained when injection takes place before saturation of the SSM ($z < 5\;\mathrm{m}$), but by electrons far from the axis instead.
In general, the optimal injection point along the plasma will be determined by the start of the saturation of the SSM, which takes place earlier with either larger $n_0$ or $N_b$. The position with the most stable phase along the bunch can also be scanned for different parameters, and it tends to be closer to the seed point for higher $n_0$ and smaller $N_b$. The increase of either of these two parameters will further lead to higher wakefield amplitudes, and hence to larger energy gains by trailing electrons. In the future, we will seek further optimization towards a higher accelerated beam quality, for example by including the witness beam emittance and beam loading effects in PIC simulations of the entire injection and acceleration process (see for example [@olsen]).
M. M. acknowledges the scientific guidance provided by Bernhard Holzer. We acknowledge the grant of computing time by the Leibniz Research Center on SuperMUC. J. V. acknowledges the support of FCT (Portugal) Grant No. SFRH/IF/01635/2015 and CERN/FIS-TEC/0032/2017.
|
---
abstract: 'We present a method for two-scale model derivation of the periodic homogenization of the one-dimensional wave equation in a bounded domain. It allows for analyzing the oscillations occurring on both microscopic and macroscopic scales. The novelty reported here is on the asymptotic behavior of high frequency waves and especially on the boundary conditions of the homogenized equation. Numerical simulations are reported.'
author:
- 'Thi Trang Nguyen , Michel Lenczner'
- Matthieu Brassart
bibliography:
- 'wave\_reference.bib'
title: 'Homogenization of the one-dimensional wave equation'
---
**Keywords.** Homogenization, Bloch waves, wave equation, two-scale transform.
Introduction
============
The paper is devoted to the periodic homogenization of the wave equation in a one-dimensional open bounded domain where the time-independent coefficients are $\varepsilon -$periodic with small period $\varepsilon >0$. Corrector results for the low frequency waves have been published in [brahim1992correctors,francfort1992oscillations]{}. These works were not taking into account fast time oscillations, so the models reflect only a part of the physical solution. In [@brassart2009two], an homogenized model has been developed to cover the time and space oscillations occurring both at low and high frequencies. Unfortunately, the boundary conditions of the homogenized model was not found. Therefore, establishing the boundary conditions of the homogenized model is critical and is the goal of the present work which also extends [@kader2000contributions].
To this end, the wave equation is written under the form of a first order formulation and the modulated two-scale transform $W_{k}^{\varepsilon }$ is applied to the solution $U^{\varepsilon }$ as in [@brassart2009two]. For $n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast }$ and $k\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$ the $n^{th}$ eigenvalue $\lambda _{n}^{k}$ of the Bloch wave problem with $k$-quasi-periodic boundary conditions satisfies $\lambda
_{n}^{k}=\lambda _{n}^{-k}$, in addition $\lambda _{m}^{k}=\lambda _{n}^{k}$ for $k\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
/2$, so the corresponding waves are oscillating with the same frequency. The homogenized model is thus derived for pairs of fibers $\{-k,k\}$ if $k\neq 0$ and for fiber $\left\{ 0\right\} $ otherwise which allows to derive the expected boundary conditions. The weak limit of $\sum\nolimits_{\sigma \in
\left\{ -k,k\right\} }W_{\sigma }^{\varepsilon }U^{\varepsilon }$ includes low and high frequency waves, the former being solution of the homogenized model derived in [@brahim1992correctors; @francfort1992oscillations] and the latter are associated to Bloch wave expansions. Numerical results comparing solutions of the wave equation with solution of the two-scale model for fixed $\varepsilon $ and $k$ are reported in the last section.
The physical problem and elementary properties[Statement\_problem]{}
====================================================================
**The physical problem** We consider $I=\left( 0,T\right) \subset
\mathbb{R}^{+}$ a finite time interval and $\Omega =\left( 0,\alpha \right)
\subset \mathbb{R}^{+}$ a space interval, which boundary is denoted by $%
\partial \Omega $. Here, as usual $\varepsilon >0$ denotes a small parameter intended to go to zero. Two functions $\left( a^{\varepsilon },\rho
^{\varepsilon }\right) $ are assumed to obey a prescribed profile $%
a^{\varepsilon }:=a\left( {\frac{x}{\varepsilon }}\right) $ and $\rho
^{\varepsilon }:=\rho \left( {\frac{x}{\varepsilon }}\right) $ where $\rho
\in L^{\infty }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $, $a\in W^{1,\infty }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ are both $Y-$periodic where $Y=\left( 0,1\right) $. Moreover, they are required to satisfy the standard uniform positivity and ellipticity conditions, $0<\rho ^{0}\leq \rho \leq \rho ^{1}$ and $0<a^{0}\leq a\leq
a^{1},$ for some given strictly positive numbers $\rho ^{0}$, $\rho ^{1}$, $%
a^{0}$ and $a^{1}$. We consider $u^{\varepsilon }\left( t,x\right) $ solution to the wave equation with the source term $f^{\varepsilon }\in
L^{2}\left( I\times \Omega \right) $, initial conditions $%
(u_{0}^{\varepsilon },v_{0}^{\varepsilon })\in L^{2}\left( \Omega \right)
^{2}$ and homogeneous Dirichlet boundary conditions,$$\begin{array}{l}
\rho ^{\varepsilon }\partial _{tt}{u^{\varepsilon }}-{\partial _{x}}\left( {{%
a^{\varepsilon }}{\partial _{x}}u^{\varepsilon }}\right) ={f^{\varepsilon }}%
\,\ \text{in}\,\ I\times \Omega , \\
{u^{\varepsilon }}\left( {t=0,.}\right) =u_{0}^{\varepsilon }\,\ \text{and}%
\,\ {\partial _{t}}{u^{\varepsilon }}\left( {t=0,.}\right)
=v_{0}^{\varepsilon }\ \text{in}\,\ \Omega , \\
{u^{\varepsilon }}={0}\,\ \text{on}\,\ I\times \partial \Omega .%
\end{array}
\label{1D-wave-equation}$$By setting: ${U^{\varepsilon }}:=({\sqrt{{a^{\varepsilon }}}{\partial _{x}}{%
u^{\varepsilon },}\sqrt{{\rho ^{\varepsilon }}}{\partial _{t}}{%
u^{\varepsilon })}},$ ${A^{\varepsilon }}=\left( {%
\begin{array}{cc}
0 & {\sqrt{{a^{\varepsilon }}}{\partial _{x}}\left( {\frac{1}{\sqrt{{\rho
^{\varepsilon }}}}.}\right) } \\
{\frac{1}{\sqrt{{\rho ^{\varepsilon }}}}{\partial _{x}}\left( {\sqrt{{%
a^{\varepsilon }}}.}\right) } & 0%
\end{array}%
}\right) ,\,$ $U_{0}^{\varepsilon }:=({\sqrt{{a^{\varepsilon }}}{\partial
_{x}}u_{0}^{\varepsilon },\sqrt{{\rho ^{\varepsilon }}}v_{0}^{\varepsilon })}
$ and ${F^{\varepsilon }}:=(0,{{f^{\varepsilon }/}}\sqrt{{\rho ^{\varepsilon
}}})$, we reformulate the wave equation (\[1D-wave-equation\]) as an equivalent system,$$\left( {{\partial _{t}}-{A^{\varepsilon }}}\right) {U^{\varepsilon }}={%
F^{\varepsilon }}\text{ in }I\times \Omega ,{U^{\varepsilon }}\left( {t=0}%
\right) =U_{0}^{\varepsilon }\text{ in }\Omega \text{ and }{{{%
U_{2}^{\varepsilon }}}}={0}\text{ on }I\times \partial \Omega$$where ${{{U_{2}^{\varepsilon }}}}$ is the second component of $%
U^{\varepsilon }$. From now on, this system will be referred to as the physical problem and taken in the distributional sense,$$\int\nolimits_{I\times \Omega }{{F^{\varepsilon }}\cdot {\Psi \,}+{%
U^{\varepsilon }}\cdot \left( {{\partial _{t}}-{A^{\varepsilon }}}\right)
\,\Psi dtdx}+\int\nolimits_{\Omega }{U_{0}^{\varepsilon }\cdot \Psi \left( {%
t=0}\right) \,dx}=0, \label{1D-1st-weak-formulation}$$for all the admissible test functions ${\Psi \in {H^{1}}{{\left( {I\times
\Omega }\right) }^{2}}}$ such that ${{\Psi \left( {t,.}\right) \in D\left( {{%
A^{\varepsilon }}}\right) }}$ for a.e. ${t\in I}$ where the domain $%
D(A^{\varepsilon }):=\{{\left( {\varphi ,\phi }\right) \in {L^{2}}\left(
\Omega \right) }^{2}|{{\sqrt{{a^{\varepsilon }}}\varphi \in {H^{1}}\left(
\Omega \right) ,}}$ ${{{\phi /\rho }\in {H_{0}^{1}}\left( \Omega \right) }\}}
$. As proved in [@brassart2009two], the operator $iA^{\varepsilon }$ with the domain $D(A^{\varepsilon })$ is self-adjoint on $L^{2}(\Omega )^{2}$. We assume that the data are bounded ${\left\Vert {f^{\varepsilon }}%
\right\Vert _{{L^{2}}\left( {I\times \Omega }\right) }}+{\left\Vert {{%
\partial _{x}}u_{0}^{\varepsilon }}\right\Vert _{{L^{2}}\left( \Omega
\right) }}+{\left\Vert {v_{0}^{\varepsilon }}\right\Vert _{{L^{2}}\left(
\Omega \right) }}\leq c_{0}$, then $U^{\varepsilon }$ is uniformly bounded [in ]{}${L^{2}}\left( {I\times \Omega }\right) .$
**Bloch waves** We introduce the dual $Y^{\ast }=\left( -\frac{1}{2},%
\frac{1}{2}\right) $ of $Y$. For any $k\in Y^{\ast }$, we define the space of $k-$quasi-periodic functions $L_{k}^{2}:=\{u\in L_{loc}^{2}(\mathbb{R})$ $%
{|} $ $u(x+\ell )=u(x)e^{2i\pi k\ell }$ a.e. in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ for all $\ell \in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\}$ and set $H_{k}^{s}:=L_{k}^{2}\cap H_{loc}^{s}\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ for $s\geq 0.$ The periodic functions correspond to $k=0$. For a given $k\in Y^{\ast }$, we denote by $(\lambda _{n}^{k},\phi _{n}^{k})_{n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast }}$ the Bloch wave eigenelements that are solution to$$\mathcal{P}(k):-\partial _{y}\left( a\partial _{y}\phi _{n}^{k}\right)
=\lambda _{n}^{k}\rho \phi _{n}^{k}\text{ in }Y\text{ with }\phi _{n}^{k}\in
H_{k}^{2}(Y)\text{ and }\left\Vert \phi _{n}^{k}\right\Vert _{L^{2}\left(
Y\right) }=1.$$The asymptotic spectral problem $\mathcal{P}(k)$ is also restated as a first order system by setting $A_{k}:=\left( {%
\begin{array}{cc}
0 & {\sqrt{a}\partial _{y}\left( \frac{1}{\sqrt{\rho }}.\right) } \\
\frac{1}{\sqrt{\rho }}{\partial _{y}\left( {\sqrt{a}.}\right) } & 0%
\end{array}%
}\right) $, $n_{A_{k}}=\frac{1}{\sqrt{\rho }}\left(
\begin{array}{cc}
0 & {\sqrt{a}n}_{Y} \\
{\sqrt{a}n}_{Y} & 0%
\end{array}%
\right) $ and $e_{n}^{k}:=\frac{1}{\sqrt{2}}\left(
\begin{array}{c}
-i{s_{n}/\sqrt{\lambda _{\left\vert n\right\vert }^{k}}}\sqrt{a}\partial
_{y}\left( {\phi _{\left\vert n\right\vert }^{k}}\right) \\
\sqrt{\rho }\phi _{\left\vert n\right\vert }^{k}%
\end{array}%
\right) $ where $s_{n}$ and $n_{Y}$ denote the sign of $n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }$ and the outer unit normal of $\partial Y$ respectively. As proved in [@brassart2009two], $iA_{k}$ is self-adjoint on the domain $D\left( {%
A_{k}}\right) :=\{{\left( {\varphi ,\phi }\right) \in L^{2}\left( Y\right)
^{2}|\sqrt{a}\varphi \in H_{k}^{1}\left( Y\right) ,}$ ${{\phi /}\sqrt{\rho }%
\in H_{k}^{1}\left( Y\right) }\subset L^{2}\left( Y\right) ^{2}\}.$ The Bloch wave spectral problem $\mathcal{P(}k\mathcal{)}$ is equivalent to finding pairs $\left( \mu _{n}^{k},e_{n}^{k}\right) $ indexed by $n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }$ solution to $\mathcal{Q(}k\mathcal{)}:A_{k}e_{n}^{k}=is_{n}\sqrt{%
\lambda _{\left\vert n\right\vert }^{k}}e_{n}^{k}$ in $Y$ with $%
e_{n}^{k}\in H_{k}^{1}\left( Y\right) ^{2}$. We pose $M_{n}^{k}:=\{m{\in {%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
}}^{\ast }{|}\lambda _{m}^{k}{=}\lambda _{n}^{k}$ and $s_{m}=s_{n}\}$ and introduce the coefficients $b(k,n,m)=\int_{Y}\rho \phi _{\left\vert
n\right\vert }^{k}\cdot \phi _{\left\vert m\right\vert }^{k}dy$ and $%
c(k,n,m)=i{s_{n}/}\left( {2\sqrt{\lambda _{\left\vert n\right\vert }^{k}}}%
\right) \int_{Y}{\phi _{\left\vert n\right\vert }^{k}\cdot a{{\partial _{y}}}%
\phi _{\left\vert m\right\vert }^{k}}-a\partial _{y}{\phi _{\left\vert
n\right\vert }^{k}}\cdot {\phi _{\left\vert m\right\vert }^{k}}dy$ for $%
n,m\in M_{n}^{k}.$
**The modulated two-scale transform** Let us** **assume from now that the domain $\Omega $ is the union of a finite number of entire cells of size $\varepsilon $ or equivalently that the sequence $\varepsilon $ is exactly $\varepsilon _{n}=\frac{\alpha }{n}$ for $n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast }$. For any $k\in Y^{\ast }$, we define $I^{k}=\left\{ -k,k\right\} $ if $k\neq 0$ and $I^{0}=\left\{ 0\right\} $. By choosing $\Lambda =\left(
0,1\right) $ as a time unit cell, we introduce the operator $%
W_{k}^{\varepsilon }:L^{2}\left( I\times \Omega \right) ^{2}\rightarrow
L^{2}\left( I\times \Lambda \times \Omega \times Y\right) ^{2}$ acting in all time and space variables, $$W_{k}^{\varepsilon }:=\left( 1-\sum\nolimits_{n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }}\Pi _{n}^{k}\right) S_{k}^{\varepsilon }+\sum\nolimits_{n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }}{{T^{\varepsilon \alpha _{n}^{k}}\Pi }}_{n}^{k}{S_{k}^{\varepsilon }%
} \label{def-W}$$where the time and space two-scale transforms ${{T^{\varepsilon \alpha
_{n}^{k}}}}$ and ${S_{k}^{\varepsilon }}$, and the orthogonal projector $\Pi
_{n}^{k}$ onto $e_{n}^{k}$ are defined in [@brassart2009two], see pages 11,15 and 17, with $\alpha _{n}^{k}=2\pi /\sqrt{\lambda _{n}^{k}}$, and where it is proved that,$$\left\Vert W_{k}^{\varepsilon }u\right\Vert _{L^{2}\left( I\times \Lambda
\times \Omega \times Y\right) }^{2}=\left\Vert u\right\Vert _{L^{2}\left(
I\times \Omega \right) }^{2}. \label{two-scale-boundness}$$We define $(\mathfrak{B}_{n}^{k}v)(t,x)=v(t,\frac{t}{\varepsilon \alpha
_{n}^{k}},x,\frac{x}{\varepsilon })$ the operator that operates on functions $v(t,{\tau },x,y)$ defined in $I\times \mathbb{R\times }\Omega \times
\mathbb{R}$. The notation $O\left( \varepsilon \right) $ refers to numbers or functions tending to zero when $\varepsilon \rightarrow 0$ in a sense made precise in each case. The next Lemma shows that $\mathfrak{B}_{n}^{k}$ is an approximation of $T^{\varepsilon \alpha _{n}^{k}\ast
}S_{k}^{\varepsilon \ast }$ for a function which is periodic in $\tau $ and $%
k-$quasi-periodic in $y$, where $T^{\varepsilon \alpha _{n}^{k}\ast
}:L^{2}\left( I\times \Lambda \right) \rightarrow L^{2}\left( I\right) $ and $S_{k}^{\varepsilon \ast }:L^{2}\left( \Omega \times Y\right) \rightarrow
L^{2}\left( \Omega \right) $ are adjoint of $T^{\varepsilon \alpha _{n}^{k}}$ and $S_{k}^{\varepsilon }$ respectively.
\[conver\] Let $v\in C^{1}\left(
I\times \Lambda \times \Omega \times Y\right) $ a periodic function in $\tau $ and $k-$quasi-periodic in $y$, then $%
T^{\varepsilon \alpha _{n}^{k}\ast }S_{k}^{\varepsilon \ast }v=\mathfrak{B}_{n}^{k}%
v+O\left( \varepsilon \right) $ in the $L^{2}\left( I \times\Omega
\right) $ sense. Consequently, for any sequence $u^{\varepsilon }$ bounded in $L^{2}\left( I\times \Omega \right) $ such that $T^{\varepsilon \alpha
_{n}^{k}}S_{k}^{\varepsilon }u^{\varepsilon }$ converges to $u$ in $%
L^{2}(I\times \Lambda \times \Omega \times Y)$ weakly when $\varepsilon
\rightarrow 0$,$$\int_{I\times \Omega }u^{\varepsilon }\cdot \mathfrak{B}_{n}^{k}v\text{ }%
dtdx\rightarrow \int_{I\times \Lambda \times \Omega \times Y}u\cdot v\text{ }%
dtd\tau dxdy\text{ \ when }\varepsilon \rightarrow 0. \label{convergence}$$
Note that for $k=0$, the convergence ([convergence]{}) regarding each variable corresponds to the definition of two-scale convergence in [@allaire1992homogenization]. The proof is carried out in three steps. First the explicit expression of $T^{\varepsilon
\alpha _{n}^{k}\ast }S_{k}^{\varepsilon \ast }v$ is derived, second the approximation of $T^{\varepsilon \alpha _{n}^{k}\ast }S_{k}^{\varepsilon
\ast }v$ is deduced, finally the convergence (\[convergence\]) follows. For a function $v\left( t,\tau ,x,y\right) $ defined in $I\times \Lambda
\times \Omega \times Y,$ we observe that$$A^{\varepsilon }\mathfrak{B}_{n}^{k}v=\mathfrak{B}_{n}^{k}\left( \left(
\frac{A_{k}}{\varepsilon }+B\right) v\right) \text{ and }{{\partial _{t}}}%
\left( {\mathfrak{B}}_{n}^{k}v\right) ={\mathfrak{B}}_{n}^{k}\left( \left(
\frac{\partial _{\tau }}{\varepsilon \alpha _{n}^{k}}+\partial _{t}\right)
v\right) \text{,} \label{derivative_t}$$where the operator $B$ is defined as the result of the formal substitution of $x-$derivatives by $y-$derivatives in $A_{k}$.
Homogenized results and their proof\[model\]
============================================
For $k\in Y^{\ast }$, we decompose $$\frac{\alpha k}{\varepsilon }=h_{\varepsilon }^{k}+l_{\varepsilon }^{k}\text{
with }h_{\varepsilon }^{k}=\left[ \frac{\alpha k}{\varepsilon }\right] \text{
and }l_{\varepsilon }^{k}\in \left[ 0,1\right) , \label{epsilon_m}$$and assume that the sequence $\varepsilon $ is varying in a set $%
E_{k}\subset \mathbb{R}^{+\ast }$ depending on $k$ so that$$l_{\varepsilon }^{k}\rightarrow l^{k}\text{ when }\varepsilon \rightarrow 0%
\text{ and }\varepsilon \in E_{k}\text{ with }l^{k}\in \left[ 0,1\right).
\label{l}$$We note that for $k=0$, $h_{\varepsilon }^{k}=0,$ $l_{\varepsilon }^{k}=0$, so $l^{k}=0$ and $E_{0}=\mathbb{R}^{+\ast }$. After extraction of a subsequence, we introduce the weak limits of the relevant projections along $%
e_{n}^{k}$ for any $n\in {%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
}^{\ast }$,$$F_{n}^{k}:=\lim_{\varepsilon \rightarrow 0}\int\nolimits_{\Lambda \times Y}{T%
}^{\varepsilon \alpha _{n}^{k}}{S{_{k}^{\varepsilon }{F^{\varepsilon }}\cdot
{e^{2i\pi {s_{n}}\tau }}e_{n}^{k}dyd\tau }}\text{ and }U_{0,n}^{k}:=\lim_{%
\varepsilon \rightarrow 0}\int\nolimits_{Y}{S_{k}^{\varepsilon
}U_{0}^{\varepsilon }\cdot e_{n}^{k}dy}. \label{data}$$The next lemmas state the microscopic equation for each mode and the corresponding macroscopic equation.
\[lemma\_micro\]For $k\in Y^{\ast }$ and $n\in {%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
}^{\ast }$, let $U^{\varepsilon }$ be a bounded solution of ([1D-1st-weak-formulation]{}), there exists at least a subsequence of $%
T^{\varepsilon \alpha _{n}^{k}}{{S}_{k}^{\varepsilon }U^{\varepsilon }}$ converging weakly towards a limit $U_{n}^{k}$ in $L^{2}(I\times \Lambda
\times \Omega \times Y)^{2}$ when $\varepsilon $ tends to zero. Then $%
U_{n}^{k}$ is a solution of the weak formulation of the microscopic equation$$\left( \frac{{{\partial _{\tau
}}}}{\alpha _{n}^{k}}{-A}_{k}\right) U_{n}^{k}=0\text{ in }I\times
\Lambda \times \Omega \times Y \label{strong-form}$$and is periodic in $\tau $ and $k-$quasi-periodic in $y$. Moreover, it can be decomposed as $$U_{n}^{k}\left( t,\tau ,x,y\right) ={\sum\limits_{p\in \mathbb{M}_{n}^{k}}}%
u_{p}^{k}\left( t,x\right) e^{{2{i\pi s}}_{p}\tau }e_{p}^{k}\left( y\right)
\text{ with }u_{p}^{k}\in L^{2}\left( I\times \Omega \right) .
\label{decompose_U}$$
\[lemma\_macro\]For each $k\in Y^{\ast }$, $n\in {%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
}^{\ast }$, for each $\sigma \in I^{k}$ and $q\in M_{n}^{\sigma }$, the macroscopic equation is stated by$$\begin{array}{l}
\sum\limits_{p\in {M}_{n}^{\sigma }}b\left( \sigma ,p,q\right) {\partial _{t}%
}u_{p}^{\sigma }-\sum\limits_{p\in {M}_{n}^{\sigma }}c\left( \sigma
,p,q\right) {\partial _{x}}u_{p}^{\sigma }=F_{q}^{\sigma }\,\ \text{in}\,\
I\times \Omega , \\
\sum\limits_{p\in {M}_{n}^{\sigma }}b\left( \sigma ,p,q\right) u_{p}^{\sigma
}\left( {t=0}\right) =U_{0,q}^{\sigma }\,\ \text{in}\,\ \Omega ,%
\end{array}
\label{macro}$$with the boundary conditions in case where there exist $p\in
{M}_{n}^{k}$ such that $c\left( k,p,q\right) \neq 0$ and $\phi
_{\left\vert p\right\vert }^{k}(0)\ne0$ $$\sum\limits_{\sigma \in
I^{k}}\sum\limits_{p\in {M}_{n}^{\sigma
}}u_{p}^{\sigma }\phi _{\left\vert p\right\vert }^{\sigma }\left( 0\right) {%
e^{sign\left( \sigma \right) 2i\pi \frac{l^{k}x}{\alpha }}}=0\,\ \text{on}%
\,\ I\times \partial \Omega . \label{boundary}$$
The low frequency part $U_{H}^{0}$ relates to the weak limit in $L^{2}\left(
I\times \Omega \times Y\right) ^{2}$ of the kernel part of $%
S_{k}^{\varepsilon }$ in \[def-W\]. It has been treated completely, in [@brahim1992correctors; @brassart2009two]. Here, we focus on the non-kernel part of $S_{k}^{\varepsilon }$, it relates to the high frequency waves and microscopic and macroscopic scales. In order to obtain the solution of the model, we analyze the asymptotic behaviour of each mode through ${{T^{\varepsilon \alpha
_{n}^{k}}}S_{k}^{\varepsilon }}$ as in Lemma \[lemma\_micro\] and Lemma \[lemma\_macro\]. Then the full solution is the sum of all modes. We introduce the characteristic function $\chi _{0}\left(
k\right) =1$ if $k=0$ and $=0$ otherwise. The main Theorem states as follows.
\[theorem\]For a given $k\in Y^{\ast
}$, let $U^{\varepsilon } $ be a solution of (\[1D-1st-weak-formulation\]) bounded in $L^{2}\left(
I\times \Omega \right) $, for $\varepsilon \in E_{k},$ as in (\[epsilon\_m\], \[l\]), the limit $G_{k}$ of any weakly converging extracted subsequence of $\sum\limits_{\sigma \in I^{k}}W_{\sigma
}^{\varepsilon }U^{\varepsilon }$ in $L^{2}\left( I\times \Lambda
\times \Omega \times Y\right) ^{2}$ can be decomposed as$$G^{k}\left( t,\tau ,x,y\right) =\chi
_{0}\left( k\right) U_{H}^{0}\left( t,x,y\right)
+\sum\limits_{\sigma \in I^{k}}\sum\limits_{n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }}{u_{n}^{\sigma }\left( {t,x}\right) {e^{2i\pi {s_{n}}\tau }}%
e_{n}^{\sigma }\left( y\right) } \label{decompose}$$where $%
\left( u_{n}^{\sigma }\right) _{n,\sigma }$ are solutions of the macroscopic equation (\[macro\], \[boundary\]).
Thus, it follows from (\[decompose\]) that the physical solution $U^{\varepsilon }$ is approximated by two-scale modes$$U^{\varepsilon }\left( t,x\right) \simeq \chi _{0}\left( k\right)
U_{H}^{k}\left( t,x,\frac{x}{\varepsilon }\right) +\sum\nolimits_{\sigma \in
I^{k}}\sum\nolimits_{n\in \mathbb{Z}^{\ast }}u_{n}^{\sigma }\left(
t,x\right) e^{is_{n}\sqrt{\lambda _{n}^{\sigma }}t/\varepsilon
}e_{n}^{\sigma }\left( \frac{x}{\varepsilon }\right) .
\label{physical_approximation}$$The remain of this section provides the proofs of results.
**Proof of Lemma \[lemma\_micro\]**. The test functions of the weak formulation (\[1D-1st-weak-formulation\]) are chosen as $\Psi ^{\varepsilon }=\mathfrak{B}_{n}^{k}\Psi \left( {t,x}%
\right) $ for $k\in Y^{\ast }$, $n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }$ where $\Psi \in {C^{\infty } }\left( I\times
\Lambda\right.$$\left. \times \Omega
\times Y\right)^{2}$ is periodic in $\tau $ and $k-$quasi-periodic in $y$. From (\[derivative\_t\]) multiplied by $\varepsilon $, since ${\left( {\frac{{{%
\partial _{\tau }}}}{{\alpha _{n}^{k}}}-{A_{k}}}\right) \Psi }$ is periodic in $\tau $ and $k-$quasi-periodic in $y$ and ${T^{\varepsilon \alpha
_{n}^{k}}S_{k}^{\varepsilon }{U^{\varepsilon }\rightarrow U}}_{n}^{k}$ in $%
L^{2}\left( I\times \Lambda \times \Omega \times Y\right) ^{2}$ weakly, Lemma \[conver\] allows to pass to the limit in the weak formulation, $%
\int\nolimits_{I\times \Lambda \times \Omega \times Y}{U}_{n}^{k}{\cdot
\left( \frac{{{\partial _{\tau }}}}{{\alpha _{n}^{k}}}{-}A_{k}\right) \Psi
dtd\tau dxdy}=0$. Using the assumption $U_{n}^{k}\in D\left( A_{k}\right)
\cap L^{2}\left( I\times \Omega \times Y;H^{1}\left( \Lambda \right) \right)
$ and applying an integration by parts,$$\begin{gathered}
\int\nolimits_{I\times \Lambda \times \Omega \times Y}\left( -\frac{{{%
\partial _{\tau }}}}{{\alpha _{n}^{k}}}+A_{k}\right) {U}_{n}^{k}{\cdot {\Psi
}dtd\tau dxdy+}\int_{{I\times \partial \Lambda \times \Omega \times Y}%
}U_{n}^{k}\cdot \Psi {dtd\tau dxdy} \\
-\int_{{I\times \Lambda \times \Omega \times \partial Y}}U_{n}^{k}\cdot
n_{A_{k}}\Psi {dtd\tau dxdy}=0.\end{gathered}$$Then, choosing ${\Psi \in L}^{2}\left( I\times \Omega
;H_{0}^{1}\left( \Lambda \times Y\right) \right) $ comes the strong form (\[strong-form\]). Since the product of a periodic function by a $k-$quasi-periodic function is $k-$quasi-periodic then $n_{A_{k}}\Psi $ is $k-$quasi-periodic in $y$. Therefore, $U_{n}^{k}$ is periodic in $\tau $ and $k-$quasi-periodic in $y.$ Moreover, (\[decompose\_U\]) is obtained, by projection. **Proof of Lemma \[lemma\_macro\]** For $k\in Y^{\ast }$, let $\left( \lambda _{p}^{\sigma },e_{p}^{\sigma
}\right) _{p\in M_{n}^{\sigma },\sigma \in I^{k}}$ be the Bloch eigenmodes of the spectral equation $\mathcal{Q}\left( \sigma \right) $ corresponding to the eigenvalue $\lambda _{n}^{k}$. We pose $\Psi ^{\varepsilon }\left(
t,x\right) =\sum\nolimits_{\sigma \in I^{k}}{\mathfrak{B}}_{n}^{k}\Psi
_{\varepsilon }^{\sigma }\in H^{1}\left( I\times \Omega \right) ^{2}$ as a test function in the weak formulation (\[1D-1st-weak-formulation\]) with each $\Psi _{\varepsilon }^{\sigma }\left( t,\tau ,x,y\right)
=\sum\nolimits_{q\in M_{n}^{k}}\varphi _{q,\varepsilon }^{\sigma }\left(
t,x\right) e^{2i\pi s_{q}\tau }e_{q}^{\sigma }\left( y\right) $ where $%
\varphi _{q,\varepsilon }^{\sigma }\in H^{1}\left( I\times \Omega \right) $ and satisfies the boundary conditions
$\sum\nolimits_{\sigma \in I^{k},q\in M_{n}^{\sigma }}e^{2i\pi
s_{q}t/(\varepsilon \alpha _{q}^{\sigma })}\varphi _{q,\varepsilon }^{\sigma
}\left( t,x\right) \phi _{\left\vert q\right\vert }^{\sigma }\left( \frac{x}{%
\varepsilon }\right) =O\left( \varepsilon \right) $ on $I\times \partial
\Omega .$ Note that this condition is related to the second component of $%
\Psi ^{\varepsilon }$ only. Since $\alpha _{q}^{\sigma }=\alpha _{n}^{k}$ and $s_{q}=s_{n}$ for all $q\in M_{n}^{\sigma }$ and $\sigma \in I^{k}$, so $%
e^{2i\pi s_{q}t/(\varepsilon \alpha _{q}^{\sigma })}\neq 0$ can be eliminated. Extracting a subsequence $\varepsilon \in E_{k}$, using the $%
\sigma -$quasi-periodicity of $\phi _{\left\vert q\right\vert }^{\sigma }$ and (\[epsilon\_m\],\[l\]), $\varphi _{q,\varepsilon }^{\sigma }$ converges strongly to some $\varphi _{q}^{\sigma }$ in $H^{1}\left( I\times
\Omega \right) $, then the boundary conditions are$$\sum\nolimits_{\sigma \in I^{k}}\sum\nolimits_{q\in M_{n}^{\sigma }}\varphi
_{q}^{\sigma }\left( t,x\right) \phi _{\left\vert q\right\vert }^{\sigma
}\left( 0\right) e^{sign\left( \sigma \right) 2i\pi \frac{l^{k}x}{\alpha }%
}=0\,\text{on }I\times \partial \Omega . \label{boundary-test}$$Applying (\[derivative\_t\]) and since ${\left( {\frac{{{\partial _{\tau }}}%
}{{\alpha _{n}^{\sigma }}}-{{{A_{\sigma }}}}}\right) \Psi ^{\sigma }}=0$ for $\sigma \in $ $I^{k}$, then in the weak formulation it remains$$\sum\limits_{\sigma \in I^{k}}\int\nolimits_{I\times \Omega }{{%
F^{\varepsilon }\cdot \mathfrak{B}}}_{n}^{k}{{\Psi _{\varepsilon }^{\sigma
}+U^{\varepsilon }}\cdot \mathfrak{B}}_{n}^{k}{\left( {{\partial _{t}}-B}%
\right) \Psi _{\varepsilon }^{\sigma }dtdx}-\int\nolimits_{\Omega }{{%
U_{0}^{\varepsilon }}\cdot \mathfrak{B}}_{n}^{k}{\Psi _{\varepsilon
}^{\sigma }}\left( t=0\right) {dx=0}.$$Since ${\left( {{\partial _{t}}-B}\right) \Psi _{\varepsilon }^{\sigma }}$ is $\sigma -$quasi-periodic, so passing to the limit thanks to Lemma [conver]{}, after using (\[data\]) and replacing the decomposition of $%
U_{n}^{\sigma }$,$$\begin{gathered}
\sum\limits_{\sigma \in I^{k},\{p,q\}\in M_{n}^{\sigma }}\left(
\int\nolimits_{I\times \Omega }b\left( \sigma ,p,q\right) u_{p}^{\sigma
}\cdot {\partial }_{t}\varphi _{q}^{\sigma }-{c}\left( \sigma ,p,q\right) {{%
u_{p}^{\sigma }\cdot {{\partial _{x}}}}\varphi _{q}^{\sigma }-F_{q}^{\sigma }%
}\cdot {\varphi _{q}^{\sigma }\,dtdx}\right. \\
\left. {-}\int_{\Omega }{U}_{0,q}^{\sigma }{\cdot \varphi _{q}^{\sigma }}%
\left( t=0\right) {dx}\right) {=0}\text{ for all }{\varphi _{q}^{\sigma }\in
}H^{1}\left( I\times \Omega \right) \text{ fulfilling (\ref{boundary-test})}.\end{gathered}$$Moreover, if $u_{q}^{\sigma }{\in }H^{1}\left( I\times \Omega \right) $ then it satisfies the strong form of the internal equations (\[macro\]) for each $\sigma \in I^{k}$, $q\in M_{n}^{\sigma }$ and the boundary conditions$$\sum\nolimits_{\sigma ,p,q}{{c\left( \sigma ,p,q\right) u_{p}^{\sigma }}}%
\overline{\varphi _{q}^{\sigma }}=0\text{ on }I\times \partial \Omega \text{
for }\varphi _{q}^{\sigma }\text{ satisfies (\ref{boundary-test})}.
\label{boudanry}$$In order to find the boundary conditions of $\left( {{u_{p}^{\sigma }}}%
\right) _{\sigma ,p}$, we distinguish between the two cases $k\neq 0$ and $%
k=0$. First, for $k\neq 0$, $\lambda _{n}^{k}$ is simple so $%
M_{n}^{k}=\left\{ n\right\} $. Introducing $C=diag\left( c\left( \sigma
,n,n\right) \right) _{\sigma }$, $B=diag\left( b\left( \sigma ,n,n\right)
\right) _{\sigma }$, $U=\left( u_{n}^{\sigma }\right) _{\sigma }$, $F=\left(
F_{n}^{\sigma }\right) _{\sigma }$, $U_{0}=\left( U_{0,n}^{\sigma }\right)
_{\sigma }$, $\Psi =\left( \varphi _{n}^{\sigma }\right) _{\sigma }$, $\Phi
=\left( \phi _{\left\vert n\right\vert }^{\sigma }\left( 0\right)
e^{sign\left( \sigma \right) 2i\pi l^{k}x/\alpha }\right) _{\sigma }$, Equation (\[macro\]) states under matrix form$$B{\partial }_{t}U+C{\partial _{x}}U=F\text{ in }I\times \Omega \text{ and }%
BU\left( t=0\right) =U_{0}\text{ in }\Omega , \label{matrixform}$$which boundary condition (\[boudanry\]) is rewritten as $CU\left(
t,x\right) .\overline{\Psi }\left( t,x\right) =0$ on $I\times \partial
\Omega $ for all $\Psi $ such that $\overline{\Phi }(x).\overline{\Psi }%
(t,x)=0$ on $I\times \partial \Omega .$ Equivalently, $CU\left( t,x\right) $ is collinear with $\overline{\Phi }(x)$ yielding the boundary condition $%
u_{n}^{k}\phi _{\left\vert n\right\vert }^{k}\left( 0\right) {e^{2i\pi \frac{%
l^{k}x}{\alpha }}}$ ${+}$ $u_{n}^{-k}\phi _{\left\vert n\right\vert
}^{-k}\left( 0\right) {e^{-2i\pi \frac{l^{k}x}{\alpha }}}=0$ on $\ I\times
\partial \Omega $ after remarking that $c\left( k,n,n\right) \neq 0$ and $%
c\left( k,n,n\right) =-c\left( -k,n,n\right) $.
Second, for $k=0$, $\lambda _{n}^{0}$ is double $\lambda _{n}^{0}=\lambda
_{m}^{0}$ so $M_{n}^{k}=\left\{ n,m\right\} $. With $C=\left( c\left(
0,p,q\right) \right) _{p,q}$, $B=\left( b\left( 0,p,q\right) \right) _{p,q}$, $U=\left( u_{p}^{0}\right) _{p}$, $F=\left( F_{q}^{0}\right) _{q}$, $%
U_{0}=\left( U_{0,q}^{0}\right) _{q}$, $\Psi =\left( \varphi _{q}^{0}\right)
_{q}$, $\Phi =\left( \phi _{\left\vert q\right\vert }^{0}\left( 0\right)
\right) _{q}$, the matrix form is still stated as (\[matrixform\]). Here, the eigenvectors are chosen as real functions then $c\left( 0,p,p\right) =0.$ Since $c\left( 0,n,m\right) \neq 0$, so the boundary condition is $%
u_{n}^{0}\phi _{\left\vert n\right\vert }^{0}\left( 0\right) {+}%
u_{m}^{0}\phi _{\left\vert m\right\vert }^{0}\left( 0\right) =0\,\text{on}\
I\times \partial \Omega \text{.}$ **Proof of Theorem** For a given $k\in Y^{\ast }$, let $U^{\varepsilon }$ be solution of ([1D-1st-weak-formulation]{}) which is bounded in $L^{2}(I\times \Omega )$, the property (\[two-scale-boundness\]) yields the boundness of $\left\Vert
W_{\sigma }^{\varepsilon }U^{\varepsilon }\right\Vert _{L^{2}\left( I\times
\Lambda \times \Omega \times Y\right) }$ for $\sigma \in I^{k}$. So there exists $G^{k}\in L^{2}\left( I\times \Lambda \times \Omega \times Y\right)
^{2}$ such that, up to the extraction of a subsequence, $\sum\nolimits_{%
\sigma \in I^{k}}W_{\sigma }^{\varepsilon }U^{\varepsilon }$ tends weakly to $G^{k}=\chi _{0}\left( k\right) U_{H}^{0}+\sum\nolimits_{\sigma \in
I^{k}}\sum\nolimits_{n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }}U_{n}^{k}$ in $L^{2}\left( I\times \Lambda \times \Omega \times
Y\right) ^{2}$. The high frequency part is based on the decomposition ([decompose\_U]{}) and Lemma \[lemma\_macro\].
This method allows to complete the homogenized model of the wave equation in [@brassart2009two] for the one-dimensional case. Let $K\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast }$, we decompose $\frac{\alpha }{\varepsilon K}=\left[ \frac{\alpha }{%
\varepsilon K}\right] +l_{\varepsilon }^{1}$ with $l_{\varepsilon
}^{1}\in \lbrack 0,1)$ and assume that the sequence $\varepsilon $ is varying in a set $E_{K}\subset
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+\ast }$ depending on $K$ so that $l_{\varepsilon }^{1}\rightarrow
l^{1}$ when $\varepsilon \rightarrow 0$ with $l^{1}\in \lbrack
0,1)$. For any $k\in L_{K}^{\ast }$, defined in [@brassart2009two], we denote $p_{k}=kK\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$, so $\frac{\alpha p_{k}}{\varepsilon K}=p_{k}\left[ \frac{\alpha }{%
\varepsilon K}\right] +p_{k}l_{\varepsilon }^{1}$ and $p_{k}l_{\varepsilon }^{1}$ $\rightarrow l^{k}:=p_{k}l^{1}$ when $\varepsilon \rightarrow 0$ with the same sequence of $\varepsilon
\in E_{K}$.
Numerical examples
==================
We report simulations regarding comparison of physical solution and its approximation for $I=\left( 0,1\right) ,$ $\Omega =\left( 0,1\right) $, $%
\rho =1$, $a=\frac{1}{3}\left( \sin \left( 2\pi y\right) +2\right) $, $%
f^{\varepsilon }=0$, $v_{0}^{\varepsilon }=0$, $\varepsilon =%
\frac{1}{10}$ and $k=0.16$. Since $k\neq 0$, so the approximation ([physical\_approximation]{}) comes$$U^{\varepsilon }\left( t,x\right) \simeq \sum\nolimits_{\sigma \in
I^{k}}\sum\nolimits_{n\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }}u_{n}^{\sigma }\left( t,x\right) e^{is_{n}\sqrt{\lambda
_{n}^{\sigma }}t/\varepsilon }e_{n}^{\sigma }\left( \frac{x}{\varepsilon }%
\right) . \label{num}$$The validation of the approximation is based on the modal decomposition of any solution $U^{\varepsilon }=\sum_{l\in
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
^{\ast }}R_{l}^{\varepsilon }\left( t\right) V_{l}^{\varepsilon }\left(
x\right) $ where the modes $V_{l}^{\varepsilon }$ are built from the solutions $v_{l}^{\varepsilon }$ of the spectral problem $\partial
_{x}\left( a^{\varepsilon }\partial _{x}v_{l}^{\varepsilon }\right) =\lambda
_{l}^{\varepsilon }v_{l}^{\varepsilon }$ in $\Omega $ with $%
v_{l}^{\varepsilon }=0$ on $\partial \Omega $. Moreover, in [nguyen2013homogenization]{}, two-scale approximations of modes have been derived on the form of linear combinations $\sum\nolimits_{\sigma \in
I^{k}}\theta _{n}^{\sigma }\left( x\right) \phi _{n}^{\sigma }\left( \frac{x%
}{\varepsilon }\right) $ of Bloch modes, so the initial conditions of the physical problem are taken on the form $$u_{0}^{\varepsilon }\left( x\right) =\sum\nolimits_{n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast }}\sum\nolimits_{\sigma \in I^{k}}\theta _{n}^{\sigma }\left(
x\right) \phi _{n}^{\sigma }\left( \frac{x}{\varepsilon }\right) .
\label{initial condition}$$Two simulations are reported, one for an initial condition $%
u_{0}^{\varepsilon }$ spanned by the pair of Bloch modes corresponding to $%
n=2$ when the other is spanned by three pairs $n\in \{2,3,4\}$. In the first case, the first component of $U_{0}^{\varepsilon }$ approximates the first component of a single eigenvector $V_{l}^{\varepsilon }$ approximated by (\[num\]) where all coefficients $u_{n}^{\sigma }=0$ for $n\neq \pm 2$. Fig. \[simulation\] $(a)$ shows the initial condition $u_{0}^{\varepsilon
} $. Fig. \[simulation\] $(b)$ presents the real part (solid line) and the imaginary part (dashed-dotted line) of the macroscopic solution $%
u_{n}^{k}$ and also the real part (dotted line) and the imaginary part (dashed line) of $u_{n}^{-k}$ at space step $x=0.699$ when Fig. [simulation]{} $\left( c,d\right) $ plot the real part of the first component $%
U_{1}^{\varepsilon }$ of physical solution and the relative error vector of $%
U_{1}^{\varepsilon }$ with its approximation which $L^{2}(\Omega )$-norm is equal to** **7e-3 at $t=0.466$**.** For the second case where $%
u_{n}^{\sigma }=0$ for $n\notin \{\pm 2,\pm 3,\pm 4\}$, the first component $%
U_{1}^{\varepsilon }$ and the relative error vector of $U_{1}^{\varepsilon }$ with its approximation which $L^{2}(\Omega )$-norm is 3.8e-3 are plotted in Fig. \[simulation\] $(e,f)$. Finally, for the two cases the $L^{2}(I)$-relative errors at $x=0.699$ on the first component are 8e-3 and 3.5e-3 respectively.
![Numerical results[]{data-label="simulation"}](initial_condition_1 "fig:")![Numerical results[]{data-label="simulation"}](macro_solution "fig:")![Numerical results[]{data-label="simulation"}](homogenized_solution_1 "fig:")
![Numerical results[]{data-label="simulation"}](error_1 "fig:")![Numerical results[]{data-label="simulation"}](homogenized_solution_2 "fig:")![Numerical results[]{data-label="simulation"}](error_2 "fig:")
|
---
abstract: 'The *Rossi X-ray Timing Explorer (RXTE) satellite was launched on 30 December 1995. It has made substantial contributions pertaining to compact objects and their environs. Broad-band spectral and short-time-scale temporal studies are exploring the effects of General Relativity in the regime of strong gravity. We present a brief outline of the principal contributions and then give a general overview of two new areas of x-ray astronomy that have proven by RXTE to be very fruitful: accreting neutron stars with millisecond spin periods and microquasars. The former pertains to the spin evolution of low-mass x-ray binaries and the equations of state of neutron stars while the latter is lends insight to disk-jet interactions in galactic black-hole binary systems.*'
---
=-1.2truecm =-2.5truecm
=cmbx10 =cmr10 =cmti10 =cmbx10 scaled1 =cmr10 scaled1 =cmti10 scaled1 =cmbx9 =cmr9 =cmti9 =cmbx8 =cmr8 =cmti8 =cmr7 =cmti7 \[\]
**HIGHLIGHTS from RXTE after 2.5 YEARS:**
**NEUTRON-STAR SPINS at KILOHERTZ FREQUENCIES,**
**MICROQUASARS and MORE**
HALE V. BRADT
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Room 37–587, Cambridge MA 02139–4307, USA
E-mail: bradt@mit.edu
Mission status
==============
The *Rossi X-ray Timing Explorer (Bradt, Rothschild & Swank 1993) was launched on 30 December 1995. Since then it has carried out a diverse observing program that has been open to the entire astronomical community since a month after launch. It carries an All-Sky Monitor (Levine et al. 1996; Levine 1998) which, together with a flexible spacecraft pointing capability, permits rapid (hours) acquisition of new or recurrent transient sources, sources entering new or interesting states, and gamma-ray burst afterglows. The pointed instruments are a large Proportional Counter Array (PCA; 2 – 60 keV; Jahoda et al. 1996) and a rocking High Energy X-ray Timing Experiment (HEXTE; 15 – 200 keV; Rothschild et al. 1998). All three instruments continue to operate close to their design state. It is hoped that operations can continue for several more years. There are no on-board expendables which would limit the spacecraft life.*
Information about the mission as well as data products may be accessed on the web through: http://heasarc.gsfc.nasa.gov/docs/xte/. The source intensities from the ASM are posted every few hours on: http://heasarc.gsfc.nasa.gov/xte\_weather/.
Scientific accomplishments: overview
====================================
The *RXTE was designed to study compact objects and the material in their environs with emphasis on temporal studies with high statistics together with broad band-band spectroscopy. Over 150 papers had been accepted in the refereed literature by the summer of 1998. *RXTE has been highly influential in conduct of science in many wavebands. Over 120 IAU Circulars had announced *RXTE discoveries of immediate interest and these have been followed by numerous reports from other observatories, gamma-ray, x-ray, radio and optical. At http://heasarc.gsfc.nasa.gov/whatsnew/xte/papers.html, circulars and papers may be found.***
The areas in which *RXTE has made important contributions are listed here, with a few sample references. They are extracted from the Proposal to the 1998 Senior Review of NASA Astrophysics Missions Operations and Data Analysis authored by J. Swank, F. Marshall & the *RXTE Users’ Group. Thereafter, I will give brief overviews, from my perspective, of two areas wherein *RXTE has broken substantial new ground, namely, kiloHertz oscillations and microquasars.***
1\. Behavior of matter in regimes of strong gravity through the temporal and spectral signatures of kiloHertz pulsars and variability in microquasars (see refs. below).
2\. Spinup evolution of neutron stars through the characteristics of kHz pulsars and the discovery of the first accretion powered millisecond pulsar (see refs. below).
3\. Formation of relativistic astrophysical jets through the multiwavelength study of (galactic) microquasars (see refs. below).
4\. AGN unified models and emission mechanisms through multiwavelength (esp. TeV) studies (*e.g., Cantanese et al. 1997), detection of iron line and reflection components in individual Sy1 and Sy2 galaxies (*e.g., Weaver, Krolik & Pier 1998, Nandra et al. 1998), and temporal studies with long term sampling with both the PCA and the ASM instruments. See for example the variability of the BL Lac objects Mkn 501 and Mkn 421 in Fig. 1.**
5\. High magnetic fields in neutron stars through the study of (1) the magnetosphere/disk boundary with low-frequency QPOs (*e.g., Kommers, Chakrabarty & Lewin 1998), Type II bursts, the propeller effect (*e.g., Cui 1997), and cyclotron lines (Kreykenbohm et al. 1999); (2) the discovery of the fastest rotation powered pulsar ($P =
16$ ms; Marshall et al. 1998), and (3) the discovery of x-ray pulsations supporting the identification of a ÔÔmagnetarÕÕ, a neutron star with an extraordinarily high magnetic field ($2 \times 10^{10}$ T), as a soft gamma-ray repeater (Kouveliotou et al. 1998).**
6\. Transient sources through PCA slews and ASM monitoring which have revealed $\sim$15 previously unknown sources and numerous recoveries of previously known sources, together with follow-on studies with the PCA and other observatories. Several new examples of radio-jet systems have been revealed (see below). Sample light curves from the RXTE/ASM extending over 2.5–yr are shown in Fig. 1.
7\. State changes in binary systems through temporal/spectral tracking during major changes in x-ray flux and spectrum. In the case of Cyg X–1, a change of corona size is indicated (Cui et al. 1997).
8\. Superorbital quasi periodicities in high and low mass binaries as well as in black-hole binaries through their discovery or confirmation with the ASM. (*e.g., $P \approx $ 60 d in SMC X1; See Fig. 1). Some are most likely due to precessing accretion disks similar to the 35–d period in the well known Her X–1. However, the evolution of wave forms and periods indicates relatively complex underlying physics (Levine 1998).*
9\. Gamma-ray burst afterglows through rapid position determinations with the ASM and PCA. Five burst positions with positions accurate to a few arcminutes in one or two dimensions have been reported to the community within hours of the event (Smith et al. 1999). One of the three known GRB with measured extragalactic red shifts (GRB 980703) was first located with the ASM on *RXTE (Levine, Morgan & Muno 1998, Djorgovski et al. 1998). Another (GRB 970828) had a very bright afterglow in x rays but no discernable optical or radio afterglow (Remillard et al. 1997; Groot et al. 1998).*
10\. X-ray emission regions in cataclysmic variables through PCA tracking of eclipse transitions with precisions of tens of kilometers (*e.g., Hellier 1997), and wind-wind collisions in Eta Carina through repeated PCA spectral observations (Corcoran et al. 1997).*
11\. Diffuse source spectra from the galactic plane (Valinia & Marshall 1998), supernova remnants (*e.g., Allen et al. 1997), and clusters of galaxies (*e.g., Rephaeli & Gruber 1999) to high energies ($>$ 10 keV) with PCA and HEXTE.**
KiloHertz oscillations in low-mass x-ray binaries
=================================================
The most prominent area of *RXTE accomplishment is that of kiloHertz oscillations. Relatively high-Q quasiperiodic oscillations (QPO) at kHz frequencies (up to 1230 Hz) in Low-Mass X-ray Binaries (LMXB) have been found in the persistent flux of 18 sources (as of this writing, Dec. 1998). Five of these, and one other, exhibit quite coherent, but transient, oscillations in the frequency range 290 – 590 Hz during Type I (thermonuclear) bursts. One additional source, a transient, exhibited sustained coherent pulsations at 401 Hz with Doppler shifts characteristic of a binary orbit. Reviews of the field may be found in van der Klis (1998, 1999).*
These new phenomena are probing the processes taking place close to the neutron stars where the effects of General Relativity are important. For example, if the highest-frequency kHz QPO observed in a given source is interpreted as the Kepler frequency of the inner accretion disk, it should be limited by the frequency of the innermost stable orbit allowed in general relativity ($r=6GM/c^2$ for Schwarzschild geometry). Since 17 of the sources exhibit maximum frequencies in the relatively narrow 1000–1200 Hz range, these frequencies may indeed represent the innermost stable orbits. The phenomenon is also placing constraints on the equations of state of neutron stars as we illustrate below.
The first discovered examples of the quasi-periodic oscillations at kHz frequencies were in Sco X-1 (van der Klis et al. 1996) and 4U 1728–34 (Strohmayer et al. 1996). These QPO usually occur in pairs. As the source intensity increases, the two QPO generally increase in frequency with a frequency difference that remains approximately constant (Strohmayer et al. 1996); see Fig. 2.
If the higher-frequency peak of the pair is the Kepler velocity of a blob of orbiting disk material, the lower-frequency peak could arise from the interaction of the blob with the magnetosphere which is co-rotating with the spinning neutron star. The observed lower frequency would thus be a beat frequency, such as that postulated for QPOs at much lower frequencies ($\le 60$ Hz) in the 1980’s (Alpar and Shaham 1985, Lamb et al. 1985). The difference of the two frequencies would be the neutron-star spin frequency. For the 17 sources that exhibit two frequencies, the differences range from $\sim$250 to $\sim$350 Hz, indicating neutron-star spins in this range. (See Table 1 in van der Klis 1999.)
The increase of frequency of the oscillations with intensity (Fig. 2) can be understood in this picture as being due to an increase in the Kepler frequency arising from a decrease in the size of the magnetosphere caused, in turn, by the increased ram pressure of the accreting material (Ghosh & Lamb 1992). In fact, the source 4U 1820–30 exhibits frequencies that increase with flux until they saturate at about 1060 Hz and 800 Hz (Fig. 3). This suggests that the innermost stable orbit has been reached (Zhang et al. 1998b). But since this plot is a compilation of data from different observations that might have differing intensity-frequency relations (see below), the effect could be an artifact (Mendez et al. 1999).
If indeed the maximal frequencies represent the innermost stable orbits, the highest observed frequency seen to date (1228 Hz) yields a neutron star mass of $\sim$2.0 $M_{\odot}$, which is significantly above the canonical 1.4 $M_{\odot}$. Further, the radius of the neutron star must not exceed the radius of the Kepler orbit corresponding to the maximum observed frequency in a given source. This limit, together with the marginally stable orbit just discussed above, places constraints on allowed equations of state for neutron stars (Miller, Lamb & Psaltis 1998, see Fig. 4). The current limits do not yet distinguish among the plotted equations of state, but the potential for doing so is clearly there.
The interpretation that the difference frequency is the neutron-star spin frequency gains credence from the discovery of nearly coherent pulsing during x-ray bursts in several sources at about, or at about twice, the difference frequency (*e.g., Strohmayer et al. 1996). The frequency of the burst oscillations in 4U 1728–34 is stable from burst to burst within about 0.01% over a period of 1.6 years (Strohmayer et al. 1998; Fig. 5). This stability is a strong indicator that these pulsations directly represent the neutron-star spin. They could arise from a transient hot spot (or hot spots) in the runaway thermonuclear burning on the neutron-star surface (Bildsten 1995). If there were a hot spot at each of two opposed magnetic poles, the detected frequency would be twice the spin frequency.*
This picture needs refinement or modification for several reasons: (1) the frequency difference as a function of intensity (or of one of the two frequencies) is not constant in Sco X–1 (van der Klis et al. 1997), 4U 1608 (Mendez et al. 1998), 4U 1735–44 (Ford et al. 1998), and possibly in all sources (Psaltis et al. 1998), (2) the frequencies in bursts are not strictly 1.0 or 2.0 times the difference frequencies in all sources, especially in 4U 1636–536 (Mendez, van der Klis & van Paradijs 1998), (3) there are small frequency drifts during a single burst (Fig 5), and (4) although there are correlations between source intensity and QPO frequency on short time scales (hours), the correlations do not hold up over long periods (days); very different intensities can yield the same QPO frequency, *e.g., in Aql X–1 (Zhang et al. 1998a). These problems are not necessarily fatal to the basic picture; there are various proposed scenarios to explain the discrepancies. Alternatively, the answers could lie in very different directions, see, *e.g., Stella & Vietri (1999).**
The so-called discrepancies are actually excellent probes with which to verify or discard theoretical models. For example, the frequency difference as a function of intensity in Sco X–1 requires quantitative understanding; see, for example, Stella & Vietri 1999. Also, the frequency *vs intensity dilemma (item 4 above) has been clarified by the discovery of monotonic mapping of frequency with position in the color-color diagram of 4U 1608–52 (Mendez et al. 1999).*
Finally, as noted above, the beat-frequency model was originally introduced to explain some of the lower-frequency quasi-periodic oscillations ($\sim$ 6 – 60 Hz) in LMXB in the 1980’s. One cannot explain *both kinds of oscillations (low-frequency and kHz) in the same source with this one model. Attempts to rationalize these phenomena include (1) a sonic-point model to explain the kHz oscillations as arising from interactions between radiation and orbiting blobs at the sonic-point radius (Miller, Lamb & Psaltis 1998), (2) nodal precession of the inner disk, dominated by the Lense-Thirring effect, to explain the lower-frequency oscillations (Stella & Vietri 1998), and (3) periastron precession to explain the lower-frequency peak of the kHz twin peaks (Stella & Vietri 1999). The latter model can reproduce the changes in the frequency difference but does not attempt to explain the apparent coincidences with the frequencies of the kHz QPO during bursts in some sources. Strong-field GR is required to calculate the periastron-precession frequency. Thus, as pointed out by the authors, if their model is validated, the kHz QPO phenomenon provides an unprecedented testbed for strong-field General Relativity.*
These indicators of the neutron-star spin are only indirect (through the beat frequency) or fleeting (during bursts). The detection of coherent, persistent, accretion-powered pulsing at millisecond periods had so far eluded *RXTE researchers. This elusive goal was reached with the recent (April 1998) *RXTE discovery of highly coherent 401–Hz x-ray pulsing in the persistent flux of a transient source (Wijnands & van der Klis 1998). A binary orbit was easily tracked with the Doppler shifts of the 401–Hz pulsations (Chakrabarty & Morgan 1998, Fig. 6). The orbital period is 2.01 hr which indicates a companion mass less than 0.1 $M_{\odot}$.**
The Doppler variation demonstrated without doubt that the pulsations arose from the neutron-star spin. The source thus became the first known *accretion-powered millisecond pulsar. The source had been detected in a previous transient episode (September 1996) with the SAX Wide Field Camera (and also in RXTE/ASM data retrospectively) and two x-ray bursts were observed from it (in ’t Zand et al. 1998). It is known as SAX J1808.4–3658. The rapid pulsing discovery occurred during a later outburst (April 1998) that was revealed in RXTE/PCA data during a spacecraft slew (Marshall 1998).*
These discoveries of millisecond-period x-ray sources fill an important link in the spin evolution of neutron stars. It had long been postulated that millisecond radio pulsars are spun up in x-ray binaries (Radhakrishnan & Srinivasan 1982, Alpar et al. 1982), and LMXB were prime candidates because of their lack of coherent pulsations at lower frequencies. This long-sought evolutionary link has now been established. This is the successful attainment of one of *RXTE’s major goals.*
Microquasars
============
Another area of major accomplishment by *RXTE is that of “microquasars”. The discovery of transient galactic x-ray emitting objects with superluminal radio jets, GRO 1655–40 and GRS 1915+105 (*e.g., Tingay et al. 1995, Mirabel & Rodriguez 1994), and the well-determined high mass ($7.0 \pm 0.2$ $M_{\odot}$) of the compact object in GRO 1655–40 (Orosz & Bailyn 1997) focused attention on the fact that counterparts of (black-hole) quasars are close by in the Galaxy. Their proximity allows studies with much higher statistics, and their lower (stellar) masses lead to much smaller time constants for motions of matter in the vicinity of the compact object. The time constants scale linearly with mass, *e.g., the orbital period of Kepler matter in the innermost stable orbit. Thus a 1–year intensity variation in the vicinity of a $10^8$–$M_{\odot}$ quasar would occur in 3 s in a galactic 10–$M_{\odot}$ microquasar.***
Other galactic x-ray sources are known to exhibit evidence for radio jets through episodic non-thermal radio emission and/or diffuse emission or resolved jets. These include the long-known Cyg X–3, GX 339–4, Cir X–1, SS433 and Cyg X–1 (see van Paradijs 1995 for references), and also the *RXTE discovered or recovered transients XTE J1748–288 (Rupen & Hjellming 1998), GRS 1739–278 (Hjellming et al. 1996, Durouchoux et al 1996) and CI Cam (Hjellming & Mioduszcwski 1998). One of these sources, Cir X–1 most likely contains a neutron star (Tennant, Fabian & Shafer 1986, Shirey, Bradt & Levine 1999), and another, CI Cam, is a symbiotic system. Altogether these sources are a rich resource for the understanding of the role accretion disks play in jet formation.*
It is fortunate that the superluminal sources GRS 1915+105 and GRO 1655–40 have exhibited extensive activity during the *RXTE mission. GRO 1655–40 was active for about 16 months beginning in April 1996, and GRS 1915+105 has been active since the beginning of the mission. The latter source exhibits a variety of states in its long-term variability as measured with the RXTE/ASM (Fig. 7). In the high-statistics data from the RXTE/PCA, its x-ray variability is dramatic and varied, including rapid oscillations, sudden dips, sharp spikes, etc., all accompanied with spectral changes (Fig. 8; Greiner, Morgan & Remillard 1996, Morgan, Remillard & Greiner 1997, Taam, Chen & Swank 1997).*
Next, I present briefly three areas of substantive progress in microquasar studies.
Initiation of accretion
-----------------------
The sudden turn-on of GRO 1655–40 in x rays was fortuitously monitored in BVRI during the week just prior to the x-ray turn on (Fig. 9; Orosz et al. 1997). A linear increase of flux was seen in all four bands. It began first in the I band, 6.1 d before the commencement of the linear x-ray rise. The increases in the R, V and B bands commenced systematically later with the latter occurring 5.0 days before the x-ray commencement. This sequence suggests that the initiating event of a transient outburst was a wave of instability propagating inward in the disk (Lasota, Narayan & Yi 1996). This is an important breakthrough in the determination of the causes of x-ray nova outbursts.
Accretion-jet correlations
--------------------------
There have been clear coincidences between radio/infrared non-thermal flares and x-ray events, both on the longer time scales of the ASM data (Pooley & Fender 1997) and shorter-term events in the PCA data (Pooley & Fender 1997, Eikenberry et al. 1998, Mirabel et al. 1998). The latter type of x-ray event consists of a large x-ray dip ($\sim$15 minutes) that contains a pronounced spike (Fig. 8, bottom panel). Such an event is associated with an infrared flare and a delayed radio flare as shown in an event captured by Mirabel et al. (1998, Fig. 10). Five and possibly six IR/x-ray coincidences of this type were reported by Eikenberry et al. (1998) and in no case was the coincidence violated!
These x-ray dips are repetitive, occurring irregularly at intervals of a half hour or so (Fig. 8). The source may reside in this state for hours to days. This is only one of several oscillatory states in which the source can find itself; see Fig. 8. The spectral evolution of an infrared/radio flare has been shown to represent a single relativistically expanding plasmoid (Eikenberry & Fazio 1997, Mirabel et al. 1998, Fender & Pooley 1998). These IR/radio events are small, *i.e., mini flares. A series of them emitted when the source is in this state could give rise to a single large superluminal outburst. It thus appears that the jets are quantized, not continuous, and that *RXTE is seeing the “pump” that creates them!**
X-ray spectral fits (Fig. 11, Swank et al. 1997) during these events show a softening of the disk-black body component, which can be interpreted as the disappearance of the inner part of the disk as proposed by Belloni et al. (1997a,b). Thereafter, the gradually increasing temperature and decreasing radius of the disk component would represent the refilling of the disk. The power-law component suddenly softens at a sharp x-ray spike near time 1600 s when the disk is nearly full. Mirabel et al. (1998) suggest that this spike is the initiating event of the flare (see the IR flare in Fig. 10). The frequency of the associated low- frequency QPO (Fig. 11) appears qualitatively to track the disk radius as if it were the Kepler frequency at this or an associated radius. But the situation is not this simple given the existence of other QPOs, *e.g., 67 Hz, in the system (see Remillard et al. 1999).*
As noted, GRS 1915+105 exhibits some half dozen states with different temporal/spectral variability, not all of which fit this simple disk-depletion picture. Additional multifrequency studies are needed as are more comprehensive models.
High-frequency QPO in Microquasars
----------------------------------
The microquasars exhibit quasi periodic oscillations (QPO) that are quite variable in frequency and also some that are relatively stable. These QPO have a large potential for probing the physics of the systems. The highest frequencies (Fig. 12), namely 67 Hz in GRS 1915+105 (Morgan, Remillard & Greiner 1997) and 300 Hz in GRO 1655–40 (Remillard et al. 1999) do not drift in frequency. They have led to intriguing speculation about their origins. The high frequencies place them close to the central black hole, and models usually invoke General Relativity. Suggested origins include the innermost stable orbit (Morgan et al. 1987), Lense-Thirring precession (Cui, Zhang & Chen 1998), diskoseismic oscillations (Nowak, et al. 1997), and oscillations in the centrifugal barrier (Titarchuk, Lapidus, Muslimov 1998).
Some investigators are using these data and models to arrive at the angular momentum of the central black hole. The black-hole mass, $7.0 \pm 0.2$ $M_{\odot}$, of GRO 1655–40 and the 300–Hz oscillations in this source suggest negligible black-hole angular momentum *if the oscillations are the Kepler frequency of the innermost stable orbit. On the other hand, if the 300 Hz oscillations are due to Lense-Thirring precession in the inner disk, they imply a maximally rotating black hole (Cui, Zhang & Chen 1998). The latter view gains some support from the measured high disk temperature which is indicative of the small inner disk radius expected for prograde orbital motion of a maximally rotating black hole (Zhang, Cui & Chen 1997). These conclusions are highly model dependent and therefore uncertain. Nevertheless, it is impressive to that the angular momentum of black holes is now being addressed by the community with data from *RXTE . This was not dreamed of even a few years ago.**
All in all, it is clear that the jet formation processes, the conditions of disk stability, and the formation of the power-law component are being explored with a powerful and effective tool, namely the temporal/spectral/statistical power of *RXTE. The behavioral detail now being acquired from microquasars extends well beyond that which can be obtained from the much more distant extragalactic quasars.*
Conclusions
===========
The *RXTE is making important strides in the study of compact objects, both galactic and extragalactic, in a wide variety of studies by a large international community of observers.*
The discovery of 401–Hz kHz coherent pulsations in a low-mass x-ray binary has established definitively an important link in the evolution of neutron stars. The kHz QPO in 18 systems and coherent pulsations during bursts give additional strong indications of neutron-star spins at frequencies of a few hundred Hz. These QPO provide information about the behavior of matter in the immediate vicinity of the neutron star and are placing limits on the possible equations of state of neutron stars.
The temporal/spectral signatures of the various behaviors in microquasars are diverse, yet repeatable and well described with high statistics. They are powerful probes of these systems and should serve as powerful discriminators of models. At the same time, the complexity makes difficult the construction of a comprehensive model of the emission processes. The results currently point toward black-hole masses and angular momenta, the nature of disk instabilities, and the precise events that initiate the jets signified by radio/IR flares. These results clearly have applicability to extragalactic quasars.
The temporal variability of x-ray spectra can, in principle, track the changing geometry of the several physical components of the system (*e.g., disk and corona). However this requires that these physical components be securely identified with the spectral components. This is a major challenge now confronting microquasar researchers.*
*RXTE studies are probing phenomena where strong General Relativity is important because of the proximity of the emitting plasmas to the central gravitational object. For example, frame dragging has been invoked for some high frequency QPOs, and the orbital frequency at the innermost stable orbit may have been encountered in LMXB systems. Measurements of these and other GR effects are now within the realm of *RXTE capabilities.**
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful for the efforts of the entire *RXTE team and the many observers whose work has contributed to the productivity of *RXTE . He is especially grateful to the staff and students of the *RXTE group at M.I.T. for many helpful and stimulating conversations. Helpful comments for this manuscript were provided by R. Remillard and L. Stella. This work was supported in part by NASA under contract NAS5–30612. The author further acknowledges with gratitude the support and hospitality provided to him during his sabbatical year at the Osservatorio Astronomico di Roma. This report was completed while overlooking the “Pines of Rome”.***
References
==========
Allen, G. E. et al. 1997, ApJ, 487, L97
Alpar, M. & Shaham, J. 1985, Nature, 316, 239
Alpar, M. Cheng, A., Ruderman, M. & Shaham, J. 1982, Nature, 300, 728
Belloni, T., Mendez, M., King, A. R., van der Klis, M. & van Paradijs, J. 1997a, ApJ, 479, L145
Belloni, T., Mendez, M., King, A. R., van der Klis, M. & van Paradijs, J. 1997b, ApJ, 488, L109
Bildsten, L. 1995, ApJ, 438, 852
Bradt, H. V., Rothschild, R. E. & Swank, J. H. 1993, A&AS, 97, 355
Cantanese, M. et al. 1997, ApJ, 487, L143
Chakrabarty, D. & Morgan, E. 1998 Nature, 394, 346
Corcoran, M. F., Ishibashi, K., Swank, J., Davidson, K., Petre, R. & Schmitt, M. 1997, Nature, 390, 587
Cui, W. 1997, ApJ, 482, L163
Cui, W., Zhang, S. N., Focke, W. & Swank, J. H. 1997, ApJ, 484, 383
Cui, W., Zhang, S, N. & Chen, W. 1998, ApJ, 492, 53
Djorgovski, S. G. et al. 1998, ApJ, 508, L17
Durouchoux P. et al. 1996, IAU Circ. 6383
Eikenberry, S.S. & Fazio, G. G. 1997, ApJ, 475, L53
Eikenberry S. S., Matthews, K., Morgan, E. H., Remillard, R. A. & Nelson, R. W. 1998, ApJ, 494, L61
Fender, G. G. & Pooley, R. P. 1998, MNRAS 300, 573
Ghosh, P. & Lamb, F. K. 1992, in X-ray Binaries & Recycled Pulsars, ed. E. P. J. van den Heuvel & S. A. Rappaport (Dordrecht: Kluwer) 487
Ford, E. C., van der Klis, M., van Paradijs, J., Mendez, M., Wijands, R. & Kaaret, P. 1998, ApJ, 508, L155
Greiner, J., Morgan, E. H. & Remillard, R. A. 1996, ApJ, 473, L107
Groot, P. J., et al. 1998, ApJ, 493, L27
Hellier, C. 1997, MNRAS, 291, 71
Hjellming, R. M. 1996, IAU Circ. 6383
Hjellming, R. M. & Mioduszcwski, A. J. 1998, IAU Circs. 6857, 6862, 6872
in ’t Zand, J. J., Heise, J., Muller, J. M., Bazzano, A., Cocchi, M., Natalucci, L. & Ubertini, P. 1998, Astr. Astrophys., 331, L25
Jahoda, K. et al. 1996, in EUV, X-ray, and Gamma-ray Instrumentation for Space Astronomy VII, ed. O. H. W. Sigmund & M. A. Grummin, Proc. SPIE 2808, 59
Kommers, J. M., Chakrabarty, D. & Lewin, W. H. G. 1998, ApJ, 497, L33
Kouveliotou, C., et al. 1998, Nature, 393, 235
Kreykenbohm, I., et al. 1999, A&A, submitted (astro-ph 9810282)
Lasota, J. P., Narayan, R. & Yi, I. 1996, A&A, 314, 813
Lamb, F. K., Shibazaki, N., Alpar, M. & Shaham, J. 1985, Nature, 317, 681
Levine, A. M. 1998, Nucl. Phys. B (Proc. Suppl.), 69/1–3, 196 \[Proc. of ÔÔThe Active X-ray SkyÕÕ, eds. L. Scarsi, H. Bradt, P. Giommi & F. Fiore, North-Holland\]
Levine, A. M. et al. 1996, ApJ, 469, L33
Levine, A. M., Morgan, E. & Muno, M. 1998, IAU Circ. 6966
Marshall, F. E. 1998, IAU Circ. 6876
Marshall, F. E., Gotthelf, E. V., Zhang, W., Middleditch, J. & Wang, Q. D. 1998, ApJ, 499, L179
Mendez, M., van der Klis, M., Ford, E. C., Wijnands, R. & van Paradijs, J. 1999 ApJ Letters (in press)
Mendez, M., van der Klis, M. & van Paradijs 1998, ApJ, 506, L117
Mendez, M., van der Klis, M., Wijnands, R., Ford, E., van Paradijs, J. & Vaughan, B. 1998, ApJ. 505 , L23.
Miller, M. C., Lamb, F. K. & Psaltis, D. 1998, ApJ, 508, 791
Mirabel, I. F. & Rodriguez, L.F. 1994, Nature, 371, 46
Mirabel, I. F., Dhawan, V., Chaty, S., Rodriguez, L. F., Marti, J., Robinson, C. R., Swank, J. H. & Geballe, T. 1998, A&A, 330, L9
Morgan, E. H., Remillard, R. A. & Greiner, J. 1997, ApJ, 482, 993
Nandra, K. et al. 1998, ApJ, 505, 594
Nowak, M. A., Wagoner, R. V., Begelman, M. C. & Lehr, D. E. 1997, ApJ, 477, L91
Orosz, J. A. & Bailyn, C. D. 1997, ApJ, 477, 876
Orosz, J. A., Remillard, R. A., Bailyn, C. D., McClintock, J. E. 1997, ApJ, 478, L83
Pooley, G. G. & Fender, R. P. 1997, MNRAS, 292, 925
Psaltis, D. et al. 1998, ApJ, 501, L95
Radhakrishnan, V. & Srinivasan, G. 1982, Curr. Sci. 51, 1096
Remillard, R. A., Wood, A., Smith, D. & Levine, A. 1997, IAU Circ. 6726; see also IAU Circ. 6728
Remillard, R. A., Morgan, E. M., McClintock, J. E., Bailyn, C. D. & Orosz, J. A. 1999, ApJ, in press (astro-ph 9806049)
Rothschild, R. E. et al. 1998, ApJ, 496, 538
Rephaeli, Y. & Gruber, D. 1999, in preparation
Rupen, M. P. & Hjellming, R. M. 1998, IAU Circ. 6938
Shirey, R. E., Bradt, H. V. & Levine, A. M. 1999, ApJ (in press)
Smith, D. A. et al. 1999, ApJ, in preparation
Stella, L. & Vietri, M. 1998, ApJ, 492, L59
Stella, L. & Vietri, M. 1999, PRL, 82, 17
Strohmayer, T. E., Zhang, W., Swank, J. H., Smale, A., Titarchuk, L., Day, C. & Lee, U. 1996, ApJ, 469, L9
Strohmayer, T., Zhang, W., Swank, J. & Lapidus, I. 1998, ApJ, 503, L147
Swank, J. H., Chen, X., Markwardt, C. & Taam, R. 1997, in Proceedings of ÔÔAccretion Processes in Astrophysics: Some Like it HotÕÕ, U. of Md. Oct. 1997, eds. S. Holt & T. Kallman; astro-ph 9801220
Taam, R. E., Chen, X. & Swank, J. H. 1997, ApJ, 485, L83
Titarchuk, L., Lapidus, I., Muslimov, A. 1998, ApJ, submitted, astro-ph 9712348
Tennant, A. F., Fabian, A. C. & Shafer, R. A. 1986, MNRAS, 221, 27p
Tingay, S. J. et al. 1995, Nature, 374, 141
Valinia, A. & Marshall, F. E. 1998, ApJ, 505, 134
van der Klis, M. et al. 1996, ApJ, 469, L1
van der Klis, M. et al. 1997, ApJ, 481, L97
van der Klis, M. 1998, Nucl. Phys. B (Proc. Suppl.) 69/1–3(1998)103. \[Proc. of ÔÔThe Active X-ray SkyÕÕ, eds. L. Scarsi, H. Bradt, P. Giommi & F. Fiore, North-Holland\].
van der Klis, M. 1999, Proceedings of the Third William Fairbank, Rome, June 1998; astro-ph 9812395.
van Paradijs, J. 1995, in X-ray Binaries, eds. W. H. G. Lewin, J. van Paradijs & E. P. J. van den Heuvel (Cambridge: Cambridge Univ. Press), 536
Weaver, K. A., Krolik, J. H. & Pier, E. A. 1998, ApJ, 498, 213
Wijnands, R. & van der Klis, M. 1998, Nature, 394, 346
Zhang, S. N., Cui, W., Chen, W. 1997, ApJ, 482, L155
Zhang, W. et al. 1998a, ApJ, 495, L9
Zhang, W., Smale, A., Strohmayer, T. & Swank, J. 1998b, ApJ. 500, L171
Figure Captions {#figure-captions .unnumbered}
===============
Fig. 1. Sample of All-Sky Monitor light curves from Mar. 1996 to June 1998 showing, top to bottom, a microquasar, the flare star CI Cam, two black-hole binaries, probable disk precession in a neutron-star binary, and two faint BL Lac objects. The ordinate is count rate adjusted to the center of the field of view of a single ASM camera; the Crab nebula would yield $\sim$75 c/s. (A. Levine, pvt. comm.)
Fig. 2. Power density spectra of 4U 1728–34 in three intensity states. Low frequency QPO are evident at 20–40 Hz as are two peaks at $\sim$1 kHz which move to higher frequencies as the intensity increases. (From Strohmayer et al. 1996)
Fig. 3. Frequency of the two QPO’s at kHz frequencies in 4U 1820–30 as a function of intensity. The saturation suggests that the innermost stable orbit has been reached, but this conclusion has been questioned — see text. (Zhang et al. 1998)
Fig. 4. Constraints on mass and radius of neutron star in the non-rotating approximation. The highest frequencies detected limit the neutron star mass to $\sim$1.8 $M_{\odot}$. If rotation is taken into account, the limits will increase up to at most 2.2 $M_{\odot}$. (Miller, Lamb & Psaltis 1998)
Fig. 5. Dynamic power spectra of two bursts from 4U 1728–34 separated in time by 1.6 y. In each case, the frequency settles to a stable period at 364.0 Hz with frequencies that agree within 0.03 Hz. (From Strohmayer et al. 1998)
Fig. 6. Doppler curve for the 401–Hz pulsar discovered with RXTE. The maximum delay is 63 ms, and the binary orbital period is 2.01 h. (From Chakrabarty & Morgan 1998)
Fig. 7. RXTE/ASM light curve of GRS 1915+105 with hardness ratio (5–12)/(3–5) from Mar. 1996 through Sept. 1998. The marks at the top indicate the times of PCA pointings. (R. Remillard, pvt. comm.)
Fig. 8. Three types of variability of GRS 1915+105 in RXTE/PCA data (E. Morgan, pvt. comm.)
Fig. 9. Precursor outburst activity in GRO 1655–40. The source intensity is shown in the optical (BVRI bands) and in the delayed x-ray flux. The onset times are progressively later and later as the radiation band hardens. (Orosz et al. 1997)
Fig. 10. Large x-ray dip with spike with simultaneous radio and infrared flares in GRS 1915+105. Other x-ray dips of this type are shown in the bottom panel of Fig. 8. (Mirabel et al. 1998)
Fig. 11. X-ray character of GRS 1915+105 during and near a dip+spike event, from RXTE data. Top to bottom: x-ray light curve, inner disk temperature, inner-disk radius, photon index of power-law component, dynamic power spectrum. (Swank et al. 1998)
Fig. 12. Power density spectra of two microquasars showing the high frequency and apparently stable QPOs at 67 Hz and 300 Hz. (Morgan, et al. 1997, Remillard et al. 1999)
|
---
abstract: 'We show that Coecke’s compositionality theorem for quantum information flow follows by the universal property of tensor products from the case in which all relevant states are totally disentangled, for which the proof is almost trivial. With the same technique we deduce a PROP structure behind general multipartite quantum information processing and show that all such are equivalent to a canonical teleportation-type form. Some philosophical issues concerning quantum information are also touched upon.'
author:
- 'George Svetlichny[^1]'
title: 'Tensor Universality, Quantum Information Flow, Coecke’s Theorem, and Generalizations'
---
\#1[| \#1]{} \#1[\#1|]{} \#1\#2[.\#1|\#2]{} \#1
\[Fig.\#1\]
Introduction
============
Bob Coecke recently proved a remarkable theorem[@coecke1; @coecke1.5] concerning quantum information processing in multipartite entangled states under successive bipartite measurements. The [*description*]{} (no claim is made as to the [*reality*]{}) is a process taking place in stages with the information flow between some of the stages necessarily being backward in time. We show here that the theorem follows readily and easily from the universal property of the tensor product and the truth of the statement when all the relevant states are totally disentangled and the measurements are unipartite; that is, in the case when all the parts coexist independently and one would not say that there was any information flow between them. Besides giving a simple proof of the theorem, this allows us to easily deduce a series of results concerning quantum state processing. For general multipartite measurements the information flow is described by an algebraic system known as a PROP, a composition scheme of many-input-many-output maps. Any process of this type is also equivalent to one of a teleportation-type, provided classical information use can be ignored. Our method of proof raises interesting philosophical questions about the nature of quantum information, and we discuss these briefly. See [@coecke1.6; @coecke2; @abdu; @zhan] for recent literature on related themes.
Tensor Universality
===================
A multipartite quantum state-vector resides in a tensor product Hilbert space $\cH_1\cotimes\cH_n$ where each $\cH_i$ is the Hilbert space of states of the $i$-th part. States of the form $\ket{\phi_1}\ket{\phi_2}\cdots\ket{\phi_n}$ are called [*product*]{}, or [*disentangled*]{} states while all states that cannot be put into this form are called [*entangled*]{}. We recall the basic [*defining*]{} property of the tensor product. Let $V_1,\dots,V_n$ be vector spaces, their [*tensor product*]{} is a vector space usually denoted by $V_1\cotimes V_n$ along with an $n$-linear map $J:V_1\times \cdots \times V_n\to V_1\cotimes V_n$ such that [*any*]{} $n$-linear map $\alpha:V_1\times \cdots \times V_n\to W$ to yet another vector space $W$ factors [*uniquely*]{} though a [*linear*]{} map $\hat\alpha:V_1\otimes \cdots \otimes V_n\to W$, that is $\alpha=\hat\alpha\circ J$. In other words $J$ is a [*universal*]{} $n$-linear map and any other differs from it by a [*unique*]{} subsequent [*linear*]{} factor. One generally writes $v_1\cotimes v_n$ for $J(v_1,\dots,v_n)$.
This basic defining mathematical property, called [*universality*]{}, has at least two interesting consequences: (1) any linear construct on entangled states is uniquely determined by what it does on disentangled states; (2) any theorem that uses only linearity on entangled states is true if it is true on disentangled states. These facts can considerably simplify construct and proofs.
All of the above is also true if we systematically replace the word “linear" by “antilinear" (with $J$ still $n$-[*linear*]{}).
Bipartite processing
====================
Consider an $n$-partite state $\Phi$ and subject it to a sequence of measurements by observables $A_1,A_2,\dots,A_m$, where each $A_i$ is assumed to possibly act only on some of the parts, on which they are non-degenerate. We also assume that the time evolution betweens the measurements if trivial, that is the states do not change. Concretely each $\cH_i$ could be describing internal degrees of freedom (such as photon polarization) that are disentangled from the spatial degrees, the former not evolving between measurements, while the latter are. The resulting state is $$\label{ppphi}
\Psi=P_mP_{m-1}\cdots P_2P_1\Phi,$$ where each $P_i$ is some spectral projection (of rank $1$) of $A_i$. Of course with each new execution of the series of measurements, the spectral projections will in general be different and the outcome state also. The outcome of each measurement is classical information which may be available before subsequent measurements are carried out and be used to change that measurement, or otherwise subject the state to unitary transformations, but here we focus on just the state transformation indicated by (\[ppphi\]).
Since each projector has rank $1$ it is uniquely determined by a state in its range, say $\Omega$, and if it is normalized we have $P=(\Omega,\cdot)\Omega$, or in Dirac notation, $P=\ket\Omega\bra\Omega$. We shall be switching between notations for convenience and clarity’s sake.
Assume, for initial simplicity, that all observations are bipartite. Thus each $\Omega$ belongs to a subproduct Hilbert space $\cH_a\otimes\cH_b$. Given any Hilbert space $\cH$ let $\cH^*$ denote the dual space, that is if $\cH$ is the space of kets, $\ket\phi$, then $\cH^*$ is the space of the corresponding bras $\bra\phi$. Give any state $\Omega\in \cH\otimes \cK$ in the tensor product of two Hilbert spaces one can uniquely define by universality two linear maps $g_\Omega:\cH\to \cK^*$ and $f_\Omega:\cH^*\to \cK$ from the case that $\Omega=\alpha\otimes \beta$ is a product state. In this case we have $$\begin{aligned}
\label{gPsi}
g_\Omega: \ket \phi &\mapsto& \bracket\alpha\phi \bra\beta, \\ \label{fPsi}
f_\Omega: \bra \phi &\mapsto& \bracket\phi\alpha \ket\beta.\end{aligned}$$
Note that $g_\Omega$ is an antilinear function of $\Omega$ while $f_\Omega$ is a linear one. Coecke makes use of [*antilinear*]{} maps $G_\Omega:\cH^*\to \cK^*$ and $F_\Omega:\cH\to \cK$, the first depending antilinearly on $\Omega$ and the second linearly, and which are defined by $$\begin{aligned}
\label{CoeckeG}
G_\Omega \bra\phi &=& \bracket\alpha\phi\bra\beta \\ \label{CoeckeF}
F_\Omega \ket\phi &=& \bracket\phi\alpha\ket\beta.\end{aligned}$$
If we denote by a superscript dagger the Riesz correspondence $\phi^\dagger =\ket\phi^\dagger=\bra\phi=(\phi,\cdot)$ and $(\phi,\cdot)^\dagger=\bra\phi^\dagger= \ket\phi =\phi$ then we have $$g\ket\phi=F(\phi)^\dagger=G(\phi^\dagger)$$ $$f\bra\phi=F(\phi)=G(\phi^\dagger)^\dagger,$$ and if $\Lambda\in \cK\otimes \cL$ then a simple calculation shows $$\begin{aligned}
\label{fgisFF}
f_\Lambda\circ g_\Omega &=& F_\Lambda \circ F_\Omega, \\ \nonumber
g_\Lambda\circ f_\Omega &=& G_\Lambda \circ G_\Omega\end{aligned}$$
Since $\cH\otimes\cK\simeq \cK\otimes\cH$, given $\Omega\in\cH\otimes\cK$ there are also functions going in the opposite direction to the ones given by (\[gPsi\]), (\[fPsi\]) and (\[CoeckeF\]), thus there is $g_\Omega$-type function $\cK\to \cH^*$. In order to distinguish the two directions we shall use the superscript “$\hbox{op}$" whenever the order of the Hilbert spaces is taken opposite to the one written. Thus, for $\Omega=\alpha\otimes \beta$ one has $g^{\hbox{op}}_\Omega:\ket\phi\mapsto \bracket\beta\phi \bra\alpha$, and similarly for the others. The relation between the corresponding functions is a form of duality. Let $\phi\in \cH$, $\psi\in\cK$. One has (dropping the indices on the $g,\,f,\,G,\,F$ functions): $$\begin{aligned}
\bracket{g(\phi)}{\psi}&=&\bracket{g^{\hbox{op}}(\psi)}{\phi}, \\
\bracket{\phi}{f(\psi^\dagger)}&=&\bracket{\psi}{f^{\hbox{op}}(\phi^\dagger)},\\
\bracket{G(\phi^\dagger)}{\psi}&=&\bracket{G^{\hbox{op}}(\psi^\dagger)}{\phi}, \\
\bracket{\phi}{F(\psi)}&=&\bracket{\psi}{F^{\hbox{op}}(\phi)}.\end{aligned}$$
We now introduce a graphical description of the process of applying successive projections as in (\[ppphi\]). The tensor factors $\cH_i$ are rendered by vertical lines with the vertical direction, bottom to top, indicating increase of time:
(70,100)(0,0) (0,0)[(0,0)[$\cH_1$]{}]{} (25,0)[(0,0)[$\cH_2$]{}]{} (65,0)[(0,0)[$\cH_n$]{}]{} (0,18)[(0,1)[82]{}]{} (25,18)[(0,1)[82]{}]{} (65,18)[(0,1)[82]{}]{} (44,60)[(0,0)[…]{}]{}
We indicate each bipartite projection $P_i$ by a box intercepting the two tensor factor lines, which for pictorial simplicity we assume here to be contiguous. For example:
(110,100)(0,0) (0,0)[(0,1)[58]{}]{}(0,72)[(0,1)[28]{}]{} (25,0)[(0,1)[34]{}]{}(25,48)[(0,1)[10]{}]{}(25,72)[(0,1)[28]{}]{} (50,0)[(0,1)[34]{}]{}(50,48)[(0,1)[24]{}]{}(50,86)[(0,1)[14]{}]{} (75,0)[(0,1)[10]{}]{}(75,24)[(0,1)[48]{}]{}(75,86)[(0,1)[14]{}]{} (100,0)[(0,1)[10]{}]{}(100,24)[(0,1)[76]{}]{} (72,10)[(31,14)[$P_1$]{}]{} (22,34)[(31,14)[$P_2$]{}]{} (-3,58)[(31,14)[$P_3$]{}]{} (47,72)[(31,14)[$P_4$]{}]{}
Coeke’s theorem now says that if the initial state is of the form $$\phi^{\hbox{in}}_1\otimes \Phi^{\hbox{in}}_{2345}\in\cH_1\otimes(\cH_2\otimes\cH_3\otimes\cH_4\otimes\cH_5),$$ then the final state is $$\Phi^{\hbox{out}}_{1234}\otimes\phi^{\hbox{out}}_5\in (\cH_1\otimes\cH_2\otimes\cH_3\otimes\cH_4)\otimes\cH_5,$$ where $$\label{process}
\phi^{\hbox{out}}_4=F_1\circ F_4\circ F_2\circ F_3(\phi^{\hbox{in}}_1),$$ and where each $F_i$ is Coeke’s function (\[CoeckeF\]) in relation to the normalized bipartite state defining each projector $P_i=\ket{\Omega_i}\bra{\Omega_i}$. Note that the “processing order" implied in (\[process\]) is [*not*]{} the temporal order of the sequence of actual measurements applied to the initial states that produced the final state. In particular the [*first*]{} projector in time, $P_1$, is the [*last*]{} to process according to (\[process\]). It is as though information has to travel backwards in time to be able to process the state $\phi^{\hbox{in}}_1$. There is some flexibility though in the temporal order. Since operators acting on disjoint subfactors of a tensor product commute, one could have used different temporal order of measurements to get identical outcomes. In pictorial terms this means that one can slide each box up and down as though the vertical lines were rails, provided if two boxes meet on a common rail they cannot pass each other. Thus $P_3$ and $P_4$ must always be later than $P_2$, and $P_4$ must always be later than $P_1$ but any temporal order that respects these conditions is allowed. A change of temporal order will of course change the times at which classical information concerning the outcomes of measurements becomes available. The order in (\[process\]) is independent of the allowed temporal orders as it only depends on the mentioned constraints.
To understand the processing order we define what we call a [*path*]{} in the diagram. This is an oriented path following the vertical lines and across the boxes which starts at the bottom (top) of the diagram at one of the vertical lines going upward (downward) and then continues through the diagram with the proviso that if it encounters a box, it must cross over to the other line entering the box and then follow it in the reverse direction to the previous one, stopping finally at either the top or bottom of the diagram. In relation to our example we have the path that starts at the bottom on the $\cH_1$ line.
(110,100)(0,0) (0,0)[(0,1)[58]{}]{}(0,72)[(0,1)[28]{}]{} (25,0)[(0,1)[34]{}]{}(25,48)[(0,1)[10]{}]{}(25,72)[(0,1)[28]{}]{} (50,0)[(0,1)[34]{}]{}(50,48)[(0,1)[24]{}]{}(50,86)[(0,1)[14]{}]{} (75,0)[(0,1)[10]{}]{}(75,24)[(0,1)[48]{}]{}(75,86)[(0,1)[14]{}]{} (100,0)[(0,1)[10]{}]{}(100,24)[(0,1)[76]{}]{} (72,10)[(31,14)]{} (22,34)[(31,14)]{} (-3,58)[(31,14)]{} (47,72)[(31,14)]{} (0,29)[(0,1)[0]{}]{} (15,58)[(1,0)[0]{}]{} (25,50)[(0,-1)[0]{}]{} (40,48)[(1,0)[0]{}]{} (50,60)[(0,1)[0]{}]{} (75,48)[(0,-1)[0]{}]{} (65,72)[(1,0)[0]{}]{} (90,24)[(1,0)[0]{}]{} (100,62)[(0,1)[0]{}]{}
The processing order is now precisely the order by which the path encounters the boxes corresponding to the projections. Coeke’s theorem is that this statement is true for any arrangement of bipartite projections on a Hilbert space of any number of tensor factors. We call attention to the fact that the function $F$ has to be computed considering the order of the tensor factors $\cH_a\otimes \cH_b$ as being that given by the orientation of the path going through the box.
Because of (\[fgisFF\]) we can write (\[process\]) equally as $$\phi^{\hbox{out}}_4=f_1\circ g_4\circ f_2\circ g_3(\phi^{\hbox{in}}_1).$$
In this version, the initial ket $\ket{\phi^{\hbox{in}}_1}$ is transformed into a bra $\bra{g_3(\phi^{\hbox{in}}_1)}$ travelling on the downward leg, then again into a ket, and so on. Thus metaphorically one has kets travelling forward in time and bras backward:
(90,75)(0,0) (0,0)[(0,1)[75]{}]{} (75,0)[(0,1)[75]{}]{} (0,38)[(0,1)[0]{}]{} (75,34)[(0,-1)[0]{}]{} (5,32)[$\ket\phi$]{} (80,32)[$\bra\psi$]{}
Proof of Coecke’s theorem {#section.proof}
=========================
The problem in trying to prove Coecke’s theorem by tensor universality is that even if $\Omega$ is not normalized, the operator $\ket\Omega\bra\Omega$ depends quadratically on $\Omega$. We circumvent this by polarization and consider a general rank one operator $$Q_{\Lambda,\Omega}=(\Omega,\cdot)\Lambda =\ket\Lambda\bra\Omega.$$
With normalized vectors such an operator can be written as $U\ket\Lambda\bra\Lambda$ or $\ket\Omega\bra\Omega V$ where $U$ and $V$ are unitary. Thus these rank one operators are physically realizable by intercalating unitary transformations between the measurements. Once we prove Coecke’s theorem for general rank-one operators, which we shall call the [*polarized*]{} Coecke’s therem, we will automatically have a proof for the version of the theorem in which unipartite unitaries are also placed on the vertical lines of the diagram between the projection boxes, as is necessary for instance for teleportation. The action of these in state processing is indicated in the following diagram:
(100,90)(0,0) (0,0)[(0,1)[40]{}]{}(0,50)[(0,1)[40]{}]{} (75,0)[(0,1)[40]{}]{}(75,50)[(0,1)[40]{}]{} (-5,40)[(10,10)[$U$]{}]{} (70,40)[(10,10)[$V$]{}]{} (0,22)[(0,1)[0]{}]{} (0,72)[(0,1)[0]{}]{} (5,18)[$\ket\phi$]{} (5,68)[$U\ket\phi$]{} (75,20)[(0,-1)[0]{}]{} (75,68)[(0,-1)[0]{}]{} (80,20)[$\bra\psi V$]{} (80,68)[$\bra\psi$]{}
Given a tensor product Hilbert space $\cH\otimes \cK$ and a vector $\Omega\in \cH$ we define by universality the [*partial inner product*]{}, or [*contraction*]{} $\Omega\rfloor\cdot:\cH\otimes \cK\to \cH$ as $$\Omega\rfloor \alpha\otimes \beta=(\Omega,\alpha)\beta.$$
The action of $Q_{\Lambda,\Omega}$ can now be written as $$Q_{\Lambda,\Omega}\Phi = \Lambda\otimes \Omega\rfloor \Phi.$$ where it must be understood that the contraction and the tensor product is in relation to that subfactor (assumed bipartite for now) of the full tensor product upon which the operator $Q$ acts.
Now instead of (\[ppphi\]) we now consider $$\label{qqphi}
\Psi=Q_mQ_{m-1}\cdots Q_2Q_1\Phi$$ where we have $Q_j=Q_{\Lambda_j,\Omega_j}$. Now we see that $\Psi$ depends linearly on $\Phi$ and each $\Lambda_j$, and antilinearly on each $\Omega_j$. We can now prove a polarized version of Coecke’s theorem by universality by showing it is true when all the above mentioned vectors are product vectors. In this case the theorem is almost trivial. Coecke’s theorem, with or without unipartite unitaries, will then follow by specializing the operators $Q$. For convenient future reference we rewrite (\[qqphi\]) as $$\Psi=\Lambda_m\otimes \Omega_m\rfloor\,\Lambda_{m-1}\otimes \Omega_{m-1}\rfloor \cdots \Lambda_2\otimes \Omega_2\rfloor\,\Lambda_1\otimes \Omega_1\rfloor\Phi$$ which explicitly exhibits the dependence of $\Psi$ on all of the relevant vectors.
We first illustrate the argument by a simple case treated diagrammatically. A box representing an operator $Q=\Lambda\otimes\Omega\rfloor\cdot$ will be one split in the middle horizontally with $\Lambda$ on top and $\Omega$ on bottom:
(31,28) (0,0)[(31,14)[$\Omega$]{}]{} (0,14)[(31,14)[$\Lambda$]{}]{}
We shall call the upper half of a $Q$-box a $\Lambda$-box, and the lower half an $\Omega$-box, even if not labelled by these letters.
Consider now the following diagram for the process $\Psi=Q_3Q_2Q_1\Phi$, which in long-hand is $$\label{3olphi}
\Psi=\Lambda_3\otimes \Omega_3\rfloor\, \Lambda_2\otimes \Omega_2\rfloor\,\Lambda_1\otimes \Omega_1\rfloor\Phi:$$
(53,144) (0,0)[(0,1)[58]{}]{}(0,86)[(0,1)[58]{}]{} (25,0)[(0,1)[10]{}]{}(25,38)[(0,1)[20]{}]{} (25,86)[(0,1)[20]{}]{}(25,134)[(0,1)[10]{}]{} (50,0)[(0,1)[10]{}]{}(50,38)[(0,1)[68]{}]{}(50,134)[(0,1)[10]{}]{} (22,10)[(31,14)[$\Omega_1$]{}]{}(22,24)[(31,14)[$\Lambda_1$]{}]{} (-3,58)[(31,14)[$\Omega_2$]{}]{}(-3,72)[(31,14)[$\Lambda_2$]{}]{} (22,106)[(31,14)[$\Omega_3$]{}]{}(22,120)[(31,14)[$\Lambda_3$]{}]{} (0,29)[(0,1)[0]{}]{} (15,58)[(1,0)[0]{}]{} (25,48)[(0,-1)[0]{}]{} (40,38)[(1,0)[0]{}]{} (50,77)[(0,1)[0]{}]{} (36,106)[(-1,0)[0]{}]{} (25,96)[(0,-1)[0]{}]{} (12,86)[(-1,0)[0]{}]{} (0,115)[(0,1)[0]{}]{}
and where we have already indicated a path that is of interest to us.
Now the right-hand side of (\[3olphi\]) depends linearly on $\Phi$ and the $\Lambda_j$ and anti-linearly on the $\Omega_j$ so we can deduce the result by tensor universality from the case when all of these vectors are completely disentangled. Hence assume: $$\begin{aligned}
\Phi&=&\phi_1\otimes\phi_2\otimes\phi_3,\\
\Lambda_j&=&\mu_j\otimes\nu_j,\\
\Omega_j&=&\sigma_j\otimes\tau_j.\end{aligned}$$
Diagrammatically the situation now looks as follows:
(53,144) (0,0)[(0,1)[58]{}]{}(0,86)[(0,1)[58]{}]{} (25,0)[(0,1)[10]{}]{}(25,38)[(0,1)[20]{}]{} (25,86)[(0,1)[20]{}]{}(25,134)[(0,1)[10]{}]{} (50,0)[(0,1)[10]{}]{}(50,38)[(0,1)[68]{}]{}(50,134)[(0,1)[10]{}]{} (18,10)[(14,14)[$\sigma_1$]{}]{}(43,10)[(14,14)[$\tau_1$]{}]{} (18,24)[(14,14)[$\mu_1$]{}]{}(43,24)[(14,14)[$\nu_1$]{}]{} (-7,58)[(14,14)[$\sigma_2$]{}]{}(18,58)[(14,14)[$\tau_2$]{}]{} (-7,72)[(14,14)[$\mu_2$]{}]{}(18,72)[(14,14)[$\nu_2$]{}]{} (18,106)[(14,14)[$\sigma_3$]{}]{}(43,106)[(14,14)[$\tau_3$]{}]{} (18,120)[(14,14)[$\mu_3$]{}]{}(43,120)[(14,14)[$\nu_3$]{}]{}
All the rank-one operators are unipartite and what we have are three completely independent quantum processes (taking place, say, on Mars, Earth, and Venus). The outcome state of course is: $$\begin{aligned}
(\sigma_2,\phi_1)\mu_2 \otimes &&\\
(\sigma_1,\phi_2)(\tau_2,\mu_1)(\sigma_3,\nu_2)\mu_3\otimes && \\
(\tau_1,\phi_3)(\tau_3,\nu_1)\nu_3.\end{aligned}$$
Now the various inner products that appear in each of the tensor factors are just complex numbers and so can be passed to any other tensor factor. We rewrite the outcome state now as: $$\begin{aligned}
\nonumber
(\sigma_2,\phi_1)(\tau_2,\mu_1)(\tau_3,\nu_1)(\sigma_3,\nu_2)\mu_2 \otimes &&\\ \label{disgood}
\mu_3\otimes \nu_3(\sigma_1,\phi_2)(\tau_1,\phi_3),\end{aligned}$$ where we have placed on the first line of (\[disgood\]) the inner products that come form the vertical segments of the indicated path, where metaphorically an upward moving ket meets a downward moving bra and forms an inner product. A simple exercise shows that (\[disgood\]) can be written as $$\label{golphi}
f_{\Lambda_2}^{\hbox{op}}\circ g_{\Omega_3}^{\hbox{op}} \circ f_{\Lambda_1}\circ g_{\Omega_2}(\phi_1)\otimes (\Lambda_3\otimes \Omega_1\rfloor \Phi_{23}),$$ where $\Phi_{23}=\phi_2\otimes\phi_3$.
Now assuming that the initial state is of the form $\Phi=\phi_1\otimes \Phi_{23}$, then (\[golphi\]) and the right-hand side of (\[3olphi\]) coincide when $\Phi_{23}$, the $\Lambda_j$, and the $\Omega_j$ are all product states. On the other hand (\[golphi\]) makes perfect sense even if these states are entangled and it depends linearly or antilinearly on these states. By tensor universality therefore the two expressions are [*always*]{} equal and define the same output state $\Psi$. Notice that the way the state $\phi_1$ is processed in the first factor in (\[golphi\]) is precisely according to the boxes traversed by the path.
We see from this example that the polarized Coecke’s theorem depends only on tensor universality and the fact that scalar multiples on tensor factors can be moved freely to other factors.
While the above example makes the truth of Coecke’s theorem almost convincing, a further elaboration will make it obvious. Disregarding state-vector normalization, the rank-one operator $Q=\Lambda\otimes \Omega\rfloor \cdot$ is realizable by some physical procedure that transforms a given state $\Phi$ into $\Lambda\otimes (\Omega\rfloor\Phi)$. Now $\Phi$ is some multipartite state and $Q$ acts only on some subproduct of the full tensor product Hilbert space. Note that after the action of $Q$ the output state is a factor in which the state $\Lambda$ coexists disentangled with the state $(\Omega\rfloor\Phi)$ which belongs to the complementary subproduct. Given this, there is another physical procedure to produce the same transformed state. We first perform the projection $\ket\Omega\bra\Omega$ whose result is $\Omega\otimes (\Omega\rfloor\Phi)$. This represents two coexisting independent systems. We now [*destroy*]{} the parts that correspond to state $\Omega$ leaving us just with $(\Omega\rfloor\Phi)$. We now by an independent physical process prepare the state $\Lambda$ which becomes coexistent with the state $(\Omega\rfloor\Phi)$ and the resulting state is again $\Lambda\otimes (\Omega\rfloor\Phi)$. The above procedure carried out for each $Q$-box of course succeeds only with a certain probability, but when it does, it produces the same output state $\Psi$. Since such destroy-and-create processes, or equivalently, state transformations by [*substitution*]{}, can always be carried out, they must necessarily be incorporated into any formalization of scientific activity[@svet1; @svet2].
We can now think of the $Q$-box as composed of two separate and physically independent parts and the new box looks like:
(31,28) (0,0)[(31,12)[$\Omega$]{}]{} (0,18)[(31,12)[$\Lambda$]{}]{}
Doing this systematically in a diagram separates the diagram into disconnected pieces. For our example this becomes:
(53,144) (0,0)[(0,1)[58]{}]{}(0,86)[(0,1)[58]{}]{} (25,0)[(0,1)[10]{}]{}(25,38)[(0,1)[20]{}]{} (25,86)[(0,1)[20]{}]{}(25,134)[(0,1)[10]{}]{} (50,0)[(0,1)[10]{}]{}(50,38)[(0,1)[68]{}]{}(50,134)[(0,1)[10]{}]{} (22,10)[(31,12)[$\Omega_1$]{}]{}(22,26)[(31,12)[$\Lambda_1$]{}]{} (-3,58)[(31,12)[$\Omega_2$]{}]{}(-3,74)[(31,12)[$\Lambda_2$]{}]{} (22,106)[(31,12)[$\Omega_3$]{}]{}(22,122)[(31,12)[$\Lambda_3$]{}]{}
where we readily recognize the three parts of (\[disgood\]), the state processor acting on the first tensor factor, the operation $\Omega_1\rfloor\cdot$ and the operation $\Lambda_3\otimes\cdot$.
Consider now a general diagram with bipartite $Q$ operators representing the process (\[qqphi\]). Separating the $Q$-boxes into their two parts, the diagram decomposes into a set of connected components. If there is a path that goes from the bottom to the top, then it comprises one such component and passes successively through a number of pairs of a lower ($\Omega$) part of one $Q$-box followed by an upper ($\Lambda$) part of another $Q$-box. We now introduce some notation and conventions. Number these states as $\Omega_j$ and $\Lambda_j $, $j=1,2,\dots,k$ [*in the order that the path traverses the corresponding boxes*]{} and not in any temporal order. Assume $\Omega_j\in \cH_{\alpha(j)}\otimes \cH_{\alpha'(j)}$ and $\Lambda_j\in\cH_{\beta(j)}\otimes \cH_{\beta'(j)}$ where the order of tensor factors here coincide with the order that the path passes through the relevant box. One has $\beta(j)=\alpha'(j)$ where the path goes downward and $\beta'(j)=\alpha(j+1)$ where the path goes upward. The following diagram illustrates this:
(212,154) (0,20)[(0,1)[76]{}]{}(0,144)[(0,1)[10]{}]{} (50,20)[(0,1)[10]{}]{}(50,78)[(0,1)[18]{}]{}(50,144)[(0,1)[10]{}]{} (100,20)[(0,1)[10]{}]{}(100,78)[(0,1)[18]{}]{}(100,144)[(0,1)[10]{}]{} (150,20)[(0,1)[10]{}]{}(150,78)[(0,1)[18]{}]{}(150,144)[(0,1)[10]{}]{} (200,20)[(0,1)[10]{}]{}(200,78)[(0,1)[76]{}]{} (-6,96)[(62,22)[$\Omega_j$]{}]{}(-6,122)[(62,22)[$\Lambda$]{}]{} (42,30)[(62,22)[$\Omega$]{}]{}(42,56)[(62,22)[$\Lambda_j$]{}]{} (92,96)[(62,22)[$\Omega_{j+1}$]{}]{}(92,122)[(62,22)[$\Lambda'$]{}]{} (142,30)[(62,22)[$\Omega'$]{}]{}(142,56)[(62,22)[$\Lambda_{j+1}$]{}]{} (0,68)[(0,1)[0]{}]{} (26,96)[(1,0)[0]{}]{} (50,84)[(0,-1)[0]{}]{} (76,78)[(1,0)[0]{}]{} (100,88)[(0,1)[0]{}]{} (126,96)[(1,0)[0]{}]{} (150,84)[(0,-1)[0]{}]{} (176,78)[(1,0)[0]{}]{} (200,128)[(0,1)[0]{}]{} (0,10)[(0,0)[$\cH_{\alpha(j)}$]{}]{} (50,10)[(0,0)[$\cH_{\alpha'(j)}$]{}]{} (50,0)[(0,0)[$\cH_{\beta(j)}$]{}]{} (100,10)[(0,0)[$\cH_{\alpha(j+1)}$]{}]{} (100,0)[(0,0)[$\cH_{\beta'(j)}$]{}]{} (150,10)[(0,0)[$\cH_{\alpha'(j+1)}$]{}]{} (150,0)[(0,0)[$\cH_{\beta(j+1)}$]{}]{} (200,0)[(0,0)[$\cH_{\beta'(j+1)}$]{}]{}
where the unindexed $\Lambda$ and $\Omega$ vectors belong to boxes that may or may not belong to the path for other index values (see Fig. \[Fig.10\] where such a box does belong). Thus the Hilbert spaces labelled by the $\alpha,\,\alpha',\,\beta,\,\beta'$ functions may not all be distinct, as can be seen again from Fig. \[Fig.10\] (for which $\cH_{\alpha(1)}=\cH_{\beta'(2)}$). Also the temporal order here may not be faithfully displayed except for that between two successively indexed boxes. None of these observations however have any bearing on the immediate argument.
Assume now that all the $\Omega$ and $\Lambda$ states associated to this path are disentangled, say $\Omega_j=\omega_j\otimes\tilde\omega_j$ and $\Lambda_j=\lambda_j\otimes\tilde\lambda_j$. Assume also that the $\alpha(1)$ part of the initial state is disentangled from the rest; that is, $\Phi=\phi\otimes\Phi'$ where $\phi\in\cH_{\alpha(1)}$ and $\Phi'$ belongs to the complementary subproduct. In the expression for the final state (\[qqphi\]) each tensor factor corresponding to $\cH_{\beta(j)}$ receives a scalar factor $(\tilde \omega_j,\lambda_j)$ coming from a downward part of the path, and for $j\neq1,\,k$ each tensor factor corresponding to $\cH_{\beta'(j)}$ a scalar factor $(\omega_{j+1},\tilde \lambda_j)$ coming from an upward part of the path. For $j=1$ we have a scalar factor $(\omega_1,\phi)$ and for $j=k$ we have an output state tensor factor $\tilde\lambda_k$ which is disentangled from the rest of the output state; that is, $\Psi=\tilde\lambda_k\otimes \Psi'$. Now the scalar factors can be moved freely among the tensor factors, so let us move all of them to be multiplying the output part $\tilde\lambda_k$. Let $M$ denote the product of all these scalar factor and so $\Psi=(M\tilde\lambda_k)\otimes \tilde\Psi$. In this form $\tilde\Psi$ is independent of the vectors $\Omega_j$ and $\Lambda_j$ as all their contribution was passed on to the (disentangled) output part in $\cH_{\beta'(k)}$. We have explicitly $$M=(\omega_1,\phi)(\tilde \omega_1,\lambda_1)(\omega_2,\tilde \lambda_1)\cdots
(\tilde \omega_{k-1},\lambda_{k-1})(\omega_k,\tilde \lambda_k).$$
A simple verification now shows that $$M\tilde\lambda_k = f_{\Lambda_k}\circ g_{\Omega_k}\circ \cdots\circ f_{\Lambda_1}\circ g_{\Omega_1}(\phi),$$ and the polarized version of Coecke’s theorem follows from tensor universality.
The above proof illustrates the general strategy: In the totally disentangled case, we associate the inner products produced by the vertical line segments of a connected part of the diagram (originally constructed for the entangled case) to the tensor product of the unipartite states associated to the output lines of this connected part. The resulting product of the number and the product state is then reinterpreted as a construct which depends linearly on all the $\Lambda$ states and antilinearly on all the $\Omega$ states of the corresponding boxes of this part, and also linearly on the state represented by the input lines of this part. Tensor universality then implies that this construct is correct in all cases. In section \[section.multi\] we use this method to deduce a generalization of Coecke’s theorem for general multipartite processing and show that any such can be placed in and equivalent form as a single $f\circ g$ composition.
Whence the flow?
================
The above exposition raises some questions concerning the nature of so called “quantum information flow". To make these clear consider the case where $\cH_j=\lC^2$ is the space of qubits. There are (unnormalized) bipartite states $\Theta$ for which $f_\Theta \circ g_\Theta = \hbox{Id}$. Explicitly if $\ket0,\,\ket1$ is any qubit basis, one can take $\Theta=\ket0\otimes\ket0+\ket1\otimes\ket1$. Consider now the following diagram
(53,96) (0,0)[(0,1)[58]{}]{}(0,86)[(0,1)[10]{}]{} (25,0)[(0,1)[10]{}]{}(25,38)[(0,1)[20]{}]{} (25,86)[(0,1)[10]{}]{} (50,0)[(0,1)[10]{}]{}(50,38)[(0,1)[58]{}]{} (22,10)[(31,12)]{}(22,26)[(31,12)[$\Theta$]{}]{} (-3,58)[(31,12)[$\Theta$]{}]{}(-3,74)[(31,12)]{}
By Coecke’s theorem this teleports a state from $\cH_1$ to $\cH_3$ provided the rank one operators don’t annihilate the state. In a true teleportation setup the right $\Theta$ represents a source of entangled qubits and we can disregard the lower part of this $Q$-box. The left $Q$-box would be one of four spectral projections of a non-degenerate observable. The classical information (two c-bit’s worth) this provides (which outcome is realized) is then transmitted to an agent that receives the $\cH_3$ output state, who then subjects the output state to a unitary transformation depending on the c-bits received and teleportation is achieved in all cases.
That teleportation is possible can be [*deduced*]{} by tensor universality from the fully disentangles situation:
(53,96) (0,0)[(0,1)[58]{}]{}(0,86)[(0,1)[10]{}]{} (25,0)[(0,1)[10]{}]{}(25,38)[(0,1)[20]{}]{} (25,86)[(0,1)[10]{}]{} (50,0)[(0,1)[10]{}]{}(50,38)[(0,1)[58]{}]{} (18,10)[(14,12)]{}(43,10)[(14,12)]{} (18,26)[(14,12)[$\mu$]{}]{}(43,26)[(14,12)[$\nu$]{}]{} (-7,58)[(14,12)[$\sigma$]{}]{}(18,58)[(14,12)[$\tau$]{}]{} (-7,74)[(14,12)]{}(18,74)[(14,12)]{}
where we’ve indicated only the elements needed in the deduction. It is truly remarkable that three independent quantum processes, on Venus, Earth, and Mars, say, provide enough information to deduce that one can teleport a photon polarization state form Rio de Janeiro to Kiev provided one has good enough optical fibers and a source of entangles photons in, say, the Canary Islands.
One is accustomed to hear that entanglement provides a channel for quantum information flow and this is understandable. After all, in teleportation a state in one location is duplicated at another, so [*something*]{}, it seems, must have passed from one place to another. This something has been dubbed “quantum information". Since a qubit contains infinite classical information being determined by a point on a two-dimensional manifold (a sphere), what passes from one location to the other, it seems, must be more than just the two c-bits of classical information. Examine now the fully disentangled case. One is inclined to say that there is no information flow between the three processes. To deduce the possibility of teleportation however one must arrange the results of the experiments as to [*mimic*]{} such a flow. In the disentangled case one has in $\cH_3$ the output state $\nu$ which for some perverse reason we now write as $(\sigma,\phi)(\tau,\mu)\nu$. Nobody should object as we’ve always been told a multiple of a state vector still defines the same state and so scalar factors don’t matter. Still more perversely we now write this as $$f_{\mu\otimes\nu}\circ g_{\sigma\otimes\tau}(\phi)=(\sigma,\phi)(\tau,\mu)\nu$$ and compare it to $$f_\Theta\circ g_\Theta(\phi)=\phi$$ in the entangled teleportation case. Both expressions seem to describe state processing (quantum information flow). One could in the first case contend that the processing is fictitious, or virtual, and in the second case real, but then, given tensor universality, one would have to maintain that fiction (or virtuality) logically implies reality. Defending this position would be an interesting philosophical challenge. A more balanced view would be to assert a common ontological grounding in both cases. If so, the words “flow" and “channel" would have to be deemed inadequate to describe this reality. A possible better notion could be “availability" and we could say: [*entangled systems define conditions for quantum state availability, subject to possession of classical information*]{}. In teleportation, due to the non-local correlations in the entangles states, availability is also correlated non-locally, thus the two c-bits of classical information transmitted by a conventional channel is just an instruction as to how to bring out an available state that was also available at a far location. No need to conceptualize a “flow of quantum information" in this view, and this phrase becomes a mere metaphor that helps us perceive certain mathematical relations and design experiments. With another imprecise picture, one could conceptualize the source of entangled photons as “broadcasting quantum information" (correlated availability) and the two agents involved in “teleportation" as making clever use of this broadcast. In the disentangled case the only available states in $\cH_3$ are multiples of the locally prepared state $\nu$, which is exactly what the first expression says. Our alternative implication is then [*uncorrelated availability logically implies correlated availability*]{}. Not too bad, though still a bit mysterious. One should not forget that these implications take place in a certain context, the essential aspects of which are: (1) coexistence of quantum systems is described by the tensor product of Hilbert spaces and (2) quantum mechanical processes (evolution and projection) are [*linear*]{}. Deviations from these aspects would seriously compromise all of what was discussed. With nonlinear quantum mechanics for instance, entangled systems become true causal channels, raising the by now familiar issues of relativistic causality.[@svetNLQG] There are clearly very interesting philosophical issued to be explored here and we shall say more in the last section.
Multipartite processing {#section.multi}
=======================
In a sense multipartite processing is very similar to bipartite, involving a more complicated composition of the $f$ and $g$ maps. An analysis of this structure will also allow us to prove that any processing is equivalent to a single appropriately defined $f_\Lambda\circ g_\Omega$ composition. To introduce the argument we start with an example. Consider the following diagram, already with separated $Q$-boxes:
(150,192) (0,0)[(0,1)[58]{}]{}(0,86)[(0,1)[106]{}]{} (25,0)[(0,1)[58]{}]{}(25,86)[(0,1)[20]{}]{}(25,134)[(0,1)[58]{}]{} (50,0)[(0,1)[10]{}]{}(50,38)[(0,1)[20]{}]{}(50,86)[(0,1)[20]{}]{}(50,134)[(0,1)[58]{}]{} (75,0)[(0,1)[10]{}]{}(75,38)[(0,1)[68]{}]{}(75,134)[(0,1)[20]{}]{}(75,182)[(0,1)[10]{}]{} (100,0)[(0,1)[10]{}]{}(100,38)[(0,1)[20]{}]{}(100,86)[(0,1)[68]{}]{}(100,182)[(0,1)[10]{}]{} (150,0)[(0,1)[58]{}]{}(150,86)[(0,1)[106]{}]{} (125,0)[(0,1)[10]{}]{}(125,38)[(0,1)[18]{}]{}(125,71)[(0,1)[2]{}]{}(125,88)[(0,1)[104]{}]{} (-3,58)[(56,12)[$\Omega_1$]{}]{}(-3,74)[(56,12)[$\Lambda_1$]{}]{} (22,106)[(56,12)[$\Omega_2$]{}]{}(22,122)[(56,12)[$\Lambda_2$]{}]{} (47,10)[(81,12)[$\Omega_3$]{}]{}(47,26)[(81,12)[$\Lambda_3$]{}]{} (72,154)[(31,12)[$\Omega_4$]{}]{}(72,170)[(31,12)[$\Lambda_4$]{}]{} (97,58)[(56,12)[$\Omega_5$]{}]{}(97,74)[(56,12)[$\Lambda_5$]{}]{}
where $\Omega_5$ and $\Lambda_5$ are bipartite acting only on $\cH_5$ and $\cH_7$
This diagram has four connected parts, one with both input and output lines and should be considered a state processor, two with only output lines which should be considered creators of two disentangled output states in two separate tensor subproducts, and one with only input lines whose contribution to the final state is a scalar factor. Let us examine the processor separately, where we have put in arrows on the vertical line segments to clarify the subsequent discussion:
(150,140)(0,-10) (0,0)[(0,1)[58]{}]{}(0,29)[(0,1)[0]{}]{}(0,86)[(0,1)[44]{}]{}(0,108)[(0,1)[0]{}]{} (25,0)[(0,1)[58]{}]{}(25,29)[(0,1)[0]{}]{}(25,86)[(0,1)[20]{}]{}(25,94)[(0,-1)[0]{}]{} (50,38)[(0,1)[20]{}]{}(50,46)[(0,-1)[0]{}]{}(50,86)[(0,1)[20]{}]{}(50,94)[(0,-1)[0]{}]{} (75,38)[(0,1)[68]{}]{}(75,72)[(0,1)[0]{}]{} (100,38)[(0,1)[20]{}]{}(100,46)[(0,-1)[0]{}]{} (125,38)[(0,1)[18]{}]{}(125,72)[(0,1)[58]{}]{}(125,108)[(0,1)[0]{}]{} (150,0)[(0,1)[58]{}]{}(150,29)[(0,1)[0]{}]{} (-3,58)[(56,12)[$\Omega_1$]{}]{}(-3,74)[(56,12)[$\Lambda_1$]{}]{} (22,106)[(56,12)[$\Omega_2$]{}]{} (47,26)[(81,12)[$\Lambda_3$]{}]{} (97,58)[(56,12)[$\Omega_5$]{}]{} (0,-10)[(0,0)[$\cH_1$]{}]{} (25,-10)[(0,0)[$\cH_2$]{}]{} (50,-10)[(0,0)[$\cH_3$]{}]{} (75,-10)[(0,0)[$\cH_4$]{}]{} (100,-10)[(0,0)[$\cH_5$]{}]{} (125,-10)[(0,0)[$\cH_6$]{}]{} (150,-10)[(0,0)[$\cH_7$]{}]{}
This obviously represents a state transformation $\cH_1\otimes\cH_2\otimes\cH_7\to \cH_1\otimes\cH_6$. Now each $\Omega$ and $\Lambda$ box has both incoming and outgoing lines. Denote the tensor product of the Hilbert spaces of the incoming lines as $\cH$ and that of outgoing lines as $\cK$. The box can now be considered bipartite with incoming $\cH$ line and outgoing $\cK$ lines. Associated to this bipartite box is then an $f$ or $g$ function. Denote the set of incoming lines by $S$ and the complementary set of outgoing lines by $S'$. We indicate the identifications by placing the superscript $S'\leftarrow S$ on the corresponding function. Thus we have functions $g^{3\leftarrow12}_{\Omega_1}$ and $f^{46\leftarrow35}_{\Lambda_3}$, etc. in our diagram above. (We abbreviate $\{1,2\}$ to simply $12$ etc.; as there are less than ten lines, there’s no ambiguity.) One now associates to the whole diagram the composition: $$\label{multicomp}
L=\left( f^{1\leftarrow23}_{\Lambda_1}\otimes I^{6\leftarrow 6}\right)\circ \left(g^{23\leftarrow4}_{\Omega_2} \otimes I^{6\leftarrow 6}\right) \circ f^{46\leftarrow35}_{\Lambda_3}
\circ \left( g^{3\leftarrow12}_{\Omega_1}\otimes g^{5\leftarrow7}_{\Omega_5}\right),$$ and the statement is that if the initial state of of the form $\phi_{127}\otimes\Phi_{3456}$, then the final state is of the form $L(\phi_{127})\otimes\Psi_{23457}$ where the indices indicate to which subproduct the vector belongs to.
To understand (\[multicomp\]) better we redraw the diagram in processing rather than temporal order, resulting in something like a Feynman diagram for particle interactions:
(110,160) (25,30)(25,30)[(0,0)[$\Omega_1$]{}]{} (50,65)(50,65)[(0,0)[$\Lambda_3$]{}]{} (50,100)(50,100)[(0,0)[$\Omega_2$]{}]{} (50,135)(50,135)[(0,0)[$\Lambda_1$]{}]{} (75,30)(75,30)[(0,0)[$\Omega_5$]{}]{} (10,0)[(1,2)[10]{}]{}(40,0)[(-1,2)[10]{}]{}(75,0)[(0,1)[20]{}]{} (29,39)[(1,1)[16.6]{}]{}(69,39)[(-1,1)[16.6]{}]{} (50,75)[(0,1)[15]{}]{}(45,108.5)[(0,1)[17.5]{}]{}(55,108.5)[(0,1)[17.5]{}]{}(50,145)[(0,1)[15]{}]{} (54,74)[(2,3)[57]{}]{} (10,10)[(0,0)[$1$]{}]{}(40,11)[(0,0)[$2$]{}]{}(80,10)[(0,0)[$7$]{}]{} (35,50)[(0,0)[$3^*$]{}]{}(70,50)[(0,0)[$5^*$]{}]{} (80,120)[(0,0)[$6$]{}]{} (45,82)[(0,0)[$4$]{}]{}(45,150)[(0,0)[$1$]{}]{} (40,120)[(0,0)[$2^*$]{}]{}(62,120)[(0,0)[$3^*$]{}]{}
Here the numbers labelling the lines indicate the Hilbert space involved and we’ve put asterisks on those lines that correspond to downward arrows, i.e., to the metaphorical flow backward in time. It is now easy to read off the terms in composition (\[multicomp\]).
We now argue that the above analysis describes in essence the general situation. Consider a connected part of a diagram with both input and output lines. Assume this part is not trivial, i.e., does not consist of a single vertical line from bottom to top. We place upward arrows on the input lines. These come to rest on a set of $\Omega$-boxes. On the remaining vertical line segments attached to these boxes, we place downward arrows. These come to rest on a set of $\Lambda$-boxes. On the remaining vertical line segments attached to these boxes, we place upward arrows. Some of these line segments may already be output lines and terminate at the top. Those that are not now terminate on a new set of $\Omega$-boxes, and we continue as in the previous step until all the line segments of this part of the diagram are supplied with arrows. Figure \[Fig.15\] illustrates such a result.
Note that this divides the $\Omega$ and $\Lambda$ boxes into consecutive stages, the first set of $\Omega$-boxes, followed by the first set of $\Lambda$-boxes, followed by the second set of $\Omega$-boxes, and so on. We can now create a processing diagram by placing each subsequent set above the previous one and also placing the $\Lambda$-boxes of each stage above the $\Omega$ ones of the same stage. Considering these boxes now as vertices of a graph, we connect them by edges that correspond to the vertical line segments of the original diagram. Each such edge can be labelled by the index of the Hilbert space that the vertical line segment represents. Associate to each such vertex either an $f^{S'\leftarrow S}_\Lambda$ or a $g^{S'\leftarrow S}_\Omega$ function according to the type of box, where $S$ represents the lines coming from below to the vertex and $S'$ the lines leaving upward. At each level one forms the tensor product of all these functions (which are linear maps), including $I$ for each line not encountering a vertex at that level, and the vertical placement of the levels, bottom to top, indicates composition of these products of function, in the corresponding order of right to left. Fig. \[Fig.16\] and equation (\[multicomp\]) illustrate this.
It should by now be clear from tensor universality that this composition is precisely the way this part of the diagram processes the input state to arrive at the output. A few details should make this obvious. Consider an $\Omega$-box and a $\Lambda$-box in which the latter is connected to the former by some vertical line segments with all downward or all upward oriented arrows. Consider now the case that both states are completely disentangled and for simplicity’s sake the Hilbert spaces are indexed such that $$\begin{aligned}
\bra\Omega&=& \bra{\omega_1}\cotimes\bra{\omega_r}\otimes\bra{\omega_{r+1}}\cotimes\bra{\omega_{r+s}},\\
\ket\Lambda&=& \ket{\lambda_{r+1}}\cotimes\ket{\lambda_{r+s}}\otimes \ket{\lambda_{r+s+1}}\cotimes\ket{\lambda_{r+s+t}},\end{aligned}$$ and where the choice of the bra and ket form is motivated by the role the boxes play in the processing. Here the $\Omega$-box is connected to the $\Lambda$-box by $s$ vertical lines numbered $r+1,\dots,r+s$. Think now of the first $r$ factors of $\Omega$ as a functional on the Hilbert space $\cH_1\cotimes\cH_r$. If we now apply this part of $\bra\Omega$ to say $\phi=\phi_1\cotimes\phi_r$ one will get $$\left(\prod_{i=1}^r(\omega_i,\phi_i)\right) \bra{\omega_{r+1}}\cotimes\bra{\omega_{r+s}}.$$ This is precisely $g^{S'\leftarrow S}_\Omega(\phi)$ where $S=\{1,\dots,r\}$ and $S'=\{r+1,\dots,r+s\}$.Similarly the first $s$ factors of $\Lambda$ can be considered as a functional on $\cH_{r+1}^*\cotimes \cH_{r+s}^*$. Applying this part to $\sigma^\dagger=\bra{\sigma_{r+1}}\cotimes\bra{\sigma_{r+s}}$ one gets $$\left(\prod_{i=r+1}^{r+s}(\sigma_i,\lambda_i)\right) \ket{\lambda_{r+s+1}}\cotimes\ket{\lambda_{r+s+t}}.$$ This is precisely $f^{T'\leftarrow T}_\Lambda(\sigma^\dagger)$ where $T=\{r+1,\dots,r+s\}$ and $T'=\linebreak\{r+s+1,\dots,r+s+t\}$. Now contracting $\Omega$ with $\Lambda$ along the lines $\{r+1,\dots,r+s\}$, assuming these have down-pointing arrows, one gets $$\left(\prod_{i=r+1}^{r+s}(\omega_i,\lambda_i)\right)\bra{\omega_1}\cotimes\bra{\omega_r}\otimes \ket{\lambda_{r+s+1}}\cotimes\ket{\lambda_{r+s+t}}.$$ If we act on $\phi$, introduce above, by the first $r$ factors of the tensor product we get precisely $$f^{T'\leftarrow T}_\Lambda\circ g^{S'\leftarrow S}_\Omega(\phi),$$ and similarly, if we assume up-pointing arrows and if we act on $\tau^\dagger=\bra{\tau_1}\cotimes\bra{\tau_t}$ by the last $t$ factors of the tensor product, we get precisely $$g^{S\leftarrow S'}_\Omega\circ f^{T\leftarrow T'}_\Lambda(\tau^\dagger).$$ From this it is clear that the vertical line segments connecting $\Omega$ and $\Lambda$-boxes act as compositions of the corresponding $g$ and $f$ functions. Thus, in the case that all the relevant states are totally disentangled, the processing diagram represents exactly the correct compositions that processes the initial state to the final one. By tensor universality this is true in all cases.
The composition scheme describe above is a realization of the algebraic system know as a [*colored PROP*]{}. See Markl[@mark] for description and references. PROPs are algebraic structures that abstract the composition properties of multi-input-multi-output maps. One means by “colored" that composition is only defined if the “output" and the “input" have some common characteristic (“color"). In our case, in the processing order diagram, a line joining two vertices is a connection of the output of one function ($f$ or $g$) to the input of another and these must refer to (be “colored by") the same Hilbert space.
As a final result we now argue that [*any*]{} processing described by a connected part of a diagram with both input and output lines is equivalent to one with a [*single*]{} $f\circ g$ composition. To motivate this, examine Fig. \[Fig.14\] and recall the destroy-and-create interpretation for $Q$-boxes given in Section \[section.proof\]. Under this interpretation the various line segments in the diagram that belong to the same $\cH_i$ vertical line actually correspond to physically distinct systems. This lessens the constraints we’ve had in moving $Q$-boxes vertically imposed by commutativity. Furthermore, all the states created by the $\Lambda$-boxes could have been created independently prior to the action of all the $\Omega$-boxes and be held in readiness until called for by these. In essence all the $\Lambda$-boxes can be moved prior to all the $\Omega$ boxes and all boxes can be thought of as acting on different Hilbert spaces, except for the connections between the $\Lambda$ and $\Omega$-boxes. We now formalize this.
Consider now a general diagram with separated $Q$-boxes. Consider now all the vertical line segments, the input lines intercepted by $\Omega$-boxes, the output lines originating from $\Lambda$-boxes and the segments connecting the two types of boxes (we assume there are no free lines going straight from input to output). Number these segments arbitrarily as $\ell_\alpha, \, \alpha=1,2,\dots,K$. Note that two segments belonging to the same vertical $\cH_j$ line are to be considered different and numbered distinctly. Each $\ell_\alpha$ corresponds to some $\cH_{j(\alpha)}$. Let now $\cK_\alpha$ be [*distinct*]{} Hilbert spaces, each isomorphic to $\cH_{j(\alpha)}$ via a unitary map $V_\alpha: \cH_{j(\alpha)}\to \cK_\alpha$. Now the state of any $p$-partite $\Omega$-box representing $\Omega\rfloor\cdot$ sits on $p$ vertical segments $\ell_{\alpha_1},\dots,\ell_{\alpha_p}$, and from any $q$-partite $\Lambda$-box representing $\Lambda\otimes\cdot$ emanate $q$ vertical segments $\ell_{\beta_1},\dots,\ell_{\beta_q}$. In the tensor product $\cK_1\cotimes\cK_K$ we now consider $\Omega$-boxes associated to the subproducts $\cK_{\alpha_1}\cotimes\cK_{\alpha_p}$ with state $\hat\Omega=(V_{\alpha_1}\cotimes V_{\alpha_p})\Omega$, and $\Lambda$-boxes associated to the subproducts $\cK_{\beta_1}\cotimes\cK_{\beta_q}$ with state $\hat\Lambda=(V_{\beta_1}\cotimes V_{\beta_q})\Lambda$. This results in a new diagram in which no two $Q$-boxes and no two $\Lambda$-boxes have a common vertical line. This implies that all the $\Omega$-boxes and all the $\Lambda$-boxes can be so placed that each of the types act at the same time, with $\Lambda$ prior to $\Omega$. Call this new diagram the [*unravelled*]{} version of the original. The various connected parts of the original diagram give rise to the connected parts of the unravelled diagram and these parts can be displayed horizontally next to each other as the parts do not share any $\cK_\alpha$ Hilbert space.
As an illustrative aside, the unravelled version of Fig. \[Fig.15\], under appropriate numbering, is:
(180,68) (0,0)[(0,1)[44]{}]{} (20,0)[(0,1)[44]{}]{} (40,24)[(0,1)[20]{}]{} (60,24)[(0,1)[44]{}]{} (80,24)[(0,1)[20]{}]{} (100,0)[(0,1)[8]{}]{}(100,26)[(0,1)[18]{}]{} (120,24)[(0,1)[20]{}]{} (140,24)[(0,1)[20]{}]{} (160,24)[(0,1)[20]{}]{} (180,24)[(0,1)[44]{}]{} (37,10)[(96,14)[$\hat\Lambda_3$]{}]{} (137,10)[(46,14)[$\hat\Lambda_1$]{}]{} (-3,44)[(46,14)[$\hat\Omega_1$]{}]{} (77,44)[(26,14)[$\hat\Omega_5$]{}]{} (117,44)[(46,14)[$\hat\Omega_2$]{}]{}
To procede we need a few more mathematical results. Suppose $\Omega_1$, $\Lambda_1$ and $\Omega_2$, $\Lambda_2$ are pairs of states belonging to two disjoint subproducts of a multipartite Hilbert space, then $$Q_{\Lambda_1,\Omega_1}\otimes Q_{\Lambda_2,\Omega_2}=Q_{\Lambda_1\otimes\Lambda_2,\Omega_1\otimes\Omega_2}.$$
Assume now $\Omega_1,\,\Lambda_1\in \cH_1\otimes \cK_1$ and $\Omega_2,\,\Lambda_2\in \cH_2\otimes \cK_2$ and consider $\Omega_1\otimes \Omega_2$ and $\Lambda_1\otimes\Lambda_2$ as belonging to $(\cH_1\otimes\cH_2)\otimes(\cK_1\otimes\cK_2)$ thinking of this as a Hilbert space with [*two*]{} (composite) tensor factors, we have:
$$\begin{aligned}
f_{\Lambda_1\otimes\Lambda_2}&=&f_{\Lambda_1}\otimes f_{\Lambda_2},\\
g_{\Omega_1\otimes\Omega_2}&=&g_{\Omega_1}\otimes g_{\Omega_2}.\end{aligned}$$
All these results have very easy proof by tensor universality as they are easy to show if all the states involved are totally disentangled.
Returning to Fig. \[Fig.17\] it is now clear that by combining the $\hat\Omega$ and the $\hat\Lambda$ boxes and the incoming, the outgoing, and the intervening Hilbert spaces by tensoring, this diagram becomes an $f\circ g$ composition. We have still to argue that it is [*equivalent*]{} to the original connected part of the diagram in state processing.
Return now to the original diagram. Let us chose a numbering for a given connected part as follows: The input lines are labelled as $\ell_1,\ell_2,\dots,\ell_s$; the output lines as $\ell_{N-r},\ell_{N-r+1},\dots,\ell_N$; and for $s<\alpha<N-r$ the line $\ell_\alpha$ is then a vertical segment connecting a $\Lambda$-box with an $\Omega$-box above it. Without loss of generality we can assume that for $1\le \alpha\le s$ one has $\cK_\alpha=\cH_{j(\alpha)}$ and $V_\alpha=I$. Assume now that the initial state $\Phi$ and each $\Omega$ and $\Lambda$ in the $Q$-boxes are totally disentangled. Denote a tensor factors of an $\Omega$ by $\omega$ and of a $\Lambda$ by $\lambda$, which we shall label by the $\alpha$ index of the vertical line that then meets it. In the full original diagram (of which we are examining a connected part) there now appear numerical factors coming from each labelled vertical segment, except for the outgoing lines which contribute the tensor product $\lambda^{\hbox{out}}=\lambda_{N-r}\otimes\lambda_{N-r+1}\cotimes\lambda_N$. Thus $\Psi=\lambda^{\hbox{out}}\otimes \Psi'$. The product of all the numerical factors (inner products) of the connected part is $$M=\prod_{\alpha=1}^s(\omega_\alpha,\phi_{j(\alpha)})\,\cdot \prod_{\alpha=s+1}^{N-r-1}(\omega_\alpha,\lambda_\alpha),$$ and associating this numerical factor with the $\lambda^{\hbox{out}}$ factor of the output state we have $\Psi=M\lambda^{\hbox{out}}\otimes \tilde\Psi$ where now $\tilde \Psi$ has no contribution from the connected part of the diagram that we are analyzing. Now in the unravelled diagram the numerical factor has the same expression except that now one must put hats on the $\omega$ and $\lambda$ vectors: $\hat\omega_\alpha=V_\alpha\omega_\alpha$, $\hat\lambda=V_\alpha\lambda_\alpha$. For $\alpha\le s$ one has $(\hat\omega_\alpha,\phi_{j(\alpha)})=(\omega_\alpha,\phi_{j(\alpha)})$ as $V_\alpha=I$ in this case, and for $s<\alpha<N-r$ one has $(\hat\omega_\alpha,\hat\lambda_\alpha)=(V_\alpha\omega_\alpha,V_\alpha\lambda_\alpha)=(\omega_\alpha,\lambda_\alpha)$ so the numerical factor in the unravelled diagram is the same as in the original. The contribution therefore to the final state via the unravelled diagram is $M\hat\lambda^{\hbox{out}}$ where $\hat\lambda^{\hbox{out}}=(V_{N-r}\cotimes V_N)\lambda^{\hbox{out}}=V\lambda^{\hbox{out}}$. If we now interpret this expression in a way that is antilinear and linear in the $\Omega$ and $\Lambda$ states then the state processing in the new box is that of the original followed by the unitary $V$. Since in the unravelled diagram all the $\Omega$ boxes act at the same time and so do the $\Lambda$ boxes, we can combine them by tensoring those of each type into one corresponding box. Let $\cL_1=\bigotimes_{\alpha=1}^s\cK_\alpha$, $\cL_2=\bigotimes_{\alpha=s+1}^{N-r-1}\cK_\alpha$ and $\cL_3=\bigotimes_{\alpha=N-r}^N\cK_\alpha$, then the two combined boxes can now be considered bipartite and we have the diagram:
(50,96) (0,10)[(0,1)[48]{}]{} (25,38)[(0,1)[20]{}]{} (50,38)[(0,1)[58]{}]{} (22,26)[(31,12)[$\hat\Lambda$]{}]{} (-3,58)[(31,12)[$\hat\Omega$]{}]{} (0,0)[(0,0)[$\cL_1$]{}]{} (25,0)[(0,0)[$\cL_2$]{}]{} (50,0)[(0,0)[$\cL_3$]{}]{}
Here $\hat\Omega=\hat\Omega_1\cotimes\hat\Omega_A$ and $\hat\Lambda=\hat\Lambda_1\cotimes\hat\Lambda_B$ where we have numbered the $\Omega$ and $\Lambda$ vectors that appear in the original connected part of the diagram. One now has $M\hat\lambda^{\hbox{out}}=f_{\hat\Lambda}\circ g_{\hat\Omega}(\phi^{\hbox{in}})$ where $\phi^{\hbox{in}}=\phi_{j(1)}\cotimes\phi_{j(s)}$. Thus $$\label{allisfg}
M\lambda^{\hbox{out}}=V^{-1}f_{\hat\Lambda}\circ g_{\hat\Omega}(\phi^{\hbox{in}}).$$
Now just as before, the right-hand side of (\[allisfg\]) depends linearly on the $\Lambda$ states and antilinearly on the $\Omega$ states in its construction, and linearly on $\phi^{\hbox{in}}$. By tensor universality therefore the state processing by the connected part of the original diagram is [*always*]{} given by the right-hand side of (\[allisfg\]). A single $f\circ g$ transform is therefore the universal quantum processor.
Of course, in the context of the original sequence of measurements where classical information is to be exchanged between the measurement acts, temporal order is important as classical information need always flow toward the future. The single $f\circ g$ form cannot express such situations. Classical information has not been taken into account in our analysis which focuses just on the quantum aspects.
Gottesman and Chuang[@goch] have shown that a generalized quantum teleportation protocol is a universal computational primitive. Our result can be considered a generalization.
One can using the same method above also show that any connected part with only input lines is equivalent to a single $\Omega$-box, with $\Omega=\hat\Lambda\rfloor\hat\Omega$ and any connected part with only output lines is equivalent to a single $\Lambda$-box with $\Lambda=\hat\Omega\rfloor\hat\Lambda$ and one with neither input or output lines is equivalent to the scalar factor $(\hat \Omega, \hat\Lambda)$. This is readily seen from Fig. \[Fig.18\] assuming that one or both of the mentioned lines is missing
As a final aside, we should mention that one can always formally add any number of input and output lines using any number of one-dimensional $\lC$ factors on which, metaphorically, complex numbers can travel forward and backward in time. Any $\Omega$ or $\Lambda$ box can be extended to intercept any number of $\lC$ lines. One has (among others) the following equivalences:
(300,176) (0,10)[(0,1)[34]{}]{} (25,10)[(0,1)[34]{}]{} (50,17)[(0,0)[$\cdots$]{}]{} (-3,44)[(81,14)[$\Omega$]{}]{} (75,10)[(0,1)[34]{}]{} (125,29)[(0,0)[$\simeq$]{}]{} (175,10)[(0,1)[34]{}]{} (200,10)[(0,1)[34]{}]{} (225,17)[(0,0)[$\cdots$]{}]{} (250,10)[(0,1)[34]{}]{} (275,34)[(0,1)[10]{}]{} (300,34)[(0,1)[34]{}]{} (172,44)[(106,14)[$\Omega\otimes1$]{}]{} (272,20)[(31,14)[$1$]{}]{} (275,0)[(0,0)[$\lC$]{}]{}(300,0)[(0,0)[$\lC$]{}]{} (0,98)
(300,78) (0,34)[(0,1)[34]{}]{} (25,34)[(0,1)[34]{}]{} (50,59)[(0,0)[$\cdots$]{}]{} (-3,20)[(81,14)[$\Lambda$]{}]{} (75,34)[(0,1)[34]{}]{} (125,39)[(0,0)[$\simeq$]{}]{} (175,34)[(0,1)[34]{}]{} (200,34)[(0,1)[34]{}]{} (225,59)[(0,0)[$\cdots$]{}]{} (250,34)[(0,1)[34]{}]{} (275,34)[(0,1)[10]{}]{} (300,10)[(0,1)[34]{}]{} (172,20)[(106,14)[$\Lambda\otimes1$]{}]{} (272,44)[(31,14)[$1$]{}]{} (275,0)[(0,0)[$\lC$]{}]{}(300,0)[(0,0)[$\lC$]{}]{}
Thus one can formally add a $\lC$ input and/or a $\lC$ output line and and reduce any argument to the case when both input and output lines are present.
Whither the flow?
=================
In the diagrammatic representation of (\[qqphi\]) we have not been representing the exact form of the incoming state $\Phi$ as we had to consider various form of it and were examining the state processing mechanism itself. To represent such an incoming state we can place a set of disjoint $\Lambda$-boxes from which all the input lines originate. By this we mean that $\Phi=\Lambda_1\cotimes\Lambda_N$ where each $\Lambda_i$ belongs to a tensor subproduct of $\cH_1\cotimes\cH_n$. The connected components of the resulting diagram now indicate the independent state processing that takes place. This would now be a true graphical representation of (\[qqphi\]) including the input state. We may want to compute the inner product of the output state $\Psi$ with some state $\Theta$ as a typical transition amplitude reminiscent of particle scattering theory. To represent this amplitude we cap off the diagram with a set of disjoint $\Omega$-boxes within which all the output lines terminate. By this we mean $\Theta=\Omega_1\cotimes\Omega_M$ with the same interpretation for this form as for the $\Lambda$-boxes. The resulting amplitude is the product of amplitudes represented by the connected components of this final diagram. Now $$(\Theta,\Psi)=(\Theta,Q_m\cdots Q_1\Phi) =\overline{(\Phi,Q_1^*\cdots Q_m^*\Theta)} =\overline{(\Psi,\Theta)}.$$ and since $Q_{\Lambda,\Omega}^*=Q_{\Omega,\Lambda}$ we see that the amplitude $\overline{(\Psi,\Theta)}$ is represented by the same diagram turned upside down with the $\Omega$ and $\Lambda$-boxes switching their roles. The upside down diagram is the same type of object as the original. The temporal “flow" in the upside down diagram represents the reverse of that of the original. Quantum processing is thus time-reversal invariant. There is much more to this however. If we analyze the diagram when all the relevant states, $\Theta$, $\Phi$, $\Lambda$, and $\Omega$ are completely disentangled, the resulting amplitude is nothing but a product of inner products corresponding to each vertical line segment in the diagram. For a typical such inner product $\bracket\sigma\tau$, it is indifferent if we think of it as the bra $\bra\sigma$ travelling backward in time to meet the ket $\ket\tau$, or the ket moving forward to meet the bra, or the two meeting head-on on their paths. Since the totally disentangled situation determines by tensor universality the entangled one, one is induced to assert that in all cases it is totally indifferent how we distribute the arrows on the vertical lines in the diagrams. Thus quantum processing is [*locally*]{} time reversal indifferent and we can time reverse any part at will, changing of course the interpretation, the metaphor. A $\Omega$-box with some in-pointing and some out-pointing lines is a transformer of kets to bras. Just with in-pointing lines it is a sink of kets producing a number, and with just out-pointing a source of bras. Similarly for a $\Lambda$-box with “ket" and “bra" interchanged. Trying to combine these views with the time-asymmetric classical world creates some enigmatic circumstances. Let us return to the teleportation situation:
(65,110)(-20,0) (0,44)[(15,14)[$A$]{}]{} (25,10)[(15,14)[$S$]{}]{} (30,86)[(10,14)[$U$]{}]{}(65,93)[(0,0)[Bob]{}]{} (5,0)[(0,1)[44]{}]{} (10,44)[(1,-1)[20]{}]{} (35,24)[(0,1)[62]{}]{} (35,100)[(0,1)[10]{}]{} (5.3,58.4)(14,70)(29,90)(17,74)[(3,4)[0]{}]{} (-20,51)[(0,0)[Alice]{}]{}
Here, in conventional terms, $A$ is Alice’s measuring device, $S$ is a source of entangled pairs and $U$ is Bob’s unitary device, the dotted line represent the two c-bits that Alice sends to Bob. Concerning the quantum lines, there are now eight possibilities for distributing arrows as being either upward (u) or downward (d). Each possibility requires a different metaphor. Such metaphors should not be considered as representing reality, since reality is indifferent to the existence of time orientation, but as a means of conceptualizing the situation to be able to deal with it more readily. For some of the possibilities the notions of “quantum information" (which we now abbreviate by “QI") and its “flow" provides a convenient enough picture that these notions have become widely used in the literature, for other possibilities such pictures are hard to come by. Cocke’s processing metaphor (udu) has QI arriving at Alice’s measuring device and producing a result; the QI gets transformed and then travels backward in time to $S$ which again transforms it now into a time-forward flow. Meanwhile, Alice sends to Bob the two c-bits of information concerning the measurement outcome which arrives at Bob’s place before the QI. Thus informed, he chooses the unitary $U$ thereby converting the incoming QI to identical form that arrived at Alice’s location. The time reverse of this (dud) has QI arriving from the future and passing through Bob’s unitary device $U$ as a bra travelling backward in time. It then proceeds to $S$ which transforms it into a time-forward flow. Arriving at Alice’s measuring device it produces precisely the result that is consistent with the choice of Bob’s unitary device, and gets transformed into its original form proceeding backward in time from Alice’s location. The forms (udd) and (uud) answer the question: “What device captures two identical QI moving in opposite time directions?"; and for the time reverse forms (duu) and (ddu) replace “captures" with “emits". The “inner workings" of these devices are somewhat enigmatic. The “broadcast" metaphor (uuu), which is a bit strained, has Alice capture QI from the “transmitter" $S$ and incoming ket with her measuring device, and advise Bob to “tune to the same channel" to capture QI of identical content. The time reversal of this, (ddd), is probably the most enigmatic. It teleports QI moving to the past from Bob’s to Alice’s location but is not clear how to describe the process in terms of a “flow" of QI as is the case for (dud). Our inability to form convenient metaphors for all the situation is most likely a lack of imagination, keeping us from a better understanding of quantum reality. If we are to take all eight of the possibilities as equally legitimate, as suggested by tensor universality, then quantum information is time-direction indifferent; it doesn’t “flow" nor “gets transferred", and if it refers to several space-time locations it is simply [*co-present*]{} at each, the co-presence being determined by the degree of entanglement. The classical world is time oriented and coupled to the quantum substratum. The two coexist without contradiction and the seeming conflicts with causality in some of the above metaphors are merely apparent. Linearity of quantum mechanics and its innate indeterminism precludes any causal paradoxes. Applied to the unverse as a whole, one metaphor would be that of a universal quantum broadcast, indifferent to time (just as there is no time in many quantum gravity theories) and place (through entanglement), to which the classical systems can “tune in" and thereby condition their temporal behavior. Of course according to modern thought, this must be taken merely as an effective picture; the (semi-)classical world should emerge from the underlying quantum substrate, but once again we are faced with the famous “problem of time" in fundamental quantum theory.
Returning to the mathematical picture, given the time-orientation indifference of the quantum substrate, the algebraic structure of a PROP does not truly capture the situation. One needs a PROP-like structure that makes no distinction between “input" and “output" and which would allow for “feedback", that is, a directed path from output that leads back to the input of the same unit. The simplest example of this would be:
(40,48) (0,0)[(40,14)[$\Lambda$]{}]{} (0,34)[(40,14)[$\Omega$]{}]{} (10,14)[(0,1)[20]{}]{}(10,26)[(0,1)[0]{}]{} (30,14)[(0,1)[20]{}]{}(30,22)[(0,-1)[0]{}]{}
A metaphor for this would be “QI caught in a time loop". Mathematically this can be thought of as a “composition’ $$C:\hbox{Hom}(\cH,\cK^*)\otimes\hbox{Hom}(\cK^*,\cH)\to \lC$$ defined by tensor universality for $\Omega=\omega_1\otimes\omega_2$ and $\Lambda=\lambda_1\otimes\lambda_2$ by $$\label{timeloop}
C(g_\Omega\otimes f^{\hbox{op}}_\Lambda)=(\omega_1,\lambda_1)(\omega_2,\lambda_2).$$
The right hand side of (\[timeloop\]) also defines the “composition" for the other three choices for the arrow directions (du), (dd) and (uu). These would be for $\hbox{Hom}(\cK,\cH^*)\otimes\hbox{Hom}(\cH^*,\cK)\to \lC$, $\hbox{Hom}(\lC,\cH^*\otimes\cK^*)\otimes\hbox{Hom}(\cH^*\otimes\cK^*,\lC)\to \lC$, and $\hbox{Hom}(\lC,\cH\otimes\cK)\otimes\hbox{Hom}(\cH\otimes\cK,\lC)\to \lC$ respectively. An algebraic structure unbiased by these time orientations would be a more proper description of “quantum information".
Acknowledgements {#acknowledgements .unnumbered}
----------------
I wish to thank Débora Freire Mondaini, a graduate student at the Mathematics Department of the Pontifícia Universidade Católica, Rio de Janeiro, whose interest in quantum information motivated a search for a simple proof of Coecke’s theorem. This research received partial financial support from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).
[xxx]{} Bob Coecke, “The logic of entanglement", quant-ph/0402014.
Bob Coecke, “The logic of entanglement. An invitation", Research Report PRG-RR-03-12 Oxford University Computing Laboratory. <http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-03-12.html>
Bob Coecke, “Quantum information-flow, concretely, and axiomatically", quant-ph/0506132.
Bob Coecke, “Kindergarten Quantum Mechanics", quant-ph/0510032.
Samson Abramsky and Ross Duncan, “A Categorical Quantum Logic" quant-ph/0512114.
Yong Zhang “Teleportation, Braid Group and Temperley–Lieb Algebra", quant-ph/0601050.
George Svetlichny, [*Foundations of Physics*]{}, **11** 741 (1981).
George Svetlichny, “On the Foundations of Experimental Statistical Sciences", unpublished monograph available on the author’s home page. <http://www.mat.puc-rio.br/~svetlich/files/statsci.pdf>
Martin Markl “Operads and PROPs", math/0601129.
Daniel Gottesman and Isaac L. Chuang, Nature, **402**, 390 (1999).
George Svetlichny, “Nonlinear Quantum Gravity", quant-ph/0602012.
[^1]: Departamento de Matemática, Pontifícia Universidade Católica, Rio de Janeiro, Brazil svetlich@mat.puc-rio.br <http://www.mat.puc-rio.br/~svetlich>
|
---
abstract: |
We have defined and established a theory of cofinite connectedness of a cofinite graph. Many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. We have seen that if $G$ is a cofinite group and $\Gamma=\Gamma(G,X)$ is the Cayley graph. Then $\Gamma$ can be given a suitable cofinite uniform topological structure so that $X$ generates $G$, topologically iff $\Gamma$ is cofinitely connected.
Our immediate next concern is developing group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we are able to characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the afore said action becomes uniformly continuous.
address:
- |
Department of Mathematics and Statistics\
University of Toledo, Main Campus\
Toledo, OH 43606-3390
- |
Department of Mathematics\
University of Alabama\
Tuscaloosa, AL 35487-0350
- |
Department of Mathematics\
University of North Georgia, Gainesville Campus\
Oakwood, Ga. 30566
author:
- Amrita Acharyya
- 'Jon M. Corson'
- Bikash Das
title: Cofinite Connectedness and Cofinite Group Actions
---
Introduction {#s:Intro}
============
A cofinite graph $\Gamma$ is said to be [*cofinitely connected*]{} if for each compatible cofinite equivalence relation $R$ on $\Gamma$, the quotient graph $\Gamma/R$ is path connected.
Similar to the standard connectedness arguements for finite graphs or general topological spaces we were able to establish that the following statements are equivalent for any cofinite graph $\Gamma$:
1. $\Gamma$ is cofinitely connected;
2. $\Gamma$ is not the union of two disjoint nonempty subgraphs.
As an immediate consequence we obtained the following generalized characterization of connected Cayley graphs of cofinite groups:
Let $G$ be a cofinite group and let $\Gamma=\Gamma(G,X)$ be the Cayley graph. Then $\Gamma$ can be given a suitable cofinite topological graph structure so that $X$ generates $G$ $($topologically$)$ iff $\Gamma$ is cofinitely connected.
Our final section is concerned with cofinite group actions on cofinite graphs.
A group $G$ is said to act uniformly equicontinuously over a cofinite graph $\Gamma$ if and only if for each entourage $W$ over $\Gamma$ there exists an entourage $V$ over $\Gamma$ such that for all $g$ in $G, (g\times g)[V]\subseteq W$. In this case the group action induces a (Hausdorff) cofinite uniformity over $G$ if and only if the aforesaid action is faithful.
We say that a group $G$ acts on a cofinite graph $\Gamma$ residually freely, if there exists a fundamental system of $G$-invariant compatible cofinite entourages $R$ over $\Gamma$ such that the induced group action of $G/N_R$ over $\Gamma/R$ is a free action, where $N_R$ is the Kernel of the action of $G$ on $\Gamma/R$.
Suppose that $G$ is a group acting faithfully and uniformly equicontinuously on a cofinite graph $\Gamma$, then the action $G\times\Gamma\to\Gamma$ is uniformly continuous. Also in that case $\widehat{G}$ acts on $\widehat{\Gamma}$ uniformly equicontinuously.
Connected Cofinite Graphs
=========================
A [*path*]{} in a graph $\Gamma$ is a finite string of edges $p=e_1\cdots e_n\in E(\Gamma)^*$ such that $t(e_i)=s(e_{i+1})$ for $1\le i\le n-1$. The [*source*]{} and [*target*]{} of this path $p$ are the vertices $s(p)=s(e_1)$ and $t(p)=t(e_n)$. We say that $\Gamma$ is [*path connected*]{} if there is a path in $\Gamma$ joining any two vertices.
A cofinite graph $\Gamma$ is [*cofinitely connected*]{} if for each compatible cofinite equivalence relation $R$ on $\Gamma$, the quotient graph $\Gamma/R$ is path connected.
The following statements are equivalent for any cofinite graph $\Gamma$:
1. $\Gamma$ is cofinitely connected;
2. $\Gamma$ is not the uniform sum of two disjoint nonempty subgraphs. We then note that if $\Gamma$ is a profinite graph then we can restate the condition as $\Gamma$ is not the disjoint union of two nonempty closed subgraphs.
3. the completion $\overline\Gamma$ of $\Gamma$ is cofinitely connected.
\(1) $\Rightarrow$ (2): If possible, let us assume that $\Gamma$ is the uniform sum of two disjoint subgraphs $\Gamma_1$ and $\Gamma_2$. Let $R_{\Gamma_1}$ be a compatible cofinite entourage over $\Gamma_1$ and $S_{\Gamma_2}$ be another compatible cofinite entourage over $\Gamma_2$. Then $W = R_{\Gamma_1}\cup S_{\Gamma_2}$ is a compatible cofinite entourage over $\Gamma$. Moreover $\Gamma/W$ is not path connected, a contradiction.
\(2) $\Rightarrow$ (3): If possible, let us assume that $\overline{\Gamma}$ is not cofinitely connected. Hence there exists a compatible cofinite entourage $W$ over $\overline{\Gamma}$ such that $\overline{\Gamma}/W$ is not path connected.
Let $\Sigma$ be a path connected component of $\overline{\Gamma}/W$. Hence $\Sigma$ is a subgraph of $\overline{\Gamma}/W$ and thus $(\overline{\Gamma}/W)\setminus\Sigma$ is a subgraph of $\overline{\Gamma}/W$ as well. Let $\Gamma_1 = \varphi^{-1}(\Sigma)$ and $\Gamma_2 = \varphi^{-1}(\overline{\Gamma}\setminus \Sigma)$, where $\varphi{\colon}\overline{\Gamma}\to\overline{\Gamma}/W$ is the canonical quotient map. Then $\Gamma_1, \Gamma_2$ are closed subgraphs of $\overline{\Gamma}
$ such that $\overline{\Gamma}$ is equal to the disjoint union of two closed subgraphs of $\overline{\Gamma}$ and then $\overline{\Gamma}$ is equal to the uniform sum of two disjoint subgraphs of $\overline{\Gamma}$, a contradiction.
\(3) $\Rightarrow$ (1): If possible assume that $\Gamma$ is not cofinitely connected. Then there exists a cofinite entourage $R$ over $\Gamma$ such that $\Gamma/R$ is not path connected. But, $\overline R$ is a compatible cofinite entourage over $\overline{\Gamma}$ such that $\Gamma/R$ is graph isomorphic to $\overline{\Gamma}/\overline R$. Hence $\overline{\Gamma}/\overline R$ is not path connected as well, a contradiction.
Many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. Next we list a few of them.
\[p:properties of connectedness\] Let $\Gamma$ be a cofinite graph and let $\Sigma$ be a uniform subgraph.
1. If $\Sigma$ is path connected, then it is also cofinitely connected.
2. If $\Sigma$ is cofinitely connected, then so is the cofinite subgraph $\overline\Sigma$.
3. If $\Sigma$ is cofinitely connected and $f{\colon}\Gamma\to\Delta$ a uniformly continuous map of graphs, then $f(\Sigma)$ is also cofinitely connected $($as a cofinite subgraph of $\Delta)$.
Note that $\Sigma$ is a also a cofinite graph
1. If $\Sigma$ is path connected then any quotient graph of $\Sigma$ is path connected as well and thus our claim follows.
2. We will first see that $\overline{\Sigma} = \overline{V(\Sigma)\cup E(\Sigma)} = \overline{V(\Sigma})\cup\overline{E(\Sigma)}$ and that equals $V(\overline{\Sigma})\cup E(\overline{\Sigma})$ and thus is a cofinite subgraph of $\Gamma$ as well. Now, if possible suppose $\overline{\Sigma} = \Sigma_1\coprod\Sigma_2$, where $\Sigma_1, \Sigma_2$ are two disjoint nonempty cofinite subgraphs of $\overline{\Sigma}$. Then $\Sigma_1\cap\Sigma, \Sigma_2\cap\Sigma$ are two disjoint connected cofinite subgraphs of $\Sigma$. Let $R_1, R_2$ be two compatible cofinite entourage over $\Sigma_1\cap\Sigma, \Sigma_2\cap\Sigma$ respectively. Then there exist two compatible cofinite entourages $\tilde{R_1}, \tilde{R_2}$ over $\Sigma_1, \Sigma_2$ respectively such that $R_1 \supseteq \tilde{R_1}\cap(\Sigma\times\Sigma)$ and $ R_2$ contains $\tilde{R_2}\cap(\Sigma\times\Sigma)$. But as $\tilde{R_1}\cup \tilde{R_2}$ is a compatible cofinite entourage over $\overline{\Sigma}$, then $(\tilde{R_1}\cup\tilde{R_2})\cap(\Sigma\times\Sigma)$ is equal to $\tilde{R_1}\cap(\Sigma\times\Sigma)\cup \tilde{R_2}\cap(\Sigma\times\Sigma)$ which is a subset of $R_1\cup R_2$. So $R_1\cup R_2$ is a compatible entourage over $\Sigma$. Hence $\Sigma = (\Sigma_1\cap \Sigma)\coprod(\Sigma_2\cap\Sigma)$. Now suppose $\Sigma_1\cap\Sigma = \emptyset$. Then $\Sigma\subseteq \Sigma_2$. However $\Sigma_2$ is closed in $\overline{\Sigma}$ and hence closed in $\Gamma$. Thus $\overline{\Sigma}\subseteq \Sigma_2$ and therefore $\Sigma_1 = \emptyset$, a contradiction. Thus $\overline{\Sigma}$ is cofinitely connected.
3. Let $S$ be a compatible cofinite entourage over $f(\Sigma)$. Then as $f|_{\Sigma}{\colon}\Sigma\to f(\Sigma)$ is uniformly continuous there is a compatible cofinite entourage $R$ over $\Sigma$ such that $R\subseteq (f\times f)^{-1}[S]$. Let us define $g{\colon}\Sigma/R\to f(\Sigma)/S$ via $g(R[a]) = S[f(a)]$, for all $a\in\Sigma$. Now if $R[a] = R[b]$, then $(a,b)\in R$. Hence $(f(a),f(b))$ is in $S$ which implies that $ S[f(a)] = S[f(b)]$. Therefore $g$ is well defined and as $f$ is a map of graphs and both of $\Sigma/R, f(\Sigma)/S$ are discrete, $g$ is a surjective uniformly continuous map of graphs. Since $\Sigma/R$ is path connected then so is $g(\Sigma/R) = f(\Sigma)/S$.
Cofinite Groups and their Cayley Graphs
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Let $G$ be an abstract group and $X = \{*\}\stackrel{\cdot}{\cup}E(X)$ be an abstract graph such that there is a map of sets $\alpha{\colon}X\to G$ with $\alpha(*) = 1_G$, $(\alpha(e))^{-1} = \alpha(\overline e)$, for all $e\in E(X)$. Then the Cayley Graph $\Gamma(G,X)$ is defined as follows:
1. $V(\Gamma(G,X)) = G\times \{*\}, E(\Gamma(G,X)) = G\times E(X)$.
2. $s(g,e) = (g,*)$, $t(g,e) = (g\alpha(e),*)$, $\overline{(g,e)} = (g\alpha(e),\overline{e})$.
Thus it follows that
1. $\Gamma(G,X) = V(\Gamma(G,X)) \stackrel{\cdot}{\cup} E(\Gamma(G,X))$.
2. $s, t, \overline{\phantom e}$ are well defined and $t(\overline{(g,e)}) = t(g\alpha(e),\overline{e}) = (g\alpha(e)\alpha(\overline{e}),*)$\
$= (g\alpha(e)\alpha(e)^{-1}),*) = (g,*) = s(g,e)$; $s(\overline{(g,e)}) = s(g\alpha(e),\overline{e})$\
$= (g\alpha(e),*) = t(g,e)$.
3. If possible, let $(g,e) = \overline{(g,e)} = ((g\alpha(e),\overline{e})$ and thus $e = \overline e$, a contradiction. Finally, $\overline{\overline{(g,e)}} = \overline{(g\alpha(e),\overline{e})}$\
$= (g\alpha(e)\alpha(\overline{e}),\overline{\overline e}) = (g\alpha(e)\alpha(e)^{-1}),e) = (g,e)$. Hence $\Gamma(G,X)$ is indeed a graph.
We say that $\alpha{\colon}X\to G$ generates $G$ algebraically if $\langle\alpha(X)\rangle = G$. Equivalently, $\alpha{\colon}X\to G$ generates $G$ algebraically if the unique extension to $\alpha{\colon}E(X)^*\to G$ is onto.
The Cayley graph $\Gamma(G,X)$ is path connected if and only if $\alpha{\colon}X\to G$ generates $G$ algebraically.
Let $G$ be a cofinite group and $X = \{*\}\stackrel{\cdot}{\cup}E(X)$ be a cofinite graph such that there is a uniform continuous map of spaces $\alpha{\colon}X\to G$ with $\alpha(*) = 1_G$, $(\alpha(e))^{-1} = \alpha(\overline e)$, for all $e\in E(X)$. Then the cofinite Cayley Graph $\Gamma(G,X)$ is defined as follows:
1. $V(\Gamma(G,X)) = G\times \{*\}, E(\Gamma(G,X)) = G\times E(X)$.
2. $s(g,e) = (g,*)$, $t(g,e) = (g\alpha(e),*)$, $\overline{(g,e)} = (g\alpha(e),\overline{e})$.
$\Gamma(G,X)$ is endowed with the product uniform topological structure obtained from $G\times X = G\times V(\Gamma(G,X))\stackrel{\cdot}{\cup}G\times E(\Gamma(G,X))$.
We have already seen that $\Gamma(G,X)$ is an abstract graph. Also being the product of Hausdorff, cofinite spaces, $\Gamma(G,X)$ is a Hausdorff, cofinite space as well. So in order to check that $\Gamma(G,X)$ is a cofinite graph it remains to prove that the compatible cofinite entourages over $\Gamma(G,X)$ forms a fundamental system of entourages. So it suffices to show that the family of cofinite entourages of the form $R\times S$, where $R$ is a cofinite congruence over $G$ and $S$ is a compatible cofinite entourage over $X$ such that $(\alpha\times \alpha)[S]\subseteq R$ forms a fundamental system of entourages.
To establish the above claim let us first see that the cofinite entourages of the form $R\times S$ are indeed compatible.
1. Let $((x,y),(p,q))\in R\times S$. So $(x,p)\in R\subseteq G\times G$ and $(y,q)$ is in $S$. Thus either $(y,q)\in S_V$ or $(y,q)\in S_E$ which implies that $y = * =q$ or $(y,q)\in S_E$. Hence $(x,y), (p,q)\in V(\Gamma(G,X))$ or $(x,y), (p,q)\in E(\Gamma(G,X))$. Hence $R\times S\subseteq (R\times S)_V\stackrel{\cdot}{\cup}(R\times S)_E$. The other direction of the inclusion follows more immediately.
2. Let $((g_1,e_1),(g_2,e_2))\in R\times S$. Then $(g_1,g_2)\in R$ and $(e_1,e_2)$ is in $S$. This implies that $(\alpha\times \alpha)(e_1,e_2) = (\alpha(e_1),\alpha(e_2))\in R$ and $(\overline{e_1},\overline{e_2})\in S$. Hence $(g_1\alpha(e_1)$, $g_2\alpha(e_2))\in R$, which implies $((g_1,*), (g_2,*))$ and $((g_1\alpha(e_1),*), (g_2\alpha(e_2),*))$ as well as $((g_1\alpha(e_1),\overline{e_1}), (g_2\alpha(e_2)\overline{e_2}))$ is in $R\times S$. Hence $(s(g_1,e_1), s(g_2,e_2))$, $(t(g_1,e_1), t(g_2,e_2))$ and $(\overline{(g_1,e_1)}$, $\overline{(g_2,e_2)})\in R\times S$.
3. If possible let $(\overline{(g_1,e_1)},(g_1,e_1))\in R\times S$ so $((g_1\alpha(e_1),\overline{e_1}),(g_1,e_1))$ is in $R\times S$. Thus $(\overline{e_1},e_1)\in S$, a contradiction.
Now let $R\times T$ be any cofinite entourage over $G\times X$. Note that since $\alpha$ is uniformly continuous and $R$ is a cofinite congruence over $G$, $T$ is a cofinite entourage over $X$, $(\alpha\times \alpha)^{-1}[R]\cap T$ is a cofinite entourage over $X$ and $(\alpha\times \alpha)[(\alpha\times \alpha)^{-1}[R]\cap T]\subseteq R$. So in particular one can take $S$ to be a compatible cofinite entourage over $X$ such that $S\subseteq (\alpha\times \alpha)^{-1}[R]\cap T$. Then $(\alpha\times \alpha)[S]\subseteq R$ and $R\times S\subseteq R\times T$. This proves that $\Gamma(G,X)$ is a cofinite graph. We say that $\alpha{\colon}X\to G$ generates $G$ topologically if $\overline{\langle\alpha(X)\rangle} = G$.
Let $\Gamma=\Gamma(G,X)$ be the cofinite Cayley graph. $\alpha$ from $X$ to $G$ generates $G$ topologically iff $\Gamma$ is cofinitely connected.
Let us first assume that $\alpha{\colon}X\to G$ topologically generates $G$ and let $T$ be a compatible cofinite entourage over $\Gamma$, say $T$ is equal to $R\times S$ where $R$ is a cofinite congruence over $G$ and $S$ is a compatible cofinite entourage over $X$ where $S\subseteq (\alpha\times\alpha)^{-1}[R]$. Let us define $\alpha_{RS}{\colon}X/S\to G/R$ via $\alpha_{RS}(S[x]) = R[\alpha(x)]$. Clearly, $\alpha_{RS}$ is well defined and $\alpha_{RS}(S[*]) = R[1_G]$ and $\alpha_{RS}(\overline{S[e]}) = R[\alpha(\overline e)] = R[(\alpha(e))^{-1}]$\
$= (R[\alpha(e)])^{-1} = (\alpha_{RS}(S[e]))^{-1}$, for all $S[e]\in E(X/S)$. Let us now see that $\Gamma/T\cong\Gamma(G/R,X/S)$. Define $\theta{\colon}\Gamma/T\to\Gamma(G/R,X/S)$ via $\theta(T[(g,x)]) = (R[g],S[x])$ for all $x$ in $X$ and all $g$ in $G$. Clearly, it is well defined injection as $T[(h,y)] = T[(g,x)]$ if and only if $((h,y),(g,x))\in T$ if and only if $(h,g)\in R, (y,x)\in S$ if and only if $R[h] = R[g]$ and $S[x] = S[y]$ if and only if $(R[h],S[y]) = (R[g],S[x])$. Also for all $(R[g],S[x])\in\Gamma(G/R,X/S)$, there exists $T[(g,x)]\in \Gamma/T$ such that $\theta(T[(g,x)]) = (R[g],S[x])$. Moreover it can easily be seen that $\theta$ is a map of graphs as $\theta(T[(g,*)])$ which is equal to $(R[g],S[*])$ belongs to $V(\Gamma(G/R,X/S))$ and $\theta(T[(g,e)])$ which equals to $(R[g],S[e])$ belongs to $E(\Gamma(G/R,X/S))$. Further more for all $(T[(g,e)])$ in $E(\Gamma/T)$ we see that $\theta(s(T[(g,e)])) = \theta(T[s(g,e)])$ which also equals to $\theta(T[(g,*)])$ equal to $(R[g],S[*])$ and that equals to $s(R[g],S[e])$. We also notice that $\theta(t(T[(g,e)])) = \theta(T[t(g,e)]) = \theta(T[(g\alpha(e),*)]) = (R[g\alpha(e)],S[*])$ which we know is equal to $(R[g]R[\alpha(e)],S[*]) = (R[g]\alpha_{RS}(S[e]),S[*])$ and that is equal to $t(R[g],S[e])$ Finally, $\theta(\overline{(T[(g,e)]}) = \theta(T[\overline{(g,e)}])$ and that equals $\theta(T[(g\alpha(e),\overline e)]) = (R[g\alpha(e)],S[\overline e])$ which can be written as $(R[g]R[\alpha(e)],S[\overline e]) = (R[g]\alpha_{RS}(S[e]),S[\overline e]) = (R[g]\alpha_{RS}(S[e]), \overline{S[e]})$ and that equals $\overline{(R[g],S[e])}$. Since $\Gamma/T, \Gamma(G/R, X/S)$ are discrete cofinite graphs, our claim follows.
Now we wish to prove that $\langle\alpha_{RS}(X/S)\rangle = G/R$. Let $R[g]\in G/R$. Then as $\overline{\langle\alpha(X\rangle)} = G$, we have $R[g]\cap\langle\alpha(X)\rangle\neq\emptyset$. Let $a\in R[g]\cap \langle\alpha(X)\rangle$. So, $ R[g] = R[a]$. Also, since $a\in \langle\alpha(X)\rangle$, $a = \alpha(e_1)\alpha(e_2)\cdots \alpha(e_n)$, for some $e_1, e_2, \cdots, e_n\in E(X)$. Hence $R[a] = R[\alpha(e_1)]R[\alpha(e_2)]\cdots R[\alpha(e_n)]$, and one can represent this as $\alpha_{RS}(S[e_1])\alpha_{RS}(S[e_1])\cdots \alpha_{RS}(S[e_1])$. Thus $R[g] = R[a]\in\langle\alpha_{RS}(X/S)\rangle$. Therefore $\langle\alpha_{RS}(X/S)\rangle = G/R$ and consequently, $\Gamma/T = \Gamma(G/R,X/S)$ is path connected. Hence $\Gamma$ is cofinitely connected.
Conversely, let us now take $\Gamma$ to be cofinitely connected. We want to show that $\overline{\langle\alpha(X)\rangle} = G$. So we intend to show that for any $g$ in $G$ and any open set $R[g]$ on $G, R[g]\cap\langle\alpha(X)\rangle\ne\emptyset$. We can form a compatible cofinite entourage $T = R\times S$ where $S$ is a compatible cofinite entourage over $X$ and $S\subseteq (\alpha\times\alpha)^{-1}[R]$. As earlier we can form the Cayley graph $\Gamma/T = \Gamma(G/R,X/S)$ and as $\Gamma$ is cofinitely connected, $\Gamma/T$ and therefore $\Gamma(G/R, X/S)$, is path connected. This implies $\langle\alpha_{RS}(X/S)\rangle = G/R$. So there is $e_1, e_2,\cdots,e_n$ in $E(X)$ such that $\alpha_{RS}(S[e_1])\alpha_{RS}(S[e_2])\cdots\alpha_{RS}(S[e_n]) = R[g]$. Thus we can finally say that $\alpha(e_1)\alpha(e_2)\cdots\alpha(e_n)\in R[g]$ which means $\langle\alpha(X)\rangle\cap R[g]\ne\emptyset$ and thus $\overline{\langle\alpha(X)\rangle} = G$. Hence $\alpha{\colon}X\to G$ topologically generates $G$.
Groups Acting on Cofinite Graphs {#Group Action}
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Let $G$ be a group and $\Gamma$ be a cofinite graph. We say that the group $G$ acts over $\Gamma$ if and only if
1. For all $x$ in $\Gamma$, for all $g$ in $G, g.x$ is in $\Gamma$
2. For all $x$ in $\Gamma$, for all $g_1, g_2$ in $G, g_1.( g_2.x)=( g_1 g_2).x$
3. For all $x$ in $\Gamma$, $1.x=x$
4. For all $v$ in $V(\Gamma)$, for all $g$ in $G, g.v$ is in $V(\Gamma)$ and for all $e$ in $E(\Gamma)$, for all $g$ in $G, g.e$ is in $E(\Gamma)$.
5. For all $e$ in $E(\Gamma)$, for all $g$ in $G, g.s(e)=s(g.e), g.t(e)=t(ge), g.(\overline{e})=\overline{g.e}$
6. There exists a $G-$invariant orientation $E^+(\Gamma)$ of $\Gamma$.
Note that the aforesaid group action restricted to a singleton group element $g\in G$ can be treated as a well defined map of graphs, $\Gamma\to \Gamma$ taking $x\mapsto g.x$.
A group $G$ is said to act uniformly equicontinuously over a cofinite graph $\Gamma$, if and only if for each entourage $W$ over $\Gamma$ there exists an entourage $V$ over $\Gamma$ such that for all $g$ in $G, (g\times g)[V]$ is a subset of $W$.
If $G$ acts uniformly equicontinuously over a cofinite graph $\Gamma$, then there exists a fundamental system of entourages consisting of $G$-invariant compatible cofinite entourages over $\Gamma$, i.e. for any entourage $U$ over $\Gamma$ there exists a compatible cofinite entourage $R$ over $\Gamma$ such that for all $g\in G, (g\times g)[R]\subseteq R\subseteq U$.
Let $U$ be any cofinite entourage over $\Gamma$. Then as $G$ acts uniformly equi continuously over $\Gamma$, there exists a compatible cofinite entourage $S$ over $\Gamma$ such that forall $g\in G, (g\times g)[S]\subseteq U$. Choose a $G$-invariant orientation $E^+(\Gamma)$ of $\Gamma$. Without loss of generality, we can assume that our compatible equivalence relation $S$ on $\Gamma$ is [*orientation preserving*]{} i.e. whenever $(e,e')\in R$ and $e\in E^+(\Gamma)$, then also $e'\in E^+(\Gamma)$. Clearly, $S\subseteq\cup_{g\in G}(g\times g)[S]\subseteq U$. Now if $S_0=\cup_{g\in G}(g\times g)[S]$ and $T=\langle S_0\rangle$, note that $S\subseteq T\subseteq U$. Since for all $h\in G, (h\times h)[S_0]=S_0$ and $S_0^{-1} = S_0$ it follows that $T$ is in the transitive closure of $S_0$. Let $(x,y)\in T$. Then there exists a finite sequence $x_0,x_1,..,x_n$ such that $(x_i,x_{i+1})\in S_0$, for all $i=0,1,2,...,n-1$ and $x = x_0, y = x_n$. Hence $(gx_i,gx_{i+1})\in S_0$, for all $i=0,1,2,...,n-1$, for all $g\in G$. Thus $(gx_0,gx_n)=(gx,gy)\in T$, for all $g\in G$. Hence for all $g\in G, (g\times g)[T]\subseteq T$ and our claim that $T$ is a $G$-invariant cofinite entourage, follows. It remains to check that $T$ is compatible. Let $(x,y)\in T$. If $(x,y)\in S_0$, then there is $(t,s)\in S = S_V\cup S_E$ and $g\in G$ such that $(gt,gs) = (x,y)$. Without loss of generality let $(t,s)\in S_V$. Then $(t,s)\in V(\Gamma)\times V(\Gamma)$ which implies that $ (x,y)\in T_V$. Now let $(x,y)\in T\setminus S_0$. Then there exists a finite sequence $x_0,x_1,..,x_n$ such that $(x_i,x_{i+1})\in S_0$, for all $i=0,1,2,...,n-1$ and $x = x_0, y = x_n$. Hence by the previous argument if $(x_0,x_1)\in T_V$ then $(x_i,x_{i+1})\in T_V$, for all $i=1,2,...,n-1$. Thus $(x,y)\in T_V$. If $(x_0,x_1)\in T_E$ then $(x_i,x_{i+1})\in T_E$, for all $i=1,2,...,n-1$, which implies $ (x,y)\in T_E$. Let $(e_1,e_2)\in T$. If $(x,y)\in S_0$, then there is $(p,q)\in S$ and $g\in G$ such that $(gp,gq) = (e_1,e_2)$. Then $(s(p),s(q))\in S$. So $(s(e_1),s(e_2))$ which equals to $(gs(p),gs(q))$ is in $(g\times g)[S]\subseteq S_0$ so that $ (s(e_1),s(e_2))\in T$. Now let $(e_1,e_2)\in T\setminus S_0$. Then there exists a finite sequence $x_0,x_1,..,x_n$ such that $(x_i,x_{i+1})\in S_0, \forall i=0,1,2,...,n-1$ and $e_1 = x_0, e_2 = x_n$. Hence by the previous argument $(s(x_i),s(x_{i+1}))\in T, \forall i=0, 1,2,...,n-1$ and thus $(s(e_1),s(e_2))\in T$. Similarly, $(t(e_1),t(e_2))\in T$ and $(\overline{e_1},\overline{e_2})\in T$. Finally, to show that for any $e\in E^+(\Gamma), (e\overline e)\in T$ it suffices to note that $T$ is orientation preserving. Alternatively, if possible let $(e,\overline e)\in T$. If $(e,\overline e)\in S_0$, then there is $(p,q)\in S$ and $g\in G$ such that $(gp,gq) = (e,\overline e)$. Then $\overline e = \overline{gp} = g\overline p = gq$ which implies that $ \overline p = q$, so $ (p,\overline p)\in S$, a contradiction. Now let $(e,\overline e)\in T\setminus S_0$. Then there exists a finite sequence $x_0,x_1,..,x_n$ such that $(x_i,x_{i+1})\in S_0$, for all $i=0,1,2,...,n-1$ and $e = x_0, \overline e = x_n$. Now let there is $(p,q)\in S$ and $g\in G$ such that $(gp,gq) = (x_0,x_1)$. Without loss of generality we may assume $(p,q)\in E^+(\Gamma)\times E^+(\Gamma)$. Then $(gp,gq) = (x_0,x_1)\in E^+(\Gamma)\times E^+(\Gamma)$. Hence $(x_i,x_{i+1})\in E^+(\Gamma)\times E^+(\Gamma)$, for all $i=1,2,...,n-1$ which implies that $ (e,\overline e)\in E^+(\Gamma)\times E^+(\Gamma)$, a contradiction. Our claim follows.
We say a group $G$ acts on a cofinite space $\Gamma$ faithfully, if for all $g$ in $G\setminus\{1\}$ there exists $x$ in $\Gamma$ such that $gx$ is not equal to $x$ in $\Gamma$.
\[equi continuous\] Let $G$ acts on a cofinite graph $\Gamma$ uniformly equicontinuously. Then $G$ acts on $\Gamma/R$ and $G/N_R$ acts on $\Gamma/R$ as well, where $R$ is a $G$-invariant compatible cofinite entourage over $\Gamma$. If $\{R\mid R\in I\}$ is a fundamental system of $G$-invariant compatible cofinite entourages over $\Gamma$, then $\{N_R\mid R\in I\}$ forms a fundamental system of cofinite congruences for some uniformity over $G$.
Let $R$ be a $G$-invariant compatible cofinite entourage over $\Gamma$. Let us define $G\times\Gamma/R\to\Gamma/R$ via $g.R[x]=R[g.x]$, for all $g\in G$, for all $x\in \Gamma$. Now let $R[x]=R[y]$ so $(x,y)\in R$ which implies that $(g.x,g.y)\in R$. Then $R[g.x]=R[g.y]$. Hence the induced group action is well defined.
Let us now consider the group action $G/N_R\times \Gamma/R\to \Gamma/R,$ defined via $N_R[g].R[x]=R[g.x]$,for all $x\in \Gamma$, for all $g\in G$. Now let $(N_R[g],R[x])=(N_R[h],R[y])$ which implies that $ (g,h)\in N_R, (x,y)$ is in $R$. Then $(g.x,h.x)\in R$, as $h^{-1}\in G, (h^{-1}g.x,h^{-1}h.x)\in R$. So $(h^{-1}g.x,y)\in R$. Thus $(g.x,h.y)\in R$ which implies that $R[g.x]$ equals to $R[h.y]$. Hence the induced group action is well defined. Let us now show that $N_R$ is an equivalence relation over $G$, for all $G$-invariant compatible cofinite entourage $R$ over $\Gamma$.
1. for all $g\in G$, for all $x\in \Gamma, (g.x,g.x)\in R$. Hence $(g,g)\in N_{R}$, for all $g\in G$ which implies that $D(G)\subseteq N_R$.
2. Now $(h,g)\in N_R^{-1}\Leftrightarrow (g,h)\in N_R\Leftrightarrow (g.x,h.x)\in R$, for all $x\in \Gamma$. Thus $(g.x,h.x)\in R\Leftrightarrow (h.x,g.x)\in R$, for all $x\in \Gamma$. Hence $(h.x,g.x)\in R\Leftrightarrow (h,g)\in N_R$. Thus $N_R^{-1}=N_R$.
3. Let $(g,h), (h,k)\in N_R$. This implies $ (g.x,h.x),(h.x,k.x)$ is in $R, \forall x\in \Gamma$. Hence $(g.x,k.x)\in R$, for all $x\in \Gamma$. So $(g,k)\in N_R$ which implies that $(N_R)^2\subseteq N_R$.
Also we now check that $N_R$ is a congruence over $G$. For, let us take $(g_1,g_2),(g_3,g_4)\in N_R$. Then for all $x\in \Gamma, (g_1.x,g_2.x),(g_3.x,g_4.x)\in R$; for all $x\in \Gamma, g_3.x\in \Gamma$ and so $(g_1g_3.x,g_2g_3.x)\in R$ and $(g_2g_3.x,g_2g_4.x)$ is in $R$, since $R$ is $G$-invariant. Thus $(g_1g_3.x,g_2g_4.x)\in R$, for all $x\in \Gamma$ so that $(g_1g_3,g_2g_4)\in N_R$. Thus our claim follows. Let us now show that $G/N_R$ is finite. Furthermore, define $g{\colon}\Gamma/R\to\Gamma/R$ as $g$ maps $(R[x])$ into $R[g.x]$. Now, $R[x]=R[y]\Longleftrightarrow (x,y)\in R$ if and only if $(g.x,g.y)\in R\Longleftrightarrow R[g.x]=R[g.y]$. Hence the map $g$ is a well defined injection. Now for all $R[x]\in\Gamma/R$ there exists $g^{-1}R[x]\in\Gamma/R$ such that $g(g^{-1}R[x])$ equals to $R[x]$. Hence $g\in Sym(\Gamma/R)$. Now let us define a map $\theta{\colon}G/N_R\to Sym(\Gamma/R)$ via $\theta(N_R[g])=g.$ Now $N_R[g_1]$ equals to $N_R[g_2]$ if and only if $(g_1,g_2)\in N_R$ if and only if $(g_1.x,g_2.x)\in R$ for all $x\in\Gamma$. Hence $(g_1.x,g_2.x)\in R$ if and only if $R[g_1.x]=R[g_2.x]$ if and only if $g_1(R[x])=g_2(R[x])$ $g_1=g_2$ in $Sym(\Gamma/R)$. Hence $\theta$ is a well defined injection. Thus $\left|G/N_R\right|\leq\left|Sym(\Gamma/R)\right|<\infty$ as $\left|\Gamma/R\right|<\infty$. So, next we will like to show that $\{N_R\mid R\in I\}$ forms a fundamental system of cofinite congruences over $G$.
1. $D(G)\subseteq N_R$, for all $R\in I$, as $N_R$ is reflexive.
2. Now for some $R,S\in I, (g_1, g_2)\in N_R\bigcap N_S$ if and only if $(g_1.x, g_2.x)\in R\bigcap S$, for all $x\in \Gamma\Leftrightarrow (g_1, g_2)\in N_{R\bigcap S}$. Thus $N_R\bigcap N_S=N_{R\bigcap S}$.
3. For all $N_R, N_R^2= N_R$, as $N_R$ is transitive.
4. For all $N_R, N_R^{-1}=N_R$, as $N_R$ is symmetric.
Hence our claim follows.
We say that a group $G$ acts on a cofinite graph $\Gamma$ residually freely, if there exists a fundamental system of $G$-invariant compatible cofinite entourages $R$ over $\Gamma$ such that the induced group action of $G/N_R$ over $\Gamma/R$ is a free action.
$N_R[1]$ is a finite index normal subgroup of $G$ and $G/N_R[1]$ is isomorphic with $G/N_R$. More generally, if $N$ is a congruence on $G$, then $N[1]$ is a normal subgroup of $G$ and $G/N[1]\cong G/N$.
Let us first see that $N_R[1]\triangleleft_fG$ for all $G$-invariant compatible cofinite entourage $R$ over $\Gamma$. Let $g,h\in N_R[1]$. This implies $(1,g)\in N_R$ and hence $(g,1), (1,h) \in N_R$. Thus $(g,h)\in N_R$. This implies $(g.x,h.x)$ is in $R$, for all $x\in \Gamma$ and so $(x,g^{-1}h.x)\in R$, for all $x\in \Gamma$. Hence, $(1,g^{-1}h)$ is in $N_R$ and thus $g^{-1}h\in N_R[1]$. So, $N_R[1]\leq G$. For all $g\in G$, for all $x\in \Gamma, g.x\in \Gamma$. Hence for all $k\in N_R[1], (x,k.x)\in R$, hence $(k.x,x)$ is in $R$. Thus $(kg.x,g.x)\in R$ and $(g^{-1}kg.x,g^{-1}g.x)=(g^{-1}kg.x,x)\in R$. Hence $(g^{-1}kg,1)\in N_R$. So, $g^{-1}kg\in N_R[1]$ and thus $N_R[1]\triangleleft G$. Now let us define $\eta$ from $G/N_R[1]$ to $G/N_R$ via $\eta(gN_R[1]) = N_R[g]$. Then, $gN_R[1]$ is equal to $hN_R[1]$ if and only if $h^{-1}g\in N_R[1]$ if and only if $(1,h^{-1}g)\in N_R$ if and only if $(x,h^{-1}g.x)\in R$ if and only if $(h.x,g.x)\in R$ if and only if $(h,g)\in N_R$ if and only if $N_R[h]=N_R[g]$, for all $x$ in $\Gamma$. Thus $\eta$ is a well defined injection and hence $\left|G/N_R[1]\right|\leq \left|G/N_R\right|<\infty$. Hence $N_R[1]\triangleleft_fG$. Let us check that $G/N_R$ is a group. For, let $N_R[g_i]$ is in $G/N_R, i=1,2$. Then $N_R[g_1]N_R[g_2]=N_R[g_1g_2]\in G/N_R$. Let $N_R[g_i]$ in $G/N_R$, for $i=1,2,3$. Then $(N_R[g_1]N_R[g_2])N_R[g_3]$ which is equal to $N_R[g_1g_2]N_R[g_3]$ and that equals to $N_R[g_1g_2g_3] = N_R[g_1]N_R[g_2g_3]$ which is equal to $N_R[g_1](N_R[g_2]N_R[g_3])$. For all $N_R[g]\in G/N_R$, there exists $N_R[1]$ in $G/N_R$, such that $N_R[1] N_R[g] = N_R[g] = N_R[g] N_R[1]$. For all $N_R[g]$ in $G/N_R$, there exists $N_R[g^{-1}]$ in $G/N_R$, such that $N_R[g^{-1}]N_R[g]$ equals to $N_R[g^{-1}g] = N_R[1] = N_R[gg^{-1}] = N_R[g]N_R[g^{-1}]$. Hence our claim. Now let us define $\zeta{\colon}G/N_R[1]\to G/N_R$ via $\zeta(gN_R[1])=N_R[g]$. Then for $g_1, g_2$ in $G, g_1N_R[1] = g_2N_R[1]$ if and only if $g_2^{-1}g_1\in N_R[1]$ if and only if $(1,g_2^{-1}g_1)\in N_R$ if and only if $(x,g_2^{-1}g_1.x)\in R$ if and only if $(g_2.x,g_1.x)\in R$ if and only if $(g_2,g_1)\in N_R$ if and only if $N_R[g_2]$ equals to $N_R[g_1]$. Hence $\zeta$ is a well defined injection. Also for all $N_R[g]$ in $G/N_R$, there exists $gN_R[1]\in G/N_R[1]$ such that $\zeta(gN_R[1]) = N_R[g]$. Thus $\zeta$ is surjective as well. Also for $g_1N_R[1], g_2N_R[1]\in G/N_R[1]$, we have $\zeta(g_1N_R[1]g_2N_R[1]) = \zeta(g_1g_2N_R[1])$ and that equals to $N_R[g_1g_2]$ which equals to $N_R[g_1]N_R[g_2] = \zeta(g_1N_R[1])\zeta(g_2N_R[1])$. Hence $\zeta$ is a group homomorphism and thus a group isomorphism. Also, both $G/N_R[1], G/N_R$, are finite discrete topological groups, so $\zeta$ is an isomorphism of uniform cofinite groups as well.
The induced uniform topology over $G$ as in Lemma \[equi continuous\] is Hausdorff if and only if $G$ acts faithfully over $\Gamma$.
Let us first assume that $G$ acts faithfully over $\Gamma$. Now let $g\neq h$ in $G$. Then $h^{-1}g\neq 1$. So there exists $x\in \Gamma$ such that $h^{-1}g.x\neq x$ implying that $ g.x\neq h.x$. Then there exists a $G$-invariant compatible cofinite entourage $R$ over $\Gamma$ such that $(g.x,h.x)\notin R$, as $\Gamma$ is Hausdorff. Hence $(g,h)\notin N_R$. Thus $G$ is Hausdorff.
Conversely, let us assume that $G$ is Hausdorff and let $g\neq 1$ in $G$. Then there exists some $G$-invariant compatible cofinite entourage $R$ over $\Gamma$ such that $(1,g)\notin N_R$. Hence there exists $x\in\Gamma$ such that $(x,g.x)\notin R$. Hence $R[x]\neq R[g.x]$ so that $x\neq g.x$. Our claim follows.
\[uniform continuous group action\] Suppose that $G$ is a group acting uniformly equicontinuously on a cofinite graph $\Gamma$ and give $G$ the induced uniformity as in Lemma \[equi continuous\]. Then the action $G\times\Gamma\to\Gamma$ is uniformly continuous.
Let $R$ be a $G$-invariant cofinite entourage over $\Gamma$. Now let $((g,x),(h,y))\in N_R\times R$, i.e. $(g,h)\in N_R, (x,y)\in R$. Now $x$ in $\Gamma$ and $(gx,hx)\in R$ this implies $(h^{-1}gx,x)\in R$. We have $(h^{-1}gx,y)\in R$ and hence $(gx,hy)\in R$. Thus our claim.
Now if $R\leq S$ in $I$, then $S\subseteq R$. Let $(g_1,g_2)\in N_S$. Then $(g_1x,g_2x)\in S$, for all $x\in\Gamma$ and hence $(g_1x,g_2x)\in R$,for all $x\in \Gamma $ which implies $ (g_1,g_2)\in N_R$. Thus $ N_S\subseteq N_R$. For all $R\leq S$, in $I$, let us define $\psi_{RS}{\colon}G/N_S\to G/N_R$ via $\psi_{RS}(N_S[g])=N_R[g]$. Then $\psi_{RS}$ is a well defined uniformly continuous group isomorphism, as each of $G/N_R, G/N_S$ are finite discrete groups. If $R=S$, then $\psi_{RR}=id_{G/N_R}$. And if $R\leq S\leq T$, then $\psi_{RS}\psi_{ST}=\psi_{RT}$. Then $\{G/N_R\mid R\in I, \psi_{RS}, R\leq S\in I\}$, forms an inverse system of finite discrete groups. Let $\widehat{\Gamma}=\varprojlim_{R\in I}\Gamma/R$ and $\widehat{G}=\varprojlim_{R\in I}G/N_R$, where $\psi_R{\colon}\widehat{G}\to G/N_R$ is the corresponding canonical projection map. Now if $I_{1}, I_{2}$ are two fundamental systems of $G$-invariant cofinite entourages over $\Gamma$, clearly $I_{1}, I_{2}$ will form fundamental systems of cofinite congruences, for two induced uniformities, over $G$. Now let $N_{R_1}$ be a cofinite congruence over $G$ for some $R_1\in I_{1}$. Then there exists a $R_2$, cofinite entourage over $\Gamma$, such that $R_2\in I_{2}$ and $R_2\subseteq R_1$. Hence $N_{R_2}\subseteq N_{R_1}$. Now let $N_{S_2}$ be a cofinite congruence over $G$ for some $S_2\in I_{2}$. Then there exists $S_1$, cofinite entourage over $\Gamma$, such that $S_1\in I_{1}$ and $S_1\subseteq S_2$. Hence $N_{S_1}\subseteq N_{S_2}$. Thus any cofinite congruence corresponding to the directed set $I_{1}$ is a cofinite congruence corresponding to the directed set $I_{2}$ and vice versa. Thus the two induced uniform structures over $G$ are equivalent and so the completion of $G$ with respect to the induced uniformity, from the cofinite graph $\Gamma$, is unique up to both algebraic and topological isomorphism.
If $G$ acts on $\Gamma$, as in Lemma \[equi continuous\], faithfully then $\widehat{G}$ acts on $\widehat{\Gamma}$ uniformly equicontinuously.
The group $G$ acts on $\Gamma$ uniformly equicontinuously. We fix a $G$-invariant orientation $E^+(\Gamma)$ of $\Gamma$. By Lemma \[uniform continuous group action\] the action is uniformly continuous as well. Let $\chi{\colon}G\times\Gamma\to \Gamma$ be this group action. Now since $\Gamma$ is topologically embedded in $\widehat{\Gamma}$ by the inclusion map, say, $i$, the map $i\circ\chi{\colon}G\times\Gamma\to\widehat{\Gamma}$ is a uniformly continuous. Then there exists a unique uniformly continuous map $\widehat{\chi}{\colon}\widehat{G}\times\widehat{\Gamma}\to\widehat{\Gamma}$ that extends $\chi$. We claim that $\widehat{\chi}$ is the required group action. We can take $\widehat{\Gamma} = \varprojlim\Gamma/R$ and $\widehat{G} = \varprojlim G/N_R$, where $R$ runs throughout all $G$-invariant compatible cofinite entourages of $\Gamma$ that are orientation preserving. Then $\widehat{G}\times \widehat{\Gamma} = \varprojlim (G/N_R\times \Gamma/R)$ and $G\times\Gamma$ is defined coordinatewise via $(N_R[g_R])_R.(R[x_R])_R=(R[g_R.x_R])_R$. If possible let, $((N_R[g_R])_R,(R[x_R])_R) = ((N_R[h_R])_R,(R[y_R])_R)$. So, $N_R[g_R]$ equals to $N_R[h_R]$ and $R[x_R]=R[y_R], \forall R\in I, (g_R,h_R)\in N_R$ and $(x_R,y_R) \in R$. This implies that $(g_R.x_R,h_R.x_R)\in R$ which further ensures that $(h_R^{-1}g_R.x_R,x_R)\in R$. Then $(h_R^{-1}g_R.x_R,y_R)\in R$ and $(g_R.x_R,h_R.y_R)\in R$. Hence $(R[g_R.x_R])_R = (R[h_R.y_R])_R$. So, the action is well defined. Let $g=(N_R[g_R])_R$ and $h=(N_R[h_R])_R$ in $\widehat{G}$, $x=(R[x_R])_R\in \widehat{\Gamma}$. Now $h.(g.x) = h.(R[g_R.x_R])_R = (R[h_Rg_R.x_R])_R$ which then equals to$(N_R[h_Rg_R])_R.x = (hg).x$. Hence the action is associative. Now $(N_R[1])_R.(R[x_R])_R = (R[1x_R])_R=(R[x_R])_R$. Furthermore for all $v$ equal to $(R[v_R])_R\in V(\widehat{\Gamma})$ and for all $g$ equal to $(N_R[g_R])_R\in \widehat{G}$ one can say that $g.v = (R[g_R.v_R])_R\in V(\widehat{\Gamma})$ as each $g_R.v_R\in V(\Gamma)$. Similarly, for all $e$ equal to $(R[e_R])_R$ in $E(\widehat{\Gamma})$ and for all $g$ equal to $(N_R[g_R])_R$ in $\widehat{G}$, $g.e = (R[g_Re_R])_R$ in $E(\widehat{\Gamma})$. For all $e$ equal to $(R[e_R])_R$ in $E(\widehat{\Gamma})$, for all $g$ equal to $(N_R[g_R])_R$ in $\widehat{G}$, we have $s(g.e) = s((R[g_Re_R])_R)$ and so $(R[g_Rs(e_R)])_R$ equals to $(g.(R[s(e_R)])_R$ and that equals to $g.s(e)$. Hence the properties $t(g.e)=g.t(e)$ and $\overline{g.e}=g.\overline{e}$ follow similarly. Finally, let $E^+(\widehat{\Gamma})$ consists of all the edges $(R[e_R])_R$, where $e_R\in E^+(\Gamma)$. Since each $R$ is orientation preserving, it follows that $E^+(\widehat{\Gamma})$ is an orientation of $\widehat{\Gamma}$. Since $E^+(\Gamma)$ is $G$-invariant, we see that $E^+(\widehat{\Gamma})$ is $\widehat{\Gamma}$-invariant. Hence this is a well defined group action. Also for all $g\in G$, and $x\in \Gamma$, $(N_R[g])_R.(R[x])_R$ equals to $(R[g.x])_R$ which equals to $g.x$ in $\Gamma$. Thus the restriction of this group action agrees with the group action $\chi$. Now $\{R\mid R\in I\}, \{N_R\mid R\in I\}$ is a fundamental system of cofinite entourages over $\Gamma$, is a fundamental system of cofinite congruences over $G$. Hence $\{\overline{R}\mid R\in I\}$ is a fundamental system of cofinite entourages over $\widehat{\Gamma}$ and $\{\overline{N_R}\mid R\in I\}$ is a fundamental system of cofinite congruences over $\widehat{G}$ respectively. Let us now see that the aforesaid group action is uniformly continuous. For let us consider the group action $G/N_R\times\Gamma/R\to\Gamma/R$ defined via $N_R[g]R[x]=R[g.x]$, which is uniformly continuous as both $G/N_R\times\Gamma/R$ and $\Gamma/R$ are finite discrete uniform topological spaces. Hence the group action, $\widehat{G}\times \widehat{\Gamma}\to\widehat{\Gamma}$ is uniformly continuous. Thus the aforesaid group action is our choice of $\widehat{\chi}$, by the uniqueness of $\widehat{\chi}$. So the restriction of the aforesaid action $\{\widehat{g}\}\times\widehat{\Gamma}\to \widehat{\Gamma}$ is a uniformly continuous map of graphs, for all $\widehat{g}\in \widehat{G}$. We check that for all $(x,y)\in R$ and for all $\widehat{g}\in \widehat{G}$ the ordered pair $(\widehat{g}.x,\widehat{g}.y)\in \overline{R}$ . For, let $\widehat{g}=(N_R[g_R])_R\in \widehat{G}$ and for $x,y\in \Gamma,((R[x])_R, (R[y])_R)\in R$. Now $\overline{R}[(R[g_R.x])_R]=\overline{R}[g_R.x]$ becomes equal to $\overline{R}[g_R.y]=\overline{R}[(R[g_R.y])_R]$. So, $((N_R[g_R])_R(R[x])_R,(N_R[g_R])_R(R[y])_R)\in \overline{R}$. This implies $(\widehat{g}\times\widehat{g})[R]$ is a subset of $\overline{R}$. Thus for all $\widehat{g}\in \widehat{G}$ we observe that $(\widehat{g}\times\widehat{g})[\overline{R}]$ is a sub set of $\overline{\widehat{g}\times\widehat{g}[R]}$ which is a sub set of $\overline{\overline{R}} = \overline{R}$. Hence $\overline{R}$ is $\widehat{G}$ invariant.
Thus $\Phi_1=\{N_{\overline{R}}\mid R\in I\}$ and $\Phi_2=\{\overline{N_R}\mid R\in I\}$ form fundamental systems of cofinite congruences over $\widehat{G}$. Let $\tau_{\Phi_1}, \tau_{\Phi_2}$ be the topologies induced by $\Phi_1, \Phi_2$ respectively.
The uniformities on $\widehat{G}$ obtained by $\Phi_1$ and $\Phi_2$ are equivalent.
Let us first show that $N_{\overline{R}}\cap G\times G = N_R$. For, let $(g,h)\in N_R$. Then for all $x\in \Gamma, (g.x,h.x)\in R\subseteq \overline{R}$. Now let $(R[x_R])_R\in \widehat{\Gamma}$. Then $\overline{R}[g(R[x_R])_R]=\overline{R}[g.x_R]=\overline{R}[h.x_R]=\overline{R}[h(R[x_R])_R]$ which implies that $(g,h)\in N_{\overline{R}}\cap G\times G$. Thus, $N_R \subseteq N_{\overline{R}}\cap G\times G$. Again, if $(g,h)$ belongs to $N_{\overline{R}}\cap G\times G$, then for all $x\in \Gamma\subseteq\widehat{\Gamma}$, and so $(g.x,h.x)\in\overline{R}\cap \Gamma\times \Gamma = R$ and this implies $(g,h)\in N_R$. Our claim follows. Then as uniform subgraphs $(G,\tau_{\Phi_1})\cong(G,\tau_{\Phi_2})$, both algebraically and topologically, their corresponding completions $(\widehat{G},\tau_{\Phi_1})\cong(\widehat{G},\tau_{\Phi_2})$, both algebraically and topologically. Since for all $S\in I$, $\psi_S{\colon}G\to G/N_S$ is a uniform continuous group homomorphism and $G/N_S$ is discrete, there exists a unique uniform continuous extension of $\psi_S$, namely, $\widehat{\psi_S}{\colon}\widehat{G}\to G/N_S$. Let us define $\lambda_S{\colon}\widehat{G}\to G/N_S$ via $\lambda_S(g)=N_S[g_S]$, where $g=(N_R[g_R])_R$. Now let $g=(N_R[g_R])_R, h=(N_R[h_R])_R\in\widehat{G}$ be such that $g=h$ which implies that $ N_S[g_S]=N_S[h_S]$ and hence $\lambda_S$ is well defined. Now let $(g,h)\in N_{\overline{S}}$. First of all $N_{\overline{S}}[g_S]=N_{\overline{S}}[g]=N_{\overline{S}}[h]=N_{\overline{S}}[h_S]$. So, $(g_S,h_S)\in N_{\overline{S}}\bigcap G\times G=N_S$. Hence $N_S[g_S]=N_S[h_S]$ which implies that $ \lambda_S(g)=\lambda_S(h)$, so $(\lambda_S(g),\lambda_S(h))\in D(G/N_R)$. Thus $N_{\overline{S}}$ is a sub set of $(\lambda_S\times \lambda_S)^{-1}D(G/N_R)$. Hence $\lambda_S$ is uniformly continuous. Now for all $g, h\in\widehat{G}, \lambda_S(gh)=N_S[g_Sh_S]=N_S[g_S]N_S[h_S]=\lambda_S(g)\lambda_S(h)$ and for all $g\in G, \lambda_S(g)=\lambda_S((N_R[g])_R)=N_S[g]=\psi_S(g)$. Thus $\lambda_S$ is an well defined uniformly continuous group homomorphism that extends $\psi_S$. Then by the uniqueness of the extension, $\widehat{\psi_S}=\lambda_S$. Now $N_{\overline{S}}$ is a closed subspace of $\widehat{G}$, then $\overline{N_{\overline{S}}\cap G\times G} = \overline{N_S}$ which implies that $\overline{N_S}$ is a sub set of $\overline{N_{\overline{S}}}$ which equals to $N_{\overline{S}}$. Let us define $\theta$ from $\widehat{G}/N_{\overline{S}}$ to $G/N_S$ as $\theta$ takes $N_{\overline{S}}[g]$ into $N_S[g_S]$, where $g=(N_R[g_R])_R$. Now $N_{\overline{S}}[g]=N_{\overline{S}}[h]$ in $\widehat{G}/N_{\overline{S}}$ will imply $(g_S,h_S)$ is in $N_{\overline{S}}$ and this implies for all $x$ in $X$ the ordered pair $(g_Sx,h_Sx)$ is in $\overline{S}\bigcap\Gamma\times \Gamma$ which is eventually equal to $S$. Thus $(g_S,h_S)\in N_S$. Then $\theta(N_{\overline{S}}[g])$ equals to $N_S[g_S]$ which is equal to $N_S[h_S]$ and that equals $\theta(N_{\overline{S}}[h])$. Hence $\theta$ is well defined. On the other hand let $N_{\overline{S}}[g]$, $N_{\overline{S}}[h]$ be such that $\theta(N_{\overline{S}}[g])$ equals $\theta(N_{\overline{S}}[h])$. Thus $N_S[g_S]$ equal to $N_S[h_S]$ implies that $(g_S,h_S)\in N_S\subseteq N_{\overline{S}}$. Hence $N_{\overline{S}}[g]=N_{\overline{S}}[g_S] = N_{\overline{S}}[h_S]=N_{\overline{S}}[h]$. So, $\theta$ is injective as well. Also for all $N_S[g]\in G/N_S$ there exists $N_{\overline{S}}[g]\in\widehat{G}/N_{\overline{S}}$ such that $\theta(N_{\overline{S}}[g])=N_S[g]$. So $\theta$ is surjective. Finally, $\theta(N_{\overline{S}}[g]N_{\overline{S}}[h])$ equals to $\theta(N_{\overline{S}}[gh])$ and that equals to $N_S[g_Sh_S]$ which is $N_S[g_S]N_S[h_S]$ and finally that equals to $\theta(N_{\overline{S}}[g])\theta(N_{\overline{S}}[h])$. So $\theta$ is an well defined group isomorphism, both algebraically and topologically. Hence $\widehat{G}/N_{\overline{S}}\cong G/N_S\cong\widehat{G}/\overline{N_S}$ which implies that $ \left|\widehat{G}/N_{\overline{S}}[1]\right|$ is equal to $\left|\widehat{G}/\overline{N_S}[1]\right|$. But since $\overline{N_S}\subseteq N_{\overline{S}}$ one obtains $\overline{N_S}[1]\leq N_{\overline{S}}[1]\leq\widehat{G}$ and thus $\left|\widehat{G}/N_{\overline{S}}[1]\right|\left|N_{\overline{S}}[1]:\overline{N_S}[1]\right|$ equals to $\left|\widehat{G}/\overline{N_S}[1]\right|$. Hence $\left|N_{\overline{S}}[1]:\overline{N_S}[1]\right|=1$ which implies that $ N_{\overline{S}}[1]=\overline{N_S}[1]$ and thus $ N_{\overline{S}}=\overline{N_S}$ as each of them are congruences. Thus our claim.
[1]{}
N. Bourbaki, *General [T]{}opology*, Elements of [M]{}athematics, Addison-Wesley, Reading Mass., 1966.
B. Hartley, *Profinite and residually finite groups*, Rocky Mountain J. Math **7** (1977), 193–217.
J. Kelley, *General [T]{}opology*, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955.
J.-P. Serre, *[T]{}rees*, Springer Verlag, Berlin, Heidelberg, New York, 1980.
J. R. Stallings, *Topology of finite graphs*, Invent. Math. **71** (1983), 551–565.
J. Wilson, *[P]{}rofinite [G]{}roups*, Oxford University Press, Oxford, 1998.
B. Das, *[C]{}ofinite [G]{}raphs and [T]{}heir [P]{}rofinite [C]{}ompletions*, proquest, acumen.lib.ua.edu, 2013.
P. A. Zalesskii, *Geometric characterization of free constructions of profinite groups*, Siberian Math J. **30** (1989), 227–235.
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abstract: 'Astrocytes express a large variety of G protein-coupled receptors (GPCRs) which mediate the transduction of extracellular signals into intracellular calcium responses. This transduction is provided by a complex network of biochemical reactions which mobilizes a wealth of possible calcium-mobilizing second messenger molecules. Inositol 1,4,5-trisphosphate is probably the best known of these molecules whose enzymes for its production and degradation are nonetheless calcium-dependent. We present a biophysical modeling approach based on the assumption of Michaelis-Menten enzyme kinetics, to effectively describe GPCR-mediated astrocytic calcium signals. Our model is then used to study different mechanisms at play in stimulus encoding by shape and frequency of calcium oscillations in astrocytes.'
author:
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Maurizio De Pittà\
EPI BEAGLE, INRIA Rhône-Alpes, Villeurbanne, France
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Eshel Ben-Jacob[^1]\
School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Israel
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Hugues Berry\
EPI BEAGLE, INRIA Rhône-Alpes, Villeurbanne, France
bibliography:
- './ch5\_depitta.bib'
title: 'G protein-coupled receptor-mediated calcium signaling in astrocytes'
---
Introduction
============
Calcium signaling is the most common measured readout of astrocyte activity in response to stimulation, be it by synaptic activity, by neuromodulators diffusing in the extracellular milieu, or by exogenous chemical, mechanical or optical stimuli. In this perspective, the individual astrocytic transient is thought, to some extent, as an integration of the triggering stimulus [@PereaAraque_JNT2005], and is thus regarded as an encoding or decoding of this stimulus, depending on the point of view [@Carmignoto2000; @DePitta_FCN13].
Multiple and varied are the spatiotemporal patterns of elevations recorded from astrocytes in response to stimulation, each possibly carrying its own encoding [@Bindocci_Science2017]. Insofar as different encoding modes could correspond to different downstream signaling, including gliotransmission and thereby regulation of synaptic function, understanding the biophysical mechanisms underlying rich dynamics in astrocytes is crucial.
Calcium-induced release (CICR) from the endoplasmic reticulum (ER) is arguably the best characterized mechanism of signaling in astrocytes [@Zorec_ASN2012]. It ensues from nonlinear properties of channels which are found on the ER membrane and are gated by the combined action of cytosolic and the second messenger molecule inositol 1,4,5-trisphosphate (3) [@Shinohara_PNAS2011 see also ]. This second messenger molecule can be produced by the astrocyte either spontaneously or, notably, in response to activation by extracellular insults activation of G protein-coupled receptors (GPCRs) found on the cell’s plasma membrane [@ParriCrunelli_Neuroscience2003; @Panatier_etal_Cell2011; @Volterra_NRN2014]. Hence, 3 together with these receptors, can be regarded as integral components of the interface whereby an astrocyte transduces extracellular insults into responses [@Marinissen_TiPS2001]. Characterizing this interface is thus an essential step in our understanding of the emerging complexity of signals, and we devote this chapter to this purpose. In the first part of the chapter, we will present a concise framework to model intracellular 3 signaling in astrocytes. This framework is general and can easily be extended to include additional biological details, such as for example, the regulation of GPCR binding efficiency by protein kinase C. Some of the models presented in this chapter are also subjected to revision and comparison with other astrocyte models in and .
Modeling intracellular 3 dynamics {#sec:ip3-modeling}
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Agonist-mediated 3 production {#sec:PLCb-production}
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G protein-coupled receptors form a large family of receptors which owe their name to their extensively studied interaction with heterotrimeric G proteins (composed of an $\upalpha$, $\upbeta$ and $\upgamma$ subunit) which undergo conformational changes that lead to the exchange of GDP for GTP, bound to the $\upalpha$-subunit, following receptor activation. Consequently, the G$\upalpha$- and G$\upbeta \upgamma$-subunits stimulate enzymes thereby activating or inhibiting the production of a variety of second messengers [@Marinissen_TiPS2001].
Among all GPCRs, those that contain the G$\upalpha_\mathrm{q}$ subunit are linked with the cascade of chemical reactions that leads to 3 synthesis. There, the G$\upalpha_\mathrm{q}$ subunit promotes activation of the enzyme pospholipase C$\upbeta$ () which hydrolizes the plasma membrane lipid phosphatidylinositol 4,5-bisphosphate (2) into diacylglycerol (DAG) and 3 [@RebecchiPentyala2000]. Examples of such receptors expressed by astrocytes ex vivo and in vivo are the type I metabotropic glutamate receptor 1 and 5 (mGluR1/5) [@WangNedergaard2006; @Sun_Science2013], the purinergic receptor [@JourdainVolterra2007; @DiCastro_Volterra_NatNeurosci2011; @Sun_Science2013], the muscarinic receptor mAchR1$\upalpha$ [@Takata_JN2011; @Chen_PNAS2012; @Navarrete_PB2012] and the adrenergic $\upalpha_1$ receptor [@Bekar_CC2008; @Ding_CC2013]. While these receptors bind different agonists, and likely display receptor-specific binding kinetics, they all share the same downstream signaling pathway and therefore may be modeled in a similar fashion.
Several are the available models for G$\upalpha_\mathrm{q}$-containing receptors, and the choice of what model to use rather than another depends on the level of biological detail and the questions one is interested in. Here our focus is on the rate of 3 production upon activation of these receptors, so we wish to keep as simple as possible the description of the reactions that regulate the activation of by $\upalpha_\mathrm{q}$, $\upbeta$ and $\upgamma$ subunits. This is possible, assuming that these reactions are much faster than the downstream ones that result in 3 production. In this case, a *quasi steady-state approximation* (QSSA) holds whereby, in the series of reactions that leads from receptor agonist binding to activation of , the intermediate reactions involving the three receptor’s subunits are at equilibrium on the time scale of the production of activated . Accordingly, assuming that on average the receptor at rest () requires $n$ molecules of an agonist () to promote activation of () at rate $O_{N}$, we can write $$\ce{R + $n$A ->T[$O_{N}$] R$^*$}\label{rc:R-activation}$$
We further make another assumption: that the cascade of reactions that leads to GPCR-mediated 3 synthesis has a Michaelis-Menten kinetics (see [Appendix \[app:Michaelis-Menten-kinetics\]]{}), so the 3 production by ($J_{\beta}$) can be taken proportional to the fraction of bound receptors, defined as $\Gamma_{A}=\ce{[R^{*}]/[R]_{T}}$, with being the total receptor concentration at the site of 3 production, i.e., $$\label{eq:Jbeta}
J_{\beta} = O_\beta \cdot \Gamma_{A}$$ In the above equation $O_\beta$ is the maximal rate of 3 production by and lumps information on receptor surface density as well as on the size of the 2 reservoir. Importantly, these two quantities may not be fixed, insofar as receptors are subjected to desensitization, internalization and recycling, and the reservoir of 2 could also be modulated by cytosolic and 3 [@RheeBaeRev1997]. The reader interested in modeling these aspects may refer to @Lemon_JTB2003. In the following, we will assume $O_\beta$ constant for simplicity.
To seek an expression for $J_{\beta}$, termination of signaling has to be considered. With this regard, as illustrated in [ \[fig:ip3-production\]]{}A, there are two possible pathways whereby 3 production by ends [@RebecchiPentyala2000]. One is by reconstitution of the inactive G protein heterotrimer, and coincides with unbinding of the agonist from the receptor, due to the intrinsic GTPase activity of the activated G$\upalpha_\mathrm{q}$ subunit. The other is by phosphorylation of the receptor, the G$\upalpha_\mathrm{q}$ subunit, or some combination thereof by conventional protein kinases C () [@RyuRhee_JBC1990; @CodazziTeruelMeyer2001]. This phosphorylation modulates either receptor affinity for agonist binding, or coupling of the bound receptor with the G protein, or coupling of the activated G protein with , ultimately resulting in receptor desensitization [@Fisher_RevEJP1995].
Denoting by the active, receptor-phosphorylating kinase C, termination of -mediated 3 production can then be modeled by the following pair of chemical reactions: $$\begin{aligned}
\ce{R^* & ->T[$\Omega_N$] R + $n$A \label{rc:R-unbinding}\\
cPKC^* + R^* & <=>T[$O_{KR}$][$\Omega_{KR}$] cPKC^*-R^* ->T[$\Omega_{K}$] cPKC^* + R + $n$A} \label{rc:R-PKC*-reaction}\end{aligned}$$ From equations \[rc:R-unbinding\]–\[rc:R-PKC\*-reaction\] we have: $$\begin{aligned}
{\frac{\mathrm{d}\ce{R^*}}{\mathrm{d}t}} &= O_N \ce{[A]^n} \ce{[R]} - \Omega_N \ce{[R^*]} - O_{KR} \ce{[cPKC^*][R^*]} + \Omega_{KR} \ce{[cPKC^*-R^*]} \label{eq:R*-ode}\\
{\frac{\mathrm{d}\ce{[cPKC^*-R^*]}}{\mathrm{d}t}} &= O_{KR} \ce{[cPKC^*][R^*]} - (\Omega_{KR} + \Omega_K) \ce{[cPKC^*-R^*]} \label{eq:PR-ode}\end{aligned}$$ Assuming that production of the intermediate kinase-receptor complex is at quasi steady state in reaction \[rc:R-PKC\*-reaction\], i.e. ${\ensuremath{\mathrm{d}\ce{[cPKC^*-R^*]}}}/{\ensuremath{\mathrm{d}t}} \approx 0$, provides ([equation \[eq:PR\]]{}) $$\ce{[cPKC^*-R^*]} = \frac{O_{KR}}{\Omega_{KR} + \Omega_K} \ce{[cPKC^*][R^*]} \label{eq:PR-qssa}$$ Then, substituting this latter equation in [equation \[eq:R\*-ode\]]{} gives $$\begin{aligned}
{\frac{\mathrm{d}\ce{R^*}}{\mathrm{d}t}} &= O_N \ce{[A]^n} \ce{[R]} - \Omega_N \ce{[R^*]} - O_{KR} {\left(1-\frac{\Omega_{KR}}{\Omega_{KR} + \Omega_K}\right)} \ce{[cPKC^*][R^*]} \nonumber \\
&= O_N \ce{[A]^n} \ce{[R]} - \Omega_N \ce{[R^*]} - O_K \ce{[cPKC^*][R^*]} \label{eq:R*-ode-qssa}\end{aligned}$$ where we defined $O_K = O_{KR} {\left(1-\Omega_{KR}/{\left(\Omega_{KR} + \Omega_K\right)}\right)}$.
To retrieve an equation for , we consider the fact that activation of requires binding to the kinase of free cytosolic ($C$) and DAG, but only if binds first, can get sensibly activated by DAG [@Oancea_Cell1998]. Accordingly, the following sequential binding reaction scheme for activation may be assumed: $$\begin{aligned}
\ce{cPKC + Ca^{2+}} &\ce{<=>T[$O_{KC}$][$\Omega_{KC}$] cPKC$\,'$}\label{rc:PKC*-reaction-ca}\\
\ce{cPKC$\,'$ + DAG}&\ce{<=>T[$O_{KD}$][$\Omega_{KD}$] cPKC^*}\label{rc:PKC*-reaction-dag}\end{aligned}$$ where is the inactive kinase, and denotes the -bound kinase complex. By QSSA in reaction \[rc:R-PKC\*-reaction\] it follows that the available activated kinase approximately equals to . Moreover, it can be assumed that only a small fraction of is bound by DAG so that $\ce{[cPKC^*] \ll \ce{[cPKC$\,'$]}}$. In this fashion, the available , denoted by , can be approximated by . Accordingly, solving reactions \[rc:PKC\*-reaction-ca\] and \[rc:PKC\*-reaction-dag\] for provides $$\begin{aligned}
\ce{[cPKC^*]} &= {\left(\ce{[cPKC^*] + [cPKC$\,'$]}\right)}\cdot {\ensuremath{\mathcal{H}_{1}\left(\ce{[DAG]},K_{KD}\right)}}\nonumber\\
&\approx \ce{[cPKC$\,'$]} \cdot {\ensuremath{\mathcal{H}_{1}\left(\ce{[DAG]},K_{KD}\right)}}\nonumber\\
&= \ce{[cPKC]_T}\cdot {\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}} \cdot {\ensuremath{\mathcal{H}_{1}\left(\ce{[DAG]},K_{KD}\right)}}\label{eq:PKC*-solution}\end{aligned}$$ where $K_{KD}=\Omega_{KD}/O_{KD}$ and $K_{KC}=\Omega_{KC}/O_{KC}$, and ${\ensuremath{\mathcal{H}_{1}\left(x,K\right)}}$ denotes the Hill function $x/(x+K)$ ([Appendix \[app:Hill-function\]]{}). In practice the activation of the kinase consists of two sequential translocations to the plasma membrane of its C2 and C1$_2$ domains [@Oancea_Cell1998]. The translocation of C2 is regulated by whereas that of C1$_2$ is by DAG. In this process however, experiments showed that the initial translocation of C2 is the rate limiting step for kinase activation [@ShinomuraNishizuka1991], inasmuch as C1$_2$ translocation rapidly follows that of C2 [@CodazziTeruelMeyer2001]. This agrees with the notion that the affinity for DAG is regarded to be much higher than the affinity of the kinase for , i.e. $K_{KD}\ll K_{KC}$ [@NishizukaRev1995]. Since the product of two Hill functions with widely separated constants can be approximated by the Hill function with the largest constant [@DePitta_JOBP2009], [equation \[eq:PKC\*-solution\]]{} can be rewritten as $$\ce{[cPKC^*]} \approx \ce{[cPKC]_T} \cdot {\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}}$$ which, once replaced in [equation \[eq:R\*-ode-qssa\]]{}, gives: $$\begin{aligned}
{\frac{\mathrm{d}\ce{[R^*]}}{\mathrm{d}t}} &= O_{N}\ce{[A]$^{n}$[R]} - \Omega_{N}{\left(1+\frac{O_{K}\ce{[cPKC]_T}}{\Omega_{N}}\,{\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}}\right)}\ce{[R^*]} \label{eq:dRdt}\end{aligned}$$ Finally, dividing both left and right terms in the above equation by , [equation \[eq:dRdt\]]{} can be rewritten as $$\label{eq:GammaA}
{\frac{\mathrm{d}\Gamma_{A}}{\mathrm{d}t}} = O_{N}\ce{[A]$^{n}$}\,{\left(1-\Gamma_{A}\right)} - \Omega_{N}{\left(1+\zeta \cdot {\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}}\right)}\,\Gamma_{A}$$ where $\zeta=O_{KC}\ce{[cPKC]_T}/\Omega_N$ quantifies the maximal receptor desensitization by cPKC. In the approximation that receptor binding and activation is much faster than the effective -mediated 3 production, $\Gamma_{A}$ can be solved for the steady state. In this fashion, 3 production by in [equation \[eq:Jbeta\]]{} becomes $$\begin{aligned}
J_{\beta} &= O_{\beta} \cdot {\ensuremath{\mathcal{H}_{n}\left(\ce{[A]},{\left(K_{N}{\left(1+\zeta\,{\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}}\right)}\right)}^\frac{1}{n}\right)}}\label{eq:Jbeta-ssa}\end{aligned}$$ where $K_{N}=\Omega_N/ O_N$. The Hill coefficient $n$ denotes cooperativity of the binding reaction of the agonist with the receptor and is both receptor and agonist specific. For example, glutamate binding to subtype 1 mGluRs, such as those expressed by astrocytes [@GalloGhiani2000], is characterized by negative cooperativity and found in association with a Hill coefficient of $n=0.48-0.88$ [@Suzuki2004]. On the contrary, binding of ATP to P$_{2}$Y$_{1}$Rs of dorsal spinal cord astrocytes from rats is characterized instead by almost no cooperativity and $n=0.9-1$ [@Fam_JN2000].
3 production by receptors with $\upalpha$ subunits other than q-type
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A series of other astrocytic GPCRs, that traditionally associate with non-$\upalpha_\mathrm{q}$ subunits, have also been reported to mediate 3-triggered CICR, both in situ and in vivo. These include G$\upalpha_{\mathrm{i/o}}$-coupled receptors [@KangNedergaard1998; @Serrano_etal_JN2006; @Mariotti_Glia2016], endocannabinoid receptors [@NavarreteAraque_Neuron2008; @MinNevian_NatNeurosci2012], adenosinergic receptors [@Cristovao-Ferreira_PS2013], adrenergic $\upalpha_2$ receptors [@Bekar_CC2008], and dopaminergic receptors [@Jennings_Glia2017]; as well as G$\upalpha_{\mathrm{s}}$-coupled receptors like adenosine receptors [@Cristovao-Ferreira_PS2013], and dopamine receptors [@Jennings_Glia2017]. $\upalpha_\mathrm{i/o}$ and $\upalpha_\mathrm{s}$ subunits are not expected to be linked with 3 synthesis [@Marinissen_TiPS2001], rather they respectively inhibit or stimulate intracellular production of cAMP. Therefore the mechanism whereby these receptors could also promote mobilization of from 3-sensitive ER stores remains a matter of investigation.
One obvious possibility is that some of these receptors could be atypical in astrocytes and also be coupled with G$\upalpha_\mathrm{q}$, as it seems the case for example of astrocytic in the hippocampus [@NavarreteAraque_Neuron2008] and in the basal ganglia [@Martin_Science2015]. Biased agonism could also be another possibility since the spatiotemporal pattern of agonist action on GPCRs could be quite different depending on agonist-binding kinetics of the receptor, especially if agonists differentially engage dynamic signalling and regulatory processes [@Overington_NRDD2006], such as in the likely scenario of synapse-astrocyte interactions [@Heller_Glia2015]. However, there is not yet direct structural evidence for distinct receptor conformations linked to specific signals such as distinct G protein classes, and future studies are required to compare crystal structures of astrocytic GPCRs bound to biased and unbiased ligands to establish these relationships [@Violin_TiPS2014].
Alternatively, other signaling pathways mediated by cAMP that result in CICR could also be envisaged. In particular, @Doengi_PNAS2009 reported that GABA-evoked astrocytic events in the olfactory bulb are fully prevented by blockers of astrocytic GABA transporters (GATs), but only partially by antagonists. GAT activation leads to an increase of intracellular , since this ion is cotransported with GABA, and such increase indirectly inhibits the exchanger on the plasma membrane. In turn, the ensuing increase could be sufficient to induce release from internal stores by stimulation of endogenous 3 production [@Losi_PTRSB2014 see the following Section]. This possibility is further corroborated by the observation that astrocytic GATs could indeed be inhibited or stimulated respectively by or [@Cristovao-Ferreira_PS2013].
Yet other mechanisms could be at play for different receptors. Dopaminergic receptors for example could either increase ( receptors) or decrease ( receptors) intracellular levels in astrocytes [@Jennings_Glia2017]. This could indeed be explained assuming a possible action of these receptors on GATs which, similarly to adenosinergic receptors, could respectively increase or reduce cotransport into the cell, ultimately promoting or inhibiting CICR according to what was suggested for . However there is also evidence that nontoxic levels of dopamine could be metabolized by monoamine-oxidase in cultured astrocytes, resulting in the production of hydrogen peroxide [@Vaarmann_JBC2010]. This reactive oxygen species ultimately activates lipid peroxidation in the neighboring membranes which in turn triggers PLC-mediated 3 production and CICR. Overall these different scenarios unravel additional complexity in the possible mechanisms of GPCR-mediated CICR in astrocytes and call for future modeling efforts that are beyond the scope of this chapter.
Endogenous 3 production
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Phospholipase C$\updelta$ () is the enzyme responsible of endogenous 3 production in astrocytes, that is 3 production that does not require external (i.e. exogenous) stimulation [@Ochocka_ABP2003; @Suh_BMB2008]. The specific catalytic activity of this enzyme in the presence of cytosolic is 50- to 100-fold greater than -stimulated activity of in the absence of activating G protein subunits [@RebecchiPentyala2000], suggesting that is prominently activated by increases of intracellular [@RheeBaeRev1997].
[ \[fig:ip3-production\]]{}B exemplifies the biochemical network associated with activation. Structural and mutational studies of complexes with and 3, revealed complex interactions of with several negatively charged residues within the catalytic domain [@EssenWilliams_Nature1996; @EssenWilliams_Biochem1997; @RheeBaeRev1997], hinting cooperative binding of at least two ions with this enzyme [@EssenWilliams_Biochem1997]. In agreement with these experimental findings, we model -mediated 3 production ($J_{\delta}$) as [@PawelczykMatecki_EurJBiochem1997; @HoferGiaume2002]: $$\label{eq:Jdelta}
J_{\delta} = \hat{J}_{\delta}(I)\cdot{\ensuremath{\mathcal{H}_{2}\left(C,K_{\delta}\right)}}$$ where [$\mathcal{H}_{2}\left(C,K_{\delta}\right)$]{} denotes the Hill function of $C$ with coefficient 2 and affinity $K_{\delta}$ ([Appendix \[app:Parameters\]]{}), and $\hat{J}_{\delta}(I)$ is the maximal rate of 3 production by which depends on intracellular 3 ($I$). Experiments revealed that high 3 concentrations, i.e. $>\SI{1}{\micro \Molar}$, inhibit activity by competing with 2 binding to the enzyme [@AllenBarres_Nature2009]. Accordingly, the maximal dependent 3 production rate can be modeled by $$\label{eq:Od-max}
\hat{J}_{\delta}(I)=\frac{O_{\delta}}{1+\frac{I}{\kappa_{\delta}}}=O_{\delta}{\left(1-{\ensuremath{\mathcal{H}_{1}\left(I,\kappa_\delta\right)}}\right)}$$ where $O_{\delta}$ is the maximal rate of 3 production by and $\kappa_{\delta}$ is the inhibition constant of activity.
3 degradation
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There are two pathways for 3 degradation in astrocytes. The first one is by dephosphorylation of 3 by inositol polyphosphate 5-phosphatase (p). The other one occurs through phosphorylation of 3 by the 3 3-kinase (k). Both pathways could be dependent but in opposite ways: while the activity of k is stimulated by cytosolic [@CommuniErneux1997], p is inhibited instead [@CommuniErneux2001] ([ \[fig:ip3-degradation\]]{}A). Thus, depending on the concentration in the cytoplasm, different mechanisms of 3 degradation could exist [@SimsAllbritton1998]. Moreover, p-mediated 3 degradation could also be inhibited by competitive binding of inositol 1,3,4,5-tetrakisphosphate (IP$_{4}$) produced by k-mediated 3 phosphorylation [@Connolly_JBC1987; @Erneux_BBA1998], thereby making the two degradation pathways interdependent [@Hermosura_Nature2000]. However, we will not consider this aspect any further, since modeling of this reaction pathway requires a detailed consideration of the complex metabolic network underpinning degradation of the large family of inositol phosphates [@CommuniErneux2001; @Irvine_NRMCB2001]. The reader interested in these aspects may refer to @DupontErneux1997 for a sample modeling approach to the problem.
Both p-mediated dephosphorylation ($J_{5P}$) and k-mediated phosphorylation of 3 ($J_{3K}$) can be described by Michaelis-Menten kinetics [@IrvineBerridge_Nat1986; @TogashiOnaya_BiochemJ1997], i.e., $$\begin{aligned}
J_{5P} &= \hat{J}_{5P}\cdot {\ensuremath{\mathcal{H}_{1}\left(I,K_{5}\right)}}\label{eq:J5P-generic}\\
J_{3K} &= \hat{J}_{3K}(C)\cdot {\ensuremath{\mathcal{H}_{1}\left(I,K_{3}\right)}}\label{eq:J3K-generic}\end{aligned}$$ Since $K_{5P}>\SI{10}{\micro \Molar}$ [@VerjansErneux1992; @SimsAllbritton1998], and such high 3 concentrations are unlikely to be physiological [@Lemon_JTB2003; @KangOthmer_Chaos2009], the activity of p can be assumed far from saturation. Accordingly, the 3 degradation rate by p can be linearly approximated by [@StryerBiochemistryBOOK]: $$\label{eq:J5P-linear-approx}
J_{5P} \approx \Omega_{5P}\cdot I$$ where $\Omega_{5P}=\hat{J}_{5P}/K_{5}$ is the maximal rate of p-mediated 3 degradation in the linear approximation.
3 phosphorylation by k is regulated in a complex fashion ([ \[fig:ip3-degradation\]]{}A). For resting conditions, when intracellular 3 and concentrations are below 0.1 $\mu$M, [@ParpuraHaydon2000; @MishraBhalla2002; @KangOthmer_Chaos2009], it is very slow. On the other hand, as increases, k activity is substantially stimulated by its phosphorylation by CaMKII in a /calmodulin (CaM)–dependent fashion [@CommuniErneux1997]. A further possibility could eventually be that k is also inhibited by -dependent PKC phosphorylation [@SimRhee_JBC1990], however, since evidence for the existence of such inhibitory pathway is contradictory [@CommuniErneux_Rev1995], this possibility will not be taken into further consideration in this study.
Phosphorylation of k by active (i.e. ) only occurs at a single threonine residue [@CommuniErneux1997; @CommuniErneux_JBC1999], so that it can be assumed that the rate of k phosphorylation is $J_{3K}^*(C)\propto \ce{[CaMKII^*]}$. On the other hand, activation of is /CaM-dependent and occurs in a complex fashion because of the unique structure of this kinase, which is composed of $\sim$12 subunits, with three to four phosphorylation sites each [@KolodziejStoops_JBC2000]. Briefly, increases lead to the formation of a complex () that may induce phosphorylation of some of the sites of each subunit. However, only when two of these sites at neighboring subunits are phosphorylated, CaMKII quickly and fully activates [@HansonSchulman_Neuron1994]. Despite the multiple binding reactions in the inactive kinase, experiments showed that activation by can be approximated by a Hill equation with unitary coefficient [@DeKonickSchulman1998]. Hence, the following kinetic reaction scheme for phosphorylation can be assumed: $$\begin{aligned}
\ce{4 Ca^{2+} + CaM} & \ce{<=>T[$O_{0}$][$\Omega_{0}$] CaM^+}\label{rc:Ca-binding}\\
\ce{KII + CaM^+} & \ce{<=>T[$O_{b}$][$\Omega_{b}$] CaMKII <=>T[$\Omega_{a}$][$\Omega_{i}$] CaMKII^*} \label{rc:KII-activation}\end{aligned}$$ Consider then first the binding reaction in \[rc:KII-activation\]. Assuming that the second step is very rapid with respect to the first one [@ThielGreengard_PNAS1988; @DeKonickSchulman1998], the generation of is in equilibrium with consumption, i.e., $$\begin{aligned}
\label{eq:CaMKII*}
\ce{[CaMKII^*]} &\approx \frac{\Omega_{a}}{\Omega_{i}}\ce{[CaMKII]}\end{aligned}$$ Then, under the hypothesis of quasi-steady state for CaMKII, $$\begin{aligned}
{\frac{\mathrm{d}\ce{[CaMKII]}}{\mathrm{d}t}} &= O_{b}\,\ce{[KII][CaM^+]}-{\left(\Omega_{a}+\Omega_{b}\right)}\ce{[CaMKII]}+\Omega_{i}\,\ce{[CaMKII^*]}\approx 0\end{aligned}$$ Replacing from [equation \[eq:CaMKII\*\]]{} in the latter equation provides $$\begin{aligned}
\label{eq:CaMKII*-equation-2}
\ce{[CaMKII^*]} &= K_{a}K_{b}\ce{[KII][CaM^+]}\end{aligned}$$ where $K_{a}=\Omega_{a}/ \Omega_{i}$ and $K_{b}= O_{b}/ \Omega_{b}$. Defining the total kinase II concentration as and assuming it constant, [equation \[eq:CaMKII\*-equation-2\]]{} can be rewritten as $$\begin{aligned}
\label{eq:CaMKII*-equation-3}
\ce{[CaMKII$^{*}$]} &= \frac{K_{a}\ce{[KII]$_{T}$}}{1+K_{a}}\cdot {\ensuremath{\mathcal{H}_{1}\left(\ce{[CaM$^{+}$]},K_{m}\right)}}\end{aligned}$$ with $K_{m}={\left(K_{b}{\left(1+K_{a}\right)}\right)}^{-1}$.
The substrate concentration for the enzyme-catalyzed reaction \[rc:KII-activation\] is provided by reaction \[rc:Ca-binding\] and reads (by QSSA) $$\begin{aligned}
\label{eq:CaM+}
\ce{[CaM$^{+}$]} &= \ce{[CaM]}\cdot {\ensuremath{\mathcal{H}_{4}\left(C,K_0\right)}}\end{aligned}$$ with $K_{0}=O_{0}/ \Omega_{0}$. Therefore, replacing the latter expression for in [equation \[eq:CaMKII\*-equation-3\]]{}, finally provides $$\begin{aligned}
\ce{[CaMKII^*]} &= \frac{K_{a}\ce{[KII]_T}}{1+K_a}\,{\left(1+\frac{K_m}{\ce{[CaM]}}\right)}^{-1}\cdot {\ensuremath{\mathcal{H}_{4}\left(C,\frac{K_0 K_m}{K_m + \ce{[CaM]}}\right)}}\end{aligned}$$ Defining the affinity constant of k as $K_{D}=K_0 K_m /{\left(K_m +\ce{[CaM]}\right)}$, the above calculations show that, despite its complexity, the reaction cascade underlying the activation of can be concisely described by a Hill function of the concentration ($C$) so that $\ce{[CaMKII^*]}\propto {\ensuremath{\mathcal{H}_{4}\left(C,K_D\right)}}$. Accordingly, it is also $\hat{J}_{3K}(C)\propto {\ensuremath{\mathcal{H}_{4}\left(C,K_D\right)}}$, and [equation \[eq:J3K-generic\]]{} for k-mediated 3 degradation can be rewritten as $$\label{eq:J3K-exact}
J_{3K} = O_{3K}\cdot {\ensuremath{\mathcal{H}_{4}\left(C,K_{D}\right)}} {\ensuremath{\mathcal{H}_{1}\left(I,K_{3}\right)}}$$ where $O_{3K}$ is the maximal rate of 3 degradation by k.
Encoding of stimulation by combined 3 and dynamics
===================================================
The model for 3/ signaling
---------------------------
A corollary of the biological and modeling arguments exposed in the previous section is that and 3 signals are, generally speaking, dynamically coupled in astrocytes. This implies that a complete model that mimics astrocytic 3 signaling must also include a description of CICR. An example of such models is the so-called $ChI$ model originally introduced by @DePitta_JOBP2009, which is constituted by three ODEs respectively for intracellular ($C$), the 3R gating variable $h$ and the mass-balance equation for intracellular 3 lumping terms, (\[eq:Jdelta\]), (\[eq:J5P-linear-approx\]) and (\[eq:J3K-exact\]), i.e. $$\begin{aligned}
{\frac{\mathrm{d}C}{\mathrm{d}t}} &= J_r(C,h,I) + J_l(C) - J_p(C) \label{eq:C}\\
{\frac{\mathrm{d}h}{\mathrm{d}t}} &= \Omega_h(C,I){\left(h_\infty(C,I)-h\right)} \label{eq:h}\\
{\frac{\mathrm{d}I}{\mathrm{d}t}} &= O_{\delta}{\ensuremath{\mathcal{H}_{2}\left(C,K_\delta\right)}}{\left(1-{\ensuremath{\mathcal{H}_{1}\left(I,\kappa_\delta\right)}}\right)} -O_{3K}\, {\ensuremath{\mathcal{H}_{4}\left(C,K_{D}\right)}} {\ensuremath{\mathcal{H}_{1}\left(I,K_{3}\right)}} -\Omega_{5P}\, I \label{eq:I}\end{aligned}$$ The above model can be extended to explicitly modeling of GPCR dynamics by a model. To this aim, we add to the right-hand side of [equation \[eq:I\]]{} the contribution of GPCR-mediated 3 synthesis given by [equation \[eq:Jbeta-ssa\]]{}. However, if one is interested in how GPCR kinetics evolves with 3 and dynamics, then the formula for $J_\beta$ given by [equation \[eq:Jbeta\]]{} must be used instead of [equation \[eq:Jbeta-ssa\]]{}. Accordingly, the above system of equations must be completed by [equation \[eq:GammaA\]]{} for astrocytic receptor activation, i.e. $$\begin{aligned}
{\frac{\mathrm{d}\Gamma_A}{\mathrm{d}t}} &=\ldots \tag{\ref{eq:GammaA}}\\
{\frac{\mathrm{d}C}{\mathrm{d}t}} &= \ldots \tag{\ref{eq:C}}\\
{\frac{\mathrm{d}h}{\mathrm{d}t}} &= \ldots \tag{\ref{eq:h}}\\
{\frac{\mathrm{d}I}{\mathrm{d}t}} &= O_\beta\Gamma_A + O_{\delta}{\ensuremath{\mathcal{H}_{2}\left(C,K_\delta\right)}}{\left(1-{\ensuremath{\mathcal{H}_{1}\left(I,\kappa_\delta\right)}}\right)} -O_{3K}\, {\ensuremath{\mathcal{H}_{4}\left(C,K_{D}\right)}} {\ensuremath{\mathcal{H}_{1}\left(I,K_{3}\right)}} -\Omega_{5P}\, I \label{eq:gchi-I}\end{aligned}$$ Regarding the differential equations for the variables $C$ and $h$ above, the original formulation of the model considered the Li-Rinzel description for CICR previously introduced in [@LiRinzel1994]. In the following, we will refer to this formulation. In practice however, it must be noted that any suitable model of and 3R dynamics discussed in and can be adopted in lieu of the Li-Rinzel description, and accordingly different models of type may be developed, each possibly customized to study specific aspects of coupled 3 and signaling in astrocytes.
[ \[fig:chi\]]{} illustrates some characteristics of 3 and dynamics reproduced by the model. In the left panel of this figure, 3R kinetic parameters are chosen to fit, as closely as possible, experimental data points for the steady-state open probabilities of type-2 3Rs at fixed (*solid line*) and 3 concentrations (*dashed line*). In the right panel, the remainder of the parameters of the model are then set to reproduce (*solid black line*) a sample trace imaged by confocal microscopy on cultured astrocytes (*gray data points*). It may be observed how the associated 3 and $h$ oscillations predicted by the model, are almost out of phase with respect to the ones. For $h$, this is due to 3R kinetics, whereby an increase of cytosolic promotes receptor inactivation. For 3 instead, this dynamics is a direct consequence of the -dependent rate of degradation of this molecule by the k enzyme. This is a crucial aspect of intracellular 3 regulation in astrocytes which is addressed more in detail below.
Different regimes of 3 signaling
--------------------------------
To develop the model in [Section \[sec:ip3-modeling\]]{}, we stressed on the molecular details of the dependence of the different enzymes involved in 3 signaling, yet how this dependence shapes and 3 oscillations remains to be elucidated. With this purpose, we consider in [ \[fig:gchi-dynamics\]]{} the simple scenario of oscillations triggered by repetitive stimulation of an astrocyte by puffs of extracellular glutamate (*top three panels*), and look at the different contributions to 3 production and degradation underpinning the ensuing and 3 dynamics (*lower panels*). With this regard, it may be noted how the total rate of 3 production (*dashed line* in the *fourth panel* from top) almost resembles the dynamics of activation of astrocyte receptors ($\Gamma_A$, *second panel* from top) except for little bumps in correspondence of pulse-like elevations (*solid trace*, *third panel* from top). Consideration of the different contributions to 3 by (*orange trace*) and (*blue trace*) reveals that, while most of 3 production is driven by mGluR-mediated activation, those bumps are instead caused by , whose activation is substantially boosted during intracellular elevations.
Similar arguments also hold for 3 degradation (*bottom panel*). In this case, the total rate of 3 degradation (*dashed line*) closely mimics 3 dynamics in between elevations (*green trace*, *third panel* from top), and is mostly contributed by -independent p-mediated degradation (*violet trace*). This scenario however changes during elevations, when k activation becomes significant and promotes faster rates of 3 degradation, as mirrored by the *dashed line* which peaks in correspondence of oscillations.
Overall, these observations suggest that -independent activity of and p vs. -dependent activation of and k account for different regimes of 3 signaling. One regime corresponds to low intracellular close to resting concentrations, whereby 3 is mainly produced by receptor-mediated activation of against degradation by p. The other regime significantly adds to the former for sufficiently high elevations, where 3 production is boosted by , but also 3 degradation is faster by k activation.
The contribution to 3 production and degradation by each enzyme clearly depends on their intracellular expression as reflected by the values of the rate constants $O_\beta,\, O_\delta,\,O_{3K}$ and $\Omega_{5P}$ in [equation \[eq:gchi-I\]]{}. Nonetheless, it should be noted that the existence of different regimes of 3 production and degradation is regardless of these rate values, insofar as it is set by the values of the Michaelis-Menten constants of the underpinning reactions, mostly $K_\delta$ and $K_D$. Remarkably, estimates of these two constants are in the range of $~0.1-\SI{1.0}{\micro \Molar}$, that is well within the range of elevations expected for an astrocyte, whose average resting concentration is reported to be $<\SI{0.15}{\micro \Molar}$ [@Zheng_Neuron2015]. This assures that activation of and k is effective only when intracellular approaches to, or increases beyond $K_\delta$ and $K_D$, as expected by the occurrence of CICR.
Signal integration
------------------
The existence of different regimes of 3 signaling shapes the time evolution of 3 with respect to stimulation in a peculiar fashion. From [ \[fig:gchi-dynamics\]]{} (*third panel*), it may indeed be noted that, starting from resting values, 3 increases for each glutamate puff almost stepwise, till it reaches a peak (or threshold) concentration (normalized to $\sim 1$) that triggers CICR, thereby triggering a pulse-like elevation. This elevation promotes 3 degradation to some concentration between its peak and baseline values, in a sort of reset mechanism, leaving 3 to increase back again to the CICR threshold until the next elevation. In between each elevation, counting from the first one ending at $t\approx \SI{4}{s}$, we may appreciate how 3 increases almost proportionally to the number of glutamate puffs, akin to an integrator of the stimulus.
This may readily be proved by analytical arguments approximating, for simplicity, each glutamate puff occurring at $t_k$ by a Dirac’s delta $\updelta(t-t_k)$, so that the external stimulus impinging on the astrocyte is modeled by $Y(t) = G\cdot\Delta\sum_k \updelta(t-t_k)$, where $G\cdot \Delta$ represents the glutamate concentration delivered in the time unit per puff (i.e. its dimensions are ). Then, assuming that in between oscillations, intracellular concentration is close to basal levels, i.e. $C\approx C_0$, with $C_0 < (\ll)\, K_{KC},\, K_\delta,\,K_3$ and $h\approx h_\infty$, it is possible to reduce equations \[eq:GammaA\] and \[eq:gchi-I\] to $$\begin{aligned}
{\frac{\mathrm{d}\Gamma_A}{\mathrm{d}t}} &\approx -(O_N Y(t)+\Omega_N) \Gamma_A + O_N Y(t)\label{eq:GammaA-approx}\\
{\frac{\mathrm{d}I}{\mathrm{d}t}} &\approx -J_{5P} + J_\beta = - \Omega_{5P} I + O_\beta \Gamma_A \label{eq:I-approx}\end{aligned}$$ Using the fact that for puffs delivered at rate $\nu$ the identity $\int_{t\,'}^{t\,''} \sum_k \updelta(t-t_k) {\ensuremath{\mathrm{d}t}} = \nu (t\,'' - t\,')$ holds, we can solve [equation \[eq:GammaA-approx\]]{} for $\Gamma_A$ obtaining $$\begin{aligned}
\Gamma_A(t) & = \int_{-\infty}^{t}O_N Y(t\,')\,e^{-\int_{t\,'}^{t}(\Omega_N+O_N Y(t\,'')){\ensuremath{\mathrm{d}t}}\,''}{\ensuremath{\mathrm{d}t}}\,' \nonumber\\
& = \int_{-\infty}^{t}O_N Y(t\,')\,e^{-\Omega_N(t-t\,')}\,e^{-O_N\int_{t\,'}^{t}Y(t\,''){\ensuremath{\mathrm{d}t}}\,''}{\ensuremath{\mathrm{d}t}}\,' \nonumber\\
& = \int_{-\infty}^{t}O_N Y(t\,')\,e^{-{\left(\Omega_N + O_N G \Delta \nu\right)} (t-t\,')}{\ensuremath{\mathrm{d}t}}\,' \nonumber\\
& = O_N Y(t\,') \ast Z_{\Gamma_A}(t) \label{eq:GammaA-solution}\end{aligned}$$ where “$\ast$" denotes the convolution operator. It is thus apparent that the fraction of activated receptors $\Gamma_A(t)$ is an integral transform of the stimulus $Y(t)$ by convolution with the kernel $Z_{\Gamma_A}(t)$. Specifically, $Z_{\Gamma_A}(t)$ may be regarded as the fraction of astrocyte receptors stimulated by one extracellular glutamate puff – or equivalently, by synaptic release triggered by an action potential –, and characterizes the encoding of the stimulus by the astrocyte via its activated receptors.
The 3 signal resulting from the activated receptors then evolves according to $$\begin{aligned}
I(t) & = \int_{-\infty}^{t} O_\beta \Gamma_A (t\,')\,e^{-\int_{t\,'}^{t}\Omega_{5P}{\ensuremath{\mathrm{d}t}}\,''}{\ensuremath{\mathrm{d}t}}\,' = \int_{-\infty}^{t} O_\beta \Gamma_A (t\,')\,e^{-\Omega_{5P}(t-t\,')}{\ensuremath{\mathrm{d}t}}\,'\nonumber\\
& = O_\beta\Gamma_A (t) \ast Z_{I}(t) \label{eq:I-solution}\end{aligned}$$ That is the 3 signal is also an integral transform of the input stimuli through the fraction of activated receptors $\Gamma_A(t)$, by convolution with the kernel $Z_{I}(t)=e^{-\Omega_{5P}t}$. In particular, experimental evidence hints that the rate constant $\Omega_{5P}$ is often small compared to the rate of incoming stimulation ([Appendix \[app:Parameters\]]{}), so that $Z_{I}(t)\approx 1$. In this case then, [equation \[eq:I-solution\]]{} predicts that $I(t)\approx \int_{-\infty}^{t}O_\beta \Gamma_A(t\,'){\ensuremath{\mathrm{d}t}}\,'$, namely that the 3 signal effectively corresponds to the integral of the fraction of activated astrocyte receptors.\
It is also worth understanding the nature of the threshold concentration that 3 must reach in order to trigger CICR. In the model, based on the Li-Rinzel description of CICR, this threshold may be not well-defined and generally varies with the parameter choice as well as with the shape and amplitude of the delivered stimulation [@DePitta_JOBP2009]. Consider for example [ \[fig:gchi-threshold\]]{}A where the response of an astrocyte (*bottom panel*) is simulated for different *color-coded* step increases of extracellular glutamate (*top panel*). It may be noted that CICR, reflected by one or multiple pulse-like increases, is triggered by glutamate concentrations greater or equal to the *orange trace*. However, the 3 threshold for CICR (*central panel*) appears to grow with the extracellular glutamate concentration. This is reflected by the first ’knee’ of the 3 curves which reaches progressively higher values of 3 concentration as extracellular glutamate increases from *orange* to *lime* levels. At the same time, as shown by the *black dashed curve* in the *top panel* of [ \[fig:gchi-threshold\]]{}B, the latency for emergence of CICR since stimulus onset (*black marks* at $t=0$) decreases. This can be explained by equations \[eq:GammaA-approx\] and \[eq:I-approx\], noting that, while larger glutamate concentrations promote larger receptor-mediated 3 production, this increased production is also counteracted by faster degradation by p, since this latter linearly increases with 3. Thus while larger 3 production assures shorter delays in the onset of CICR, a larger 3 level must be reached to compensate for its faster degradation.
The *top panel* of [ \[fig:gchi-threshold\]]{}B further illustrates how the latency period for CICR onset depends on the activity of the different enzymes regulating 3 production and degradation. Here the different *colored curves* were obtained repeating the simulations of [ \[fig:gchi-threshold\]]{}A for a 50% increase of the activity respectively of (*orange trace*), (*blue trace*), k (*red trace*) and p (*violet trace*). In agreement with our previous analysis, and p have the largest impact on respectively reducing or increasing the latency period, given that they are the main enzymes at play in 3 signaling before CICR onset. The effect of an increase of 3 production by is instead mainly significant for low glutamate concentrations, such that they could promote an activation of this enzyme that is comparable to that of . Conversely, k does not have any role in the control of CICR latency since its activation effectively requires CICR to onset first.
The variability of 3 concentrations attained to trigger CICR by different glutamate concentrations, and its correlation with the latency for CICR onset, suggest that the mere 3 concentration is not an effective indicator of the CICR threshold, rather we should consider instead the total 3 amount produced in the astrocyte cytosol during the latency period that precedes CICR onset, that is the integral in time of 3 concentration during such period. This is exemplified in the *bottom panel* of [ \[fig:gchi-threshold\]]{}B where such integral is plotted as a function of the different latency values computed in the *top panel*. It may be appreciated how this integral is essentially similar for different enzyme expressions (*colored curves*) yet associated with the same latency value.
Taken together these results put emphasis on the crucial role exerted by 3 signaling in the genesis of agonist-mediated elevations. In particular they suggest that the expression of different enzymes responsible of 3 production and degradation, which is likely heterogeneous across an astrocyte, could locally set different requirements for integration and encoding of external stimuli by the same cell.
Role of cPKCs and beyond
------------------------
Different mechanisms of production and degradation of 3 are only one example of the possible many signaling pathways that could shape the nature of signaling in astrocytes. There is also compelling evidence in vitro that shape and duration of oscillations could be controlled by astrocyte receptor phosphorylation by [@CodazziTeruelMeyer2001]. To better understand this aspect of astrocyte signaling, we relax the quasi steady-state approximation on phsophorylation and thus rewrite [equation \[eq:R\*-ode-qssa\]]{} as $${\frac{\mathrm{d}\Gamma_A}{\mathrm{d}t}} = O_N \ce{[A]$^{n}$}\,{\left(1-\Gamma_A\right)} - {\left(\Omega_N + O_K P\right)}\Gamma_A \label{eq:GammaA-cpkc}$$ where $P$ denotes the concentration at the receptors’ site. This in turn, requires to also consider a description of dynamics, whereby at least two additional equations in the model must be included: one that takes into account $P$ dynamics, but also a further one that describes dynamics ($D$), which is responsible for activation by -dependent translocation of the inactive kinase to the plasma membrane [@Oancea_Cell1998].
By QSSA, the quantity of is conserved during receptor phosphorylation in reaction \[rc:R-PKC\*-reaction\]. In this fashion, production and degradation are only controlled by the pair of reactions \[rc:PKC\*-reaction-ca\] and \[rc:PKC\*-reaction-dag\]. On the other hand, taking into account from [Section \[sec:PLCb-production\]]{} that production of depends on the availability of the -bound kinase complex , we may assume at first approximation that reaction \[rc:PKC\*-reaction-ca\] for -binding to the kinase is at equilibrium, i.e. . Accordingly, we can consider dynamics to be driven simply by reaction \[rc:PKC\*-reaction-dag\], i.e. $$\begin{aligned}
{\frac{\mathrm{d}P}{\mathrm{d}t}} &= J_{KP} - J_{KD}\nonumber\\
&= O_{KD}\ce{[cPKC$\,'$]}\cdot D - \Omega_{KD} P \nonumber\\
&= O_{KD}\ce{[cPKC]_T}{\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}}\cdot D - \Omega_{KD} P \nonumber\\
&\equiv O_{KD}{\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}}\cdot D - \Omega_{KD} P \label{eq:cPKC}\end{aligned}$$ where we re-defined $O_{KD}\leftarrow O_{KD}\ce{[cPKC]_T}$ as the maximal rate of production (in ).
To model dynamics we start instead from the consideration that PLC isoenzymes hydrolyze 2 into one molecule of 3 and one of , so that production coincides with that of 3 [@BerridgeIrvine_Nature1989 and see also [ \[fig:ip3-degradation\]]{}B]. Yet, only part of this produced is used to activate , while the rest is mainly degraded by diacylglycerol kinases (DAGKs) into phosphatidic acid [@Carrasco_TiBS2007] and, to a minor extent, by diacylglycerol lipases (DAGLs) into 2-arachidonoylglycerol (2-AG), although this latter pathway has only been linked to some types of metabotropic receptors in astrocytes [@BrunerMurphy_JNC1990; @Giaume_PNAS1991; @Walter_JN2004]. Other pathways of use of are also possible in principle, inasmuch as is a key molecule in the cell’s lipid metabolism and a basic component of membranes. Nonetheless there is evidence that levels are strictly regulated within different subcellular compartments, and generated by GPCR stimulation is not usually consumed for metabolic purposes [@van-der-Bend1994; @Carrasco_TiBS2007].
DAGK activation reflects the sequence of mediated translocation, binding and activation that is also required for cPKCs, so the two reactions may be thought to be characterized by similar kinetics, yet with an important difference. Sequence analysis of DAGK$\upalpha,\,\upgamma$ – the two isoforms of DAGKs most likely involved in astrocytic GPCR signaling [@Dominguez_CD2013] – reveals in fact the existence of two EF-hand motifs characteristics of -binding and two C1 domains for binding [@Merida_BJ2008]. In this fashion, a Hill exponent of 2 instead of 1 as in [equation \[eq:cPKC\]]{} must be considered for the DAGK activating reaction, so that DAGK-mediated DAG degradation can be modeled by $$J_D = O_{D} {\ensuremath{\mathcal{H}_{2}\left(C,K_{DC}\right)}}{\ensuremath{\mathcal{H}_{2}\left(D,K_{DD}\right)}}$$ Finally, to take into account other mechanisms of degradation ($J_A$), including but not limited to DAGLs, we assume a linear degradation rate, i.e. $J_A = \Omega_D D$. This is a crude approximation insofar as DAGL, could also be activated in a -dependent fashion [@Rosenberger_Lipids2007]. Nonetheless, the complexity of the molecular reactions likely involved in these other pathways of DAG degradation would require to consider additional equations in our model which are beyond the scope of this chapter. The reader who is interested in these further aspects, may refer to @Cui_eLife2016 for a possible modeling approach. For the purposes of our analysis instead, we will consider the following equation for dynamics: $$\begin{aligned}
{\frac{\mathrm{d}D}{\mathrm{d}t}} &= J_\beta + J_\delta - J_{KP} - J_D - J_A \nonumber\\
&= O_\beta \Gamma_A + O_{\delta}{\ensuremath{\mathcal{H}_{2}\left(C,K_\delta\right)}}{\left(1-{\ensuremath{\mathcal{H}_{1}\left(I,\kappa_\delta\right)}}\right)} + \nonumber\\
&\phantom{=} -O_{KD} {\ensuremath{\mathcal{H}_{1}\left(C,K_{KC}\right)}}\cdot D - O_{D} {\ensuremath{\mathcal{H}_{2}\left(C,K_{DC}\right)}}{\ensuremath{\mathcal{H}_{2}\left(D,K_{DD}\right)}} - \Omega_D D \label{eq:DAG}\end{aligned}$$
[ \[fig:gchidp\]]{}A shows a comparison of experimental and traces with those reproduced by the model including equations \[eq:cPKC\] and \[eq:DAG\]. For inherent limitations of the Li-Rinzel description of the gating kinetics of 3Rs, which fails to describe these receptors’ open probability for large concentration ([ \[fig:chi\]]{}) and predicts fast rates of receptor de-inactivation ($O_2 / d_2$, [ \[tab:Model-Parameters\]]{}), the model cannot generate peaks as large as those experimentally observed and shown here. Nonetheless we would like to emphasize how our model qualitatively matches experimental -dependent dynamics, accurately reproducing the phase shift between and oscillations. This phase shift is critically controlled by the constant $K_{KC}$ for binding to the kinase, along with the rates of production vs. degradation, i.e. $O_{KD}$ vs. $\Omega_{KD}$ ([equation \[eq:cPKC\]]{}), and the rate of receptor phosphorylation $O_K$ ([equation \[eq:GammaA-cpkc\]]{}).
[ \[fig:gchidp\]]{}B further reveals the role of these rate constants in the control of oscillations. In this figure, we simulated the astrocyte response for a step increase of $\sim \SI{1.5}{\micro \Molar}$ extracellular glutamate, starting from resting conditions, both in the absence of kinase-mediated receptor phosphorylation (*gray trace*) and in the presence of it, for two different $O_{K}$ rate values (*black traces*). It may be noted how receptor phosphorylation by can rescue oscillations that otherwise would vanish by saturating intracellular 3 concentrations ensuing from large receptor activation. This activation indeed is decreased by according to [equation \[eq:GammaA-cpkc\]]{}, thereby regulating intracellular 3 within the range of oscillations. Nonetheless, as the rate of receptor phosphorylation increases (*dash-dotted trace*), the period of oscillations appears to slow down and oscillations even fail to emerge, if the supply of results in a phosphorylation rate of astrocyte receptors that exceeds their agonist-mediated activation (results not shown).
These considerations can be explained considering the period of oscillations as a function of the extracellular glutamate concentration. As shown in [ \[fig:gchidp\]]{}C, -mediated receptor phosphorylation shifts (*black curves*) the range of glutamate concentrations that trigger oscillations to higher values than those otherwise expected in the absence of it (*gray curve*). In particular, and in agreement with experimental findings [@CodazziTeruelMeyer2001], the exact value of the rate $O_K$ for receptor phopshorylation sets the entity of this shift, accounting either for oscillations of period longer than without receptor phosphorylation, or for the requirement of larger glutamate concentrations to observe such oscillations. This is respectively reflected by the portions of the *black curves* that are within the range of extracellular glutamate concentrations of the *gray curve*), and those that instead are not. On the other hand, longer-period oscillations in the presence of receptor phosphorylation are likely to be observed as long as the rate of activation by DAG ($O_{KD}$) is below some critical value. A three-fold increase of this rate indeed requires glutamate concentrations beyond those needed in the absence of receptor phosphorylation to trigger oscillations, regardless of the $O_K$ value at play (*blue curves*). In this scenario in fact, the large supply of , resulting from the high $O_{KD}$ value, favors phosphorylation of receptors while hindering intracellular buildup of 3 to trigger CICR. This in turn requires a larger recruitment of astrocyte receptors by larger agonist concentrations to evoke oscillations.
Conclusions
===========
The modeling arguments introduced in this chapter overall suggest a great richness in the possible modes whereby astrocytes could translate extracellular stimuli into intracellular dynamics. These modes are brought forth by a complex network of biochemical reactions that is exquisitely nonlinearly coupled with dynamics through different second messengers, among which 3 and possibly DAG could play a paramount signaling role. In particular, the regulation of different regimes of 3 production and degradation by in parallel with the differential regulation by this latter and DAG of the activities of cPKCs and DAGKs opens to the scenario of the existence of different regimes of signal transduction that a single astrocyte could multiplex towards different intracellular targets depending on different local conditions of neuronal activity.
An interesting implication emerging from our analysis of the regulation of the period of oscillations by cPKCs and DAG-related lipid signals is the possibility that these pathways, which could be crucially linked with inflammatory responses underpinning reactive astrocytosis [@Brambilla_BJP1999; @Griner_NRC2007], could be found at different operational states, akin to what suggested for proinflammatory cytokines like TNF$\upalpha$ [@Santello_TiNS2012]. In our analysis for example, intermediate activation of cPKC activity could promote oscillations at physiological rates, while an increase of it could exacerbate fast, potentially inflammatory responses [@Sofroniew_AN2010].
Similar arguments also hold for 3 signaling. Calcium-dependent 3 production by and (via cPKC) could modulate the rate of integration of synaptic stimuli and thus dictate the threshold synaptic activity triggering CICR. On the other hand, the existence of different regimes of 3 degradation could be responsible for different cutoff frequencies of synaptic release, beyond which integration of external stimuli by the cells could cease. In particular, this cutoff frequency could be mainly set by p during low synaptic activity, possibly associated with low intracellular levels, while be dependent on k in regimes of strong astrocyte activation, and thus ultimately depend on the history of activation of the astrocyte. The following chapter looks closely at some of these aspects, focusing in particular, on the role of different 3 degradation regimes in the genesis and shaping of oscillations.
Arguments of chemical kinetics {#app:chem-kinetics}
==============================
The Hill equation {#app:Hill-function}
-----------------
In biochemistry, the binding reaction of $n$ molecules of a ligand $L$ to a receptor macromolecule $R$, i.e., $$\label{eq:binding-reaction}
\ce{R + $n$L <=>T[$k_{f}$][$k_{b}$] RL_n}$$ can be mathematically described by the differential equation $$\label{eq:binding-ode}
{\frac{\mathrm{d}\ce{[RL_n]}}{\mathrm{d}t}}=k_{f}\ce{[R][L]^n}-k_{b}\ce{[RL_n]}$$ where $k_{f}$, $k_{b}$ denote the forward (binding) and backward (unbinding) reaction rates respectively. At equilibrium, $$\label{eq:binding-equilibrium}
0 = k_{f}\ce{[R][L]^n}-k_{b}\ce{[RL_n]} \Rightarrow \ce{[RL_n]}=\frac{\ce{[R][L]^n}}{K_d}$$ where $K_{d}=k_{b}/k_{f}$ is the *dissociation constant* of the binding reaction \[eq:binding-reaction\]. Then, the fraction of bound receptor macromolecules with respect to the total receptor macromolecules can be expressed by the Hill equation [@StryerBiochemistryBOOK] $$\label{eq:Hill-equation}
\frac{\textrm{Bound}}{\textrm{Total}} = \frac{\ce{[RL_{n}]}}{\ce{[R] + [RL_{n}]}} = \dfrac{\dfrac{\ce{[L]^n}}{K_{d}}}{\dfrac{[L]^{n}}{K_{d}}+1} = \dfrac{\ce{[L]^n}}{\ce{[L]^n}+K_{d}} = \dfrac{[L]^{n}}{[L]^{n}+K_{0.5}^{n}}={\ensuremath{\mathcal{H}_{n}\left(\ce{[L]},K_{0.5}\right)}}$$ where the function [$\mathcal{H}_{n}\left(\ce{[L]},K_{0.5}\right)$]{} denotes the sigmoid (Hill) function $\ce{[L]^n / ([L]^n + K_{0.5}^n})$, and $K_{0.5} = \sqrt[n]{K_{d}}$ is the receptor *affinity* for the ligand $L$, and corresponds to the ligand concentration for which half of the receptor macromolecules are bound (i.e. the midpoint of the [$\mathcal{H}_{n}\left(\ce{[L]},K_{0.5}\right)$]{} curve). The sigmoid shape of [$\mathcal{H}_{n}\left(\ce{[L]},K_{0.5}\right)$]{} denotes *saturation kinetics* in the binding reaction \[eq:binding-reaction\], that is, for $\ce{[L]}\gg K_{0.5}$ almost all the receptor molecules are bound to the ligand, so that the fraction of bound receptor molecules does not essentially change for an increase of $[L]$.
The coefficient $n$, also known as *Hill coefficient*, quantifies the cooperativity among multiple ligand binding sites. A Hill coefficient $n>1$ denotes *positively cooperative binding*, whereby once one ligand molecule is bound to the receptor macromolecule, the affinity of the latter for other ligand molecules increases. Conversely, a value of $n<1$ denotes *negatively cooperative binding*, namely when binding of one ligand molecule to the receptor decreases the affinity of the latter to bind further ligand molecules. Finally, a coefficient $n=1$ denotes completely *independent binding* when the affinity of the receptor to ligand molecules is not affected by its state of occupation by the latter.
For unimolecular reactions, $n=1$ coincides with the number of binding sites of the receptor. For multimolecular reactions involving $\eta > 1$ ligand molecules instead, the Hill coefficient in general, only loosely estimates the number of binding sites, being $n\le \eta$ [@Weiss1997]. This follows from the hypothesis of total allostery that is implicit in the reaction \[eq:binding-reaction\], whereby the Hill function is a very simplistic way to model cooperativity. It describes in fact the limit case where affinity is 0 if no ligand is bound, and infinite as soon as one receptor binds. That is, only two states are possible: free receptor and receptor with all ligand bound. More realistic descriptions are available in literature, such as for example the Monod–Wyman–Changeux (MWC) model, but they yield much more complex equations and more parameters [@Changeux_Science2005].
The Michaelis-Menten model of enzyme kinetics {#app:Michaelis-Menten-kinetics}
---------------------------------------------
The Michaelis-Menten model of enzyme kinetics is one of the simplest and best-known models to describe the kinetics of enzyme-catalyzed chemical reactions. In general enzyme-catalyzed reactions involve an initial binding reaction of an enzyme to a substrate to form a complex . The latter is then converted into a product and the free enzyme by a further reaction that is mediated by the enzyme itself and can be quite complex and involve several intermediate reactions. However, there is typically one rate-determining enzymatic step that allows this reaction to be modeled as a single catalytic step with an apparent rate constant $k_{\mathrm{cat}}$. The resulting kinetic scheme thus reads $$\label{eq:Michaelis-Menten-reaction}
\ce{E + S <=>T[$k_{f}$][$k_{b}$] ES ->T[$k_{\mathrm{cat}}$] P + E}$$ By law of mass action, the above kinetic scheme gives rise to 4 differential equations [@StryerBiochemistryBOOK]:
\[eq:Michaelis-Menten-equations\] $$\begin{aligned}
{\frac{\mathrm{d}\ce{[S]}}{\mathrm{d}t}} & = -k_{f}\ce{[E][S]} + k_{b}\ce{[ES]}\\
{\frac{\mathrm{d}\ce{[E]}}{\mathrm{d}t}} & = -k_{f}\ce{[E][S]} + k_{b}\ce{[ES]} + k_{\mathrm{cat}}\ce{[ES]} \label{eq:Michaelis-Menten-E}\\
{\frac{\mathrm{d}\ce{[ES]}}{\mathrm{d}t}}& = k_{f}\ce{[E][S]} - k_{b}\ce{[ES]} - k_{\mathrm{cat}}\ce{[ES]} \label{eq:Michaelis-Menten-ES}\\
{\frac{\mathrm{d}\ce{[P]}}{\mathrm{d}t}} & = k_{\mathrm{cat}}\ce{[ES]} \label{eq:Michaelis-Menten-P}\end{aligned}$$
In the Michaelis-Menten model the enzyme is a catalyst, namely it only facilitates the reaction whereby is transformed into , hence its total concentration must be preserved. This is indeed apparent by the sum of the second and the third equations above, since: ${\frac{\mathrm{d}(\ce{[E] + [ES]})}{\mathrm{d}t}}={\frac{\mathrm{d}\ce{[E]_T}}{\mathrm{d}t}}=0\Rightarrow \ce{[E]_T} = \textrm{const}$.
The system of equations \[eq:Michaelis-Menten-equations\] can be solved for the products as a function of the concentration of the substrate . A first solution assumes instantaneous chemical equilibrium between the substrate and the complex , i.e. ${\frac{\mathrm{d}\ce{[S]}}{\mathrm{d}t}}=0$, whereby the initial binding reaction can be equivalently described by a Hill equation [@KeenerSneyd_2008_Book], i.e., $$\label{eq:Michaelis-Menten-ChemEq}
\frac{\ce{[ES]}}{\ce{[E]_T}} = \frac{\ce{[S]}}{\ce{[S]} + K_{d}} \Rightarrow \ce{[ES]} = \frac{\ce{[E]_T [S]}}{\ce{[S]} + K_{d}}$$ Alternatively, the *quasi-steady-state assumption* (QSSA) that does not change on the time scale of product formation can be made, so that $\frac{d}{dt}\ce{[ES]} = 0 \Rightarrow k_{f}\ce{[E][S]} = k_{b}\ce{[ES]} + k_{\mathrm{cat}}\ce{[ES]}$ [@KeenerSneyd_2008_Book], and $$\begin{aligned}
\label{eq:Michaelis-Menten-QSSA}
k_{f}\ce{[E][S]} = k_{b}\ce{[ES]} + k_{\mathrm{cat}}\ce{[ES]} & \Rightarrow k_{f}
\ce{{\left(\ce{[E]_T} - \ce{[ES]}\right)}[S]} = k_{b}\ce{[ES]} + k_{\mathrm{cat}}\ce{[ES]} \nonumber\\
& \Rightarrow k_{f}\ce{[E]_T [S]} = {\left(k_{f}\ce{[ES]}\ce{[S]}+k_{b}\ce{[ES]} + k_{\mathrm{cat}}\ce{[ES]}\right)} \nonumber\\
& \Rightarrow \ce{[ES]} = \ce{[E]_T}\frac{\ce{[S]}}{\ce{[S]}+K_{\mathrm{M}}}\end{aligned}$$ where $K_{\mathrm{M}}={\left(k_{b}+k_{\mathrm{cat}}\right)}/k_{f}$ is the *Michaelis-Menten constant* of the reaction which quantifies the affinity of the enzyme to bind to the substrate.
Regardless of the hypothesis made to find an expression for , the rate $v_{P}$ of production of can be always written as $$\label{eq:Michaelis-Menten-Hill}
v_{P} = {\frac{\mathrm{d}\ce{[P]}}{\mathrm{d}t}} = k_{\mathrm{cat}}\ce{[ES]} = k_{\mathrm{cat}}\ce{[E]_T}\frac{\ce{[S]}}{\ce{[S]}+K_{0.5}} = v_{max}\frac{\ce{[S]}}{\ce{[S]}+K_{0.5}}$$ where $v_{max}=k_{\mathrm{cat}}\ce{[E]_T}$ is the maximal rate of production of in the presence of enzyme saturation, when all the available enzyme takes part in the reaction; and the affinity constant $K_{0.5}$ equals the dissociation constant $K_{d}$ of the initial binding reaction in the chemical equilibrium approximation ([equation \[eq:Michaelis-Menten-ChemEq\]]{}), or the Michaelis-Menten constant in the QSSA ([equation \[eq:Michaelis-Menten-QSSA\]]{}).
An important corollary of the Michaelis-Menten model of enzyme kinetics is that the fraction of the total enzyme that forms the intermediate complex can be expressed by a Hill equation of the type $$\label{eq:Michaelis-Menten-Hill}
\frac{\ce{[ES]}}{\ce{[E]_T}} = \frac{\ce{[S]}}{\ce{[S]}+K_{0.5}} = {\ensuremath{\mathcal{H}_{1}\left([S],K_{0.5}\right)}}$$ and $K_{0.5}$ can be regarded as the half-saturating substrate concentration of the reaction. Similarly, the effective reaction rate $v_{P}$ ([equation \[eq:Michaelis-Menten-Hill\]]{}) is proportional to the maximal reaction rate by a Hill-like term ${\ensuremath{\mathcal{H}_{1}\left([S],K_{0.5}\right)}}$.
Parameter estimation {#app:Parameters}
====================
Metabotropic receptors
----------------------
Rate constants $O_{N},\,\Omega_{N}$ ([equation \[eq:GammaA\]]{}) lump information on astrocytic metabotropic receptors’ activation and inactivation, namely how long it takes for these receptors, once bound by the agonist, to trigger -mediated 3 production and how long this latter lasts. Since 3 production mediated by agonist binding with the receptors controls the initial intracellular surge, these two rate constants may be estimated by rise times of agonist-triggered signals. With this regard, experiments reported that application of DHPG – a potent agonist of mGluR5 which are the main type of metabotropic glutamate receptors expressed by astrocytes [@AronicaTroost2003] –, triggers submembrane signals characterized by a rise time $\tau_{r}=0.272\pm \SI{0.095}{s}$. Because mGluR5 affinity ($K_{0.5}$) for DHPG is $\sim \SI{2}{\micro \Molar}$ [@Brabet_NP1995], that is much smaller than the applied agonist concentration, receptor saturation may be assumed in those experiments whereby the receptor activation rate by DHPG ($O_{\textrm{DHPG}}$) can be expressed as a function of $\tau_{r}$ [@Barbour_JN2001], i.e. $O_{\textrm{DHPG}}\approx \tau_{r}/(\SI{50}{\micro \Molar})=0.055-\SI{0.113}{\micro \Molar^{-1} s^{-1}}$, so that $\Omega_{\textrm{DHPG}}=O_{\textrm{DHPG}}K_{0.5}\approx 0.11-\SI{0.22}{s^{-1}}$. Corresponding rate constants for glutamate may then be estimated assuming similar kinetics, yet with $K_{0.5}=K_N=\Omega_N/O_N \approx 3-\SI{10}{\micro \Molar}$ [@Daggett_NP1995], that is 1.5–5-fold larger than $K_{0.5}$ for DHPG. Moreover, since rise times of signals triggered by non-saturating physiological stimulation are faster than in the case of DHPG [@Panatier_etal_Cell2011], it may be assumed that $O_N > O_\mathrm{DHPG}$. With this regard, for a choice of $O_N \approx 3\times O_\textrm{DHPG}= \SI{0.3}{\micro \Molar^{-1} s^{-1}}$, with $K_{N}=\SI{6}{\micro \Molar}$ such that $\Omega_N = (\SI{0.3}{\micro \Molar^{-1}s^{-1}})(\SI{6}{\micro \Molar})=\SI{1.8}{s^{-1}}$, a peak of extracellular glutamate concentration of $\SI{250}{\micro \Molar}$, delivered at $t=0$ and exponentially decaying at rate $\Omega_{c}=\SI{40}{s^{-1}}$ [@Clements1992], is consistent with a peak fraction of bound receptors of $\sim 0.75$ within $\sim \SI{70}{\milli s}$ from stimulation ([equation \[eq:GammaA\]]{}), which is in good agreement with experimental rise times.
3R kinetics
-----------
We consider a steady-state receptor open probability in the form of $p_\mathrm{open}(C,I) = \mathcal{H}_1^3(I,d_1)\cdot \cdot \mathcal{H}_1^3(C,d_5) (1-{\ensuremath{\mathcal{H}_{1}\left(C,Q_2\right)}})^3$ with $Q_2=d_2(I+d_1)/(I+d_3)$ (see ) and choose parameters to fit corresponding experimental data by @Ramos_BJ2000 for (i) different concentrations ($\hat{C}$ at a fixed 3 level of $\bar{I}=\SI{1}{\micro \Molar}$, i.e. $\hat{p}(\hat{C})$; and (ii) for different 3 concentrations ($\hat{I}$) at an intracellular concentration of $\bar{C}=\SI{25}{n \Molar}$, i.e. $\hat{p}(\hat{I})$. To reduce the problem dimensionality while retaining essential dynamical features of 3 gating kinetics we set $d_1=d_3$ [@LiRinzel1994]. Accordingly, defining the vector parameter ${\ensuremath{\mathbf{x}}}_p = {\left(d_1,d_2,d_5,O_2\right)}$, we minimize the cost function $c_p({\ensuremath{\mathbf{x}}}_p)=(p_\mathrm{open}(\hat{C},\bar{I})-\hat{p}(\hat{C}))^2 + (p_\mathrm{open}(\bar{C},\hat{I})-\hat{p}(\hat{I}))^2$ by the Artificial Bee Colony (ABC) algorithm [@Karaboga_JGO2007] considering 2000 evolutions of a colony of 100 individuals.
Ultrastructural analysis of astrocytes *in situ* revealed that the probability of ER localization in the cytoplasmic space at the soma is between $\sim$40–70% [@Pivneva_CellCalcium2008]. This suggests that the corresponding ratio between ER and cytoplasmic volumes ($\rho_{A}$) is comprised between $\sim$0.4–0.7.
To estimate the cell’s total free content $C_{T}$ we make the consideration that the resting concentration in the cytosol is $<\SI{0.15}{\micro \Molar}$ [@Zheng_Neuron2015] and can be neglected with respect to the amount of stored in the ER ($C_{ER}$) [@Berridge_etal_NatRev2003]. Hence, with $C_{ER}\ge \SI{10}{\micro \Molar}$ [@GovolinaScience1997] and a choice of $\rho_{A}\ge 0.4$, it follows that $C_{T}\approx \rho_{A}\, C_{ER} \ge \SI{4}{\micro \Molar}$. In conditions close to store depletion during oscillations [@CamelloTepikin2002], this latter value would also coincide with the peak reached in the cytoplasm, which is reported between $<\SI{5}{\micro \Molar}$ and $\sim \SI{20}{\micro \Molar}$ [@CsordasHajnoczky1999; @ParpuraHaydon2000; @KangOthmer_Chaos2009; @Shigetomi_etal_Nature2010].
In our simulations we set $\rho_A = 0.5$ while leaving arbitrary the choice of $C_T$ as far as the resulting oscillations qualitatively resemble the shape of those observed in experiments. The remaining parameters for CICR, i.e. ${\ensuremath{\mathbf{z}}}_c = {\left(\Omega_C, O_P\right)}$, were chosen to approximate the number and period of oscillations observed *on average* in experiments on cultured astrocytes that were stimulated by glutamate perfusion. By “on average” we mean that we considered the average trace resulting from $n=5$ different signals generated within the same period of time and by the same stimulus in identical experimental conditions.
3 signaling
-----------
Once set the CICR parameters, individual traces used to obtained the above-mentioned “average trace” were used to search for ${\ensuremath{\mathbf{z}}}_p = {\left(O_\beta,O_\delta,O_{3K},\Omega_{5P}\right)}$, assuming random initial conditions. The ensuing parameter values were also used in Figures \[fig:gchi-dynamics\]–\[fig:gchidp\] although $O_\beta,\,O_\delta$ and $O_{3K}$ were increased, from case to case, by a factor comprised between $1.2-2$ either to expand the oscillatory range or to promote CICR emergence (by increasing $O_\beta,\,O_\delta$) or termination (by larger $O_{3K}$ values).
and DAG signaling
------------------
Calcium-dependent cPKC-mediated phosphorylation has been documented for astrocyticmGluRs and [@CodazziTeruelMeyer2001; @Hardy_Blood2005] and results in a reduction of receptor binding affinity by a factor $\zeta \approx 2-10$ [@Hardy_Blood2005], or possibly higher depending on the cell’s expression of cPKCs [@Nakahara_JNC1997; @Shinohara_PNAS2011]. Since experiments showed that cPKC is robustly activated only when increases beyond half of the peak concentration reached during oscillations [@CodazziTeruelMeyer2001] then, considering peak values of $\sim 1-\SI{3}{\micro \Molar}$ [@Shigetomi_etal_Nature2010] allows estimating affinity of cPKC in the range of $K_{KC} \le 0.5-\SI{1.5}{\micro \Molar}$ which indeed comprises the value of $\sim \SI{700}{\nano \Molar}$ predicted experimentally [@Mosior_JBC1994]. Of the same order of magnitude also is the affinity reported for DAGK, i.e. $K_{DC} \approx 0.3-\SI{0.4}{\micro \Molar}$ [@Sakane_JBC1991; @Yamada_BJ1997].
Reported values of DAG affinities for cPKC and DAGK may considerably differ. Micellar assays of cPKCs activity, suggests values of $K_{KD}$ as low as 4.6– [@Ananthanarayanan_JBC2003], whereas studies on purified DAGK suggest a substrate affinity for this kinase of $K_{DD}\approx \SI{60}{\micro \Molar}$ [@Kanoh_JBC1983]. The differences in experimental setups and the possibility that the activity of these kinases could be widely regulated by different DAG pools make these estimate of scarce utility for our model, where the DAG concentration is of the same order of magnitude of 3 one. With this regard we choose to set these affinities to which corresponded in our simulations to the average intracellular DAG concentration during oscillations.
The remaining parameters, namely ${\ensuremath{\mathbf{z}}}_k = {\left(O_{KD}, O_K, \Omega_D, O_D, \Omega_D\right)}$ were arbitrarily chosen considering two constrains: (i) DAG concentration for damped oscillations must stabilize to a constant value; and (ii) the down phase of oscillations must follow that of ones as suggested by experimental observations by @CodazziTeruelMeyer2001.
Software
========
The Python file `figures.py` used to generate the figures of this chapter can be downloaded from the online book repository at <https://github.com/mdepitta/comp-glia-book>. The software for this chapter is organized in two folders. The `data` folder contains data to fit the model. WebPlotDigitizer 4.0 (<https://automeris.io/WebPlotDigitizer>) was used to extract experimental data by @Ramos_BJ2000 [Figures 6 and 7] and @CodazziTeruelMeyer2001 [Figure 5]. The Jupyter notebook file `data_loader.ipynb` found in this folder contains the code to load and clean experimental data used in the simulations.
The `code` folder contains instead all the routines (including `figures.py`) used for the simulations of this chapter. The two files `astrocyte_models.h` and `astrocyte_models.cpp` contains the core model implementation in C/C++11, while the class `Astrocyte` in`astrocyte_models.py` provides the Python interface to simulate the model. The model was integrated by a variable-coefficient linear multistep Adams method in Nordsieck form which proved robust to correctly solve stiff problems rising from different parameter choices [@Skeel_MC1986]. Model fitting is provided by `gchi_fit.py` and relies on the PyGMO 2.6 optimization package (<https://github.com/esa/pagmo2.git>).
The library `gchi_bifurcation.py` provides routines to estimate the period and range of oscillation as in Figures \[fig:gchidp\]. These routines use numerical continuation of the extended model by the Python module PyDSTool 0.92 [@Clewley_PCB2012 <https://github.com/robclewley/pydstool>].
Model parameters used in simulations
====================================
\[t\][l X l l]{} Symbol & Description &Value &Units\
\
$\Omega_N$ & Rate of receptor de-activation & 1.8 &\
$O_N$ & Rate of agonist-mediated receptor activation & 0.3 &\
$n$ & Agonist binding cooperativity & 1 &–\
\
$d_{1}$ & 3 binding affinity & 0.1 &\
$O_{2}$ & Inactivating binding rate & 0.325 &\
$d_{2}$ & Inactivating binding affinity & 4.5 &\
$d_{3}$ & 3 binding affinity (with inactivation) & 0.1 &\
$d_{5}$ & Activating binding affinity & 0.05 &\
\
$C_{T}$ & Total ER content & 5 &\
$\rho_{A}$ & ER-to-cytoplasm volume ratio & 0.5 &–\
$\Omega_{C}$& Maximal release rate by 3Rs & 7.759 &\
$\Omega_{L}$& leak rate & 0.1 &\
$O_{P}$ & Maximal uptake rate & 5.499 &\
$K_{P}$ & affinity of SERCA pumps & 0.1 &\
\
$O_{\beta}$ & Maximal rate of 3 production by & 0.8 &\
$O_{\delta}$& Maximal rate of 3 production by & 0.025 &\
$K_{\delta}$& affinity of & 0.5 &\
$\kappa_{\delta}$ & Inhibiting 3 affinity of & 1.0 &\
\
$\Omega_{5P}$ & Rate of 3 degradation by p & 0.86 &\
$O_{3K}$ & Maximal rate of 3 degradation by k & 0.86 &\
$K_{D}$ & affinity of k & 0.5 &\
$K_{3K}$ & 3 affinity of k & 1.0 &\
\
$\Omega_D$ & Unspecific rate of degradation & 0.26 &\
$O_D$ & Rate of degradation by DAGK & 0.45 &\
$K_{DC}$ & DAGK affinity for & 0.3 &\
$K_{DD}$ & DAGK affinity for DAG & 0.1 &\
\
$O_{KD}$ & Rate of production & 0.28 &\
$\Omega_{KD}$ & Rate of deactivation & 0.33 &\
$K_{KC}$ & affinity of PKC & 0.5 &\
$O_{K}$ & Rate of receptor phosphorylation & 1.0 &\
\
\[tab:Model-Parameters\]
[0.48]{}
![3 production. **A** Hydrolysis of the membrane lipid phosphatidylinositol 4,5-bisphosphate (2) by and isoenzymes produces 3 and diacylglycerol (). The contribution of to 3 production depends on agonist binding to astrocyte G protein-coupled receptors (GPCRs). This production pathway is inhibited via receptor phosphorylation by -dependent activation of conventional protein kinases C (cPKCs). Blue: promoting pathway; *red*: inhibitory pathway.[]{data-label="fig:ip3-production"}](scheme_production_plcb.eps){width="\textwidth"}
[0.48]{}
![3 production. **A** Hydrolysis of the membrane lipid phosphatidylinositol 4,5-bisphosphate (2) by and isoenzymes produces 3 and diacylglycerol (). The contribution of to 3 production depends on agonist binding to astrocyte G protein-coupled receptors (GPCRs). This production pathway is inhibited via receptor phosphorylation by -dependent activation of conventional protein kinases C (cPKCs). Blue: promoting pathway; *red*: inhibitory pathway.[]{data-label="fig:ip3-production"}](scheme_production_plcd.eps){width="\textwidth"}
[0.7]{}
![3 and degradation. **A** Degradation of 3 occurs by phosphorylation into inositol 1,3,4,5-tetrakisphosphate () by k and dephosphorylation into lower inositol phosphates by p. Both pathways are regulated by : k activity is stimulated by phosphorylation by /calmodulin-dependent protein kinase II (CaMKII), whereas p is inhibited thereby. Moreover k-mediated degradation could also be promoted by and -dependent cPKC-mediated phosphorylation, while p could also be inhibited by . For the sake of simplicity, p dependence on and along with k dependence on cPKC are not taken into consideration in this study (*dashed pathways*). **B** is mainly degraded into phosphatidic acid (PA) by DAG kinases (DAGK) in a -dependent fashion, and to a minor extent, into 2-arachidonoylglycerol (2-AG) by DAG lipases (DAGL). In turn 2-AG is hydrolized by monoacylglycerol lipase (MAGL) into arachidonic acid (AA). 2-AG and AA may promote activity of DAGK and (*orange patwhays*) although this scenario is not taken into consideration here. Colors of other pathways as in [ \[fig:ip3-production\]]{}.[]{data-label="fig:ip3-degradation"}](scheme_degradation_ip3.eps){width="\textwidth"}
\
[0.7]{}
![3 and degradation. **A** Degradation of 3 occurs by phosphorylation into inositol 1,3,4,5-tetrakisphosphate () by k and dephosphorylation into lower inositol phosphates by p. Both pathways are regulated by : k activity is stimulated by phosphorylation by /calmodulin-dependent protein kinase II (CaMKII), whereas p is inhibited thereby. Moreover k-mediated degradation could also be promoted by and -dependent cPKC-mediated phosphorylation, while p could also be inhibited by . For the sake of simplicity, p dependence on and along with k dependence on cPKC are not taken into consideration in this study (*dashed pathways*). **B** is mainly degraded into phosphatidic acid (PA) by DAG kinases (DAGK) in a -dependent fashion, and to a minor extent, into 2-arachidonoylglycerol (2-AG) by DAG lipases (DAGL). In turn 2-AG is hydrolized by monoacylglycerol lipase (MAGL) into arachidonic acid (AA). 2-AG and AA may promote activity of DAGK and (*orange patwhays*) although this scenario is not taken into consideration here. Colors of other pathways as in [ \[fig:ip3-production\]]{}.[]{data-label="fig:ip3-degradation"}](scheme_degradation_dag.eps){width="\textwidth"}
![** model**. (*left panel*) Fit of 3Rs kinetic parameters on experimental data of steady-state open probabilities of type-2 3Rs by @Ramos_BJ2000. In this example, and through all this chapter, we consider the Li-Rinzel description for CICR. This choice allows a reasonable fit (*solid and dashed lines*) of the receptors’ open probability as function of either intracellular 3 ($\blacktriangle$) or intracellular ($\CIRCLE$). The only exception is for concentrations $>\SI{1}{\micro \Molar}$ for which the open probability predicted by the Li-Rinzel model (*solid line*) vanishes much more quickly than experimental values. (*right panel*) Sample ($C$), 3 ($I$) and $h$ traces ensuing from a simulation of the model to reproduce experimental oscillations in cultured astrocytes (*gray data points*) triggered by application of $>\SI{5}{\micro \Molar}$ glutamate. Experimental data courtesy of Nitzan Herzog (University of Nottingham). A saturating glutamate concentration (i.e. $\Gamma_A=1$) was assumed with initial conditions $C(0) = \SI{0.098}{\micro \Molar},\,h(0)=0.972$ and $I(0)=\SI{0.190}{\micro \Molar}$. Simulated and 3 traces are reported in normalized units with respect to minimum values of $C_0=\SI{0.1}{\micro \Molar}$ and $I_0=\SI{0.16}{\micro \Molar}$ and peak values of $\hat{C}=\SI{1.42}{\micro \Molar}$ and $\hat{I}=\SI{0.19}{\micro \Molar}$. Model parameters as in [ \[tab:Model-Parameters\]]{} except for $O_\beta=\SI{0.141}{\micro \Molar s^{-1}}$ and $O_{3K}=\SI{0.163}{\micro \Molar s^{-1}}$.[]{data-label="fig:chi"}](f1_chi_model.eps){width="90.00000%"}
![Coexistence of different regimes of 3 signaling. From top to bottom: (*first panel*) Repetitive stimulation of an astrocyte by puffs of glutamate ($\SI{8}{\micro \Molar}$, rectangular pulses at rate and 15% duty cycle); (*second panel*) fraction of activated astrocytic receptors; (*third panel*) ensuing ($C$) and 3 ($I$) traces (normalized with respect to their maximum excursion: $C_0=\SI{40}{n \Molar},\,I_0=\SI{50}{n \Molar},\,\hat{C}=\SI{0.73}{\micro \Molar},\,\hat{I}=\SI{0.15}{\micro \Molar}$); (*fourth panel*) total rate of 3 production (*dashed line*) and contributions to it by ($J_\beta$) and ($J_\delta$); (*bottom panel*) total rate of 3 degradation (*dashed line*) resulting from the combination of degradation by p ($J_{5P}$) and k ($J_{3K}$). Besides pulsed-oscillations, 3 is mainly regulated by (*orange trace*) and p (*violet trace*), and its concentration tends to increase in an integrative fashion with the number of glutamate puffs. During elevations instead, activity of (*blue trace*) and k (*red trace*) become significant, with this latter responsible for a sharp drop of intracellular 3. Model parameters as in [ \[tab:Model-Parameters\]]{} except for $C_T=\SI{10}{\micro \Molar},\,O_P=\SI{10}{\micro \Molar s^{-1}}$ and $O_\delta=\SI{0.05}{\micro \Molar s^{-1}}$.[]{data-label="fig:gchi-dynamics"}](f2_gchi_model.eps){width="90.00000%"}
[0.49]{}
![Threshold for CICR. **A** (*top panel*) Step increases of extracellular glutamate (*color coded*) and resulting 3 (*central panel*) and dynamics (*bottom panel*) in a astrocyte model. *Black marks* at $t=0$ denote stimulus onset. **B** (*top panel*) Latency for the onset of CICR as a function of the applied glutamate concentration for the traces in **A** (*black dashed curve*), as well as for 50% increases in the rate of ($O_\beta$), ($O_\delta$), k ($O_{3K}$) and p ($\Omega_{5P}$) respectively. Emergence of CICR was detected for ${\frac{\mathrm{d}C}{\mathrm{d}t}} \ge \SI{0.5}{\micro \Molar / s}$. (*bottom panel*) Integral of 3 concentration as a function of the latency values computed in the top panel. This integral is a better estimator of CICR threshold than the sole 3 concentration. Model parameters as [ \[fig:gchi-dynamics\]]{}.[]{data-label="fig:gchi-threshold"}](f3_gchi_thr_0.eps){width="100.00000%"}
[0.49]{}
![Threshold for CICR. **A** (*top panel*) Step increases of extracellular glutamate (*color coded*) and resulting 3 (*central panel*) and dynamics (*bottom panel*) in a astrocyte model. *Black marks* at $t=0$ denote stimulus onset. **B** (*top panel*) Latency for the onset of CICR as a function of the applied glutamate concentration for the traces in **A** (*black dashed curve*), as well as for 50% increases in the rate of ($O_\beta$), ($O_\delta$), k ($O_{3K}$) and p ($\Omega_{5P}$) respectively. Emergence of CICR was detected for ${\frac{\mathrm{d}C}{\mathrm{d}t}} \ge \SI{0.5}{\micro \Molar / s}$. (*bottom panel*) Integral of 3 concentration as a function of the latency values computed in the top panel. This integral is a better estimator of CICR threshold than the sole 3 concentration. Model parameters as [ \[fig:gchi-dynamics\]]{}.[]{data-label="fig:gchi-threshold"}](f3_gchi_thr_1.eps){width="83.00000%"}
[0.285]{}
![Regulation of oscillations by . **A** (*top panel*) Comparison between experimental traces for (*black*) and (*red*) originally recorded in cultured astrocytes by @CodazziTeruelMeyer2001 and simulations (*bottom panel*). Despite quantitive differences in the shape and period of oscillations, the model can reproduce the essential correlation and phase shift between and dynamics observed in experiments. and oscillations were triggered assuming an extracellular glutamate concentration of , and were normalized according to their maximum excursion: $C_0=\SI{0.04}{\micro \Molar}$, $P_0=\SI{48}{\nano \Molar}$, $\hat{C}=\SI{0.49}{\micro \Molar}$ and $\hat{P}=\SI{65}{\nano \Molar}$. **B** DAG and dynamics associated with two different rates of receptor phosphorylation by ($O_K$, *black traces*) in response to a step increase of extracellular glutamate ( at $t=0$). In the absence of receptor phosphorylation (*gray traces*), oscillations would vanish due to saturating intracellular 3 levels ensued from large receptor activation. **C** Period of oscillations as a function of extracellular glutamate concentration. Receptor phosphorylation by critically controls the oscillatory range (*black* and *blue curves*) with respect to the scenario without activation (*gray curve*). Higher glutamate concentrations are required to trigger oscillations for larger rates of -dependent activation ($O_{KD}$). Parameters as in [ \[tab:Model-Parameters\]]{} except for $\Omega_C=\SI{6.207}{s^{-1}},\, \Omega_L=\SI{0.01}{s^{-1}},\, O_\beta=\SI{1}{\micro \Molar s^{-1}}$.[]{data-label="fig:gchidp"}](f4_pkc_model.eps){width="100.00000%"}
[0.35]{}
![Regulation of oscillations by . **A** (*top panel*) Comparison between experimental traces for (*black*) and (*red*) originally recorded in cultured astrocytes by @CodazziTeruelMeyer2001 and simulations (*bottom panel*). Despite quantitive differences in the shape and period of oscillations, the model can reproduce the essential correlation and phase shift between and dynamics observed in experiments. and oscillations were triggered assuming an extracellular glutamate concentration of , and were normalized according to their maximum excursion: $C_0=\SI{0.04}{\micro \Molar}$, $P_0=\SI{48}{\nano \Molar}$, $\hat{C}=\SI{0.49}{\micro \Molar}$ and $\hat{P}=\SI{65}{\nano \Molar}$. **B** DAG and dynamics associated with two different rates of receptor phosphorylation by ($O_K$, *black traces*) in response to a step increase of extracellular glutamate ( at $t=0$). In the absence of receptor phosphorylation (*gray traces*), oscillations would vanish due to saturating intracellular 3 levels ensued from large receptor activation. **C** Period of oscillations as a function of extracellular glutamate concentration. Receptor phosphorylation by critically controls the oscillatory range (*black* and *blue curves*) with respect to the scenario without activation (*gray curve*). Higher glutamate concentrations are required to trigger oscillations for larger rates of -dependent activation ($O_{KD}$). Parameters as in [ \[tab:Model-Parameters\]]{} except for $\Omega_C=\SI{6.207}{s^{-1}},\, \Omega_L=\SI{0.01}{s^{-1}},\, O_\beta=\SI{1}{\micro \Molar s^{-1}}$.[]{data-label="fig:gchidp"}](f4_pkc_osc_1.eps){width="100.00000%"}
[0.35]{}
![Regulation of oscillations by . **A** (*top panel*) Comparison between experimental traces for (*black*) and (*red*) originally recorded in cultured astrocytes by @CodazziTeruelMeyer2001 and simulations (*bottom panel*). Despite quantitive differences in the shape and period of oscillations, the model can reproduce the essential correlation and phase shift between and dynamics observed in experiments. and oscillations were triggered assuming an extracellular glutamate concentration of , and were normalized according to their maximum excursion: $C_0=\SI{0.04}{\micro \Molar}$, $P_0=\SI{48}{\nano \Molar}$, $\hat{C}=\SI{0.49}{\micro \Molar}$ and $\hat{P}=\SI{65}{\nano \Molar}$. **B** DAG and dynamics associated with two different rates of receptor phosphorylation by ($O_K$, *black traces*) in response to a step increase of extracellular glutamate ( at $t=0$). In the absence of receptor phosphorylation (*gray traces*), oscillations would vanish due to saturating intracellular 3 levels ensued from large receptor activation. **C** Period of oscillations as a function of extracellular glutamate concentration. Receptor phosphorylation by critically controls the oscillatory range (*black* and *blue curves*) with respect to the scenario without activation (*gray curve*). Higher glutamate concentrations are required to trigger oscillations for larger rates of -dependent activation ($O_{KD}$). Parameters as in [ \[tab:Model-Parameters\]]{} except for $\Omega_C=\SI{6.207}{s^{-1}},\, \Omega_L=\SI{0.01}{s^{-1}},\, O_\beta=\SI{1}{\micro \Molar s^{-1}}$.[]{data-label="fig:gchidp"}](f4_pkc_osc_2.eps){width="100.00000%"}
[^1]: Deceased June 5, 2015.
|
University of Barcelona {#university-of-barcelona .unnumbered}
-----------------------
### PhD Thesis in Astronomy {#phd-thesis-in-astronomy .unnumbered}
New observational techniques and analysis tools for wide field CCD surveys and high resolution astrometry {#new-observational-techniques-and-analysis-tools-for-wide-field-ccd-surveys-and-high-resolution-astrometry .unnumbered}
=========================================================================================================
### by {#by .unnumbered}
Octavi Fors Aldrich {#octavi-fors-aldrich .unnumbered}
===================
### March 7th, 2006 {#march-7th-2006 .unnumbered}
Abstract {#abstract .unnumbered}
========
The aim of this thesis is two-fold. First it provides a general methodology for applying image deconvolution to wide-field CCD imagery. Second, two new CCD observational techniques and two data analysis tools are proposed for the first time in the context of high resolution astrometry, in particular for lunar occultations and speckle interferometry observations.\
In the first part of the thesis a wavelet-based adaptive image deconvolution algorithm ([AWMLE]{}) has been applied to two sets of survey type CCD data: QUasar Equatorial Survey Team project ([QUEST]{}) and Near-Earth Space Surveillance Terrestrial ([NESS-T]{}). Richardson-Lucy image deconvolution has also been used with Flagstaff Transit Telescope ([FASTT]{}) imagery. Both the obtaining and performance of those images were accomplished by following a new methodology which includes accurate image calibration, source detection and centering, and correct assessment procedures of the performance of the deconvolution. Results show that [AWMLE]{} deconvolution can increase limiting magnitude up to 0.6 mag and improve limiting resolution 1 pixel with respect to original image. These studies have been conducted in the context of programs dedicated to macrolensing search ([QUEST]{}) and NEOs discovery ([NESS-T]{}). Finally, astrometric accuracy of [FASTT]{} images have not been found to change significantly after deconvolution. In the same way, no positional bias towards the centre of the pixel has been observed.\
In the second part of the thesis a new observational technique based on CCD fast drift scanning has been proposed, implemented and assessed for lunar occultations (LO) and speckle interferometry observations.
In the case of LO, the technique yielded positive detection of binaries up to 2 mas of projected separation and stellar diameters measurements in the 7 mas regime. The proposed technique implies no optical or mechanical additional adjustments and can be applied to nearly all available full frame CCDs. Thus, it enables all kind of professional and high-end amateur observatories for LO work.
Complementary to this work, a four-year LO program (CALOP) at Calar Alto Observatory spanning 71.5 nights of observation and 388 recorded events has been conducted by means of CCD and MAGIC IR array cameras at OAN 1.5m and CAHA 2.2m telescopes. CALOP results include the detection of one triple system and 14 new and 1 known binaries in the near-IR, and one binary in the visible. Their projected separations range from 90 to 2 mas with brightness ratios up to 1:35 in the $K$ band. Several angular diameters have been also measured in the near-IR. The performance of CALOP has been calibrated in terms of limiting magnitude ($K_{lim}\sim9.0$) and limiting angular resolution ($\phi_{\rm
lim}\sim$1-3 mas). In addition, the binary detection probability of the program has found to be about 4%. Finally, a new wavelet-based method for extracting and characterizing LO lightcurves in an automated fashion was proposed, implemented and applied to CALOP database. This pipeline addresses the need of disposing of preliminary results in immediate basis for future programs which will provide larger number of events.
In the case of speckle interferometry, CCD fast drift scanning technique has been validated with the observation of four binary systems with well determined orbits. The results of separation, position angle and magnitude difference are in accordance with published measurements by other observers and predicted orbits. Error estimates for these have been found to be $0{\rlap.{''}}017$, $1{\rlap.^{\circ}}5$ and 0.34 mag, respectively. These are in the order of other authors and can be considered as successful for a first trial of this technique. Finally, a new approach for calibrating speckle transfer function from the binary power spectrum itself has been introduced. It does not require point source observations, which gives a more effective use of observation time. This new calibration method appears to be limited to zenith angles above $30{^{\circ}}$ when observing with no refraction compensation devices.
|
---
abstract: 'We discuss five ways of proving Chernoff’s bound and show how they lead to different extensions of the basic bound.'
author:
- Wolfgang Mulzer
bibliography:
- 'chernoff.bib'
title: 'Five Proofs of Chernoff’s Bound with Applications[^1]'
---
Introduction
============
Chernoff’s bound gives an estimate on the probability that a sum of independent Binomial random variables deviates from its expectation [@Hoeffding63]. It has many variants and extensions that are known under various names such as *Bernstein’s inequality* or *Hoeffding’s bound* [@Bernstein64; @Hoeffding63]. Chernoff’s bound is one of the most basic and versatile tools in the life of a theoretical computer scientist, with a seemingly endless amount of applications. Almost every contemporary textbook on algorithms or complexity theory contains a statement and a proof of the bound [@AroraBa09; @Goldreich08; @KleinbergTa06; @CormenLeRiSt09], and there are several texts that discuss its various applications in great detail (e.g., the textbooks by Alon and Spencer [@AlonSp16], Dubhashi and Panchonesi [@DubhashiPa09], Mitzenmacher and Upfal [@MitzenmacherUp17], Motwani and Raghavan [@MotwaniRa95], or the articles by Chung and Lu [@ChungLu06], Hagerup and Rüb [@HagerupRu90], or McDiarmid [@McDiarmid98]).
In the present survey, we will see five different ways of proving the basic Chernoff bound. The different techniques used in these proofs allow various generalizations and extensions, some of which we will also discuss.
The Basic Bound
===============
We begin with a statement of the basic Chernoff bound. For this, we first need a notion from information theory [@CoverTo06]. Let $P = (p_1, \dots, p_m)$ and $Q = (q_1, \dots, q_m)$ be two probability distributions on $m$ elements, i.e., $p_i, q_i \in {{\ensuremath {\mathbb {R}}}}$ with $p_i, q_i \geq 0$, for $i=1, \dots, m$, and $\sum_{i=1}^{m} p_i = \sum_{i=1}^m q_i = 1$. The *Kullback-Leibler divergence* or *relative entropy* of $P$ and $Q$ is defined as $${D_\textup{KL}}(P \| Q) {:=}\sum_{i=1}^m p_i \ln \frac{p_i}{q_i}.$$ If $m = 2$, i.e., if $P = (p, 1-p)$ and $Q = (q, 1-q)$, we write ${D_\textup{KL}}(p \| q)$ for ${D_\textup{KL}}((p, 1-p) \| (q, 1-q))$. The Kullback-Leibler divergence measures the distance between the distributions $P$ and $Q$: it represents the expected loss of efficiency if we encode an $m$-letter alphabet with distribution $P$ with a code that is optimal for distribution $Q$. Now, the basic Chernoff bound is as follows:
\[thm:chernoff\] Let $n \in {{\ensuremath {\mathbb {N}}}}$, $p \in [0,1]$, and let $X_1, \dots, X_n$ be $n$ independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$. Then, for any $t \in [0, 1-p]$, we have $$\Pr[X \geq (p+t)n ] \leq e^{-{D_\textup{KL}}(p+t \| p)n}.$$
Five Proofs for Theorem \[thm:chernoff\] {#sec:proofs}
========================================
We will now see five different ways of proving Theorem \[thm:chernoff\].
The Moment Method {#sec:moment}
-----------------
The usual textbook proof of Theorem \[thm:chernoff\] uses the exponential function $\exp$ and Markov’s inequality. It is called the *moment method*, because $\exp$ simultaneously encodes all *moments* $X, X^2, X^3, \dots$ of $X$. This trick is often attributed to Bernstein [@Bernstein64]. It is very general and can be used to obtain several variants of Theorem \[thm:chernoff\], perhaps most prominently, the Azuma-Hoeffding inequality for martingales with bounded differences [@Hoeffding63; @Azuma67].
The proof goes as follows. Let $\lambda > 0$ be a parameter to be determined later. We have $$\Pr[X \geq (p+t)n ] = \Pr[\lambda X \geq \lambda (p+t)n ] =
\Pr\bigl[e^{\lambda X} \geq e^{\lambda (p+t)n} \bigr].$$ From Markov’s inequality, we obtain $$\Pr\bigl[e^{\lambda X} \geq e^{\lambda (p+t)n} \bigr] \leq
\frac{{\mathbf {E}}[e^{\lambda X}]}{e^{\lambda (p+t)n}}.$$ Now, the independence of the $X_i$ yields $${\mathbf {E}}[e^{\lambda X}] = {\mathbf {E}}\Bigl[e^{\lambda \sum_{i=1}^n X_i}\Bigr]
= {\mathbf {E}}\Biggl[\prod_{i=1}^n e^{\lambda X_i}\Biggr]
= \prod_{i=1}^n {\mathbf {E}}\Bigl[e^{\lambda X_i}\Bigr]
= \bigl(p e^\lambda + 1-p\bigr)^n.$$ Thus, $$\label{equ:lambdabound}
\Pr[X > (p+t)n] \leq
\Bigl(\frac{pe^\lambda + 1-p}{e^{\lambda (p+t)}}\Bigr)^n,$$ for every $\lambda > 0$. Optimizing for $\lambda$ using calculus, we get that the right hand side is minimized if $$e^\lambda = \frac{(1-p)(p+t)}{p(1-p-t)}.$$ Plugging this into (\[equ:lambdabound\]), we get $$\Pr[X > (p+t)n] \leq \Biggl[\Bigl(\frac{p}{p+t}\Bigr)^{p+t}
\Bigl(\frac{1-p}{1-p-t}\Bigr)^{1-p-t}\Biggr]^n =
e^{-{D_\textup{KL}}(p+t \| p)n},$$ as desired.
Chvátal’s Method {#sec:chvatal}
----------------
The following proof of Theorem \[thm:chernoff\] is due to Chvátal [@Chvatal79]. As we will see below, it can be generalized to give tail bounds for the *hypergeometric distribution*. Let $B(n,p)$ be the random variable that gives the number of heads in $n$ independent Bernoulli trials with success probability $p$. Then, $$\Pr[B(n,p) = l] = \binom{n}{l} p^l (1-p)^{n-l},$$ for $l = 0, \dots, n$. Thus, for any $\tau \geq 1$ and $k \geq pn$, we get $$\begin{gathered}
\Pr[B(n,p)\ge k] = \sum_{i=k}^n \binom{n}{i}p^i (1-p)^{n-i}\\
\leq \sum_{i=k}^n \binom{n}{i}p^i (1-p)^{n-i}
\underbrace{\tau^{i-k}}_{\geq 1} +
\underbrace{\sum_{i=0}^{k-1} \binom{n}{i}p^i (1-p)^{n-i}
\tau^{i-k}}_{\geq 0}
= \sum_{i=0}^n \binom{n}{i}p^i (1-p)^{n-i} \tau^{i-k}.\end{gathered}$$ Using the Binomial theorem, we obtain $$\Pr[B(n,p)\ge k] \leq
\sum_{i=0}^n \binom{n}{i}p^i (1-p)^{n-i} \tau^{i-k} =
\tau^{-k}\sum_{i=0}^n \binom{n}{i}(p\tau)^{i} (1-p)^{n-i} =
\frac{(p\tau+1-p)^n}{\tau^k}.$$ If we write $k = (p+t)n$ and $\tau = e^\lambda$, we get $$\Pr[B(n,p)\ge (p+t)n] \leq
\Bigl(\frac{p e^\lambda+1-p}{e^{\lambda(p+t)}}\Bigr)^n.$$ This is the same as (\[equ:lambdabound\]), so we can complete the proof of Theorem \[thm:chernoff\] as in Section \[sec:moment\].
The Impagliazzo-Kabanets Method {#sec:ik}
-------------------------------
The third proof is due to Impagliazzo and Kabanets [@ImpagliazzoKa10], and it leads to a constructive version of the bound. Let $\lambda \in [0,1]$ be a parameter to be chosen later. Let $I \subseteq \{1, \dots, n\}$ be a random index set obtained by including each element $i \in \{1, \dots, n\}$ with probability $\lambda$. We estimate ${\mathbf {E}}\bigl[\prod_{i \in I} X_i\bigr]$ in two different ways, where the expectation is over the random choice of $X_1, \dots, X_n$ and $I$.
On the one hand, using the law of total expectation and independence, we have $$\begin{gathered}
\label{equ:ikupper}
{\mathbf {E}}\Bigl[\prod_{i \in I} X_i\Bigr]
= \sum_{S \subseteq \{1, \dots, n\}} \Pr[I = S] \cdot
{\mathbf {E}}\Bigl[\prod_{i \in S} X_i\Bigr]
= \sum_{S \subseteq \{1, \dots, n\}} \Pr[I = S] \cdot
\prod_{i \in S} \Pr[ X_i = 1]\\
= \sum_{S \subseteq \{1, \dots, n\}} \lambda^{|S|}(1-\lambda)^{n-|S|} \cdot
p^{|S|}
= (\lambda p + 1 - \lambda)^n.\end{gathered}$$ On the other hand, by the law of total expectation, $${\mathbf {E}}\Bigl[\prod_{i \in I} X_i\Bigr]
\geq
{\mathbf {E}}\Bigl[\prod_{i \in I} X_i \mid X \geq (p+t)n\Bigr]\Pr[X \geq (p+t)n].$$ Now, fix $X_1, \dots, X_n$ with $X \geq (p+t)n$. For the fixed choice of $X_1 = x_1, \dots, X_n = x_n$, the expectation ${\mathbf {E}}\bigl[\prod_{i \in I} x_i\bigr]$ is exactly the probability that $I$ avoids all the $n-X$ indices $i$ where $x_i = 0$. Thus, the conditional expectation is $${\mathbf {E}}\Bigl[\prod_{i \in I} X_i \mid X \geq (p+t)n\Bigr]
=
{\mathbf {E}}\Bigl[(1-\lambda)^{n-X} \mid X \geq (p+t)n\Bigr]
\geq (1-\lambda)^{(1-p-t)n},$$ so $${\mathbf {E}}\Bigl[\prod_{i \in I} X_i\Bigr]
\geq
(1-\lambda)^{(1-p-t)n}\Pr[X \geq (p+t)n].$$ Combining with (\[equ:ikupper\]), $$\label{equ:ikbound}
\Pr[X \geq (p+t)n] \leq
\left(\frac{\lambda p + 1 - \lambda}{(1-\lambda)^{(1-p-t)}}\right)^n.$$ Using calculus, we get that the right hand side is minimized for $\lambda = t/(1-p)(p+t)$ (note that $\lambda \leq 1$ for $t \leq 1-p$). Plugging this into (\[equ:ikbound\]), $$\Pr[X > (p+t)n] \leq
\Biggl[\Bigl(\frac{p}{p+t}\Bigr)^{p+t}
\Bigl(\frac{1-p}{1-p-t}\Bigr)^{1-p-t}\Biggr]^n =
e^{-{D_\textup{KL}}(p+t \| p)n},$$ as desired.
The Encoding Argument {#sec:encoding}
---------------------
The next proof stems from discussions with Luc Devroye, Gábor Lugosi, and Pat Morin, and it is inspired by an encoding argument [@MorinMuRe17]. A similar argument can also be derived from Xinjia Chen’s *likelihood ratio method* [@Chen13]. Let $\{0,1\}^n$ be the set of all bit strings of length $n$, and let $w: \{0, 1\}^n \rightarrow [0,1]$ be a *weight function*. We call $w$ *valid* if $\sum_{x \in \{0,1\}^n} w(x) \leq 1$. The following lemma says that for any probability distribution $p_x$ on $\{0,1\}^n$, a valid weight function is unlikely to be substantially larger than $p_x$.
\[lem:encoding\] Let $\mathcal{D}$ be a probability distribution on $\{0,1\}^n$ that assigns to each $x \in \{0,1\}^n$ a probability $p_x$, and let $w$ be a valid weight function. For any $s \geq 1$, we have $$\Pr_{x \sim \mathcal{D}}\left[w(x) \geq sp_x\right]
\leq 1/s.$$
Let $Z_{s} = \{ x \in \{0,1\}^n \mid w(x) \geq sp_x\}$. We have $$\Pr_{x \sim \mathcal{D}}\left[w(x) \geq s p_x\right]
= \sum_{\substack{x \in Z_s \\ p_x > 0}} p_x
\leq \sum_{\substack{x \in Z_s \\ p_x > 0}} p_x \frac{w(x)}{sp_x}
\leq (1/s) \sum_{x \in Z_s} w(x)
\leq 1/s,$$ since $w(x) / sp_x \geq 1$ for $x \in Z_s$, $p_x > 0$, and since $w$ is valid.
We now show that Lemma \[lem:encoding\] implies Theorem \[thm:chernoff\]. For this, we interpret the sequence $X_1, \dots, X_n$ as a bit string of length $n$. This induces a probability distribution $\mathcal{D}$ that assigns to each $x \in \{0, 1\}^n$ the probability $p_x = p^{k_x} (1-p)^{n-k_x}$, where $k_x$ denotes the number of $1$-bits in $x$. We define a weight function $w : \{0,1\}^n \rightarrow [0,1]$ by $w(x) = (p+t)^{k_x}(1-p-t)^{n-k_x}$, for $x \in \{0,1\}^n$. Then $w$ is valid, since $w(x)$ is the probability that $x$ is generated by setting each bit to $1$ independently with probability $p+t$. For $x \in \{0,1\}^n$, we have $$\frac{w(x)}{p_x}
=
\left(\frac{p+t}{p}\right)^{k_x}
\left(\frac{1-p-t}{1-p}\right)^{n - k_x}.$$ Since $((p+t)/p)((1-p)/(1-p-t)) \geq 1$, it follows that $w(x)/p_x$ is an increasing function of $k_x$. Hence, if $k_x \geq (p + t)n$, we have $$\frac{w(x)}{p_x}
\geq
\left[\left(\frac{p+t}{p}\right)^{p+t}
\left(\frac{1-p-t}{1-p}\right)^{1 - p-t}\right]^n
= e^{{D_\textup{KL}}(p+t \| p)n}.$$ We now apply Lemma \[lem:encoding\] to $\mathcal{D}$ and $w$ to get $$\Pr[X \geq (p+t)n] = \Pr_{x\sim \mathcal{D}} [ k_x \geq (p+t)n]
\leq \Pr_{x \sim \mathcal{D}} \left[ w(x) \geq
p_xe^{{D_\textup{KL}}(p+t \| p)n}\right]
\leq e^{-{D_\textup{KL}}(p+t \| p)n},$$ as claimed in Theorem \[thm:chernoff\].
See the survey [@MorinMuRe17] for a more thorough discussion of how this proof is related to coding theory.
A Proof via Differential Privacy
--------------------------------
The fifth proof of Chernoff’s bound is due to Steinke and Ullman [@SteinkeUl17], and it uses methods from the theory of differential privacy [@DworkRo14]. Unlike the previous four proofs, it seems to lead to a slightly weaker version of the bound. Let $m$ be a parameter to be determined later. The main idea is to bound the expectation of $m - 1$ independent copies of $X$.
Let $m \in {{\ensuremath {\mathbb {N}}}}$ and $m \leq e^{n}$. Let $X^{(1)}, \dots, X^{(m - 1)}$ be $m - 1$ independent copies of $X$, and set $X^{(m)} = {\mathbf {E}}[X]$. Then, $${\mathbf {E}}\big[\max\{X^{(1)}, \dots, X^{(m)}\}\big]
\leq pn +
5\sqrt{n \ln m}.$$ \[lem:maxexp\]
We will give a proof of Lemma \[lem:maxexp\] below. First, however, we will see how we can use Lemma \[lem:maxexp\] to derive the following weaker version of Theorem \[thm:chernoff\].[^2]
\[thm:weakchernoff\] Let $n \in {{\ensuremath {\mathbb {N}}}}$, $p \in [0,1]$, and let $X_1, \dots, X_n$ be $n$ independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$. Then, for any $t \in [0, 1-p]$, we have $$\Pr[X \geq (p+t)n ] \leq e^{1-\frac{1}{64}t^2 n}.$$
We may assume that $t \geq 8/\sqrt{n}$, since otherwise the lemma holds trivially. Set $\alpha = \Pr[X \geq (p+t)n ]$. Let $X^{(1)}, \dots, X^{(m - 1)}$ be $m - 1$ independent copies of $X$ and let $X^{(m)} = {\mathbf {E}}[X]$. Then, $$\label{equ:maxXlb}
\Pr\big[\max\{X^{(1)}, \dots, X^{(m)}\}
\geq (p+t)n\big] = 1 - (1-\alpha)^{m - 1} \geq 1 - e^{-\alpha (m - 1)}.$$ On the other hand, Markov’s inequality gives $$\begin{gathered}
\Pr\big[\max\{X^{(1)}, \dots, X^{(m)}\} \geq (p+t)n\big] =
\Pr\big[\max\{X^{(1)}, \dots, X^{(m)}\} -pn \geq tn\big]\\
\leq
\frac{{\mathbf {E}}\big[\max\{X^{(1)}, \dots, X^{(m)}\} - pn\big]}
{tn}
\leq \frac{5\sqrt{\ln m}}{t\sqrt{n}},\end{gathered}$$ by Lemma \[lem:maxexp\]. Thus, setting $m = \exp\Big(\big(\frac{e-1}{5e}\big)^2t^2n\Big)$, and combining with (\[equ:maxXlb\]), we get $$\frac{e-1}{e} \geq 1 - e^{-\alpha (m - 1)} \Leftrightarrow
\alpha \leq \frac{1}{\exp\Big(\big(\frac{e-1}{5e}\big)^2t^2n\Big)
- 1}
\leq
\frac{1}{\exp\big(\frac{t^2n}{64}\big) - 1},$$ since $\big(\frac{e-1}{5e}\big)^2 \geq \frac{1}{64}$. Now the lemma follows from $$\frac{\exp\big(\frac{t^2n}{64}\big)}{\exp\big(\frac{t^2n}{64}\big) - 1}
\leq
\frac{e}{e-1}
\leq
e,$$ which holds as $t \geq 8/\sqrt{n}$, as $x \mapsto x/(x-1)$ is decreasing for $x \geq 0$, and as $e \geq 2$.
It remains to prove Lemma \[lem:maxexp\]. For this, we use an idea from differential privacy. Let $A \in [0, 1]^{m \times n}$, $A = (a_{ij})$, be an $(m \times n)$-matrix with entries from $[0, 1]$. For a given parameter $\gamma > 1$, we define a random variable $S_\gamma(A)$ with values in $\{1, \dots, m \}$ as follows: for $i = 1, \dots, m$, let $b_i = \sum_{j = 1, \dots, n} a_{ij}$ be the sum of the entries in the $i$-th row of $A$. Set $$C_\gamma(A) = \sum_{i = 1}^{m} \gamma^{b_i}.$$ Then, for $i = 1, \dots, m$, we define $$\Pr[S_\gamma(A) = i] = \frac{\gamma^{b_i}}{C_\gamma(A)}.$$ The random variable $S_\gamma(A)$ is called a *stable selector* for $A$ (see the work by McSherry and Talwar [@McSherryTa07] for more background). The next lemma states two interesting properties for $S_\gamma(A)$. For a matrix $A \in [0, 1]^{m \times n}$, a vector $\vec{c} \in [0,1]^m$, and a number $j \in \{1, \dots, n\}$ we denote by $(A_{-j}, \vec{c})$ the matrix obtained from $A$ by replacing the $j$-th column of $A$ with $\vec{c}$.
\[lem:stableselect\] Let $A \in [0, 1]^{m \times n}$ be an $m \times n$ matrix with entries in $[0, 1]$. We have
- **Stability**: For every vector $\vec{c} \in [0,1]^m$ and every $i \in \{1, \dots, m\}$, $$\gamma^{-2} \Pr[S_\gamma(A_{-j},\vec{c}) = i]
\leq \Pr[S_\gamma(A) = i] \leq
\gamma^{2} \Pr[S_\gamma(A_{-j},\vec{c}) = i].$$
- **Accuracy**: Let $b_i$ be the sum of the $i$-th row of $A$. Then,
$${\mathbf {E}}_{i \sim S_\gamma(A)}[b_i] \leq
\max_{i = 1}^{m} b_i \leq {\mathbf {E}}_{i \sim S_\gamma(A)}[b_i] + \log_\gamma m.$$
**Stability**: for $k \in \{1, \dots, m\}$, let $b_k$ be the sum of the $k$-th row of $A$, and let $\widetilde{b}_k$ be the sum of the $k$-th row of $(A_{-j}, \widetilde{c})$. Since $A$ and $(A_{-j}, \widetilde{c})$ differ in one column, and since the entries are from $[0, 1]$, we have $\widetilde{b}_k - 1 \leq b_k \leq \widetilde{b}_k + 1$. Hence, $$\gamma^{-1} C_\gamma(A_{-j},\vec{c}) \leq C_\gamma(A) \leq
\gamma C_\gamma(A_{-j},\vec{c})$$ and $$\gamma^{-2} \Pr[S_\gamma(A_{-j},\vec{c}) = i]
\leq \Pr[S_\gamma(A) = i] \leq
\gamma^{2} \Pr[S_\gamma(A_{-j},\vec{c}) = i],$$ as claimed.
**Accuracy**: The inequality ${\mathbf {E}}_{i \sim S_\gamma(A)}[b_i] \leq \max_{i = 1}^{m} b_i$ is obvious. For the second inequality, we observe that by definition, $$b_i = \log_\gamma (C_\gamma(A) \Pr[S_\gamma(A) = i]).$$ Thus, $$\begin{aligned}
{\mathbf {E}}_{i \sim S_\gamma(A)}[b_i]
&= \sum_{i = 1}^{m}\Pr[S_\gamma(A) = i]\log_\gamma
(C_\gamma(A) \Pr[S_\gamma(A) = i])\\
&=
\sum_{i = 1}^{m}\Pr[S_\gamma(A) = i]\log_\gamma
C_\gamma(A) -
\sum_{i = 1}^{m}\Pr[S_\gamma(A) = i]\log_\gamma
\frac{1}{\Pr[S_\gamma(A) = i]}\\
&\geq
\sum_{i = 1}^{m}\Pr[S_\gamma(A) = i]\log_\gamma
\gamma^{\max_{i = 1}^m b_i} - \log_\gamma m,\\
&=
\max_{i = 1}^m b_i- \log_\gamma m,\end{aligned}$$ since $C_\gamma(A) = \sum_{i = 1}^{m} \gamma^{b_i} \geq
\gamma^{\max_{i = 1}^m b_i}$ and since $x \mapsto - \log_\gamma(x)$ is a convex function.
Lemma \[lem:stableselect\] shows that $S_\gamma(A)$ constitutes a reasonable mechanism of estimating the maximum row sum of $A$ without revealing too much information about any single column of $A$. We can now use Lemma \[lem:stableselect\] to bound the expectation of the maximum of $m - 1$ independent copies of $X$ and ${\mathbf {E}}[X]$.
\[lem:maxgamma\] Let $m \in {{\ensuremath {\mathbb {N}}}}$. let $X^{(1)}, \dots, X^{(m - 1)}$ be $m - 1$ independent copies of $X$, and set $X^{(m)} = {\mathbf {E}}[X]$. Then, for any $\gamma > 1$, we have $${\mathbf {E}}\big[\max\{X^{(1)}, \dots, X^{(m)}\}\big] \leq
\gamma^2pn + \log_\gamma m.$$
Let $X_1^{(1)}, \dots, X_1^{(m - 1)}$ be $m - 1$ independent copies of $X_1$, and let $X_1^{(m)} = {\mathbf {E}}[X_1]$; let $X_2^{(1)}, \dots, X_2^{(m - 1)}$ be $m - 1$ independent copies of $X_2$ and let $X_2^{(m)} = {\mathbf {E}}[X_2]$; and so on. We consider the random $m \times n$ matrix $M \in \{0,1\}^{m \times n}$ whose entry in row $i$ and column $j$ is $X_j^{(i)}$. Then, we can write $X^{(i)} = \sum_{j = 1}^n X_j^{(i)}$, for $i = 1, \dots, m$. By the accuracy claim in Lemma \[lem:stableselect\], $$\label{equ:accumatrix}
{\mathbf {E}}_{M} \big[\max\{X^{(1)}, \dots, X^{(m)}\}\big]
\leq {\mathbf {E}}_{M, i \sim S_\gamma(M)}\big[X^{(i)}\big] + \log_\gamma m$$ Now we bound ${\mathbf {E}}_{M, i \sim S_\gamma(M)}\big[X^{(i)}\big]$. We unwrap the expectation for $i \sim S_\gamma(M)$ and get $${\mathbf {E}}_{M, i \sim S_\gamma(M)}[X^{(i)}]
=
{\mathbf {E}}_{M}
\Big[\sum_{i = 1}^{m} \Pr[S_\gamma(M) = i] X^{(i)}\Big]$$ Let $\widetilde{M}$ be an independent copy of $M$. Denote the entry in the $i$-th row and $j$-th column of $\widetilde{M}$ by $\widetilde{X}_j^{(i)}$, and set $\widetilde{X}^{(i)} = \sum_{j = 1}^{n} \widetilde{X}_j^{(i)}$, for $i = 1, \dots, m$. By the stability claim in Lemma \[lem:stableselect\], for every $j \in \{1, \dots, n\}$, $$\begin{aligned}
{\mathbf {E}}_{M}
\Big[\sum_{i = 1}^{m} \Pr\big[S_\gamma(M) = i\big] X^{(i)}\Big]
&\leq
\gamma^2
{\mathbf {E}}_{M,\widetilde{M}}
\Big[\sum_{i = 1}^{m}
\Pr\big[S_\gamma(M_{-j},\widetilde{M}_j) = i\big] X^{(i)}\Big].\\
\intertext{Since the random variables $X_j^{(i)}$, $\widetilde{X}_j^{(i)}$,
$1 \leq i \leq m$, $1 \leq j \leq n$,
are independent, the pairs $\big((M_{-j}, \widetilde{M}_j), X_j^{(i)}\big)$
and $\big(M, \widetilde{X}_j^{(i)}\big)$ have the same distribution.
Therefore, we can write}
{\mathbf {E}}_{M}
\Big[\sum_{i = 1}^{m}
\Pr\big[S_\gamma(M) = i\big] X^{(i)}\Big]&=
{\mathbf {E}}_{M}
\Big[\sum_{i = 1}^{m}
\sum_{j = 1}^{n}
\Pr\big[S_\gamma(M) = i\big] X_j^{(i)}\Big]\\
&\leq \gamma^2
{\mathbf {E}}_{M,\widetilde{M}}
\Big[\sum_{j = 1}^{n} \sum_{i = 1}^{m}
\Pr\big[S_\gamma(M_{-j},\widetilde{M}_j) = i\big] X_j^{(i)}\Big]\\ &=
\gamma^2
{\mathbf {E}}_{M,\widetilde{M}}
\Big[\sum_{j = 1}^{n} \sum_{i = 1}^{m}
\Pr\big[S_\gamma(M) = i\big] \widetilde{X}_j^{(i)}\Big]\\ &=
\gamma^2
{\mathbf {E}}_{M}
\Big[\sum_{i = 1}^{m} \Pr\big[S_\gamma(M) = i\big]
{\mathbf {E}}_{\widetilde{M}}\big[\widetilde{X}^{(i)}\big]\Big]\\
&=
\gamma^2
{\mathbf {E}}_{M}
\Big[\sum_{i = 1}^{m} \Pr\big[S_\gamma(M) = i\big]
pn\Big] = \gamma^2 pn.\end{aligned}$$ We can conclude the lemma by plugging this bound into (\[equ:accumatrix\]).
To obtain Lemma \[lem:maxexp\], we set $\gamma = 1+\frac{\sqrt{\ln m}}{\sqrt{n}}$. Now, Lemma \[lem:maxgamma\] gives $$\begin{aligned}
{\mathbf {E}}\big[\max\{X^{(1)}, \dots, X^{(m)}\}\big] &\leq
\left(1+\frac{\sqrt{\ln m}}{\sqrt{n}}\right)^2pn +
\frac{\ln m}{\ln\left(1+\frac{\sqrt{\ln m}}{\sqrt{n}}\right)}\\
&\leq
\left(1+\frac{3\sqrt{\ln m}}{\sqrt{n}}\right)pn +
\frac{\ln m}{\frac{\sqrt{\ln m}}{2 \sqrt{n}}},\\
\intertext{since $\frac{\sqrt{\ln m}}{\sqrt{n}} \leq 1$ by our assumption
$m \leq e^n$ and
$\ln (1+x) \geq x/2$, for $x \in [0,1]$. Hence, using $pn \leq n$,}
{\mathbf {E}}\big[\max\{X^{(1)}, \dots, X^{(m)}\}\big] &\leq
pn + 5\sqrt{n\ln m},\end{aligned}$$ as desired.
Useful Consequences {#sec:conseq}
===================
We now show several useful consequences of Theorem \[thm:chernoff\]. These results can be derived directly from Theorem \[thm:chernoff\], and therefore they also hold for variants of the theorem with slightly different assumptions.
The Lower Tail
--------------
First, we show that an analogous bound holds for the lower tail probability $\Pr[ X \leq (p-t) n]$.
\[cor:chernoff\_lower\] Let $X_1, \dots, X_n$ be independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$. Then, for any $t \in [0, p]$, we have $$\Pr[X \leq (p-t)n ] \leq e^{-{D_\textup{KL}}(p-t \| p)n}.$$
$$\begin{aligned}
\Pr[X \leq (p-t)n ] = \Pr[n-X \geq n-(p-t)n ] = \Pr[X' \geq (1-p+t)n ],\end{aligned}$$
where $X' = \sum_{i=1}^{n} X_i'$ with independent random variables $X_i' \in \{0,1\}$ such that $\Pr[X_i' = 1] = 1-p$. The result follows from ${D_\textup{KL}}(1-p+t \| 1-p) = {D_\textup{KL}}(p-t \| p)$.
Multiplicative Version
----------------------
Next, we derive a multiplicative variant of Theorem \[thm:chernoff\]. This well-known version of the bound can be found in the classic text by Motwani and Raghavan [@MotwaniRa95].
\[cor:MR\] Let $X_1, \dots, X_n$ be independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$ and $\mu = pn$. Then, for any $\delta \geq 0$, we have $$\begin{aligned}
\Pr[X \geq (1+\delta)\mu ] &\leq
\left(\frac{e^{\delta}}{(1+\delta)^{1+\delta}}\right)^\mu,
\text{ and}\\
\Pr[X \leq (1-\delta)\mu ] &\leq
\left(\frac{e^{-\delta}}{(1-\delta)^{1-\delta}}\right)^\mu.\end{aligned}$$
Setting $t = \delta\mu/n$ in Theorem \[thm:chernoff\] yields $$\begin{aligned}
\Pr[X \geq (1+\delta)\mu] &\leq
\exp\left(-n\left[p(1+\delta)\ln(1+\delta) +
p\left(\frac{1-p}{p}-\delta \right)\ln\left(1-\delta
\frac{p}{1-p}\right)\right]\right)\\
&= \left(\frac{(1-\delta p/(1-p))^{\delta - (1-p)/p}}{(1+\delta)^{1+\delta}}\right)^\mu\\
&\leq \left(\frac{e^{-\delta^2p/(1-p) + \delta}}{(1+\delta)^{1+\delta}}\right)^\mu
\leq \left(\frac{e^{\delta}}{(1+\delta)^{1+\delta}}\right)^\mu.\end{aligned}$$ Setting $t = \delta\mu/n$ in Corollary \[cor:chernoff\_lower\] yields $$\begin{aligned}
\Pr[X \leq (1-\delta)\mu] &\leq
\exp\left(-n\left[p(1-\delta)\ln(1-\delta) +
p\left(\frac{1-p}{p}+\delta \right)
\ln\left(1+\delta \frac{p}{1-p}\right)\right]\right)\\
&= \left(\frac{(1+\delta p/(1-p))^{-\delta - (1-p)/p}}
{(1-\delta)^{1-\delta}}\right)^\mu\\
&\leq \left(\frac{e^{-\delta^2p/(1-p) - \delta}}
{(1-\delta)^{1-\delta}}\right)^\mu
\leq \left(\frac{e^{-\delta}}{(1-\delta)^{1-\delta}}\right)^\mu.\end{aligned}$$
Useful Variants
---------------
The next few corollaries give some handy variants of the bound that are often more manageable in practice. First, we give a simple bound for the multiplicative lower tail.
\[cor:handy\_lower\] Let $X_1, \dots, X_n$ be independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$ and $\mu = pn$. Then, for any $\delta \in (0, 1)$, we have $$\Pr[X \leq (1-\delta)\mu ] \leq e^{-\delta^2\mu/2}.$$
By Corollary \[cor:MR\] $$\Pr[X \leq (1-\delta)\mu ] \leq
\left(\frac{e^{-\delta}}{(1-\delta)^{1-\delta}}\right)^\mu.$$ Using the power series expansion of $\ln(1-\delta)$, we get $$(1-\delta)\ln(1-\delta) = -(1 - \delta)
\sum_{i=1}^\infty \frac{\delta^i}{i}
= -\delta + \sum_{i=2}^\infty \frac{\delta^i}{(i-1)i}
\geq -\delta + \delta^2/2.$$ Thus, $$\Pr[X \leq (1-\delta)\mu ] \leq e^{[-\delta + \delta-\delta^2/2]\mu} =
e^{-\delta^2\mu/2},$$ as claimed.
An only slightly more complicated bound can be found for the multiplicative upper tail.
\[cor:handy\_upper\] Let $X_1, \dots, X_n$ be independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$ and $\mu = pn$. Then, for any $\delta \geq 0$, we have $$\Pr[X \geq (1+\delta)\mu ] \leq e^{-\min\{\delta^2,\delta\}\mu/4}.$$
We may assume that $(1+\delta)p \leq 1$. Then, Theorem \[thm:chernoff\] gives $$\Pr[X \geq (1+\delta)pn ] \leq e^{-{D_\textup{KL}}((1+\delta)p \| p)n}.$$ Define $f(\delta) {:=}{D_\textup{KL}}((1+\delta)p \| p)$. Then, $$f'(\delta) = p \ln(1+\delta) - p \ln (1-\delta p / (1-p))$$ and $$f''(\delta) = \frac{p}{(1+\delta)(1-p - \delta p)} \geq
\frac{p}{1+\delta}.$$ By Taylor’s theorem, we have $$f(\delta) = f(0) + \delta f'(0) + \frac{\delta^2}{2} f''(\xi),$$ for some $\xi \in [0,\delta]$. Since $f(0) = f'(0) = 0$, it follows that $$f(\delta) = \frac{\delta^2}{2} f''(\xi) \geq
\frac{\delta^2p}{2(1+\xi)} \geq \frac{\delta^2p}{2(1+\delta)}.$$ For $\delta \geq 1$, we have $\delta/(1+\delta) \geq 1/2$, for $\delta < 1$, we have $1/(\delta+1) \geq 1/2$. This gives, for all $\delta \geq 0$, $$f(\delta) \geq \min\{\delta^2, \delta\}p/4,$$ and the claim follows.
The following corollary combines the two bounds. This variant can be found, e.g., in the book by Arora and Barak [@AroraBa09].
Let $X_1, \dots, X_n$ be independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$ and $\mu = pn$. Then, for any $\delta > 0$, we have $$\Pr[|X - \mu| \geq \delta\mu ] \leq 2e^{-\min\{\delta^2, \delta\}\mu/4}.$$
Combine Corollaries \[cor:handy\_lower\] and \[cor:handy\_upper\].
The following corollary, which appears, e.g., in the book by Motwani and Raghavan [@MotwaniRa95], is also sometimes useful.
Let $X_1, \dots, X_n$ be independent random variables with $X_i \in \{0,1\}$ and $\Pr[X_i = 1] = p$, for $i = 1, \dots n$. Set $X {:=}\sum_{i=1}^n X_i$ and $\mu = pn$. For $t \geq 2e\mu$, we have $$\Pr[X \geq t ] \leq 2^{-t}.$$
By Corollary \[cor:MR\] $$\Pr[X \geq (1+\delta)\mu ] \leq \left(\frac{e^{\delta}}{(1+\delta)^{1+\delta}}\right)^\mu
\leq \left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu}.$$ For $\delta \geq 2e-1$, the denominator in the right hand side is at least $2e$, and the claim follows.
Generalizations
===============
We mention a few generalizations of the proof techniques for Section \[sec:proofs\]. Since the consequences from Section \[sec:conseq\] are based on simple algebraic manipulation of the bounds, the same consequences also hold for the generalized settings.
Hoeffding Extension
-------------------
The moment method (Section \[sec:moment\]) yields many generalizations of Theorem \[thm:chernoff\]. The following result is known as *Hoeffding’s extension* [@Hoeffding63]. It shows that the $X_i$ can actually be chosen to be continuous with varying expectations.
\[thm:chernoff\_hoeff\] Let $X_1, \dots, X_n$ be independent random variables with $X_i \in [0,1]$ and ${\mathbf {E}}[X_i] = p_i$. Set $X {:=}\sum_{i=1}^n X_i$ and $p {:=}(1/n)\sum_{i=1}^n p_i$. Then, for any $t \in [0, 1-p]$, we have $$\Pr[X \geq (p+t)n ] \leq e^{-{D_\textup{KL}}(p+t \| p)n}.$$
Let $\lambda > 0$ a parameter to be determined later. As before, Markov’s inequality yields $$\Pr\bigl[e^{\lambda X} \geq e^{\lambda (p+t)n} \bigr] \leq
\frac{{\mathbf {E}}[e^{\lambda X}]}{e^{\lambda (p+t)n}}.$$ Using independence, we get $$\label{equ:boundExp}
{\mathbf {E}}[e^{\lambda X}] = {\mathbf {E}}\Bigl[e^{\lambda \sum_{i=1}^n X_i}\Bigr]
= \prod_{i=1}^n {\mathbf {E}}\Bigl[e^{\lambda X_i}\Bigr].$$ Now we need to estimate ${\mathbf {E}}\bigl[e^{\lambda X_i}\bigr]$. The function $z \mapsto e^{\lambda z}$ is convex, so $e^{\lambda z}
\leq (1-z) e^{0 \cdot \lambda} + z e^{1 \cdot \lambda}$ for $z \in [0,1]$. Hence, $${\mathbf {E}}\bigl[e^{\lambda X_i}\bigr] \leq {\mathbf {E}}[1-X_i + X_i e^\lambda] =
1-p_i + p_ie^\lambda.$$ Going back to (\[equ:boundExp\]), $${\mathbf {E}}[e^{\lambda X}] \leq
\prod_{i=1}^n (1-p_i + p_i e^\lambda).$$ Using the arithmetic-geometric mean inequality $\prod_{i=1}^n x_i \leq
\bigl((1/n)\sum_{i=1}^n x_i\bigr)^n$, for $x_i \geq 0$, this is $${\mathbf {E}}[e^{\lambda X}] \leq (1-p+pe^\lambda)^n.$$ From here we continue as in Section \[sec:moment\].
Hypergeometric Distribution
---------------------------
Chvátals proof [@Chvatal79] from Section \[sec:chvatal\] generalizes to the *hypergeometric* distribution. We emphasize once again that this means that all the corollaries from Section \[sec:conseq\] also apply to this case.
\[thm:hyper\] Suppose we have an urn with $N$ balls, $P$ of which are red. We randomly draw $n$ balls from the urn without replacement. Let $H(N,P,n)$ denote the number of red balls in the sample. Set $p {:=}P/N$. Then, for any $t \in [0, 1-p]$, we have $$\Pr\big[H(N,P,n) \geq (p+t)n \big] \leq e^{-{D_\textup{KL}}(p+t \| p)n}.$$
It is well known that $$\Pr[H(N,P,n) = l] = \binom{P}{l}\binom{N-p}{n-l}\binom{N}{l}^{-1},$$ for $l = 0, \dots, n$.
\[clm:hyper\_bound\] For every $j \in \{0, \dots, n\}$, we have $$\binom{N}{n}^{-1} \sum_{i = j}^n \binom{P}{i}\binom{N-P}{n - i}\binom{i}{j}
\leq \binom{n}{j}p^j.$$
Consider the following random experiment: take a random permutation of the $N$ balls in the urn. Let $S$ be the sequence of the first $n$ elements in the permutation. Let $X$ be the number of $j$-subsets of $S$ that contain only red balls. We compute ${\mathbf {E}}[X]$ in two different ways. On the one hand, $$\label{equ:hyper_bound_var1}
{\mathbf {E}}[X] = \sum_{i = j}^n \Pr[\text{S contains $i$ red balls}] \binom{i}{j}
= \sum_{i = j}^n \binom{N}{n}^{-1} \binom{P}{i} \binom{N-P}{n-i} \binom{i}{j}.$$ On the other hand, let $I \subseteq \{1, \dots, n\}$ with $|I| = j$. Then the probability that all the balls in the positions indexed by $I$ are red is $$\frac{P}{N} \cdot \frac{P-1}{N-1} \cdot \cdots \cdot
\frac{P-j+1}{N-j+1} \leq \left(\frac{P}{N}\right)^j = p^j.$$ Thus, by linearity of expectation ${\mathbf {E}}[X] \leq \binom{n}{j} p^j$. Together with (\[equ:hyper\_bound\_var1\]), the claim follows.
\[clm:hyper\_bound2\] For every $\tau \geq 1$, we have $$\binom{N}{n}^{-1} \sum_{i = 0}^n \binom{P}{i}\binom{N-P}{n - i}\tau^i
\leq (1 + (\tau-1)p)^n.$$
Using Claim \[clm:hyper\_bound\] and the Binomial theorem (twice), $$\begin{aligned}
\binom{N}{n}^{-1} \sum_{i = 0}^n \binom{P}{i}\binom{N-P}{n - i}\tau^i
&= \binom{N}{n}^{-1} \sum_{i = 0}^n \binom{P}{i}\binom{N-P}{n - i}
(1-(\tau-1))^i\\
&= \binom{N}{n}^{-1} \sum_{i = 0}^n \binom{P}{i}\binom{N-P}{n - i}
\sum_{j=0}^i \binom{i}{j}(\tau-1)^j\\
&= \binom{N}{n}^{-1} \sum_{j = 0}^n (\tau-1)^j \sum_{i=j}^n
\binom{P}{i}\binom{N-P}{n - i}\binom{i}{j}\\
&\leq \sum_{j = 0}^n \binom{n}{j}((\tau-1)p)^j = (1 + (\tau-1)p)^n,\end{aligned}$$ as claimed.
Thus, for any $\tau \geq 1$ and $k \geq pn$, we get as before $$\begin{gathered}
\Pr[H(N,P,n)\geq k] = \binom{N}{n}^{-1}\sum_{i=k}^n \binom{P}{i}\binom{N-P}{n-i}\\
\leq \binom{N}{n}^{-1}\sum_{i=0}^n \binom{P}{i}\binom{N-P}{n-i}
\tau^{i-k}\leq
\frac{(p\tau+1-p)^n}{\tau^k},\end{gathered}$$ by Claim \[clm:hyper\_bound2\]. From here the proof proceeds as in Section \[sec:chvatal\].
Negative Correlations
---------------------
The proof by Impagliazzo and Kabanets [@ImpagliazzoKa10] from Section \[sec:ik\] can be used to relax the independence assumption. It now suffices that the random variables are *negatively correlated*.
\[thm:general\_ik\] Let $X_1, \dots, X_n$ be random variables with $X_i \in \{0,1\}$. Suppose there exist $p_i \in [0,1]$, $i = 1, \dots, n$, such that for every index set $I \subseteq \{1, \dots, n\}$, we have ${\mathbf {E}}\big[\prod_{i \in I} X_i \big] \leq \prod_{i \in I} p_i$. Set $X {:=}\sum_{i=1}^n X_i$ and $p {:=}(1/n)\sum_{i=1}^n p_i$. Then, for any $t \in [0, 1-p]$, we have $$\Pr[X \geq (p+t)n ] \leq e^{-{D_\textup{KL}}(p+t \| p)n}.$$
Let $\lambda \in [0,1]$ be a parameter to be chosen later. Let $I \subseteq \{1, \dots, n\}$ be a random index set obtained by including each element $i \in \{1, \dots, n\}$ with probability $\lambda$. As before, we estimate the expectation ${\mathbf {E}}\bigl[\prod_{i \in I} X_i \bigr]$ in two different ways, where the expectation is over the random choice of $X_1, \dots, X_n$ and $I$. Similarly to before, $$\begin{gathered}
\label{equ:ikupper_gen}
{\mathbf {E}}\Bigl[\prod_{i \in I} X_i \Bigr]
= \sum_{S \subseteq \{1, \dots, n\}}
\Pr[I = S] \cdot {\mathbf {E}}\Bigl[ \prod_{i \in S} X_i \Bigr]
\leq \sum_{S \subseteq \{1, \dots, n\}} \lambda^{|S|}(1-\lambda)^{n-|S|}
\cdot \Big(\prod_{i \in S} p_i \Big)\\
= \sum_{S \subseteq \{1, \dots, n\}}
\Big(\prod_{i \in S} \lambda p_i\Big)
\Big(\prod_{i \in \{1, \dots, n\} \setminus S}
(1-\lambda)\Big) =
\prod_{i=1}^n (1-\lambda + p_i \lambda)
\leq (1-\lambda + p \lambda)^n,\end{gathered}$$ by the arithmetic-geometric mean inequality. The proof of the lower bound remains unchanged and yields $${\mathbf {E}}\Bigl[\prod_{i \in I} X_i\Bigr]
\geq
(1-\lambda)^{(1-p-t)n}\Pr[X \geq (p+t)n],$$ as before. Combining with (\[equ:ikupper\_gen\]) and optimizing for $\lambda$ finishes the proof, see Section \[sec:ik\].
#### Acknowledgments. {#acknowledgments. .unnumbered}
This survey is based on lecture notes for a class on advanced algorithms at Freie Universität Berlin. I would like to thank all the students who took this class for their interest and participation. I would also like to thank Nabil Mustafa and Jonathan Ullman for valuable comments that improved this survey.
[^1]: Supported in part by DFG Grants MU 3501/1 and MU 3501/2 and ERC StG 757609.
[^2]: In the published version of this paper, the proof of Theorem \[thm:weakchernoff\] is based on an incorrect application of Markov’s inequality. We have changed Lemma \[lem:maxexp\] so that $X^{(m)}$ is fixed to ${\mathbf {E}}[X]$. This ensures that Markov’s inequality is applied to a nonnegative random variable. We thank Natalia Shenkman for pointing this out to us.
|
---
author:
- 'Authors: R. Adhikari'
- 'M. Agostini'
- 'N. Anh Ky'
- 'T. Araki'
- 'M. Archidiacono'
- 'M. Bahr'
- 'J. Behrens'
- 'F. Bezrukov'
- 'P.S. Bhupal Dev'
- 'D. Borah'
- 'A. Boyarsky'
- 'A. de Gouvea'
- 'C.A. de S. Pires'
- 'H.J. de Vega$^\dagger$'
- 'A.G. Dias'
- 'P. Di Bari'
- 'Z. Djurcic'
- 'K. Dolde'
- 'H. Dorrer'
- 'M. Durero'
- 'O. Dragoun'
- 'M. Drewes'
- 'Ch.E. Düllmann'
- 'K. Eberhardt'
- 'S. Eliseev'
- 'C. Enss'
- 'N.W. Evans'
- 'A. Faessler'
- 'P. Filianin'
- 'V. Fischer'
- 'A. Fleischmann'
- 'J.A. Formaggio'
- 'J. Franse'
- 'F.M. Fraenkle'
- 'C.S. Frenk'
- 'G. Fuller'
- 'L. Gastaldo'
- 'A. Garzilli'
- 'C. Giunti'
- 'F. Glück'
- 'M.C. Goodman'
- 'M.C. Gonzalez-Garcia'
- 'D. Gorbunov'
- 'J. Hamann'
- 'V. Hannen'
- 'S. Hannestad'
- 'J. Heeck'
- 'S.H. Hansen'
- 'C. Hassel'
- 'F. Hofmann'
- 'T. Houdy'
- 'A. Huber'
- 'D. Iakubovskyi'
- 'A. Ianni'
- 'A. Ibarra'
- 'R. Jacobsson'
- 'T. Jeltema'
- 'S. Kempf'
- 'T. Kieck'
- 'M. Korzeczek'
- 'V. Kornoukhov'
- 'T. Lachenmaier'
- 'M. Laine'
- 'P. Langacker'
- 'T. Lasserre'
- 'J. Lesgourgues'
- 'D. Lhuillier'
- 'Y. F. Li'
- 'W. Liao'
- 'A.W. Long'
- 'M. Maltoni'
- 'G. Mangano'
- 'N.E. Mavromatos'
- 'N. Menci'
- 'A. Merle'
- 'S. Mertens'
- 'A. Mirizzi'
- 'B. Monreal'
- 'A. Nozik'
- 'A. Neronov'
- 'V. Niro'
- 'Y. Novikov'
- 'L. Oberauer'
- 'E. Otten'
- 'N. Palanque-Delabrouille'
- 'M. Pallavicini'
- 'V.S. Pantuev'
- 'E. Papastergis'
- 'S. Parke'
- 'S. Pastor'
- 'A. Patwardhan'
- 'A. Pilaftsis'
- 'D.C. Radford'
- 'P. C.-O.Ranitzsch'
- 'O. Rest'
- 'D.J. Robinson'
- 'P.S. Rodrigues da Silva'
- 'O. Ruchayskiy'
- 'N.G. Sanchez'
- 'M. Sasaki'
- 'N. Saviano'
- 'A. Schneider'
- 'F. Schneider'
- 'T. Schwetz'
- 'S. Schönert'
- 'F. Shankar'
- 'N. Steinbrink'
- 'L. Strigari'
- 'F. Suekane'
- 'B. Suerfu'
- 'R. Takahashi'
- 'N. Thi Hong Van'
- 'I. Tkachev'
- 'M. Totzauer'
- 'Y. Tsai'
- 'C.G. Tully'
- 'K. Valerius'
- 'J. Valle'
- 'D. Venos'
- 'M. Viel'
- 'M.Y. Wang'
- 'C. Weinheimer'
- 'K. Wendt'
- 'L. Winslow'
- 'J. Wolf'
- 'M. Wurm'
- 'Z. Xing'
- 'S. Zhou'
- 'K. Zuber'
bibliography:
- 'all.bib'
title: A White Paper on keV Sterile Neutrino Dark Matter
---
Executive Summary {#executive-summary .unnumbered}
=================
Despite decades of searching, the nature and origin of Dark Matter (DM) remains one of the biggest mysteries in modern physics. Astrophysical observations over a vast range of physical scales and epochs clearly show that the movement of celestial bodies, the gravitational distortion of light and the formation of structures in the Universe cannot be explained by the known laws of gravity and observed matter distribution [@Ade:2015xua; @Persic:1995ru; @Faber:1976sn; @Kaiser:1992ps; @Clowe:2003tk; @Percival:2007yw; @Dave:1998gm]. They can, however, be brought into very good agreement if one postulates the presence of large amounts of non-luminous DM in and between the galaxies, a substance which is much more abundant in the Universe than ordinary matter [@Ade:2015xua]. Generic ideas for what could be behind DM, such as Massive Compact Halo Objects (MACHOs) [@Paczynski:1985jf; @Griest:1990vu; @Lasserre:2000xw; @Bennett:2005at] are largely ruled out [@Clowe:2006eq; @Yoo:2003fr] or at least disfavored [@Griest:2013esa; @Pani:2014rca]. Alternative explanations based on a modification of the law of gravity [@Milgrom:1983ca] have not been able to match the observations on various different scales. Thus, the existence of one or several new elementary particles appears to be the most attractive explanation.
As a first step, the suitability of known particles within the well-tested Standard Model (SM) has been examined. Indeed, the neutral, weakly interacting, massive neutrino could in principle be a DM candidate. However, neutrinos are so light that even with the upper limit for their mass [@Kraus:2004zw; @Lobashev:1999tp] they could not make up all of the DM energy density [@Kolb:1990vq]. Moreover, neutrinos are produced with such large (relativistic) velocities that they would act as *hot* DM (HDM), preventing the formation of structures such as galaxies or galaxy clusters [@White:1984yj].
Consequently, explaining DM in terms of a new elementary particle clearly requires physics beyond the SM. There are multiple suggested extensions to the SM, providing a variety of suitable DM candidates, but to date there is no clear evidence telling us which of these is correct. Typically, extensions of the SM are sought at high energies, resulting in DM candidates with masses above the electroweak scale. In fact, there is a class of good DM candidates available at those scales, which are called Weakly Interacting Massive Particles (WIMPs). If these particles couple with a strength comparable to the SM weak interaction, they would have been produced in the early Universe via thermal freeze-out in suitable amounts [@Gondolo:1990dk] [^1] WIMPs generically avoid the structure formation problem associated with SM neutrinos, as they are much more massive and therefore non-relativistic at the time of galaxy formation. That is, WIMPs act as *cold* DM (CDM). Typical examples for WIMPs are neutralinos as predicted by supersymmetry [@Jungman:1995df; @Gelmini:2006pw; @Belanger:2005kh; @Gunion:2005rw] or Kaluza-Klein bosons as predicted by models based on extra spatial dimensions [@Servant:2002aq; @Kong:2005hn; @Bonnevier:2011km; @Melbeus:2012wi]. More minimal extensions of the SM also predict WIMPs, e.g. an inert scalar doublet [@LopezHonorez:2006gr; @Dolle:2009fn].
One of the advantages of WIMPs is that there is a variety of ways to test their existence. WIMPs could annihilate in regions of sufficiently high density, such as the center of a galaxy, thereby producing detectable (indirect) signals [@Cirelli:2012tf] in e.g. photons, antimatter, or neutrinos. The same interactions that are responsible for the annihilation of two WIMPs in outer space can also be responsible for their production at colliders [@Goodman:2010ku] or their scattering with ordinary matter in direct search experiments [@Baudis:2012ig].[^2] While a lot of experiments are currently taking data, no conclusive evidence for WIMPs has yet been found. Direct searches keep on pushing the limit on DM-matter cross sections towards smaller and smaller values [@Angloher:2014myn; @Aprile:2012nq; @Akerib:2013tjd], indirect searches yield some interesting but still inconclusive hints [@Accardo:2014lma; @Ackermann:2013uma; @Adriani:2013uda], and as of today the LHC has not discovered a hint of a DM-like particle [@ATLAS:2012ky; @Chatrchyan:2012me; @Khachatryan:2014qwa; @Aad:2014vma]. WIMPs are certainly not yet excluded, nevertheless the current experimental results suggest the thorough exploration of alternative DM candidates.
A seemingly unrelated issue arose recently in $N$-body simulations of cosmological structure formation. Advanced simulations [@Springel:2005nw] revealed some discrepancies between purely CDM scenarios and observations at small scales (a few 10 kpc or smaller). For example, there seem to be too few dwarf satellite galaxies observed compared to simulations (the missing satellite problem) [@Klypin:1999uc; @Moore:1999nt]; the density profile of galaxies is observed to be cored, whereas simulations predict a cusp profile (the cusp-core problem) [@Dubinski:1991bm; @Navarro:1995iw] and, finally, the observed dwarf satellite galaxies seem to be smaller than expected. This could possibly be explained if larger galaxies exist but are invisible due to a suppression of star formation [@Maccio':2009dx; @GarrisonKimmel:2013aq; @Geen:2011fj]. In CDM-simulations, however, these galaxies are too big to fail producing enough stars (too-big-to-fail problem) [@BoylanKolchin:2011de].
While the discrepancy between simulation and observation is apparent, its origin is not so clear. A natural possibility would be that earlier simulations did not include baryons, although we clearly know they exist. The full inclusion of baryons and their interactions is highly non-trivial and only recently has it been attempted [@Brooks:2012vi; @Zhu:2015]. Another source for the discrepancy could arise from astrophysical feedback effects [@Maccio':2009dx; @GarrisonKimmel:2013aq]. These include, for example, relatively large supernova rates in dwarf galaxies which could wipe out all the visible material so that many dwarfs are simply invisible [@Geen:2011fj]. Finally, it could also be that the DM velocity spectrum is not as cold as assumed [@Herpich:2013yga]. It has been shown that a *warm* DM (WDM) spectrum can significantly affect structure formation and strongly reduce the build-up of small objects [@Lovell:2011rd]. Even more generally, the DM spectrum need not be thermal at all. It could have various shapes depending on the production mechanism (see Sec. 5) and thereby influence structure formation in non-trivial ways. Thus, DM may be not simply *cold*, *warm*, or *hot*, but the spectra could be more complicated resembling, e.g., mixed scenarios [@Boyarsky:2008xj]. In any case, resolving the small-scale structure problem by modifying the DM spectrum would require a new DM candidate.
The candidate particle discussed in this White Paper is a *sterile neutrino with a keV-scale mass*. A sterile neutrino is a hypothetical particle which, however, is connected to and can mix with the known active neutrinos. In SM language, sterile neutrinos are right-handed fermions with zero hypercharge and no color, i.e., they are total singlets under the SM gauge group and thus perfectly neutral. These properties allow sterile neutrinos to have a mass that does not depend on the Higgs mechanism. This so-called Majorana mass [@Majorana:1937vz] can exist independently of electroweak symmetry breaking, unlike the fermion masses in the SM. In particular, the Majorana mass can have an arbitrary scale that is very different from all other fermion masses. Typically, it is assumed to be very large, but in fact it is just unrelated to the electroweak scale and could also be comparatively small. Observationally and experimentally the magnitude of the Majorana mass is almost unconstrained [@Ibarra:2011xn; @Ruchayskiy:2012si; @Abada:2012mc; @Ruchayskiy:2011aa; @Abada:2013aba; @Merle:2013gea; @Drewes:2013gca; @Hernandez:2014fha; @Fernandez-Martinez:2015hxa; @Drewes:2015jna; @Drewes:2015iva; @Antusch:2015mia; @deGouvea:2015euy; @Deppisch:2015qwa].
Depending on the choice of the Majorana mass, the implications for particle physics and cosmology are very different, , see e.g. [@Drewes:2013gca]. Two reasons motivate a keV mass scale for a sterile neutrino DM candidate. First, fermionic DM can not have an arbitrarily small mass, since in dense regions (e.g. in galaxy cores) it cannot be packed within an infinitely small volume, due to the Pauli principle. This results in a lower bound on the mass, the so-called Tremaine-Gunn bound [@Tremaine:1979we]. Second, sterile neutrinos typically have a small mixing with the active neutrinos, which would enable a DM particle to decay into an active neutrino and a mono-energetic photon. Since the decay rate scales with the fifth power of the initial state mass, a non-observation of the corresponding X-ray peak leads to an upper bound of a few tens of keV.[^3] It is these two constraints, the phase space and X-ray bounds, which enforce keV-scale masses for sterile neutrinos acting as DM.
This White Paper attempts to shed light on sterile neutrino DM from *all* perspectives: astrophysics, cosmology, nuclear, and particle physics, as well as experiments, observations, and theory. Progress in the question of sterile neutrino DM requires expertise from all these different areas. The goal of this document is thus to advance the field by stimulating fruitful discussions between these communities. Furthermore, it should provide a comprehensive compendium of the current knowledge of the topic, and serve as a future reference.[^4] The list of authors indicates that there is great interest in the subject among scientists from many areas of physics.
This White Paper is laid out as follows. First, sterile neutrinos are introduced from the particle physics (Sec. 1) and cosmology/astrophysics (Sec. 2) perspectives. Sec. 3 reviews the current tensions of CDM simulations with small-scale structure observations, and discusses attempts to tackle them. Sec. 4 gives a comprehensive summary of current constraints on keV sterile neutrino DM, arising from all accessible observables. The different sterile neutrino DM production mechanisms in the early Universe, and how they are constrained by astrophysical observations, are treated in Sec. 5. Sec. 6 turns to particle physics by reviewing attempts to explain or motivate the keV mass scale in various scenarios of physics beyond the SM. Current and future astrophysical and laboratory searches are discussed in Secs. 7 and 8, respectively, highlighting new ideas, their experimental challenges, and future perspectives for the discovery or exclusion of sterile neutrino DM. We end by giving an overall conclusion, involving all the viewpoints discussed in this paper.
Let us now start our journey into the fascinating world of keV sterile neutrino DM and address one of the biggest questions in modern science:
*What is Dark Matter and where did it come from?*
Neutrinos in the Standard Model of Particle Physics and Beyond\
===============================================================
Section Editors:\
Carlo Giunti, André de Gouvea
Neutrinos in The Standard Model of Cosmology and Beyond\
========================================================
Section Editors:\
Julien Lesgourgues, Alessandro Mirizzi
Dark Matter at Galactic Scales: Observational Constraints and Simulations\
{#sec:DMGalactic}
==========================================================================
Section Editors:\
Aurel Schneider, Francesco Shankar, Oleg Ruchayskiy
Observables Related to keV Neutrino Dark Matter\
================================================
Section Editors:\
Marco Drewes, George Fuller
Constraining keV Neutrino Production Mechanisms\
================================================
Section Editors:\
Marco Drewes, Fedor Bezrukov, George Fuller
keV Neutrino Theory and Model Building (Particle Physics)\
==========================================================
Section Editors:\
Alexander Merle, Viviana Niro
Current and Future keV Neutrino Search with Astrophysical Experiments\
======================================================================
Section Editors:\
Steen Hansen, Alexei Boyarsky
Current and Future keV Neutrino Search with Laboratory Experiment\
==================================================================
Section Editors:\
Susanne Mertens, Loredana Gastaldo
Discussion - Pro and Cons for keV Neutrino as Dark Matter and Perspectives\
===========================================================================
Section Editors:\
Marco Drewes, Thierry Lasserre, Alexander Merle, Susanne Mertens
[^1]: Note that this is true independently of the WIMP mass – up to logarithmic corrections – as long as they freeze out cold, since the main dependence on the mass drops out in the formula for the DM abundance [@Jungman:1995df].
[^2]: At the level of amplitudes, this relation between “break it”, “make it” and “shake it” can be visualized by rotating the Feynman diagram in steps of 90 degrees.
[^3]: This only holds if active-sterile mixing is not switched off or forbidden, which may be the case in certain scenarios, see Sec. 6.
[^4]: The reader should be warned that the texts contributed to this work by the different authors cannot treat the various topics in full detail. They should, however, serve as possible overview and we made a great effort to ensure that they do contain all the relevant references, so that the present White Paper can guide the inclined reader to more specific information.
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author:
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[^1]\
Joint Institute for VLBI in Europe, Dwingeloo, Netherlands\
E-mail:
bibliography:
- 'proceedings.bib'
title: Gravitational lensing
---
The idea
========
The idea that gravity might influence the propagation of light is much older than the EVN or even radio astronomy. Even @newton04 mentioned the possibility[^2], but without going into any details. Henry Cavendish was the first to calculate deflections using the simplified model that light consists of classical particles moving with the speed of light, thus being weakly deflected in a gravitational field [see @will88]. Without knowing of this unpublished work, @soldner1801 used a similar concept and wrote the first article about gravitational lensing. He found that the deflection angle (see Fig. \[fig:defl\]) between the asymptotic light paths before and after an encounter with a mass $M$ at a distance $r$ is $\alpha=2GM/(c^2r)$, where $G$ and $c$ are the gravitational constant and the speed of light, respectively. Before developing the theory of general relativity, @einstein11 used another approach that does not rely on the incorrect classical model of light but, instead, used the principle of equivalence to find the same result as Soldner. A few years later, our still best theory of gravitation was finished and @einstein15 could reconsider the deflection of light, finding that it should be *twice* as large as in non-relativistic theory: $$\alpha = \frac{4GM}{c^2 r}
\label{eq:defl}$$ This result was confirmed (with limited accuracy) by @eddington19, who determined the deflection of light caused by the Sun by measuring positions of stars close to its limb during a solar eclipse. This served as a first independent test of the new theory.
![Light deflection[]{data-label="fig:defl"}](ablenk){width="50.00000%"}
Almost all aspects of gravitational lensing theory can be derived from Eq. (\[eq:defl\]), together with the astrophysical background that is required to relate it to observable quantities. It is easy to see that in the presence of sufficiently large and compact masses, situations are possible where light from a background source reaches the observer along different light paths, which are then interpreted as multiple images of this source. Typical lenses show two or four images, which in very symmetrical systems merge to form rings around the lens, the so-called Einstein rings.
The first lensed double: 0957+561
=================================
This radio source (without knowing about its lensed nature at that time) was discovered in a 966-MHz survey made with the Jodrell Bank MkIA telescope. One of the sources in that survey, 0958+56, was a blend of the spiral galaxy NGC 3079 with the new source 0957+561. The latter had an optical identification as two point-like images with a separation of about $6''$ [@porcas80 and Fig. \[fig:0957\], left].
Optical spectra of the two images appeared like almost identical copies of a QSO spectrum with a redshift of $z=1.4$. Since a chance alignment of so similar objects with such a close separation is extremely unlikely, the favoured interpretation was (and still is) that we see two gravitationally lensed images of one and the same source [@walsh79]. Many interesting (and amusing) details of the first discovery, including a description of all coincidences that had to conspire to allow the success, are described by @walsh89.
![Interferometric observations of 0957+561 at 6cm. Left: early VLA [@roberts79]. Centre: Cambridge 5km. Right: full VLA [@harvanek97].[]{data-label="fig:0957 interf"}](roberts79-002 "fig:"){height="27.00000%"}![Interferometric observations of 0957+561 at 6cm. Left: early VLA [@roberts79]. Centre: Cambridge 5km. Right: full VLA [@harvanek97].[]{data-label="fig:0957 interf"}](pooley79-001 "fig:"){height="28.00000%"}![Interferometric observations of 0957+561 at 6cm. Left: early VLA [@roberts79]. Centre: Cambridge 5km. Right: full VLA [@harvanek97].[]{data-label="fig:0957 interf"}](harvanek97-002 "fig:"){height="27.00000%"}
Doubts were expressed by @roberts79 when first interferometric observations showed the complicated structure of the double radio source (see Fig. \[fig:0957 interf\]). The *A* image seems to show a jet and radio lobes and one could naively expect that the same should be true for *B* if both are really lensed images of one source. In the maps, on the other hand, image *B* looks perfectly point-like. However, Fig. \[fig:lenseq\] illustrates how different parts of the source may well be lensed with different multiplicity. The specific situation for 0957+561 is shown in Fig. \[fig:0957 max\].
![Left: Illustration of a typical lens mapping in one dimension, showing the true source position $z{_{\rms}}$ as a function of the apparent image position $z$. For source positions far away from the origin, we see only one image. When the source crosses the extrema of the curve, two additional images appear or disappear. Centre and right: Illustration of this effect in two dimensions. The central panel shows the source plane with the caustic curves, the right one the image plane with the corresponding critical curves. These curves correspond to the red dots in the left panel. Whenever the source crosses one of the curves, two merging images (dis)appear. At this moment, the images are magnified extremely. []{data-label="fig:lenseq"}](diag){width="\textwidth"}
![Left: Illustration of a typical lens mapping in one dimension, showing the true source position $z{_{\rms}}$ as a function of the apparent image position $z$. For source positions far away from the origin, we see only one image. When the source crosses the extrema of the curve, two additional images appear or disappear. Centre and right: Illustration of this effect in two dimensions. The central panel shows the source plane with the caustic curves, the right one the image plane with the corresponding critical curves. These curves correspond to the red dots in the left panel. Whenever the source crosses one of the curves, two merging images (dis)appear. At this moment, the images are magnified extremely. []{data-label="fig:lenseq"}](ABiggs-011){width="\textwidth"}
![Schematic source plane (left) and image plane (right) of 0957+561. Shown are caustics/critical curves, radio contours in grey and optical contours in black. We see two images of the core but only one image of the extended radio jet and lobes. [from @avruch97][]{data-label="fig:0957 max"}](avruch97_2){width="30.00000%"}
First VLBI observations of lenses
=================================
Naturally, the first discovered lens was also the first target for VLBI observations (Fig. \[fig:0957vlbi\]). In the fringe-rate spectrum, we see two peaks corresponding to the two images *A* and *B*, which thus both must have compact structure on scales below 20mas. The first multi-component model fits show that both images have a very similar structure, consisting of a compact core component and an elongated feature that probably corresponds to a jet. These jets show many details in VLBI maps produced later.
It is impressive to see how far VLBI has evolved from simple fringe-rate spectrum plots to detailed maps of gravitational lenses. The first observations also show that real science can be done with only one VLBI baseline.
![Left: Residual fringe rate spectrum of 0957+561 from the first VLBI observations; 20-min integration, Effelsberg-Jodrell baseline [from @porcas79]. Centre: First Gaussian model fit results [from @porcas81]. Right: Later detailed maps for comparison [from @campbell95]. []{data-label="fig:0957vlbi"}](porcas79-000b){width="50.00000%"}
Fields of study
===============
After this introduction, we come to a more systematic discussion of the several topics that can be studied with gravitational lensing. Fig. \[fig:fields\] schematically shows a typical lensing scenario with multiple images of one source.
![Lensing can be used to study the lensed sources (left), the lenses themselves (centre), propagation effects and extinction (anywhere between source and observer), and the properties of global spacetime (including cosmology and tests of relativity.)[]{data-label="fig:fields"}](diag2){width="80.00000%"}
Lenses as natural telescopes
----------------------------
The lens effect can be utilised to extend the capabilities of our instruments because it provides additional magnification and corresponding flux amplification, and in this way boosts the effective resolution and sensitivity of observations. Clusters of galaxies provide the largest lensing cross-sections and are thus the primary of these *natural telescopes*. Two examples are shown in Figs. \[fig:a2218\] and \[fig:alicia\]. In one case, the amplification made it possible to detect a distant galaxy at $z=7$ and observe its optical spectrum. In the others, the lensing amplification enhances weak, background star-forming galaxies that have first been found at sub-mm wavelength above the radio detection limit. Without the lenses, current radio telescopes would not be sufficiently sensitive to study these objects at all.
![The lensing cluster of galaxies A2218. Left: A $z=7$ background galaxy amplified by a factor of $\mu\approx 25$ [from @kneib04]. Right: The first radio detection of a lensed star-forming galaxy [from @garrett05].[]{data-label="fig:a2218"}](abell2218_small "fig:"){height="35.00000%"}![The lensing cluster of galaxies A2218. Left: A $z=7$ background galaxy amplified by a factor of $\mu\approx 25$ [from @kneib04]. Right: The first radio detection of a lensed star-forming galaxy [from @garrett05].[]{data-label="fig:a2218"}](hst_vla_abc_289 "fig:"){height="35.00000%"}
![The cluster MS0451.6–0305 shows a number of lensed background source components in the radio (white contours) and sub-mm (black contours) domains. Lens models predict that they all form a compact ensemble of forming or merging galaxies at high redshift. [from @berciano06][]{data-label="fig:alicia"}](alicia){width="60.00000%"}
In galaxy-scale lenses, the regions of high amplifications are very small, but in rare occasions it happens that a radio source is located so close to a caustic that it is amplified by factors of a few hundred. B2016+112 is one of those examples. In another talk at this meeting, [@more06] present new HSA observations of this system, showing source components almost merging at the critical curve with extreme magnifications. Back-projecting these components into the source plane will in the future provide $\mu$arcsec-resolution maps of this source.
Mass distributions of lenses
----------------------------
Studying the mass distributions of lenses is one of the most important fields in lensing research for several reasons. On one hand, the mass distribution governs the light deflection and image configuration, including magnifications. Without a good knowledge of these properties, applications of lenses as natural telescopes (or for other purposes) would become inaccurate and speculative.
However, this argument can be turned around and we can use the image configuration to constrain the mass distribution of the lens. This method provides the most accurate information about the mass distributions of very distant lenses, allowing the study of structure and evolution of galaxies in a systematic way. The standard approach to do this is to assume a simple parametrised mass distribution for the lens and make simple assumptions for the source structure, e.g. assume that it consists of a small number of compact components. The model parameters (of the lens and source) are then varied in a way to fit the predicted image configuration to the observed one. This can include image positions as well as shapes of compact but slightly extended components. Generally, it is advantageous to have many lensed source components, but they have to be of a sufficiently simple structure to make this standard approach viable.
Fig. \[fig:model 0957\] illustrates this for the case of 0957+561, where the source is complicated enough to make this procedure a non-trivial task. The constraints are quite good, but unfortunately the structure of the lens (which consists of a massive cD galaxy together with its surrounding cluster) is so complicated that even these constraints are not sufficient to provide a good accuracy for the lens mass distribution. One disadvantage is that all the jet components are located very close to each other so that they sample the light deflection only in small regions, basically providing the deflection angle and its derivatives at only two points.
![Constraints from the substructure of both jet images in 0957+561. The observed structure (left panels) of the jet images is parametrised as a collection of Gaussian model components (right panels). The parameters of these components are then used to determine a relative magnification matrix that linearly maps the central region of image *A* to *B* or vice versa. The mass-model parameters are then varied to fit the predicted matrix to the one determined from the observations. [from @garrett94][]{data-label="fig:model 0957"}](garrett94_1 "fig:"){height="35.00000%"} ![Constraints from the substructure of both jet images in 0957+561. The observed structure (left panels) of the jet images is parametrised as a collection of Gaussian model components (right panels). The parameters of these components are then used to determine a relative magnification matrix that linearly maps the central region of image *A* to *B* or vice versa. The mass-model parameters are then varied to fit the predicted matrix to the one determined from the observations. [from @garrett94][]{data-label="fig:model 0957"}](garrett94_2 "fig:"){height="35.00000%"}
In order to obtain more information about the global properties of the mass distribution, it is necessary to have images at widely separated positions, especially at different distances from the lens centre. B1933+503 is a unique lens system consisting of *ten* images of several source components (see Fig. \[fig:1933\]). Parametrised model fits [@cohn01] for this system show that the mass distribution must be very close to isothermal (projected surface mass density $\sigma\propto 1/r$), a property found in most accurately modelled lenses. Later on, we will discuss how to use lensed general *extended* sources to constrain lens models.
![The lens B1933+503. Left: Image configuration with critical curves. Right: Source component configuration with caustics. One component (black dot) is doubly imaged, the other two (circles) are both quadruply imaged, resulting in ten images of these three components. The images probe the lensing potential at a wide range of radii. [from @nair98][]{data-label="fig:1933"}](nair98){width="75.00000%"}
The lensed image configuration not only provides information about the global mass distribution but is also sensitive for small-scale deviations as predicted by CDM structure-formation scenarios. Fig. \[fig:0128\] shows an example where each of the four images consists of three subcomponents. The general four-image geometry can be fitted well with smooth mass distributions, but these models cannot explain the relative positions of all subcomponents in the images. An easy explanation would be the presence of a small-mass clump close to one of the images that distorts the geometry very locally without affecting the global configuration. There are other cases where it is not the image configuration but the flux density ratios that are in conflict with simple models. This is not surprising since small-scale mass substructure influences derivatives of deflection (determining the amplifications) much stronger than the deflections (and image positions) themselves.
![The quadruple lens system B0128+437. Centre: MERLIN map of the complete system [from @phillips00]. Outer fields: High-resolution VLBI maps of the four images [from @biggs04].[]{data-label="fig:0128"}](biggs04_c "fig:"){width="\textwidth"}\
![The quadruple lens system B0128+437. Centre: MERLIN map of the complete system [from @phillips00]. Outer fields: High-resolution VLBI maps of the four images [from @biggs04].[]{data-label="fig:0128"}](biggs04_d "fig:"){width="\textwidth"}
![The quadruple lens system B0128+437. Centre: MERLIN map of the complete system [from @phillips00]. Outer fields: High-resolution VLBI maps of the four images [from @biggs04].[]{data-label="fig:0128"}](biggs04_b "fig:"){width="\textwidth"}\
![The quadruple lens system B0128+437. Centre: MERLIN map of the complete system [from @phillips00]. Outer fields: High-resolution VLBI maps of the four images [from @biggs04].[]{data-label="fig:0128"}](biggs04_a "fig:"){width="\textwidth"}
Another aspect of the small-scale mass distribution that can be studied with lensing is the mass profile in the very centres of lensing galaxies. For a smooth mass profile with a non-diverging deflection angle, one would always expect an odd number of images, one of which would be located close to the lens centre. However, if the central mass concentration becomes very steep or even singular, this central image will be highly de-magnified or even completely suppressed. Currently, only one such central image is believed to be detected (see Fig. \[fig:central\], together with a case with no central image). Instead, there is evidence that isothermality of the mass profile typically extends very close to the lens galaxy centres, which would suppress the central images. There is no good explanation for this fact in standard structure-formation theories.
![Two cases of studies concerning possible central images. Left: B1030+074, where even with HSA observations no central image could be found [from @zhang06]. Right: J1632–0033, the currently most (and maybe only) convincing case with a weak central image [from @winn03].[]{data-label="fig:central"}](b1030hsa_box){width="60.00000%"}
Propagation effects: scattering, absorption etc.
------------------------------------------------
The study of propagation effects like scatter broadening, free-free absorption or extinction and reddening in the optical domain is usually plagued by the lack of knowledge of the *intrinsic* structure and spectral energy distribution of a distant source, with which the *observed* properties could be compared. Gravitational lensing kindly produces two or more copies of the same source component that are identical in their spectra and have only simple (and easy to model) differences in their total flux density and source structure. By comparing properties of these images, one can directly infer the *differential* propagation effects like differential reddening or differential scatter broadening, without assumptions about the intrinsic properties. This will be shown for one example later.
Measurement of distances and cosmology
--------------------------------------
Long before the discovery of the first lens, it was argued by @refsdal64b [@refsdal66] that the lens effect can be used to test cosmological theories and to measure distances and thus the Hubble constant in particular. The principle behind this idea is very simple. In a typical lens system, we can measure the angular separations between the observed images and the lens. Modelling the mass distribution then gives us all the other angles defining the geometry (like the true source position). Ratios of distances between observer, lens and source can be derived easily from the observed redshifts. We conclude that in such a situation the complete lensing geometry is known, *except for the scale.* If only one length in the system can be measured, all other lengths, including the distances to the lens and source, can immediately be calculated. It was the idea of @refsdal64b that the light travel-times will differ from image to image (because of different geometrical paths and different Shapiro delays) and that this light travel-time difference can be used as the defining length to scale the whole geometry. If the background source is variable, these variations will be seen in all the images, but shifted relative to each other by the corresponding time delays. Once the distances are determined, they can be used together with the redshifts to estimate the Hubble constant. Distances for given redshifts are inversely proportional to the Hubble constant, and the time delay is proportional to the distances, so that the product $H_0\Delta t$ is a constant that can be derived from the lens mass model, which, in turn, is constrained by the image geometry. This method has been applied to a number of lens systems, leading to more or less consistent results. We will discuss only one example further below.
Tests of relativity
-------------------
The most fundamental test of relativity performed with the lens effect consists of measuring the deflection angle in our solar system and comparing it with the theoretical expectations. For this purpose the deflection can be parametrised as $$\alpha = 2(1+\gamma)\frac{GM}{c^2 r}$$ and the resulting $\gamma$ can be measured. Two special values would be $\gamma=0$ as expected from Newtonian theory and $\gamma=1$ as expected from general relativity. The work of @eddington19 was only a first example, and the accuracy has improved by several orders of magnitude by using VLBI techniques. @shapiro04 found $\gamma=0.9998\pm0.0004$ from a combination of very many geodetic VLBI data sets, which confirms general relativity with high accuracy.
It has also been claimed that the deflection of light by a moving object (Jupiter in this case) can be used to measure the propagation speed of gravity [@kopeikin01]. Such an experiment has actually been carried out [@fomalont03], and the outcome is consistent with relativity. However, there are serious arguments against this interpretation of the experiment [see e.g. @will03; @carlip04] and the debate has not led to an agreement between the opponents yet.
The lens B0218+357
==================
This lens system is one of the best studied cases and is a good example for many of the applications of the lens effect discussed above. It consists of two bright images and an additional Einstein ring with the same diameter as the image separation (Fig. \[fig:0218 colour\]). The two images are resolved by VLBI and show the core and inner jet of the lensed background source [e.g. @biggs03].
![The lens B0218+357. Left: The lens plane as we see it. The central map provides an overview. The Einstein ring and the two images are shown in the upper left; the insets at the right show the magnified VLBI substructure of the two images. The magnified regions are marked by black rectangles. Right: The reconstructed (unlensed) source plane. [from @lc2] []{data-label="fig:0218 colour"}](colour1 "fig:"){width="49.00000%"}![The lens B0218+357. Left: The lens plane as we see it. The central map provides an overview. The Einstein ring and the two images are shown in the upper left; the insets at the right show the magnified VLBI substructure of the two images. The magnified regions are marked by black rectangles. Right: The reconstructed (unlensed) source plane. [from @lc2] []{data-label="fig:0218 colour"}](colour2 "fig:"){width="49.00000%"}
Lens models and Hubble constant
-------------------------------
B0218+357 has some unique properties that make it particularly well suited to determine the Hubble constant using Refsdal’s method. Firstly, the time delay is known with sufficient accuracy. @biggs99 find $\Delta
T=(10.5\pm0.4)\,\rm d$, a result that is consistent with that of @cohen00. Secondly, the lens is an isolated spiral galaxy without close neighbours or clusters nearby, which allows the use of simple models with a small number of parameters as a realistic description. Finally, the wealth of structure in the Einstein ring and the VLBI maps of the lensed jets provides a good number of valuable constraints for the modelling. The VLBI substructure is especially sensitive to the radial mass profile, which seems to be very close to isothermal but slightly shallower [@biggs03]. Unfortunately, the accurate lens position (relative to the images) was not known until recently. The image separation is only $330\,\rm mas$, the smallest of all galaxy lenses, which makes direct optical measurements very difficult. However, the structure of the Einstein ring can be used to determine the lens position indirectly. Contrary to the lens modelling methods described before, the source cannot be described as a small number of point-like sources anymore. Instead, more sophisticated methods must be used, in which the true source structure is also fitted for in a non-parametric way.
We used our own improved version of the LensClean algorithm to accomplish this task [@lc1]. LensClean iteratively builds a source model similarly to the normal Clean algorithm, but takes into account the effect of the (for the moment fixed) lens model. ‘Clean components’ are allowed only in combinations that are consistent with the effect of the lens. If, for example, a component is to be included at a position that is quadruply imaged, all four lensed images have to be included in the model, scaled with their respective amplifications. Once converged, this inner loop of LensClean has determined a source model that minimises the residuals given a certain lens model. An outer loop then varies the lens model parameters, again in order to minimise the residuals. The final result is a simultaneous fit of lens and source model. The main result in the case of B0218+357 is the lens position, which then directly translates to a value for the Hubble constant [@lc2]: $$H_0 = (78\pm6) \;\rm km\,s^{-1}\,Mpc^{-1}$$ The lens position was later confirmed by a direct optical measurement using the HST/ACS [@york05]. The result for $H_0$ is consistent with other methods that use completely independent information. Because it is determined by a direct one-step method, the systematic uncertainties inherent in the complex distance-ladder methods are avoided.
Propagation effects
-------------------
It has been known for long that the flux density ratio of the two images $A/B$ shows a strong frequency dependence. At high frequencies, it is close to 4 and decreases for lower frequencies. This is surprising since lensing is achromatic and should not change the spectra of the images. One possible explanation would be source shifts as a function of frequency, which, together with the strong magnification gradient, could explain the observed trend, provided they were of the order $10\,\rm mas$ or more. @mittal06 investigated this possibility by measuring the image positions with multi-frequency, phase-referencing VLBI observations. They found (see Fig. \[fig:mittal\]) that any image shifts are of the order of $1\,\rm mas$ or less, not enough to explain the observed effect. Similarly, changes of the source structure could be ruled out as reason for the changing flux density ratios. The most plausible explanation is a free-free absorption, mainly in the *A* image.
![Left: Image positions of $A$ and $B$ for different frequencies [from @mittal06]. Right: free-free absorption model for the flux density of $A$ derived from $B$ ($F_A^{\rm ff}$) with corresponding measurements ($F_A^{\rm obs}$). The agreement is not perfect but reasonably good [from @mittal06b].[]{data-label="fig:mittal"}](mittal06_b "fig:"){height="28.00000%"}![Left: Image positions of $A$ and $B$ for different frequencies [from @mittal06]. Right: free-free absorption model for the flux density of $A$ derived from $B$ ($F_A^{\rm ff}$) with corresponding measurements ($F_A^{\rm obs}$). The agreement is not perfect but reasonably good [from @mittal06b].[]{data-label="fig:mittal"}](mittal06_a "fig:"){height="28.00000%"}![Left: Image positions of $A$ and $B$ for different frequencies [from @mittal06]. Right: free-free absorption model for the flux density of $A$ derived from $B$ ($F_A^{\rm ff}$) with corresponding measurements ($F_A^{\rm obs}$). The agreement is not perfect but reasonably good [from @mittal06b].[]{data-label="fig:mittal"}](ffabs "fig:"){height="28.00000%"}
More VLBI aspects
-----------------
It should be noted that B0218+357 was the target of the first real-time EVN observations in 2004. The map from this small data set is shown in Fig. \[fig:0218 more VLBI\] (left). Currently, the author is working on the analysis of a 90-cm VLBI experiment conducted recently. The goal was to map more details in the Einstein ring and to obtain more information about propagation effects that are strongest at lower frequencies. Fig. \[fig:0218 more VLBI\] (right) shows a very preliminary map compared to a VLA+Pie Town map at 2-cm. Two additional correlator passes of the same 90-cm observations are used for a wide-field mapping experiment that constitutes the first wide-field VLBI mini-survey to study the high-resolution radio sky at such low frequencies. First results are presented by @lenc06 at this conference. See also @lenc07.
![Left: The first real-time EVN image produced at JIVE. Right: Preliminary 90-cm VLBI map in comparison with a 2-cm VLA+Pie Town map. This is shown only for illustration, the calibration is still very poor at this stage.[]{data-label="fig:0218 more VLBI"}](MAP2_label2){width="\textwidth"}
Outlook
=======
Like many other fields, gravitational lens research takes advantage of the technical progress in radio astronomy and particularly in VLBI. The EVN grows continuously, providing steadily improving $uv$ coverage, mapping quality and resolution. Higher bandwidths provide the increasing sensitivity that is needed to map more extended source components with high quality. Such future VLBI experiments will lead to more accurate lens mass models than currently available. This development is complemented by the upgraded EVLA and *e*-MERLIN that will come online soon. All these arrays will allow the mapping of even weaker structures on scales of the image splitting with sufficient resolution to utilise the information such sources provide for mass models and other purposes. It will finally be possible to map even normal star-forming galaxies in great detail. Such lensed sources have such a wealth of smaller structures (Fig. \[fig:1131\]) that they allow a detailed mapping of the lensing potential and thus the mass distribution of lensing galaxies.
EVLA, *e*-MERLIN, and above all LOFAR (with extended baselines) will allow future lens surveys that increase the number of radio lenses by at least an order of magnitude. This development has to be supplemented by the development of new analysis techniques that can extract all information from current and future observations. LensClean can only be one first step in this direction.
![Radio astronomy has to compete with this! Shown is the lensed star-burst galaxy J1131–1231 in the optical domain [from @claeskens06]. Future radio telescopes will map such sources with even better quality than shown in this HST image. Each lensed star-forming region will provide its own constraints on the mass distribution of the lens.[]{data-label="fig:1131"}](claeskens){width="30.00000%"}
Summary
=======
We have discussed how the lens effect can be used to study all aspects of the lensing situation. We use the lens as a natural telescope to study the sources in greater detail, we determine the mass distributions of high-redshift lens galaxies, and can even determine the Hubble constant and do cosmology. Additionally, lenses provide us with (almost) identical copies of lensed sources that can then be used to study differential propagation effects like scattering or absorption.
Radio observations are particularly useful for several reasons. Firstly, radio interferometers span the largest range of possible resolutions down to the sub-mas level. Secondly, the effects of microlensing and extinction, that make the interpretation of optical observations very difficult, can mostly be avoided at radio wavelengths. Instead of asking what radio astronomy can do for lensing, we can also ask what lensing can do for radio astronomy. It is our hope that the further development of deconvolution methods for the lensed situation will also provide new and better algorithms for a general use. Such a trigger for new developments should be very welcome.
Finally, we have to add that this review covers only a small subset of relevant topics in gravitational lens research. Other important subjects like microlensing, weak lensing, lensing surveys, measuring time delays, and a number of very interesting lens systems have not been mentioned at all. That does not imply that those fields are less exciting.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the European Community’s Sixth Framework Marie Curie Research Training Network Programme, Contract No.MRTN-CT-2004-505183 “ANGLES”.
0.8ex
[^1]: Current address: Argelander-Institut für Astronomie, Universität Bonn, Germany
[^2]: “Do not bodies act upon light at a distance, and by their action bend its rays; and is not this action strongest at the least distance?”
|
---
abstract: 'We present a thermodynamic formulation for scale-invariant systems based on the minimization with constraints of Fisher’s information measure. In such a way a clear analogy between these systems’s thermal properties and those of gases and fluids is seen to emerge in natural fashion. We focus attention on the non-interacting scenario, speaking thus of *scale-free ideal gases* (SFIGs) and present some empirical evidences regarding such disparate systems as electoral results, city populations and total citations in Physics journals, that seem to indicate that SFIGs do exist. We also illustrate the way in which Zipf’s law can be understood in a thermodynamical context as the surface of a finite system. Finally, we derive an equivalent microscopic description of our systems which totally agrees with previous numerical simulations found in the literature.'
address:
- 'Departament ECM, Facultat de Física, Universitat de Barcelona. Diagonal 647, 08028 Barcelona, Spain'
- 'Sogeti España, WTCAP 2, Plaça de la Pau s/n, 08940 Cornellà, Spain'
- 'National University La Plata, IFCP-CCT-CONICET, c.c. 727, 1900 La Plata, Argentina'
author:
- 'A. Hernando'
- 'C. Vesperinas'
- 'A. Plastino'
title: ' Fisher-information and the thermodynamics of scale-invariant systems'
---
Fisher information ,scale-invariance
Introduction
============
Scale-invariant phenomena are rather abundant in Nature and display somewhat unexpected features. Examples can be found that range from physical and biological to technological and social sciences [@uno]. One may cite, among many possibilities, that empirical data from percolation theory and nuclear multifragmentation [@perco] reflect scale-invariant behaviour, as does the abundance of genes in various organisms and tissues [@furu]. Additionally, we can speak of the frequency of words in natural languages [@zip], scientific collaboration networks [@cites], the Internet traffic [@net1], Linux packages links [@linux], as well as of electoral results [@elec1; @ccg], urban agglomerations [@ciudad; @ciudad2] and firm sizes all over the world [@firms]. What characterizes these disparate systems is the lack of a characteristic size, length or frequency for an observable $k$ under scrutiny. This fact usually leads to a power law distribution $p(k)$, valid in most of the domain of definition of $k$, $$\label{eq1}
p(k)\sim1/k^{1+\gamma},$$ with $\gamma\geq0$. Special attention deserves the class of universality defined by $\gamma=1$, which corresponds to the so-called Zipf’s law in the cumulative distribution or the rank-size distribution [@perco; @furu; @zip; @net1; @linux; @ciudad; @ciudad2; @firms; @citis]. Recently, Maillart et al. [@linux] have found that links’ distributions follow Zipf’s law as a consequence of stochastic proportional growth. In its simplest formulation such kind of growth assumes that an element of the system becomes enlarged proportionally to its size $k$, being governed by a Wiener process. The class $\gamma=1$ emerges from the condition of stationarity, i.e., when the system reaches a dynamic equilibrium [@citis]. We will as well propose to consider the case $\gamma=0$ as representative of a second class of universality, since the ensuing behavior, empirically found by Costa Filho et al. [@elec1] with regards to the vote-distribution in Brazilian electoral results, emerges as the result of multiplicative processes in complex networks [@ccg]. In this paper we attempt to formulate a thermodynamic treatment common to these systems. Our efforts are based on the minimization with appropriate constraints of Fisher’s information measure (FIM), abbreviated as the MFI approach. It is shown in [@fisher2] that MFI leads to a (real) Schreodinger-like equation whose “potential” function is given by the constraints employed to constrain the variational process. The interplay between constraints and associated Lagrange multipliers turns our to be Legendre-invariant [@fisher2] and leads to all known thermodynamic relations. Such result constitutes the essential ingredient of our considerations here. We will first consider the MFI treatment of the ideal gas (Seccion 3), not given elsewhere as far as we are aware of, since it is indispensable to deal with it in order to fully understand the methodology employed for scale-free systems, which is tackled in Section 4. Applicatioms are discussed in Section 5 and some conclusiones drawn in Section 6. We begin our considerations in Section 2 with a brief Fisher’s sketch.
Minimum Fisher Information approach (MFI) {#p2}
=========================================
The Fisher information measure $I$ for a system described by a set of coordinates $\mathbf{q}$ and physical parameters $\mathbf{\theta}$, has the form [@libro] $$\label{fish}
I(F)=\int_\Omega
d\mathbf{q}F(\mathbf{q}|\mathbf{\theta})\sum_{ij}c_{ij}\frac{\partial}{\partial\theta_i}\ln
F(\mathbf{q}|\mathbf{\theta})\frac{\partial}{\partial\theta_j}\ln
F(\mathbf{q}|\mathbf{\theta}),$$ where $F(\mathbf{q}|\mathbf{\theta})$ is the density distribution in a configuration space ($\mathbf{q}$) of volume $\Omega$ conditioned by the physical parameters collectively represented by the variable $\mathbf{\theta}$. The constants $c_{ij}$ account for dimensionality, and take the form $c_{ij}=c_i\delta_{ij}$ if $q_i$ and $q_j$ are uncorrelated, where $\delta_{ij}$ is the Kronecker delta. As shown in [@fisher2], the thermal-equilibrium state of the system can be determined by minimizing $I$ subject to adequate prior conditions (MFI), like the normalization of $F$ or by any constraint on the mean value of an observable $\langle A_i \rangle$ [@fisher2]. The MFI is then cast as a variation problem of the form $$\delta\left\{I(F)-\sum_i\mu_i\langle A_i \rangle\right\}=0,$$ where $\mu_i$ are appropriate Lagrange multipliers.
MFI treatment of the ideal gas {#aa}
==============================
As a didactic introductory example, not discussed in [@fisher2], we will here rederive, [*via MFI*]{} (something original as far as we know), the density distribution, in configuration space, of the (translational invariant) ideal gas (IG) [@termo], that describes non-interacting classical particles of mass $m$ with coordinates $\mathbf{q}=(\mathbf{r},\mathbf{p})$, where $md\mathbf{r}/dt=\mathbf{p}$. The translational invariance is described by the translational family of distributions $F(\mathbf{r},\mathbf{p}|\mathbf{\theta}_r,\mathbf{\theta}_p)=F(\mathbf{r}',\mathbf{p}')$ whose form does not change under the transformations $\mathbf{r}'=\mathbf{r}-\mathbf{\theta}_r$ and $\mathbf{p}'=\mathbf{p}-\mathbf{\theta}_p$. We assume that these coordinates are canonical [@mec] and uncorrelated. This assumption is introduced into the information measure (\[fish\]) setting $c_{ij}=c_i\delta_{ij}$, where $c_{i}=c_r$ for space coordinates, $c_{i}=c_p$ for momentum coordinates. The density can obviously be factorized in the fashion $F(\mathbf{r},\mathbf{p})=\rho(\mathbf{r})\eta(\mathbf{p})$, and then [@libro] it follows from the additivity of the information measure that $I=I_r+I_p$. If $D$ is the dimensionality we have $$\begin{array}{rl}
I_r=&\displaystyle c_r\int
d^D\mathbf{r}~\rho(\mathbf{r})\left|\mathbf{\nabla}_r \ln \rho(\mathbf{r})\right|^2\\
I_p=&\displaystyle c_p\int
d^D\mathbf{p}~\eta(\mathbf{p})\left|\mathbf{\nabla}_p\ln
\eta(\mathbf{p})\right|^2.
\end{array}$$
In extremizing FIM we constrain the normalization of $\rho(\mathbf{r})$ and $\eta(\mathbf{p})$ to the total number of particles $N$ and to $1$, respectively, i.e., $$\label{normg}
\int d^D\mathbf{r}~\rho(\mathbf{r})=N,\qquad\int
d^D\mathbf{p}~\eta(\mathbf{p})=1.$$ In addition, we penalize infinite values for the particle momentum with a constraint on the variance of $\eta(\mathbf{p})$ to a given empirically obtained value, namely, $$\label{sp2}
\int
d^D\mathbf{p}~\eta(\mathbf{p})(\mathbf{p}-\overline{\mathbf{p}})^2=D\sigma_p^2,$$ where $\overline{\mathbf{p}}$ is the mean value of $\mathbf{p}$. For each degree of freedom it is known from the Virial Theorem that the variance is related to the temperature $T$ as $\sigma_p^2=mk_BT$, with $k_B$ the Boltzmann constant. Variation thus yields $$\label{exg}
\displaystyle\delta\left\{c_r\int
d^D\mathbf{r}~\rho\left|\mathbf{\nabla}_r \ln \rho\right|^2+\mu\int
d^D\mathbf{r}~\rho\right\}=0$$ and $$\label{exh}
\delta\left\{c_p\int d^D\mathbf{p}~\eta\left|\mathbf{\nabla}_p\ln
\eta\right|^2+\lambda\int
d^D\mathbf{p}~\eta(\mathbf{p}-\overline{\mathbf{p}})^2 +\nu\int
d^D\mathbf{p}~\eta\right\}=0,$$ where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing now $\rho(\mathbf{r})=\Psi^2(\mathbf{r})$ and varying (\[exg\]) with respect to $\Psi$ leads to a Schroedinger-like equation [@fisher2; @QM] $$\left[-4\nabla_r^2+\mu'\right]\Psi(\mathbf{r})=0,$$ where $\mu'=\mu/c_r$. To fix the boundary conditions, we first assume that the $N$ particles are confined in a box of volume $V$, and next we take the thermodynamic limit $N,V\rightarrow\infty$ with $N/V$ finite. The equilibrium state compatible with this limit corresponds to the ground state solution ($\mu'=0$), which is the uniform density $\rho(\mathbf{r})=N/V$.
Introducing $\eta(\mathbf{p})=\Phi^2(\mathbf{p})$ and varying (\[exh\]) with respect to $\Phi$ leads to the quantum harmonic oscillator-like equation [@QM] $$\left[-4\nabla_p^2+\lambda'(\mathbf{p}-\overline{\mathbf{p}})^2+\nu'\right]\Phi(\mathbf{p})=0,$$ where $\lambda'=\lambda/c_p$ and $\nu'=\nu/c_p$. The equilibrium configuration corresponds to the ground state solution, which is now a gaussian distribution. Using (\[sp2\]) to identify $|\lambda'|^{-1/2}=\sigma_p^2$ we get the Maxwell-Boltzmann distribution, which leads to a density distribution in configuration space of the form $$f(\mathbf{r},\mathbf{p})=\frac{N}{V}\frac{\exp\left[-(\mathbf{p}-\overline{\mathbf{p}})^2/2\sigma_p^2\right]}{(2\pi\sigma_p^2)^{D/2}}.$$ If $H$ is the elementary volume in phase space, the total number of microstates is $Z=N!H^{DN}\prod_{i=1}^NF_1(\mathbf{r}_i,\mathbf{p}_i)$, where $F_1=F/N$ is the monoparticular distribution and $N!$ counts all possible permutations for distinguishable particles. The entropy $S=-k_B\ln Z$ gets then written in the form $$\label{SIG}
S=Nk_B\left\{\ln\frac{V}{N}\left(\frac{2\pi
\sigma_p^2}{H^2}\right)^{D/2}+\frac{2+D}{2}\right\},$$ where we have used the Stirling approximation for $N!$. This expression agrees, of course with the known value entropic expression for the IG [@termo], illustrating on the predictive power of the MFI formulation advanced in [@fisher2].\
Scale invariant systems
=======================
We pass now to the leit-motif of the present communication and consider a one-dimensional system with dynamical coordinates $\mathbf{q}=(k,v)$ where $dk/d\tau=v$, with $\tau$ the time variable. We define $k$ as a [*discrete*]{} coordinate, i.e. $k=k_1,k_2,\ldots,k_M$, where $k_i=i\Delta k$ and $M\gg1$, is the total number of bins of width $\Delta k$ in our system. In order to address the scale-invariance behaviour of $k$ we change variables passing to new coordinates $u=\ln k$ and $w=du/dt$. We work under the hypothesis that $u$ and $w$ are canonically conjugated [@mec] and uncorrelated. This assumption immediately leads to proportional growth since $$\label{dyn}
d k/d t=v=kw.$$ For constant $w$ this equation yields an exponential growth $k=k_0e^{wt}$, which represents uniform linear motion in $u$, that is, $u=wt+u_0$, with $u_0=\ln k_0$ [^1]. It is easy to check that the scale transformation $k'=k/\theta_k$ leaves invariant the coordinate $w$, whereas the coordinate $u$ transforms translationally as $u'=u-\Theta_k$, where $\Theta_k=\ln\theta_k$. Thus, the physics does not depend on scale and the system is translationally invariant with respect to the coordinates $u$ and $w$, entailing that the distribution of physical elements can be described by the monoparametric translation families $f(u,w|\Theta_k,\Theta_w)=f(u',w')$. By analogy with the IG, we will call our system a “scale-free ideal gas” (SFIG), i.e., a system of $N$ non-interacting elements. Taking into account that i) $u$ and $w$ are canonical and uncorrelated ($c_{ii}=c_i\neq0$ and $c_{uw}=c_{wu}=0$), so the density distribution can be factorized as $f(u,w)=g(u)h(w)$, and ii) that the Jacobian for our change of variables is $dkdv=e^{2u}dudw$, the information measure $I=I_u+I_w$ can be obtained in the continuous limit as $$\begin{array}{rl}
I_u=&\displaystyle c_u\int_\Omega d u~e^{2u}g(u)\left|\frac{\partial\ln g(u)}{\partial u}\right|^2\\
I_w=&\displaystyle c_w\int_{-\infty}^\infty d
w~h(w)\left|\frac{\partial\ln h(w)}{\partial w}\right|^2,
\end{array}$$ where $\Omega=\ln(k_M/k_1)=\ln M$ is the volume defined in “$u$”-space.
MFI treatment of the scale-free ideal gas {#p3}
-----------------------------------------
The constraints to the given observables $\langle A_i \rangle$ in the extremization problem determine the behaviour of the system. For the general case, we constrain the normalization of $g(u)$ and $h(w)$ to the total number of particles $N$ and to $1$, respectively $$\label{w2a}
\int_\Omega d u~e^{2u}g(u)=N,\qquad\int_{-\infty}^\infty d w~h(w)=1.$$ In addition, we penalize infinite values for $w$ with a constraint on the variance of $h(w)$ to a given measured value $$\int_{-\infty}^\infty d
w~h(w)(w-\overline{w})^2=\sigma_w^2,\label{w2b}$$ where $\overline{w}$ is the average growth. The variation yields $$\label{varg}
\delta\left\{c_u \int_\Omega d u~e^{2u}g\left|\frac{\partial\ln
g}{\partial u}\right|^2+\mu \int_\Omega d u~e^{2u}g\right\}=0$$ and $$\label{varh}
\delta\left\{c_w \int_{-\infty}^\infty d w~h\left|\frac{\partial\ln
h}{\partial w}\right|^2+\lambda \int_{-\infty}^\infty d
w~h(w-\overline{w})^2 +\nu\int_{-\infty}^\infty d w~h\right\}=0,$$ where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing $g(u)=e^{-2u}\Psi^2(u)$, and varying (\[varg\]) with respect to $\Psi$ leads, as is always the case with the MFI [@fisher2], to the Schroedinger-like equation $$\left[-4\frac{\partial^2}{\partial u^2}+4+\mu'\right]\Psi(u)=0,$$ where $\mu'=\mu/c_u$. Analogously to the IG, we impose solutions compatible with a finite normalization of $g$ in the thermodynamic limit $N,\Omega\rightarrow\infty$ with $N/\Omega=\rho_0$ finite, where $\rho_0$ is defined as the *bulk density*. Solutions compatible with the normalization of (\[w2a\]) are given by $\Psi(u)=A_\alpha e^{-\alpha u/2}$, where $A_\alpha$ is the normalization constant and $\alpha=\sqrt{4+\mu'}$. In this general case, the density distribution as a function of $k$ takes the form of a power law: $g_\alpha(\ln k)=A^2/k^{2+\alpha}$. The equilibrium is always defined for the MFI as the ground state solution [@fisher2], which corresponds to the lowest allowed value $\alpha=0$.
Introducing now $h(w)=\Phi^2(w)$ and varying (\[varh\]) with respect to $\Phi$ leads to the quantum harmonic oscillator-like equation [@fisher2; @QM] $$\left[-4\frac{\partial^2}{\partial
w^2}+\lambda'(w-\overline{w})^2+\nu'\right]\Phi(w)=0,$$ where $\lambda'=\lambda/c_w$ and $\nu'=\nu/c_w$. The equilibrium configuration corresponds to the ground state solution, which is now a Gaussian distribution. Using (\[w2b\]) to identify $|\lambda'|^{-1/2}=\sigma_w^2$ we get the Maxwell-Boltzmann distribution $$\label{hw}
h(w)=\frac{\exp\left[-(w-\overline{w})^2/2\sigma_w^2\right]}{\sqrt{2\pi}\sigma_w}.$$
The density distribution in configuration space $F(k,v)dkdv=f(u,w)e^{2u}dudw$ is then $$F(k,v)=\frac{N}{\Omega
k^2}\frac{\exp\left[-(v/k-\overline{w})^2/2\sigma_w^2\right]}{\sqrt{2\pi}\sigma_w}.\label{fkv}$$ If we define $H=\Delta k^2/\Delta\tau$ as the elementary volume in phase space, where $\Delta\tau$ is the time element, the total number of microstates is $Z=N!H^N\prod_{i=1}^NF_1(k_i,v_i)$, where $F_1=F/N$ is the monoparticular distribution function and $N!$ counts all possible permutations for distinguishable elements. The entropy equation of state $S=-\kappa\ln Z$ reads $$S=N\kappa\left\{\ln\frac{\Omega}{N}\frac{\sqrt{2\pi}\sigma_w}{H'}+\frac{3}{2}\right\},$$ where $\kappa$ is a constant that accounts for dimensionality and $H'=H/(k_M k_1)=H/(M\Delta k^2)=1/(M\Delta\tau)$. Remarkably, this expression has the same form as the one-dimensional IG ($D=1$ in (\[SIG\])); instead of the thermodynamical variables $(N,V,T)$, here we deal with the variables $(N,\Omega,\sigma_w)$, which make the entropy scale-invariant as well.
![(colour on-line) **a**, rank-size distribution of the cities of the province of Huelva, Spain (2008), sorted from largest to smallest, compared with the result of a simulation with Brownian walkers (green squares). **b**, rank-plot of the 2008 General Elections results in Spain. **c**, rank-plot of the 2005 General Elections results in the United Kingdom. (Red dots: empirical data; blue lines: fit to (\[coldr\])).\[fig1\]](fig1a.eps "fig:"){width="0.5\linewidth"}\
![(colour on-line) **a**, rank-size distribution of the cities of the province of Huelva, Spain (2008), sorted from largest to smallest, compared with the result of a simulation with Brownian walkers (green squares). **b**, rank-plot of the 2008 General Elections results in Spain. **c**, rank-plot of the 2005 General Elections results in the United Kingdom. (Red dots: empirical data; blue lines: fit to (\[coldr\])).\[fig1\]](fig1b.eps "fig:"){width="0.5\linewidth"}\
![(colour on-line) **a**, rank-size distribution of the cities of the province of Huelva, Spain (2008), sorted from largest to smallest, compared with the result of a simulation with Brownian walkers (green squares). **b**, rank-plot of the 2008 General Elections results in Spain. **c**, rank-plot of the 2005 General Elections results in the United Kingdom. (Red dots: empirical data; blue lines: fit to (\[coldr\])).\[fig1\]](fig1c.eps "fig:"){width="0.5\linewidth"}
The total density distribution for $k$ is obtained integrating for all $v$ the density distribution in configuration space. Accordingly, from (\[fkv\]) we get $$\label{cold}
F(k)=\int dvF(k,v)=\frac{N}{\Omega}\frac{1}{k}=\frac{\rho_0}{k}.$$ It can be shown that this it is just a uniform density-distribution in $u$-space of the bulk density: $F(k)dk=f(u)e^udu=N/\Omega du=\rho_0du$.
Social examples of scale-free ideal gases
=========================================
A common representation of empirical data is the so-called rank-plot or Zipf plot [@zip; @ciudad2; @mod1], where the $j$th element of the system is represented by its size, length or frequency $k_j$ against its rank, sorted from the largest to the smallest one. This process just renders the inverse function of the ensuing cumulative distribution, normalized to the number of elements. We call $r$ the rank that ranges from 1 to $N$. For large $N$, the density distribution (\[cold\]) correspond to an exponential rank-size distribution $$\label{coldr}
k(r)=k_M\exp\left[-\frac{r-1}{\rho_0}\right].$$
This behaviour, which corresponds to the class of universality $\gamma=0$ in (\[eq1\]), is that empirically found by Costa Filho et al. [@elec1] in the distribution of votes in the Brazilian electoral results. We have found such a behaviour in i) the city-size distribution of small regions (as in the province of Huelva (Spain) [@muni]) and ii) electoral results (as in the 2008 Spanish General Elections results [@elecSp]). We depict in figures 1a and 1b the pertinent rank-sizes distributions in semi-logarithmic scale, where a straight line corresponds to a distribution of type (\[coldr\]). A large portion of the distributions can be fitted to (\[coldr\]), with a correlation coefficient of $0.994$ and $0.998$, respectively. From these fits we have obtain a bulk density of $\rho_0=0.058$ for the General Elections results, and in the case of Huelva of $\rho_0=17.1$ ($N=77$, $\Omega=4.5$). Using historical data for the later [@muni], we have used the backward differentiation formula to calculate the relative growth rate of the $i$-th city as $$\label{wi}
w_i = \frac{\ln k_i^{(2008)}-\ln k_i^{(2007)}}{\Delta t}$$ where $k_i^{(2007)}$ and $k_i^{(2008)}$ are the number of inhabitants of the $i$-th city in $2007$ and $2008$, respectively while $\Delta t=1$ year. We show in figure \[fig2\] the empirical rank-plot of the relative growth, where we have obtained $\overline{w}=0.011$ years$^{-1}$ and $\sigma_w=0.030$ years$^{-1}$, compared with the rank-plot of a Maxwell-Boltzmann distribution with the same mean value and standard deviation.
![(colour on-line) Rank-plot of the growth rate $w$ of the province of Huelva between 2007 and 2008 (red dots) compared with a Boltzmann distribution with the same mean value and standard deviation (blue line).\[fig2\]](fig2.eps){width="0.5\linewidth"}
However, these regularities are not always obvious to the naked eye, as shown for the case of the most voted parties in Spain’08 or for the whole distribution of the 2005 General Elections results in the United Kingdom [@elecUK] (figure 1c). In both cases, the competition between parties seems to play an important role, and the assumption of non-interacting elements can be unrealistic [^2].
Bulk and Zipf regimes
---------------------
The situation in which $N/\Omega\rightarrow \,{\rm constant}\, \ne 0$ as $N,\Omega\rightarrow\infty$ will be referred to herefrom as the *bulk regime*. Now, in a recent communication [@algo], we show that Zipf’s law ($\gamma=1$ in (\[eq1\]) with a slope of $-1$ in the rank-plot) can be derived from the extremization of Fisher’s information with [*no*]{} constraints. In the thermodynamic context studied here, the absence of normalization can be understood as the inability of the system to reach the thermodynamic limit, i.e. $N/\Omega\rightarrow0$ as $N,\Omega\rightarrow\infty$. In this case the system can not follow (\[cold\]). Zipf’s law emerges as this behaviour of the density in what we will accordingly denominate the *Zipf regime* ($N/\Omega\rightarrow0$). We digress on the conditions for both regimes in the example discussed below.
We have studied the system formed by all Physics journals [@isi] ($N=310$) using their total number of cites as coordinate $k$. If a journal receives more cites due to its popularity, it becomes even more popular and therefore it will receive more cites. Under such conditions proportional growth and scale invariance are expected. Since we consider [*all sub-fields*]{} of Physics, correlation effects are much lower than they would be should we only consider journals pertaining to an specific sub-field. Accordingly, the non-interacting approximation seems to be realistic in this instance. In figure 3 we depict the rank-plot of the number of citations in Physic journals, and find a slope approaching $-1$ for the most-cited journals in the logarithmic representation (figure 3a) and an slope in the vicinity of r $+1$ for the less-cited journals (figure 3b). For the central part of the distribution, the bulk density reaches a value of $\rho_0\sim57$ (figure 3c).
![(colour on-line) **a**, rank-plot of the total number of cites of physics journal, from most-cited to less-cited, in logarithm scale. **b**, sorted from less-cited to most cited **c**, same as a, in semi-logarithm scale. (Red dots: empirical data; blue line: fit to (\[coldr\])).\[fig3\]](fig3a.eps "fig:"){width="0.5\linewidth"}\
![(colour on-line) **a**, rank-plot of the total number of cites of physics journal, from most-cited to less-cited, in logarithm scale. **b**, sorted from less-cited to most cited **c**, same as a, in semi-logarithm scale. (Red dots: empirical data; blue line: fit to (\[coldr\])).\[fig3\]](fig3b.eps "fig:"){width="0.5\linewidth"}\
![(colour on-line) **a**, rank-plot of the total number of cites of physics journal, from most-cited to less-cited, in logarithm scale. **b**, sorted from less-cited to most cited **c**, same as a, in semi-logarithm scale. (Red dots: empirical data; blue line: fit to (\[coldr\])).\[fig3\]](fig3c.eps "fig:"){width="0.5\linewidth"}
This distribution shows a notably symmetric behaviour under the change $k\rightarrow1/k$ ($u\rightarrow-u$). We exhibit in figure 4 the raw empirical data as compared with the distribution obtained from the transformation $k'=c/k$ ($u'=-u+\ln c$), where $c=3.3\times10^6$. The main part of the density distribution reaches the bulk density obeying (\[cold\]), whereas Zipf’s law emerges at the edges, which could be understood as constituting the *surface* of the system, since they explain how the density (exponentially) falls from its bulk-value to zero in $u$-space when the system is exposed to an infinitely empty volume. This effect is clearly visible in figure 5, where the empirical density distribution $p(u)du$ in $u$-space is compared with the “fitted” density $$\label{fit}
p(u)=\left\{
\begin{array}{ll}
\rho_Ze^{u-u_1}&\mathrm{if~} u<u_1\\
\rho_0&\mathrm{if~} u_1<u<u_2\\
\rho_Ze^{u_2-u}&\mathrm{if~} u>2
\end{array}
\right.$$ where $\rho_Z=18$, $\rho_0=57$, $u_1=5.2$ and $u_2=10$. These findings lead us to conclude that the system consisting of Physics journals, when sorted by total number of citations, is a perfect example of a finite scale-free ideal gas at equilibrium.
![(colour online) Rank-plot of the total number of cites of Physics journal, from most-cited to less-cited, compared with the distribution obtained from the inverse transformation $k'=3.3\times10^6/k$ where $k$ is the number of cites.\[fig4\]](fig4.eps){width="0.5\linewidth"}
![(colour online) Empirical density distribution in $u$-space of the total number of cites of Physics journals, compared with (\[fit\]). The bulk regime and the Zipf regime at the edges is clearly visible. \[fig5\]](fig5.eps){width="0.5\linewidth"}
An accompanying microscopic description {#p6}
---------------------------------------
The dynamics of the system under scrutiny here can be microscopically described as a stochastic process using (\[dyn\]) together with the density distribution (\[hw\]). Treating $w$ as a random variable, the pertinent stochastic equation of motion is written in the guise of a geometrical Brownian motion, i.e., [@exp] $$\label{eqd}
dk = k\overline{w}dt + k\sigma_w dW,$$ where $dW$ represents a Wiener process. In the $u$-space, this equation reads $$\label{eqd2}
du = \overline{w}dt + \sigma_w dW,$$ and is known to describe the celebrated Brownian motion (\[eqd\]), which exactly describes the dynamical condition found empirically in [@linux] and also the (stochastic) proportional growth model used in [@citis] to obtain Zipf’s law. We thus dare to suggest that we are dealing here with a sort of “equivalent” of a molecular dynamics’ simulation for gases/liquids [@DM].
Indeed, (\[eqd2\]) describes $N$ Brownian walkers moving in a fixed volume $\Omega$ with uniform density in $u$-space, a model used in the literature to describe the ideal gas [@DM]. This scenario can also be reproduced by our free-scale ideal gas merely by choosing to represent the system with the coordinates $(k,v)$. In figure 1a we show the rank-plot for $k$ of a system of $N=78$ geometrical Brownian walkers with $\overline{w}=0.011$ and $\sigma_w=0.030$ in a volume $\Omega=4.5$, with d $k_1=200$ in reduced units, which approximately describes the distribution of the population of the province of Huelva. We also show in figure 2 the rankplot of $w$ of the same random walkers, compared with the empirical data.
Conclusions {#p7}
===========
Our present considerations derive from the fact that, as shown in [@fisher2], thermodynamics can be reformulated in terms of the minimization with appropriate constraints of Fisher’s information. We have applied such reformulation in order to discuss the thermodynamics of scale-free systems and derived the density distribution in configuration space and the entropic expression for the equilibrium state of what we call SFIG: the scale-free ideal gas (in the thermodynamic limit). We have encountered convincing empirical evidences of the SFIG actual existence in sociological scenarios. Thus, we have dealt with city populations, electoral results and citations in Physics journals. In such a context it is seen that Zipf’s law emerges naturally as the equilibrium density of the non-interacting system when the volume grows without bounds, a situation that we call the Zipf regime. Using empirical data we have revealed that this regime can be understood as a density-decay at the “surface” separating the bulk from an empty and very large volume. Finally, we have shown with a simulation of city-populations that geometrical Brownian motion can describe such systems at a microscopic level.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank D. Puigdomenech, D. Villuendas, M. Barranco, R. Frieden, and B. H. Soffer for useful discussions. This work has been partially performed under grant FIS2008-00421/FIS from DGI, Spain (FEDER).
[99]{}
Fractals in Physics: Essays in Honour of Benoit B Mandelbrot, edited by A. Aharony and J. Feder, Proceedings of a Conference in honor of B.B. Mandelbrot on His 65th in Vence, France, North Holland, Amsterdam, 1989.
K. Paech, W. Bauer, and S. Pratt , [ Phys. Rev. C]{} [76]{} (2007), p. 054603; X. Campi and H. Krivine, [ Phys. Rev. C]{} [72]{} (2005) 057602; Y. G. Ma et al., [ Phys. Rev. C]{} [71]{} (2005) 054606.
C. Furusawa and K. Kaneko, [ Phys. Rev. Lett.]{} [90]{} (2003) 088102.
G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley Press, Cambridge, Mass., 1949; I. Kanter and D. A. Kessler, [ Phys. Rev. Lett.]{} [74]{} (1995) 4559.
M. E. J. Newman, [ Phys. Rev. E]{} [64]{} (2001) 016131.
A.-L. Barabasi and R. Albert, [ Rev. Mod. Phys.]{} [74]{} (2002) 47.
T. Maillart, D. Sornette, S. Spaeth, and G. von Krogh, [ Phys. Rev. Lett.]{} [101]{} (2008) 218701.
R. N. Costa Filho, M. P. Almeida, J. S. Andrade, and J. E. Moreira, [ Phys. Rev. E]{} [60]{} (1999) 1067.
A. A. Moreira, D. R. Paula, R. N. Costa Filho, and J. S. Andrade, Phys. Rev. E, 73 (2006) 065101(R).
L. C. Malacarne, R. S. Mendes, and E. K. Lenzi, [ Phys. Rev. E]{} [65]{} (2001) 017106.
M. Marsili and Yi-Cheng Zhang, [ Phys. Rev. Lett.]{} [80]{} (1998) 2741.
R. L. Axtell, [ Science]{} [293]{} (2001) 1818.
X. Gabaix, [ Quarterly Journal of Economics]{} [114]{} (1999) 739.
B. R. Frieden, A. Plastino, A. R. Plastino, and B. H. Soffer, Phys. Rev. E [60]{} (1999) 48; [66]{} (2002) 046128.
B. R. Frieden and B. H. Soffer, [ Phys. Rev. E]{} [ 52]{} (1995) 2274; B. R. Frieden, Physics from Fisher Information, Cambridge University Press, Cambridge, England, 1998; B. R. Frieden, Science from Fisher Information, Cambridge University Press, Cambridge, 2004.
M. W. Zemansky and R. H. Dittmann,Heat and Thermodynamics, McGraw-Hill, London, 1981.
H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd Ed., Addison Wesley, San Francisco, 2002.
C. Cohen-Tannoudji, B. Diu and F. Laloe, Quantum Mechanics, Wiley-Interscience, New York, 2006.
W. J. Reed, and B. D. Hughes, [ Phys. Rev. E]{} [66]{} (2002) 067103.
K. E. Kechedzhi, O. V. Usatenko, and V. A. Yampol’skii, [ Phys. Rev. E]{} [72]{}, 046138 (2005); S. Ree, [ Phys. Rev. E]{} [73]{} (2006) 026115.
National Statistics Institute, Spain, www.ine.es.
Ministry of the Interior, Spain, www.elecciones.mir.es
Electoral Commission, Government of the UK, www.electoralcommission.org.uk. A. Hernando, D. Villuendas, M. Abad and C. Vesperinas, arXiv:0905.3704v1, 2009. A. Hernando, D. Puigdomenech, D. Villuendas, C. Vesperinas and A. Platino, submitted to Phys. Lett. A.
Journal Citation Reports (JCR) for 2007, Thomson Reuters.
Gould H and Tobochnik J 1996 *An Introduction to Computer Simulation Methods: Applications to Physical Systems*, 2nd Ed. (Addison-Wesley).
[^1]: This exponential growth allows to identify the systems that we study in this work in macroscopic fashion with those addressed in [@exp].
[^2]: The effects of interaction are studied in [@nota2], where we go beyond the non-interacting system using a microscopic description based on complex networks.
|
---
abstract: 'We study characteristic (quasinormal) modes of a $D$-dimensional Schwarzshild black hole. It proves out that the real parts of the complex quasinormal modes, representing the real oscillation frequencies, are proportional to the product of the number of dimensions and inverse horizon radius $\sim$ $D$ $r_{0}^{-1}$. The asymptotic formula for large multipole number $l$ and arbitrary $D$ is derived. In addition the WKB formula for computing QN modes, developed to the 3th order beyond the eikonal approximation, is extended to the 6th order here. This gives us an accurate and economic way to compute quasinormal frequencies.'
author:
- |
R.A.Konoplya\
Department of Physics, Dniepropetrovsk National University\
St. Naukova 13, Dniepropetrovsk 49050, Ukraine\
konoplya@ff.dsu.dp.ua
title: 'Quasinormal behavior of the D-dimensional Schwarzshild black hole and higher order WKB approach'
---
Introduction
============
Within the framework of the brane world models the size of extra spatial dimensions may be much larger than the Plank’s length, and the fundamental quantum gravity scale may be very low ($\sim$ Tev). When considering models with large extra dimensions the black hole mass may be of order Tev., i.e. much smaller than the Plank’s mass. There is a possibility of production of such mini black holes in particle collisions in colliders and in cosmic ray experiments [@S.Dimopoulos]. Estimations show that these higher dimensional black holes can be described by classical solutions of vacuum Einstein equations. Thus the investigation of general properties of these black holes, including perturbations and decay of different fields around them, attracts considerable interest now (see for example [@Frolov1], [@Cardoso-Lemos2] and references therein).
It is well-known that when perturbing black hole it undergoes damping oscillations which are characterized by some complex eigenvalues of the wave equations called [*quasinormal frequencies*]{}. Their real parts represent the oscillation frequencies, while the imaginary ones determine the damping rates of the modes. The quasinormal modes (QN) of black holes (BH’s) depend only on a black hole parameters and not on a way in which they were excited. QN’s are called, therefore, ”footprints" of a black hole. Being a useful characteristic of black hole’s dynamics, quasinormal modes are studied also within different contexts now: in Anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence (see for example [@Horowitz1]- [@Moss-Norman] and references therein), because of the possibility of observing quasinormal ringing of astrophysical BH’s (see [@Kokkotas-Schmidt] for a review), when considering thermodynamic properties of black holes in loop quantum gravity [@Dreyer]-[@Motl], in the context of possible connection with critical collapse [@Horowitz1], [@Konoplya3], [@Kim], [@KonoplyaPLB1].
Thus it would be interesting to know, from different grounds, what happen with QN spectrum a black hole living in $D$ -dimensional space-time [@Motl1], [@Cardoso-Lemos2]. The present paper is two-fold: First we extend the WKB method of Schutz, Will and Iyer for computing QN modes from the 3th to the 6th order beyond the eikonal approximation (see Sec.II and Appendix I). In a lot of physical situations this allows us to compute the QNMs accurately and quickly without resorting to complicated numerical methods. In Appendix II QN modes of $D=4$ Schwarzshild black hole induced by perturbations of different spin are obtained by the 6th order WKB formula, and compared with the numerical values and 3th order WKB values. Second, motivated by the above reasons, we apply the obtained WKB formula for finding of the scalar quasinormal modes of multi-dimensional Schwarzshild black hole (Sec. III). It proves out that the real parts of the quasinormal frequencies are proportional to the product $D r_{0}^{-1}$, where $r_{0}$ is the horizon radius, and $D$ is the dimension of space-time.
Sixth order WKB analysis
========================
First semi-analytical method for calculations of BH QNMs was apparently proposed by Bahram Mashhoon who used the Poschl-Teller potential to estimate the QN frequencies [@Mashhoon]. In [@Will-Schutz] there was proposed a semi-analytical method for computing QNM’s based on the WKB treatment. Then in [@S.Iyer-C.M.Will] the first WKB order formula was extended to the third order beyond the eikonal approximation, and, afterwards, was frequently used in a lot of works (see for example [@Konoplya3] [@Kokkotas2], [@Kokkotas3], [@Simone-Will], [@Andersson], [@Piazza], [@Onozawa], [@KonoplyaGRG1] and references therein). The accuracy of the 3th order WKB formula (see eq. (1.5) in [@S.Iyer-C.M.Will]) is the better, the more multipole number $l$ and the less overtone $n$. For the Schwarzshild BH the results practically coincide with accurate numerical results of Leaver [@Leaver] at $l\geq 4$ when being restricted by lower overtones for which $l > n$. For fewer multipoles, however, accuracy is worse, and may reaches $10$ per cents at $l=0$, $n=0$. Numerical approach [@Leaver], on contrary, is very accurate, but, dealing with numerical integration or systems of recurrence relations, is very cumbersome, and, often, require modification to be applied to different effective potentials. At the same time WKB approach lets us to obtain QNM’s for a full range of parameters giving thereby some fields of work for intuition as to physical behavior of a system. Even though WKB formula gives the best accuracy at $l > n$, it includes the case of astrophysical black hole radiation where only lower overtones are significantly excited [@Stark]. Both advantages and deficiencies of the WKB approach motivated us to extend the existent 3th order WKB formula up to the 6th order.
The perturbation equations of a black hole can be reduced to the Schrodinger wave-like equation: $$\label{1}
\frac{d^2 \psi}{d x^2} + Q(x) \psi(x)=0,$$ where “the potential” $-Q(x)$ is constant at the event horizon ($x=-\infty$) and at the infinity ($x=+\infty$) and it rises to maximum at some intermediate $x= x_{0}$. Consider radiation of a given frequency $\omega$ incident on the black hole from infinity and let $R(\omega)$ and $T(\omega)$ be the reflection and transmission amplitudes respectively. Extend $R(\omega)$ to the complex frequency plane such that $Re(z)\neq 0$, and $T(z)/R(z)$ is regular. Then, the quasinormal modes correspond to the singularities of $R(z)$. We have a direct analogy with the problem of scattering near the pick of the potential barrier in quantum mechanics, where $\omega^2$ plays a role of energy, and the two turning points divide the space into three regions at which boundaries the corresponding solutions should be matched.
To extend the 3th order WKB formula of [@S.Iyer-C.M.Will] we used the technique of Iyer and Will. We shall omit here the technicalities of this approach which are described in [@S.Iyer-C.M.Will]. The only thing we should stress is that since the coefficients $M_{ij}$, that connect amplitudes near the horizon with those at infinity, depend only on $\nu$ (related to the overtone number $n$) they may be found to higher orders, simply by solving the interior (between the turning points) problem to higher orders. Thus there is no need to perform an explicit match of the solutions to WKB solutions in the exterior (outside turning points) regions to the same order. The result has the form: $$\label{2}
\frac{\imath Q_{0}}{\sqrt{2 Q_{0}''}}
-\Lambda_{2}-\Lambda_{3} -\Lambda_{4} -\Lambda_{5} -\Lambda_{6} =n+\frac{1}{2},$$ where the correction terms $\Lambda_{4}$, $\Lambda_{5}$, $\Lambda_{6}$ can be found in the Appendix I. Note that $\Lambda_{4}$ coincides with preliminary formula (A3) of [@S.Iyer-C.M.Will] in proper designations.
An alternative, pure algebraic approach to finding higher order WKB corrections was proposed by O.Zaslavskii [@O.Zaslavsky], using a quantum anharmonic oscillator problem where WKB correction terms come from perturbation theory corrections to the potential anharmonicity.
Thus we have obtained an economic and accurate formula for straightforward calculation of QNM frequencies. The 6th order formula applied to the $D=4$ Schwarzshild BH is as accurate already at $l=1$ as the 3th order formula does at $l=4$. We show it in Appendix II on example of QNM’s corresponding to perturbations of fields of different spins: scalar ($s=0$), neutrino ($s=1/2$), electromagnetic ($s=1$), gravitino ($s=3/2$), and gravitational ($s=2$). In addition, looking at the convergence of all sixth WKB values to some unknown true QN mode, we can judge, approximately, how far from the true QN value we are, staying within the framework of WKB method.
Quasinormal modes of the D-dimensional\
Schwarzshild black hole
=======================================
The metric of the Schwarzshild black hole in $D$-dimensions has the form: $$\label{3}
ds^2= f(f) dt^2 -f^{-1}(r) dr^2 + r^2 d \Omega^{2}_{D-2},$$ where $$\label{4}
f(r)=1-\left(\frac{r_{0}}{r}\right)^{D-3}= 1-\frac{16 \pi G M}{(D-2) \Omega_{D-2} r^{D-3}}.$$ Here we used the quantities $$\Omega_{D-2}=\frac{(2 \pi)^{(D-1)/2}}{\Gamma((D-1)/2)}, \quad \Gamma(1/2) =
\sqrt{\pi}, \quad \Gamma(z+1)=z \Gamma(z)$$.
The scalar perturbation equation of this black hole can be reduced to the Schrodinger wave-like equation (\[1\]) with respect to the “tortoise” coordinate $x$: $d x = \frac{d r}{f(r)}$ where “the potential” $-Q(x)$ has the form: $$\label{5}
Q(x)= \omega^2-f(r)\left(\frac{l(l+D-3)}{r^2} + \frac{(D-2) (D-4)}{4 r^2} f(r) + \frac{D-2}{2 r} f'(r)\right),$$ At some fixed $D$ we can put $r_{0}=2$ and measure $\omega$ in units $2 r_{0}^{-1}$. The quasinormal modes satisfy the boundary conditions: $$\label{6}
\phi(x) \sim c_{\pm} e^{ \pm i \omega x} \qquad as \qquad x\rightarrow \pm
\infty.$$
The 6th WKB order formula used here gives very accurate results for low overtones. The previous orders serve us to see the convergence of the WKB values of $\omega^2$ as a WKB order grows to an accurate numerical result. Namely we can observe that for $l=1,2,3,4,..$ for the fundamental overtone the 6th order values differs form its 5th order value by fractions of a percent or less at not very large $D$ (we are restricted here by $D=4, 5, ...15$).
It proves out that if one takes $r_{0}=2$ for each given $D$, then the real parts of $\omega$ for different $D$ lay on a strict line. That is, $\omega_{Re}$ is proportional to the product $r_{0}
D$ (Remember that $r_{0}$ depends on $D$ itself). Namely, for the fundamental overtone we obtain the following approximate relations: $$\label{7}
\omega_{Re} \sim 0.244 D (r_{0}/2)^{-1}, \qquad l=2$$
$$\label{8}
\omega_{Re} \sim 0.275 D (r_{0}/2)^{-1}, \qquad l=3$$
$$\label{9}
\omega_{Re} \sim 0.290 D (r_{0}/2)^{-1}, \qquad l=4.$$
Here we take $\omega= \omega_{Re}- i \omega_{Im}$. Generally, the more the multipole number $l$, the more the coefficient before the product $D r_{0}^{-1}$. The same $ \sim D r_{0}^{-1}$ relation we observed for higher overtone but not higher than $l$, for which WKB treatment is applicable. In Fig.1,2 we presented the real and imaginary parts of $\omega$ measured in $2 r_{0}^{-1}$ for different $D$. For real parts of $l=1$ modes we see the deviation from the strict line at large $D$. This however, is stipulated by a bad accuracy of the WKB approach, and we believe that the true frequencies will lay on strict line again. Indeed, one can judge about it by looking at the convergence plot Fig.3-Fig.6 where the real and imaginary parts of $\omega$ are shown as a function of the WKB order. Generally the accuracy of the WKB formula is the better, the more $l$, and the less $n$ and $D$. Note that the dependence $D r_{0}^{-1}$ for lower overtones can be recovered even within 3th order formula, provided $l$ is greater than $2$, and $D$ is not very large.
![$Re \omega$ for different dimensions $D$; $l=1$ (bottom), $2$, $3$, $4$ (top); $n=0$.[]{data-label=""}](bh_multi_sh_1)
![$Im \omega$ for different dimensions $D$; $l=1$ (bottom), $2$, $3$, $4$ (top); $n=0$.[]{data-label=""}](bh_multi_sh_2)
![$\omega_{Re}$ (bottom) and $\omega_{Im}$ (top) as a function of WKB order of the formula with which it was obtained for $l=1$, $n=2$, $D=4$ modes, and the corresponding numerical value. We see how the WKB values converge to an accurate numerical value as the WKB order increases.[]{data-label=""}](bh_multi_sh_3)
![$\omega_{Re}$ (top) and $\omega_{Im}$ (bottom) as a function of WKB order of the formula with which it was obtained for $l=0$, $n=0$, $D=12$ modes.[]{data-label=""}](bh_multi_sh_4)
![$\omega_{Re}$ (top) and $\omega_{Im}$ (bottom) as a function of WKB order of the formula with which it was obtained for $l=0$, $n=0$, $D=6$ modes.[]{data-label=""}](bh_multi_sh_5)
![$\omega_{Re}$ (top) and $\omega_{Im}$ (bottom) as a function of WKB order of the formula with which it was obtained for $l=1$, $n=0$, $D=6$ modes.[]{data-label=""}](bh_multi_sh_6)
Another point is the $l=0$ modes: in this case the lowest overtone implies $l=n$, and the WKB formula has considerable relative error. For a four-dimensional BH, for which the accurate numerical results are known, the error is about $10$ percent for $\omega_{Im}$, and $5$ percent for $\omega_{Re}$ in the third WKB order, while in the sixth order it reduces to $0$ percent for $\omega_{Re}$ and $3$ percent for $\omega_{Im}$ (see Appendix II). For greater $D$ the error increases, the difference between the fifth and sixth order WKB values grows and one cannot judge of true quasinormal behavior in this case (see Fig.3, 4, 5.). Fortunately, other field perturbations, including gravitational, have the lowest overtone with $l>n$ and the WKB treatment is of good accuracy for all $l$. In Table 1. we compare the third order WKB values of $l=0$, $n=0$ modes for different $D$ [@Cardoso-Lemos2] with those obtained through the sixth order here.
$D$ 3th WKB order 6th WKB order $1/r_{0}$
------ -------------------- --------------------- -----------
$4$ $0.1046-0.1152 i$ $0.1105 - 0.1008 i$ $0.5$
$6$ $1.0338- 0.7133 i$ $1.1808- 0.6438 i$ $1.28$
$8$ $1.9745- 1.0258 i$ $2.3004- 1.0328 i$ $1.32$
$10$ $2.7828- 1.1596 i$ $3.2214- 1.3766 i$ $1.25$
$12$ $3.4892- 1.2020 i$ $3.9384- 1.7574 i$ $1.17$
Table I. Schwarzshild QN frequencies for $l=0$, $n=0$ scalar perturbations in various $D$.
For large $l$ the well-known approximate formula reads (see [@Mashhoon2], [@Will2], [@Press] for a proof) $$\label{10}
\omega_{Re}= \frac{1}{3 \sqrt{3}}\left(l+\frac{1}{2}\right),
\qquad \omega_{Im}= \frac{1}{3 \sqrt{3}}\left(n+\frac{1}{2}\right)$$
To obtain its $D$-dimensional generalization we find a value $r_{max}$ at which the effective potential $V$ attains its maximum, provided $l$ is large $$\label{11}
r_{max} \approx 2^{\frac{D-4}{D-3}} (D-1)^\frac{1}{D-3}, \qquad D=4,5,6,
\ldots .$$ Then let us make use of this value $r_{max}$ when dealing with the first order WKB formula. After expansion in terms of small values of $1/l$, for a fixed $D$ in units of $2 r_{0}^{-1}$ we obtain $$\label{12}
\omega_{Re} \approx \frac{D+2 l -3}{4} \left(\frac{2}{D-1}\right)^{\frac{1}{D-3}} \sqrt{\frac{D-3}{D-1}}$$
$$\label{13}
\omega_{Im} \approx \frac{(D-3)}{4} \left(\frac{2}{D-1}\right)^{\frac{1}{D-3}}
\frac{2 n +1}{\sqrt{D-1}}$$
When $D=4$ these formulas go over into (\[11\]). We see that when $l$ is much larger than $D$, the $ \sim D r_{0}^{-1}$ dependence of $\omega_{Re}$ breaks down.
Conclusion
==========
We were interested here in a question how dimensionality effects on quasinormal behavior of black holes. Yet, several interesting points are left beyond our consideration of low laying quasinormal modes of multi-dimensional black holes. First of all, one would like to understand the origin of the relation $ \sim D r_{0}^{-1}$ in $\omega_{Re}$ dependence. In this question it is possible to try to explain it from the interpretations of QN modes as Breit-Wigner type resonances generated by a family of surfaces wave propagating close to the unstable circular photon orbit [@Decanini]. Second, we do not know whether $ \sim D r_{0}^{-1}$ dependence will be present for perturbations of other fields, and for more general backgrounds, such as multi-dimensional Reissner-Nordstrom or Kerr. We hope further investigations will clarify these points.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to acknowledge stimulating discussions with Vitor Cardoso and Oleg Zaslavskii.
Appendix 1: Correction terms for WKB formula
============================================
Here we shall follow the designations: $Q_{0}$ means the value of the potential $Q$ at its pick, while $Q_{i}$ is the $i$th derivative of $Q$ with respect to the tortoise coordinate $x$. Then $Q_{i}^j$ is the $j$th power of the $i$th derivative of $Q$.
$$\Lambda_{4}=\frac{1}{597196800 \sqrt{2} Q_{2}^{7} \sqrt{Q_{2}}}
(2536975 Q_{3}^{6} - 9886275 Q_{2} Q_{3}^{4} Q_{4}
+5319720 Q_{2}^{2} Q_{3}^{3} Q_{5}-$$ $$225 Q_{2}^{2} Q_{3}^{2}
(-40261 Q_{4}^{2}+9688 Q_{2} Q_{6}) + 3240 Q_{2}^{3} Q_{3}
(-1889 Q_{4} Q_{5}+220 Q_{2} Q_{7}) -$$ $$729 Q_{2}^{3} (1425 Q_{4}^{3}
-1400 Q_{2} Q_{4} Q_{6} +8 Q_{2} (-123 Q_{5}^{2} +25 Q_{2}
Q_{8})))+$$ $$\frac{(n+1/2)^2}{4976640 \sqrt{2}
Q_{2}^{7} \sqrt{Q_{2}}}
(348425 Q_{3}^{6} -1199925 Q_{2} Q_{3}^{4} Q_{4}
+57276 Q_{2}^{2} Q_{3}^{3} Q_{5}-$$ $$45 Q_{2}^{2} Q_{3}^{2}
(-20671 Q_{4}^{2}+4552 Q_{2} Q_{6}) + 1980 Q_{2}^{3} Q_{3}
(-489 Q_{4} Q_{5}+52 Q_{2} Q_{7})-$$ $$27 Q_{2}^{3} (2845 Q_{4}^{3}
-2360 Q_{2} Q_{4} Q_{6} +56 Q_{2} (-31 Q_{5}^{2} +5 Q_{2}
Q_{8})))+$$ $$\frac{ (n+1/2)^4}{2488320 \sqrt{2} Q_{2}^{7}
\sqrt{Q_{2}}}
(192925 Q_{3}^{6} -581625 Q_{2} Q_{3}^{4} Q_{4}
+234360 Q_{2}^{2} Q_{3}^{3} Q_{5}-$$ $$45 Q_{2}^{2} Q_{3}^{2}
(-8315 Q_{4}^{2}+1448 Q_{2} Q_{6}) + 1080 Q_{2}^{3} Q_{3}
(-161 Q_{4} Q_{5}+12 Q_{2} Q_{7}) -$$ $$\label{75}
27 Q_{2}^{3} (625 Q_{4}^{3}
-440 Q_{2} Q_{4} Q_{6} +8 Q_{2} (-63 Q_{5}^{2} +5 Q_{2}
Q_{8})))$$ $$\Lambda_{5}=\frac{(n+1/2)}{57330892800 Q_{2}^{10}}
(2768256 Q_{10} Q_{2}^7 -1078694575 Q_{3}^8+5357454900
Q_{2}Q_{3}^6 Q_{4}-$$ $$2768587920 Q_{2}^2 Q_{3}^5 Q_{5} +90 Q_{2}^2 Q_{3}^4
(-88333625 Q_{4}^2+ 12760664 Q_{2} Q_{6})-$$ $$4320 Q_{2}^3 Q_{3}^3
(-1451425 Q_{4} Q_{5}+91928 Q_{2} Q_{7}) - 27 Q_{2}^4
(7628525 Q_{4}^4 -9382480 Q_{2} Q_{4}^2 Q_{6} +$$ $$64 Q_{2}^2 (19277 Q_{6}^2 + 37764 Q_{5} Q_{7}) +
576 Q_{2} Q_{4} (-21577 Q_{5}^2 +2505 Q_{2} Q_{8}))+$$ $$540 Q_{2}^3 Q_{3}^2 (6515475 Q_{4}^3 -3324792 Q_{2} Q_{4} Q_{6}
+16 Q_{2} (-126468 Q_{5}^2 +12679 Q_{2} Q_{8}))-$$ $$432 Q_{2}^4 Q_{3}
(5597075 Q_{4}^2 Q_{5}-854160 Q_{2} Q_{4} Q_{7} +8 Q_{2} (-145417 Q_{5}
Q_{6}
+ 6685 Q_{2} Q_{9})))+$$ $$\frac{(n+1/2)^3}{477757440 Q_{2}^{10}}
(31104 Q_{10} Q_{2}^7 -42944825 Q_{3}^8+193106700
Q_{2}Q_{3}^6 Q_{4}-$$ $$-90039120 Q_{2}^2 Q_{3}^5 Q_{5}+30 Q_{2}^2 Q_{3}^4
(-8476205 Q_{4}^2+1102568 Q_{2} Q_{6})-$$ $$4320 Q_{2}^3 Q_{3}^3
(-41165 Q_{4} Q_{5}+2312 Q_{2} Q_{7}) - 9 Q_{2}^4
(445825 Q_{4}^4-472880 Q_{2} Q_{4}^2 Q_{6} +$$ $$64 Q_{2}^2 (829 Q_{6}^2 + 1836 Q_{5} Q_{7}) +
4032 Q_{2} Q_{4} (-179 Q_{5}^2 +15 Q_{2} Q_{8}))+$$ $$180 Q_{2}^3 Q_{3}^2 (532615 Q_{4}^3 -241224 Q_{2} Q_{4} Q_{6}
+16 Q_{2} (-9352 Q_{5}^2 +799 Q_{2} Q_{8}))-$$ $$144 Q_{2}^4 Q_{3}
(392325 Q_{4}^2 Q_{5}-51600 Q_{2} Q_{4} Q_{7} +8 Q_{2} (-8853 Q_{5} Q_{6}
+ 335 Q_{2} Q_{9})))+$$ $$\frac{(n+1/2)^5}{1194393600 Q_{2}^{10}}
(10368 Q_{10} Q_{2}^7 -66578225 Q_{3}^8+272124300
Q_{2}Q_{3}^6 Q_{4}-$$ $$112336560 Q_{2}^2 Q_{3}^5 Q_{5} +9450 Q_{2}^2 Q_{3}^4
(-33775 Q_{4}^2 + 3656 Q_{2} Q_{6})-$$ $$151200 Q_{2}^3 Q_{3}^3
(-1297 Q_{4} Q_{5}+56 Q_{2} Q_{7}) - 27 Q_{2}^4
(89075 Q_{4}^4 -83440 Q_{2} Q_{4}^2 Q_{6} +$$ $$64 Q_{2}^2 (131 Q_{6}^2 + 396 Q_{5} Q_{7}) +
576 Q_{2} Q_{4} (-343 Q_{5}^2 +15 Q_{2} Q_{8}))+$$ $$540 Q_{2}^3 Q_{3}^2 (188125 Q_{4}^3 -71400 Q_{2} Q_{4} Q_{6}
+16 Q_{2} (-3052 Q_{5}^2 +177 Q_{2} Q_{8}))-$$ $$\label{76}
432 Q_{2}^4 Q_{3}
(118825 Q_{4}^2 Q_{5}-11760 Q_{2} Q_{4} Q_{7} +8 Q_{2}
(-2303 Q_{5} Q_{6} + 55 Q_{2} Q_{9})))$$ $$\Lambda_{6}=\frac{-i}{202263389798400 Q_{2}^{12}\sqrt{2 Q_{2}}}
(-171460800 Q_{12} Q_{2}^9 +1714608000
Q_{11} Q_{2}^8 Q_{3}-$$ $$10268596800 Q_{10} Q_{2}^7 Q_{3}^2 +
970010662775 Q_{3}^{10} + 3772137600 Q_{10} Q_{2}^8 Q_{4}-$$ $$6262634175525 Q_{2} Q_{3}^8 Q_{4}
+13782983196150 Q_{2}^{2} Q_{3}^{6} Q_{4}^{2} -11954148125850 Q_{2}^3
Q_{3}^4 Q_{4}^3 +$$ $$3449170577475 Q_{2}^4 Q_{3}^2 Q_{4}^4 -144528059025 Q_{2}^5 Q_{4}^5
+ 3352602187200 Q_{2}^2 Q_{3}^7 Q_{5}-$$ $$12300730092000 Q_{2}^3 Q_{3}^5 Q_{4} Q_{5} + 11994129604800
Q_{2}^4 Q_{3}^3 Q_{4}^2 Q_{5} -2624788605600
Q_{2}^5 Q_{3} Q_{4}^3 Q_{5}+$$ $$2580769643760 Q_{2}^4 Q_{3}^4 Q_{5}^2 -3453909784416
Q_{2}^5 Q_{3}^2 Q_{4} Q_{5}^2 +438440697072 Q_{2}^6 Q_{4}^2 Q_{5}^2+$$ $$+260524397952 Q_{2}^6 Q_{3} Q_{5}^3-1475306441280 Q_{2}^3 Q_{3}^6 Q_{6}
+4329682610400 Q_{2}^4 Q_{3}^4 Q_{4} Q_{6}-$$ $$2865128172480 Q_{2}^5 Q_{3}^2 Q_{4}^2 Q_{6} +
233443879200 Q_{2}^6 Q_{4}^3 Q_{6} -1660199804928 Q_{2}^5 Q_{3}^3 Q_{5} Q_{6}+$$ $$1281705296256 Q_{2}^6 Q_{3} Q_{4} Q_{5} Q_{6} -87403857408 Q_{2}^7 Q_{5}^2 Q_{6} + 231105873600
Q_{2}^6 Q_{3}^2 Q_{6}^2-$$ $$68412859200 Q_{2}^7 Q_{4} Q_{6}^2 +552968700480 Q_{2}^4 Q_{3}^5 Q_{7}
-1231789749120 Q_{2}^5 Q_{3}^3 Q_{4} Q_{7}+$$ $$470726303040 Q_{2}^6 Q_{3} Q_{4}^2 Q_{7} +413953400448 Q_{2}^6 Q_{3}^2 Q_{5} Q_{7}
-126242178048 Q_{2}^7 Q_{4} Q_{5} Q_{7}-$$ $$91489305600Q_{2}^7 Q_{3} Q_{6} Q_{7} + 5619715200 Q_{2}^8 Q_{7}^2
-175752294480 Q_{2}^5 Q_{3}^4 Q_{8}+$$ $$271759652640 Q_{2}^6 Q_{3}^2 Q_{4} Q_{8} -39736040400 Q_{2}^7 Q_{4}^2 Q_{8}-
73378363968 Q_{2}^7 Q_{3} Q_{5} Q_{8}+$$ $$9773265600 Q_{2}^8 Q_{6} Q_{8} +47107126080 Q_{2}^6 Q_{3}^3 Q_{9}
-43345290240 Q_{2}^7 Q_{3} Q_{4} Q_{9} +7400248128 Q_{2}^8 Q_{5} Q_{9})-$$
$$\frac{(n+1/2)^2 i}{687970713600 Q_{2}^{12}\sqrt{2 Q_{2}}}
(-4551552 Q_{12} Q_{2}^9+60279552
Q_{11} Q_{2}^8 Q_{3}-$$ $$425036160 Q_{10} Q_{2}^7 Q_{3}^2 +
73727194625 Q_{3}^{10} +116743680 Q_{10} Q_{2}^8 Q_{4}-$$ $$443649208275 Q_{2} Q_{3}^8 Q_{4}+
901144103850 Q_{2}^{2} Q_{3}^{6} Q_{4}^{2} -711096726150 Q_{2}^3
Q_{3}^4 Q_{4}^3 +$$ $$182164306725 Q_{2}^4 Q_{3}^2 Q_{4}^4 -6289615575 Q_{2}^5 Q_{4}^5
+ 222467624400 Q_{2}^2 Q_{3}^7 Q_{5}-$$ $$746418445200 Q_{2}^3 Q_{3}^5 Q_{4} Q_{5} + 653423900400
Q_{2}^4 Q_{3}^3 Q_{4}^2 Q_{5} -124319674800
Q_{2}^5 Q_{3} Q_{4}^3 Q_{5}+$$ $$143980943040 Q_{2}^4 Q_{3}^4 Q_{5}^2 -169712521920
Q_{2}^5 Q_{3}^2 Q_{4} Q_{5}^2 +18188188416 Q_{2}^6 Q_{4}^2 Q_{5}^2+$$ $$11240861184 Q_{2}^6 Q_{3} Q_{5}^3-91198200240 Q_{2}^3 Q_{3}^6 Q_{6}
+241513732080 Q_{2}^4 Q_{3}^4 Q_{4} Q_{6}-$$ $$140030897040 Q_{2}^5 Q_{3}^2 Q_{4}^2 Q_{6}
+9200103120 Q_{2}^6 Q_{4}^3 Q_{6} -84218693760 Q_{2}^5 Q_{3}^3 Q_{5} Q_{6}+$$ $$55248386688 Q_{2}^6 Q_{3} Q_{4} Q_{5} Q_{6} -3173043456 Q_{2}^7 Q_{5}^2 Q_{6} + 10464952896
Q_{2}^6 Q_{3}^2 Q_{6}^2-$$ $$2403421632 Q_{2}^7 Q_{4} Q_{6}^2 +31637744640 Q_{2}^4 Q_{3}^5 Q_{7}
-62649953280 Q_{2}^5 Q_{3}^3 Q_{4} Q_{7}+$$ $$20409822720 Q_{2}^6 Q_{3} Q_{4}^2 Q_{7} + 18860532480 Q_{2}^6 Q_{3}^2 Q_{5} Q_{7}
-4693344768 Q_{2}^7 Q_{4} Q_{5} Q_{7}-$$ $$3625731072 Q_{2}^7 Q_{3} Q_{6} Q_{7} + 188054784 Q_{2}^8 Q_{7}^2
-9155635200 Q_{2}^5 Q_{3}^4 Q_{8}+$$ $$12238024320 Q_{2}^6 Q_{3}^2 Q_{4} Q_{8} -1405278720 Q_{2}^7 Q_{4}^2 Q_{8}-
2866700160 Q_{2}^7 Q_{3} Q_{5} Q_{8}+$$ $$303295104 Q_{2}^8 Q_{6} Q_{8} +2210705280 Q_{2}^6 Q_{3}^3 Q_{9}
-1685525760 Q_{2}^7 Q_{3} Q_{4} Q_{9} +235488384 Q_{2}^8 Q_{5} Q_{9})-$$
$$\frac{(n+1/2)^4 i}{20065812480 Q_{2}^{12}\sqrt{2 Q_{2}}}
(-66528 Q_{12} Q_{2}^9+1245888
Q_{11} Q_{2}^8 Q_{3}-$$ $$11158560 Q_{10} Q_{2}^7 Q_{3}^2 +
4668804525 Q_{3}^{10} +2116800 Q_{10} Q_{2}^8 Q_{4}-$$ $$25898331375 Q_{2} Q_{3}^8 Q_{4}+
47959232650 Q_{2}^{2} Q_{3}^{6} Q_{4}^{2} -33861927750 Q_{2}^3
Q_{3}^4 Q_{4}^3 +$$ $$7454763225 Q_{2}^4 Q_{3}^2 Q_{4}^4 -184988475 Q_{2}^5 Q_{4}^5
+ 11891917800 Q_{2}^2 Q_{3}^7 Q_{5}-$$ $$36105463800 Q_{2}^3 Q_{3}^5 Q_{4} Q_{5} + 27953667000
Q_{2}^4 Q_{3}^3 Q_{4}^2 Q_{5} -4457716200
Q_{2}^5 Q_{3} Q_{4}^3 Q_{5}+$$ $$6285855240 Q_{2}^4 Q_{3}^4 Q_{5}^2 -6471756144
Q_{2}^5 Q_{3}^2 Q_{4} Q_{5}^2 +565259688 Q_{2}^6 Q_{4}^2 Q_{5}^2+$$ $$380939328 Q_{2}^6 Q_{3} Q_{5}^3-4375251160 Q_{2}^3 Q_{3}^6 Q_{6}
+10317018600 Q_{2}^4 Q_{3}^4 Q_{4} Q_{6}-$$ $$5113813320 Q_{2}^5 Q_{3}^2 Q_{4}^2 Q_{6}
+238888440 Q_{2}^6 Q_{4}^3 Q_{6} -3203871552 Q_{2}^5 Q_{3}^3 Q_{5} Q_{6}+$$ $$1758685824 Q_{2}^6 Q_{3} Q_{4} Q_{5} Q_{6} -88566912 Q_{2}^7 Q_{5}^2 Q_{6} + 335466432
Q_{2}^6 Q_{3}^2 Q_{6}^2-$$ $$55073088 Q_{2}^7 Q_{4} Q_{6}^2 +1351294560 Q_{2}^4 Q_{3}^5 Q_{7}
-2341442880 Q_{2}^5 Q_{3}^3 Q_{4} Q_{7}+$$ $$626542560 Q_{2}^6 Q_{3} Q_{4}^2 Q_{7} + 619520832 Q_{2}^6 Q_{3}^2 Q_{5} Q_{7}
-123524352 Q_{2}^7 Q_{4} Q_{5} Q_{7}-$$ $$96574464 Q_{2}^7 Q_{3} Q_{6} Q_{7} + 4048704 Q_{2}^8 Q_{7}^2
-341160120 Q_{2}^5 Q_{3}^4 Q_{8}+$$ $$386210160 Q_{2}^6 Q_{3}^2 Q_{4} Q_{8} -30837240 Q_{2}^7 Q_{4}^2 Q_{8}-
78073632 Q_{2}^7 Q_{3} Q_{5} Q_{8}+$$ $$5848416 Q_{2}^8 Q_{6} Q_{8} +70415520 Q_{2}^6 Q_{3}^3 Q_{9}
-43424640 Q_{2}^7 Q_{3} Q_{4} Q_{9} +5255712 Q_{2}^8 Q_{5} Q_{9})-$$
$$\frac{(n+1/2)^6 i}{300987187200Q_{2}^{12}\sqrt{2 Q_{2}}}
(-72576 Q_{12} Q_{2}^9+1886976
Q_{11} Q_{2}^8 Q_{3}-$$ $$22135680 Q_{10} Q_{2}^7 Q_{3}^2 +
27463538375 Q_{3}^{10} +2903040 Q_{10} Q_{2}^8 Q_{4}-$$ $$141448688325 Q_{2} Q_{3}^8 Q_{4}+
240655765350 Q_{2}^{2} Q_{3}^{6} Q_{4}^{2} -152907158250 Q_{2}^3
Q_{3}^4 Q_{4}^3 +$$ $$28724479875 Q_{2}^4 Q_{3}^2 Q_{4}^4 -413669025 Q_{2}^5 Q_{4}^5
+59058073200 Q_{2}^2 Q_{3}^7 Q_{5}-$$ $$164264209200 Q_{2}^3 Q_{3}^5 Q_{4} Q_{5} +113654696400
Q_{2}^4 Q_{3}^3 Q_{4}^2 Q_{5} -15166342800
Q_{2}^5 Q_{3} Q_{4}^3 Q_{5}+$$ $$26061194880 Q_{2}^4 Q_{3}^4 Q_{5}^2 -23876233920
Q_{2}^5 Q_{3}^2 Q_{4} Q_{5}^2 +1767189312 Q_{2}^6 Q_{4}^2 Q_{5}^2+$$ $$1292433408 Q_{2}^6 Q_{3} Q_{5}^3-18902165520 Q_{2}^3 Q_{3}^6 Q_{6}
+40256773200 Q_{2}^4 Q_{3}^4 Q_{4} Q_{6}-$$ $$17116974000 Q_{2}^5 Q_{3}^2 Q_{4}^2 Q_{6}
+483582960 Q_{2}^6 Q_{4}^3 Q_{6} -11384150400 Q_{2}^5 Q_{3}^3 Q_{5} Q_{6}+$$ $$5285056896 Q_{2}^6 Q_{3} Q_{4} Q_{5} Q_{6} -246903552 Q_{2}^7 Q_{5}^2 Q_{6} + 992779200
Q_{2}^6 Q_{3}^2 Q_{6}^2-$$ $$101860416 Q_{2}^7 Q_{4} Q_{6}^2 +4966859520 Q_{2}^4 Q_{3}^5 Q_{7}
-7661606400 Q_{2}^5 Q_{3}^3 Q_{4} Q_{7}+$$ $$1683037440 Q_{2}^6 Q_{3} Q_{4}^2 Q_{7} +1861574400 Q_{2}^6 Q_{3}^2 Q_{5} Q_{7}
-316141056 Q_{2}^7 Q_{4} Q_{5} Q_{7}-$$ $$235146240 Q_{2}^7 Q_{3} Q_{6} Q_{7} +8895744 Q_{2}^8 Q_{7}^2
-1042372800 Q_{2}^5 Q_{3}^4 Q_{8}+$$ $$1016789760 Q_{2}^6 Q_{3}^2 Q_{4} Q_{8} -52436160 Q_{2}^7 Q_{4}^2 Q_{8}-
189060480 Q_{2}^7 Q_{3} Q_{5} Q_{8}+$$ $$\label{77}
9217152 Q_{2}^8 Q_{6} Q_{8} +175190400 Q_{2}^6 Q_{3}^3 Q_{9}
-87816960 Q_{2}^7 Q_{3} Q_{4} Q_{9} +10378368 Q_{2}^8 Q_{5} Q_{9})$$
All six WKB corrections printed in MATEMATICA are available from the author in electronic form upon request.
Appendix 2: QNMs of a 4-dimensional Schwarzshild black hole
===========================================================
“The potential” $Q(x)$ in case of a Schwarzshild black hole has the form $$\label{3}
Q(x)= \omega^2-\left(1-\frac{1}{r}\right)\left(\frac{l(l+1)}{r^2}
+\frac{1-s^2}{r^3}\right),$$ where $s=0$ corresponds to scalar perturbations, $s=1/2$ - neutrino perturbations, $s=1$ - electromagnetic perturbations, $s=3/2$ - gravitino perturbations, $s=2$ - gravitational perturbations. The quasinormal frequencies at 3th and 6th WKB orders and in comparison with numerical results [@Leaver] are presented in the table I.
$s=0$ numerical 3th order WKB 6th order WKB
-------------- -------------------- --------------------- ----------------------
$l=0$, $n=0$ $0.1105- 0.1049 i$ $0.1046-0.1152 i$ $0.1105 - 0.1008 i$
$l=1$, $n=0$ $0.2929- 0.0977 i$ $0.2911- 0.0980 i$ $0.2929- 0.0978 i$
$l=1$, $n=1$ $0.2645- 0.3063 i$ $0.2622- 0.3074 i$ $0.2645- 0.3065 i$
$l=2$, $n=0$ $0.4836 -0.0968 i$ $0.4832 -0.0968 i$ $0.4836 -0.0968 i$
$l=2$, $n=1$ $0.4639 -0.2956 i$ $0.4632 -0.2958 i$ $0.4638 -0.2956 i$
$l=2$, $n=2$ $0.4305 -0.5086 i$ $0.4317 -0.5034 i$ $0.4304 -0.5087 i$
$s=1/2$ numerical 3th order WKB 6th order WKB
$l=1$, $n=0$ $-$ $0.2803 - 0.0969 i$ $0.2822 - 0.0967 i$
$l=1$, $n=1$ $-$ $0.2500 - 0.3049 i$ $0.2525 - 0.3040 i$
$l=2$, $n=0$ $-$ $0.4768 - 0.9639 i$ $0.4772- 0.0963 i$
$l=2$, $n=1$ $-$ $0.4565 - 0.2947 i$ $0.4571 -0.2945 i$
$l=2$, $n=2$ $-$ $0.4244 - 0.5016 i$ $0.4231 -0.5070 i$
$l=3$, $n=0$ $-$ $0.6706 - 0.0963 i$ $0.6708 - 0.0963 i$
$l=3$, $n=1$ $-$ $0.6557 - 0.2917 i$ $0.6560 - 0.2917 i$
$l=3$, $n=2$ $-$ $0.6299 - 0.4931 i$ $0.6286 - 0.4950 i$
$l=3$, $n=3$ $-$ $0.5970 - 0.6997 i$ $0.5932 - 0.7102 i$
$s=1$ numerical 3th order WKB 6th order WKB
$l=1$, $n=0$ $0.2483-0.0925 i$ $0.2459-0.0931 i$ $0.2482-0.0926 i$
$l=1$, $n=1$ $0.2145-0.2937 i$ $0.2113-0.2958 i$ $0.2143-0.2941 i$
$l=2$, $n=0$ $0.4576-0.0950 i$ $0.4571-0.0951 i$ $0.4576-0.0950 i$
$l=2$, $n=1$ $0.4365-0.2907 i$ $0.4358-0.2910 i$ $0.4365-0.2907 i$
$l=2$, $n=2$ $0.4012-0.5016 i$ $0.4023-0.4959 i$ $0.4009-0.5017 i$
$l=3$, $n=0$ $0.6569-0.0956 i$ $0.6567-0.0956 i$ $0.6569-0.0956 i$
$l=3$, $n=1$ $0.6417-0.2897 i$ $0.6415-0.2898 i$ $0.6417-0.2897 i$
$l=3$, $n=2$ $0.6138-0.4921 i$ $0.6151-0.4901 i$ $0.6138-0.4921 i$
$l=3$, $n=3$ $0.5779-0.7063 i$ $0.5814-0.6955 i$ $0.5775-0.7065 i$
$s=3/2$ numerical 3th order WKB 6th order WKB
$l=1$, $n=0$ $-$ $0.1817 - 0.0866 i$ $0.1739 - 0.08357 i$
$l=1$, $n=1$ $-$ $0.1354 - 0.2812 i$ $0.1198 - 0.2813 i$
$l=2$, $n=0$ $-$ $0.4231 - 0.926 i$ $0.4236- 0.0925 i$
$l=2$, $n=1$ $-$ $0.4000 - 0.2842 i$ $0.4007 -0.2838 i$
$l=2$, $n=2$ $-$ $0.3636 - 0.4853 i$ $0.3618 -0.4919 i$
$l=3$, $n=0$ $-$ $0.6332 - 0.0945 i$ $0.6333 - 0.0944 i$
$l=3$, $n=1$ $-$ $0.6173 - 0.2864 i$ $0.6175 - 0.2863 i$
$l=3$, $n=2$ $-$ $0.5898 - 0.4846 i$ $0.5884 - 0.4868 i$
$l=3$, $n=3$ $-$ $0.5547 - 0.6882 i$ $0.5505 - 0.7000 i$
$s=2$ numerical 3th order WKB 6th order WKB
$l=2$, $n=0$ $0.3737-0.0890 i$ $0.3732-0.0892 i$ $0.3736-0.0890 i$
$l=2$, $n=1$ $0.3467-0.2739 i$ $0.3460-0.2749 i$ $0.3463-0.2735 i$
$l=2$, $n=2$ $0.3011-0.4783 i$ $0.3029-0.4711 i$ $0.2985-0.4776 i$
$l=3$, $n=0$ $0.5994-0.0927 i$ $0.5993-0.0927 i$ $0.5994-0.0927 i$
$l=3$, $n=1$ $0.5826-0.2813 i$ $0.5824-0.2814 i$ $0.5826-0.2813 i$
$l=3$, $n=2$ $0.5517-0.4791 i$ $0.5532-0.4767 i$ $0.5516-0.4790 i$
$l=3$, $n=3$ $0.5120-0.6903 i$ $0.5157-0.6774 i$ $0.5111-0.6905 i$
$l=4$, $n=0$ $0.8092-0.0942 i$ $0.8091-0.0942 i$ $0.8092-0.0942 i$
$l=4$, $n=1$ $0.7966-0.2843 i$ $0.7965-0.2844 i$ $0.7966-0.2843 i$
$l=4$, $n=2$ $0.7727-0.4799 i$ $0.7736-0.4790 i$ $0.7727-0.4799 i$
$l=4$, $n=3$ $0.7398-0.6839 i$ $0.7433-0.6783 i$ $0.7397-0.6839 i$
$l=4$, $n=4$ $0.7015-0.8982 i$ $0.7072-0.8813 i$ $0.7006-0.8985 i$
Table I. Schwarzshild QN frequencies for perturbations of different spin.
[99]{} S.Dimopoulos and S.Landsberg, Phys. Rev. Lett. [**[87]{}**]{} 161602 (2002) V.P.Frolov and D.Stojkovic, gr-qc 0301016 V.Cardoso, O.J.C. Dias, and J.P.S. Lemos, hep-th 0212168 G.T.Horowitz and V.Hubeny [*Phys.Rev.*]{} [**[D62]{}**]{} 024027 (2000) J.S.F.Chan and R.B.Mann [*Phys.Rev.*]{} [**[D59]{}**]{} 064025 (1999) V.Cardoso and J.P.S.Lemos [*Phys.Rev.*]{} [**[D63]{}**]{} 124015 (2001) D. Birmingham, I. Sachs, and S.N. Solodukhin, Phys. Rev. Lett. [**[88]{}**]{} 151301 (2002) D. Birmingham, I. Sachs, and S.N. Solodukhin, hep-th/0212308 (2002) R.A.Konoplya, [*Phys.Rev.*]{} [**[D66]{}**]{} 084007 (2002) R.A.Konoplya, [*Phys.Rev.*]{} [**[D66]{}**]{} 044009 (2002) A. O. Starinets, [*Phys.Rev.*]{} [**[D66]{}**]{} 124013 (2002). D. T. Son, A. O. Starinets, J.H.E.P. [**0209**]{}:042, (2002) R.Aros, C.Martinez, R.Troncoso, and J.Zanelli, hep-th/0211024; M.Musiri and G.Siopsis, hep-th/0301081. I.G.Moss and J.P.Norman, Class.Quant.Grav. [**[19]{}**]{}, 2323 (2002) (2002) K.Kokkotas and B.Schmidt, “Quasi-normal modes of stars and black holes”, Living.Reviews.Relativ., [**[2]{}**]{}, 2 (1999); H. P. Nollert, Class. Quantum Grav. [**16**]{}, R159 (2000). O. Dreyer, gr-qc/0211076. G. Kunstatter, gr-qc/0211076. V.Cardoso and J.P.S.Lemos, gr-qc/0301078. L. Motl, gr-qc/0212096. W.T.Kim and J.J.Oh, [*Phys.Lett.*]{} [**[B514]{}**]{} 155 (2001) R.A.Konoplya, Phys.Lett. [**[B 550]{}**]{}, 117 (2002) L. Motl and A. Neitzke, hep-th/0301173; H-J.Blome and B.Mashhoon, Phys.Lett. [**[100A]{}**]{} 231 (1984) B.F.Schutz and C.M.Will [*Astrophys.J.Lett*]{} [**[291]{}**]{} L33 (1985) S.Iyer and C.M.Will, [*Phys.Rev.*]{} [**[D35]{}**]{} 3621 (1987) S.Iyer, [*Phys.Rev.*]{} [**[D35]{}**]{} 3632 (1987) K.Kokkotas, and B.F.Schutz, [*Phys.Rev.*]{} [**[D37]{}**]{} 3378 (1988) K.Kokkotas, [*Nuovo.Cimento.*]{} [**[B108]{}**]{} 991 (1993) L.E.Simone and C.M.Will, Class.Quant.Grav. [**[9]{}**]{}, 963 (1992) N.Andersson and H.Onozawa, [*Phys.Rev.*]{} [**[D54]{}**]{} 7470 (1996) V.Ferrari, M.Pauri and F.Piazza, [*Phys.Rev.*]{} [**[D63]{}**]{} 064009 (2001) H.Onozawa, T.Okamura, T.Mishima, and H.Ishihara, [*Phys.Rev.*]{} [**[D53]{}**]{} 7033 (1996) R.A.Konoplya, Gen.Relativ.Grav. [**[34]{}**]{}, 329 (2002) E.Leaver, Proc.R.Soc.London [**[A402]{}**]{}, 285 (1985) R.F.Stark and T.Piran, Phys. Rev. Lett. [**[55]{}**]{} 891 (1985) O.B.Zaslavskii, [*Phys.Rev.*]{} [**[D43]{}**]{} 605 (1991) V.Ferrari and B.Mashhoon, Phys. Rev. Lett. [**[52]{}**]{} 1361 (1984). W.H.Press, [*Astrophys.J*]{} [**[170]{}**]{} L105 (1971) N.Andersson, Class.Quant.Grav. [**[11]{}**]{}, 3003 (1994); Y.Decanini, A.Folacci and B.Jensen, gr-qc/0212093.
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---
abstract: 'If the $\ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is [@Ereg Theorem 1.1]. If the model ${{\mathcal X}}$ is regular, one has a congruence $|{{\mathcal X}}(k)|\equiv 1 $ modulo $|k|$ for the number of $k$-rational points ([@Ept Theorem 1.1]). The congruence is violated if one drops the regularity assumption.'
address:
- ' Universität Duisburg-Essen, Mathematik, 45117 Essen, Germany'
- 'Department of Mathematics, Princeton University, Princeton NJ 08544, USA'
author:
- Hélène Esnault
- Chenyang Xu
date: 'June 6, 2007'
title: Congruence for rational points over finite fields and coniveau over local fields
---
[^1]
Introduction
============
Let $X$ be a projective variety defined over a local field $K$ with finite residue field $k={{\mathbb F}}_q$. Let $R$ be the ring of integers of $K$. A [*model*]{} of $X/K$ is a flat projective morphism ${{\mathcal X}}\to {\rm Spec}(R)$, with ${{\mathcal X}}$ an integral scheme, such that tensored with $K$ over $R$, it is $X\to {\rm Spec}(K)$. As in [@Ept] and [@Ereg], we consider $\ell$-adic cohomology $H^i(\bar X)$ with ${{\mathbb Q}}_\ell$-coefficents. Recall briefly that one defines the first coniveau level [ $$\begin{gathered}
\label{}N^1H^i(\bar X)=\{\alpha \in H^i(\bar X), \exists \ {\rm divisor} \ D\subset X \ {\rm s.t.} \
0=\alpha|_{X\setminus D}\in H^i(\overline{X\setminus D})\}.\notag\end{gathered}$$ ]{} As $H^i(\bar X)$ is a finite dimensional ${{\mathbb Q}}_\ell$-vector space, one has by localization [ $$\begin{gathered}
\label{}\exists D \subset X \ {\rm s.t.} \ N^1H^i(\bar X)= {\rm Im}\big(H^i_{\bar D}(\bar X)\to H^i(\bar X)\big),\notag\end{gathered}$$ ]{} where $D\subset X$ is a divisor. One says that $H^i(\bar X)$ is [*supported in codimension 1*]{} if $N^1H^i(\bar X)=H^i(\bar X)$. The purpose of this note is twofold. We show the following theorem.
\[thm1.1\] Let $X$ be a smooth, projective, absolutely irreducible variety defined over a local field $K$ with finite residue field $k$. Assume that $\ell$-adic cohomology $H^i(\bar X)$ is supported in codimension $\ge 1$ for all $i\ge 1$. Let ${{\mathcal X}}$ be a model of $X$ over the ring of integers $R$ of $K$. Then there is a projective surjective morphism $\sigma: {{\mathcal Y}}\to {{\mathcal X}}$ of $R$-schemes such that $$|{{\mathcal Y}}(k)|\equiv 1 \ {\rm mod} \ |k|.$$ In particular, any model ${{\mathcal X}}/R$ of $X/K$ has a $k$-rational point.
This generalizes [@Ereg Theorem 1.1] where the theorem is proven under the assumption that $K$ has characteristic 0. On the other hand, assuming that ${{\mathcal X}}$ is regular, we showed in [@Ept Theorem 1.1] that the number of $k$-rational points $|{{\mathcal X}}(k)|$ is congruent to $1$ modulo $|k|$. It was in fact the way to show that $k$-rational points exist on ${{\mathcal X}}$, as surely $|k|$, being a $p$-power, where $p$ is the characteristic of $k$, is $>1$. We show that if we drop the regularity assumption, there are models which, according to Theorem \[thm1.1\], have a rational point, but do not satisfy the congruence.
\[thm1.2\] Let $X_0={{\mathbb P}}^2$ over $K_0:={{\mathbb Q}}_p$ or ${{\mathbb F}}_p((t))$. Then there is a finite field extension $K\supset K_0$, which can be chosen to be unramified, and there is a normal model ${{\mathcal X}}/R$ of $X:=X_0\otimes_{K_0} K$, such that $|{{\mathcal X}}(k)|$ is not congruent to $1$ modulo $|k|$.
The proof of Theorem \[thm1.1\] follows closely the one in unequal characteristic in [@Ereg Theorem 1.1], and, aside of Deligne’s integrality theorem [@DeInt Corollaire 5.5.3] and [@Ept Appendix] and purity [@Fu], relies strongly on de Jong’s alteration theorem as expressed in [@deJ2]. However, we have to replace the trace argument we used there by a more careful analysis of the Leray spectral sequence stemming from de Jong’s construction. The construction of the examples in Theorem \[thm1.2\] uses Artin’s contraction theorem as expressed in [@artin] and is somewhat inspired by Kollár’s construction exposed in [@BE Section 3.3].\
\
[*Acknowledgements:*]{} We thank Johan de Jong for his interest.
Proof of Theorem \[thm1.1\]
===========================
This section is devoted to the proof of Theorem \[thm1.1\].
Let $K$ be a local field with finite residue field $k$. Let $R\subset K$ be its valuation ring. Let ${{\mathcal X}}\to {{\rm Spec \,}}R$ be a model of a projective variety $X\to {{\rm Spec \,}}K$. We do not assume here that $X$ is absolutely irreducible, nor do we assume that $X/K$ is smooth. Then by [@deJ2 Corollary 5.15], there is a diagram [ $$\begin{gathered}
\label{2.1}\xymatrix{\ar[drr] {{\mathcal Z}}\ar[r]^{\pi} & \ar[dr] {{\mathcal Y}}\ar[r]^{\sigma} & {{\mathcal X}}\ar[d]\\
& & {{\rm Spec \,}}R
}\end{gathered}$$ ]{} and a finite group $G$ acting on ${{\mathcal Z}}$ over ${{\mathcal Y}}$ with the properties
- ${{\mathcal Z}}\to {{\rm Spec \,}}R$ and ${{\mathcal Y}}\to {{\rm Spec \,}}R$ are flat,
- $\sigma$ is projective, surjective, $K({{\mathcal X}})\subset K({{\mathcal Y}})$ is a purely inseparable field extension,
- ${{\mathcal Y}}$ is the quotient of ${{\mathcal Z}}$ by $G$,
- ${{\mathcal Z}}$ is regular.
We want to show that this ${{\mathcal Y}}$ does it. Let us set $$Y={{\mathcal Y}}\otimes K, \ Z={{\mathcal Z}}\otimes K.$$ The only difference with [@Ereg (2.1)] is that $K({{\mathcal X}})\subset K({{\mathcal Y}})$ may be a purely inseparable extension rather than an isomorphism. Thus, the argument there breaks down as one does not have traces as in [@Ereg (2.3), (2.4)]. We do not have [@Ereg (2.5)] a priori, and we can’t conclude [@Ereg Claim 2.1].
Let us overtake the notations of [*loc. cit.*]{}: we endow all schemes considered (which are $R$-schemes) with the upper subscript $^u$ to indicate the base change $\otimes_R R^u$ or $\otimes_K K^u$, where $K^u\supset K$ is the maximal unramified extension, and $R^u\supset R$ is the normalization of $R$ in $K^u$. Likewise, we write $\overline{?}$ to indicate the base change $\otimes_R \bar R, \ \otimes_K \bar K, \ \otimes_k \bar k$, where $\bar K\supset K, \ \bar k\supset k$ are the algebraic closures and $\bar R\supset R$ is the normalization of $R$ in $\bar K$. We consider as in [@Ept (2.1)] the $F$-equivariant exact sequence ([@DeWeII 3.6(6)]) [ $$\begin{gathered}
\label{2.2}\ldots \to H^i_{\bar B}({{\mathcal Y}}^u)\xrightarrow{\iota} H^i(\bar B)=H^i({{\mathcal Y}}^u)\xrightarrow{sp^u} H^i(Y^u) \to \ldots, \end{gathered}$$ ]{} where $F \in {\rm Gal}(\bar k/k)$ is the geometric Frobenius, and $B={{\mathcal Y}}\otimes k$. We have [@Ereg Claim 2.2] unchanged:
\[claim2.1\] The eigenvalues of the geometric Frobenius $F\in {\rm Gal}(\bar k/k)$ acting on $\iota(H^i_{\bar B}({{\mathcal Y}}^u))\subset H^i(\bar B)$ lie in $q\cdot \bar{{{\mathbb Z}}}$ for all $i\ge 1$.
So the problem is to show that the eigenvalues of $F$ acting on Im$(sp^u)\subset H^i(Y^u)$ lie in $q\cdot \bar{{{\mathbb Z}}}$ as well. Let us decompose the morphism $\sigma$ as [ $$\begin{gathered}
\label{2.3}\sigma: Y\xrightarrow{\tau} X_1\xrightarrow{\epsilon} X\end{gathered}$$ ]{} where $X_1$ is the normalization of $X$ in $K(Y)$. Thus in particular, $\tau$ is birational, $\epsilon$ is finite and purely inseparable. Let us denote by $U\subset X$ a non-empty open such that $\tau|_{\epsilon^{-1}(U)}$ is an isomorphism, and let us set $D:= X\setminus U$. We define [ $$\begin{gathered}
\label{2.4}{{\mathcal C}}:={\rm cone}({{\mathbb Q}}_\ell \to R\tau_*{{\mathbb Q}}_\ell)[-1]\end{gathered}$$ ]{} as an object in the bounded derived category of ${{\mathbb Q}}_\ell$-constructible sheaves on $X_1$. Since $\tau_*{{\mathbb Q}}_\ell={{\mathbb Q}}_\ell$, the cohomology sheaves of ${{\mathcal C}}$ are in degree $\ge 1$, and have support in $D_1:=D\times_X X_1$. We conclude [ $$\begin{gathered}
\label{2.5}H^i_{D_1^u}(X_1^u, {{\mathcal C}})=H^i(X_1^u, {{\mathcal C}}) \ \forall i\ge 0.\end{gathered}$$ ]{} One has the commutative diagram of exact sequences [ $$\begin{gathered}
\label{2.6}\xymatrix{ H^{i+1}_{ D^u_1}( X_1^u) \\
\ar[u] H^i_{{ D_1^u}}( X_1^u, {{\mathcal C}}) \ar[r]^{= \eqref{2.5}} &
H^i(X_1^u, {{\mathcal C}}) \\
\ar[u] H^i_{E^u}( Y^u) \ar[r] & \ar[u] H^i(Y^u) \\
\ar[u] H^i_{D_1^u}(X_1^u) \ar[r] & \ar[u] H^i(X_1^u)
}\end{gathered}$$ ]{} where $E= \sigma^{-1}(D)$. By [@Ept Theorem 1.5 and Appendix] the eigenvalues of $F$ on $H^i( X^u)=H^i(X_1^u)$ and on $H^{i+1}_{ D^u_1}( X_1^u)=H^{i+1}_{D^u}(X^u)$ lie in $q\cdot \bar{{{\mathbb Z}}}$. For the latter cohomology, one has to argue again by purity on $X^u$ before applying [*loc. cit.*]{}: by purity one is reduced to considering cohomology of the type $H^a(\Sigma^u)(-1)$ for a regular scheme $\Sigma$ over $K$ and $a\ge 0$. It remains to consider the eigenvalues of $F$ on $H^i_{E^u}( Y^u) =H^i_{L^u}(Z^u)^G$ where $L=D\times_X Z$. This is again the argument by purity and then [*loc. cit.*]{} So we conclude
\[claim2.2\] The eigenvalues of the geometric Frobenius $F\in {\rm Gal}(\bar k/k)$ acting on $H^i(Y^u)$, and therefore on ${\rm Im}(sp^u)\subset H^i(Y^u)$, lie in $q\cdot \bar{{{\mathbb Z}}}$ for all $i\ge 1$.
So we conclude now as usual that all the eigenvalues of $F$ acting on $H^i(\bar B)$ lie in $q\cdot \bar{{{\mathbb Z}}}$ for $i\ge 1$, thus the Grothendieck-Lefschetz trace formula applied to $H^*(\bar B)$, together with the absolute irreducibility of $B$, imply the congruence. This finishes the proof of Theorem \[thm1.1\].
Construction of examples
========================
This section is devoted to the proof of Theorem \[thm1.2\].
Let us first recall that if $E$ is a smooth genus 1 curve over a finite field ${{\mathbb F}}_q$, it is always an elliptic curve, which means that it always carries a ${{\mathbb F}}_q$-rational point. Furthermore one has
\[claim3.1\] Given an elliptic curve $E/{{\mathbb F}}_q$, there is a finite field extension ${{\mathbb F}}_{q^n}\supset {{\mathbb F}}_q$ such that $|E({{\mathbb F}}_{q^n})|$ is not congruent to $1$ modulo $q^n$.
By the trace formula, $|E({{\mathbb F}}_{q^n})|$ being congruent to $1$ modulo $q^n$ for all $n\ge 1$ is equivalent to saying that the eigenvalues of $F^n$ acting on $H^i(\bar E)$ lie in $q^n\cdot \bar{{{\mathbb Z}}}$ for all $n\ge 1$ and $i\ge 1$. By purity (which in dimension 1 is Weil’s theorem), this is equivalent to saying that the eigenvalues of $F^n$ acting on $H^1(\bar E)$ lie in $q^n\cdot \bar{{{\mathbb Z}}}$ for all $n\ge 1$. On the other hand, by duality, if $\lambda$ is an eigenvalue, then $\frac{q^n}{\lambda}$ is also an eigenvalue. This is then impossible that both $\lambda$ and $\frac{q^n}{\lambda}$ be $q^n$-divisible as algebraic integers.
We now construct the following scheme. Let us set ${{\mathcal P}}_0:={{\mathbb P}}^2$ over $R_0:={{\mathbb Z}}_p$ or over ${{\mathbb F}}_p[[t]]$. Choose an elliptic curve $E_0 \subset {{\mathcal P}}\otimes {{\mathbb F}}_p={{\mathbb P}}^2_{{{\mathbb F}}_p}$ defined over ${{\mathbb F}}_p$. Let $k\supset {{\mathbb F}}_p$ be a finite field extension such that $|E_0(k)|$ is not $k$-divisible (Claim \[claim3.1\]). Set $E:=E_0\otimes_{{{\mathbb F}}_p} k, \ {{\mathcal P}}:={{\mathcal P}}_0\otimes_{R_0} R$, with $R=W(k)$ or ${{\mathbb F}}_q[[t]]$, and $K={\rm Frac}(R)$. Choose a smooth projective curve ${{\mathcal C}}\subset {{\mathcal P}}$ over $R$, of degree $\ge 4$, such that $C:={{\mathcal C}}\otimes k$ is transversal to $E$. Define $\Sigma=E\cap C \subset E$ to be the 0-dimensional intersection subscheme. It has degree $\ge 12$, thus in particular $>9$. Let $b: {{\mathcal Y}}\to {{\mathcal P}}$ be the blow up of $\Sigma \subset {{\mathcal P}}$. We denote by $P_\Sigma$ the exceptional locus, which is a trivial ${{\mathbb P}}^2$ bundle over $\Sigma$, by $Y$ the strict transform of ${{\mathbb P}}^2_{k}$, and we still denote by $E\subset Y$ the strict transform of the elliptic curve. Then the conormal bundle $N^\vee_{E/{{\mathcal Y}}}$ of $E$ in ${{\mathcal Y}}$ is an extension of the conormal bundle $N^\vee_{E/Y}$ of $E$ in $Y$ by the restriction to $E$ of the conormal bundle $N^\vee_{Y/{{\mathcal Y}}}$ of $Y$ in ${{\mathcal Y}}$, both ample line bundles on $E$ by the condition on the degree of $\Sigma$.
Let $I\subset {{\mathcal O}}_{{{\mathcal Y}}}$ be the ideal sheaf of $E$. For a coherent sheaf ${{\mathcal F}}$ on ${{\mathcal Y}}$, we denote by $I^n/I^{n+1}\cdot {{\mathcal F}}$ the image of $I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal F}}$ in ${{\mathcal F}}$, where $n\in {{\mathbb N}}$.
\[claim3.2\] For every coherent sheaf ${{\mathcal F}}$ on ${{\mathcal Y}}$, one has $H^1(E, I^n/I^{n+1}\cdot {{\mathcal F}})=0$ for all $n\in {{\mathbb N}}$ large enough.
As by definition one has a surjection $I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal F}}{\twoheadrightarrow}I^n/I^{n+1}\cdot {{\mathcal F}}$, it is enough to show $H^1(E, I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal F}})=0$ for $n$ large enough. As $I^n/I^{n+1}$ is locally free, $I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal F}}$ is an extension of $I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal F}}_0$ by $I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal T}}$, where ${{\mathcal T}}\subset {{\mathcal F}}$ is the maximal torsion subsheaf and ${{\mathcal F}}_0={{\mathcal F}}/{{\mathcal T}}$ is locally free. As $H^1(E, I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal T}})=0$, we may assume that ${{\mathcal F}}$ is locally free. As $I^n/I^{n+1}$ is a locally free filtered sheaf, with associated graded a sum of ample line bundles of strictly increasing degree as $n$ grows, we have $H^1(E, {\rm gr}(I^n/I^{n+1})\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal F}})=0$ for $n$ large enough, and thus $H^1(E, I^n/I^{n+1}\otimes_{{{\mathcal O}}_{{{\mathcal Y}}}} {{\mathcal F}})=0$ as well.
Artin’s contraction criterion [@artin Theorem 6.2] applied to $E\to {\rm Spec}(k)$, together with Artin’s existence theorem [@artin Theorem 3.1] show the existence of a contraction [ $$\begin{gathered}
\label{3.1}a_1: {{\mathcal Y}}\to {{\mathcal X}}_1\end{gathered}$$ ]{} where ${{\mathcal X}}_1$ is an algebraic space over $R$, $a_1|_{{{\mathcal Y}}\setminus E}$ is an isomorphism and $a_1(E)={\rm Spec}(k)$. Let ${{\mathcal X}}\xrightarrow{\nu} {{\mathcal X}}_1$ be the normalization of ${{\mathcal X}}_1$ in $K({{\mathcal Y}})=K({{\mathcal P}})$. This is a normal algebraic space over $R$. One has a diagram [ $$\begin{gathered}
\label{3.2}\xymatrix{\ar[d]_b {{\mathcal Y}}\ar@/^1pc/[rr]^{a_1} \ar[r]_a & {{\mathcal X}}\ar[r]_\nu & {{\mathcal X}}_1\\
{{\mathcal P}}}\end{gathered}$$ ]{}
\[claim3.3\] $|{{\mathcal X}}(k)|$ is not congruent to $1$ modulo $|k|$.
By [@Ept Theorem 1.1] (or by a simple computation in this case), $|{{\mathcal Y}}(k)|$ is congruent to $1$ modulo $|k|$. By Claim \[claim3.1\] and the choice of $E$, $|{{\mathcal X}}_1(k)|$ is not congruent to $1$ modulo $|k|$. On the other hand, as the fibers of $a_1$ are absolutely irreducible, $\nu$ has to be a homeomorphism. Thus $|{{\mathcal X}}(k)|=|{{\mathcal X}}_1(k)|$. This finishes the proof.
In order to finish the proof of Theorem \[thm1.2\], it remains to show
\[claim3.4\] ${{\mathcal X}}\to {\rm Spec}(R)$ is a model of $X={{\mathbb P}}^2/K$.
We have to show that ${{\mathcal X}}\to {\rm Spec}(R)$ is a flat projective morphism. Since ${{\mathcal X}}$ is reduced, ${\rm Spec}(R)$ is regular of dimension 1, then [@EGA IV Proposition 14.3.8] allows to conclude that ${{\mathcal X}}/R$ is flat. Thus we just have to show that ${{\mathcal X}}/R$ is projective. To this aim, we want to decend a line bundle from ${{\mathcal Y}}$ to ${{\mathcal X}}$. Let us define the line bundle ${{\mathcal M}}:=b^*{{\mathcal O}}_{{{\mathcal P}}}({{\mathcal C}})(-P_\Sigma)$ on ${{\mathcal Y}}$. By definition, one has [ $$\begin{gathered}
\label{3.3} {{\mathcal M}}|_E\cong {{\mathcal O}}_E.\end{gathered}$$ ]{}
\[claim3.5\] The line bundle ${{\mathcal M}}$ descends to ${{\mathcal X}}$, that is there is a line bundle ${{\mathcal L}}$ on ${{\mathcal X}}$ with $a^*{{\mathcal L}}={{\mathcal M}}$.
The proper morphism of algebraic spaces $a: {{\mathcal Y}}\to {{\mathcal X}}$, with $a_*{{\mathcal O}}_{{{\mathcal Y}}}={{\mathcal O}}_{{{\mathcal X}}}$, has the property that $a^{-1}a(E)=E$ set-theoritically, that $a|_{{{\mathcal Y}}\setminus E}: {{\mathcal Y}}\setminus E\to {{\mathcal X}}\setminus a(E)$ is an isomorphism, and that $H^1(E, I^n/I^{n+1})=0$ for $n\ge 1$. So Keel’s theorem [@keel Lemma 1.10] asserts that some positive power ${{\mathcal M}}^{\otimes r}$ descends to ${{\mathcal X}}$ if the following condition is fulfilled [ $$\begin{gathered}
\label{3.4}\forall m>0, \exists r(m)>0 \ {\rm s.t} \ {{\mathcal M}}^{\otimes r(m)}|_{E_m} \ {\rm descends \ to} \
a(E_m) \\ {\rm where} \ E_m:={\rm Spec}({{\mathcal O}}_{{{\mathcal Y}}}/I^{m+1}). \notag\end{gathered}$$ ]{} So we just have to check that is fulfilled with $r=1$ in our situation. The scheme $a(E_m)$ has Krull dimension $0$. Thus by Hilbert 90’s theorem (see e.g. [@Milne Corollary 11.6]) one has [ $$\begin{gathered}
\label{3.5}{\rm Pic}(a(E_m))=0.\end{gathered}$$ ]{} We conclude that to check is equivalent to checking that ${{\mathcal M}}^{\otimes r(m)}|_{E_m}\cong {{\mathcal O}}_{E_m}$ for some positive power $r(m)$. In fact one has [ $$\begin{gathered}
\label{3.6}{{\mathcal M}}|_{E_m}\cong {{\mathcal O}}_{E_m} \ \forall m\ge 1.\end{gathered}$$ ]{} For $m=1$, this is . We argue by induction and assume that for $m>1$, we have a trivializing section $s_m: {{\mathcal O}}_{E_m}\xrightarrow{\cong} {{\mathcal M}}|_{E_m}$. We want to show that it lifts to a trivializing section $s_{m+1}: {{\mathcal O}}_{E_{m+1}}\xrightarrow{\cong} {{\mathcal M}}|_{E_{m+1}}$. One has an exact sequence [ $$\begin{gathered}
\label{3.7}0\to I^{m+1}/I^{m+2}\to {{\mathcal M}}|_{E_{m+1}}\to {{\mathcal M}}|_{E_m}\to 0.\end{gathered}$$ ]{} Since $H^1(E, I^{m+1}/I^{m+2})=0$, as $m\ge 0$, the trivializing section of $s_m: {{\mathcal O}}_{E_m}\xrightarrow{\cong} {{\mathcal M}}|_{E_m}$ lifts to a section $s_{m+1}: {{\mathcal O}}_{E_{m+1}}\to {{\mathcal M}}|_{E_{m+1}}$, and likewise, its inverse $t_m: {{\mathcal M}}|_{E_m}\xrightarrow{\cong} {{\mathcal O}}_{E_m}$ lifts to $t_{m+1}: {{\mathcal M}}|_{E_{m+1}} \to {{\mathcal O}}_{E_{m+1}}$. The composite $t_{m+1}\circ s_{m+1}: {{\mathcal O}}_{E_{m+1}} \to {{\mathcal O}}_{E_{m+1}}$ lifts the identity of ${{\mathcal O}}_{E_m}$. Therefore it is invertible. This shows that $s_{m+1}$ trivializes. The proof of Keel’s theorem (see (2) after [@keel (1.10.1)]) shows then that one can take $r=1$.
In order the finish the proof of Claim \[claim3.4\], it remains to see that ${{\mathcal L}}$ on ${{\mathcal X}}$ is ample. First, ${{\mathcal L}}|_{{{\mathcal X}}\otimes k}$ is ample because by [@keel Corollary 0.3], this is enough to see that the linear system associated to ${{\mathcal L}}|_{{{\mathcal X}}\otimes k}$ does not contract any curve, which is true by construction. So by Serre vanishing theorem, for sufficiently large $m$, $H^1({{\mathcal X}}\otimes k, {{\mathcal L}}|_{{{\mathcal X}}\otimes k}^{\otimes m})=0$. Base change implies $H^1({{\mathcal X}},{{\mathcal L}}^{\otimes m})\otimes k=0$ ([@EGA III Theorem 7.7.5]), thus by Nakayama’s lemma, one has [ $$\begin{gathered}
\label{3.8}H^1({{\mathcal X}},{{\mathcal L}}^{\otimes m})=0 \ {\rm for \ } m \ {\rm large \ enough}.\end{gathered}$$ ]{} As ${{\mathcal L}}$ is invertible, multiplication ${{\mathcal L}}^{\otimes m}\xrightarrow{\pi} {{\mathcal L}}^{\otimes m}$ by the uniformizer $\pi$ is injective, with quotient ${{\mathcal L}}|_{{{\mathcal X}}\otimes k}^{\otimes m}$. Thus implies surjectivity $H^0({{\mathcal X}}, {{\mathcal L}}^{\otimes m}){\twoheadrightarrow}H^0({{\mathcal X}}\otimes k, {{\mathcal L}}|_{{{\mathcal X}}\otimes k}^{\otimes m})$ for $m$ large enough. Thus $H^0({{\mathcal X}}, {{\mathcal L}}^{\otimes m})$ is a free $R$-module, and the linear system $H^0({{\mathcal X}}, {{\mathcal L}}^{\otimes m})$ maps base point freely ${{\mathcal X}}$ to ${{\mathbb P}}^N_R$, with $N+1={\rm rank}_R H^0({{\mathcal X}}, {{\mathcal L}}^{\otimes m})$. As it embedds ${{\mathcal X}}\otimes k$, it embedds ${{\mathcal X}}$ as well. This finishes the proof.
Dimension 1
===========
\[rmk4.1\] In Theorem \[thm1.1\], if $X/K$ has dimesnion 1, which means concretely if $X/K={{\mathbb P}}^1/K$, then any normal model ${{\mathcal X}}/R$ satisfies the congruence $|{{\mathcal X}}(k)| \equiv 1$ modulo $|k|$. Thus the examples of Theorem \[thm1.2\] have the smallest possible dimension.
Indeed, using , the only thing to check is that $H^1(\bar A)$, which is equal to $H^1({{\mathcal X}}^u)$, injects via $\sigma^*$ into $H^1(\bar B)=H^1({{\mathcal Y}}^u)$. Here $A:={{\mathcal X}}\otimes_R k$. Let us denote by ${{\mathcal X}}'$ the normalization of ${{\mathcal X}}$ in $K({{\mathcal Y}})$, with factorization [ $$\begin{gathered}
\label{4.1}\xymatrix{ {{\mathcal Y}}\ar@/^1pc/[rr]^{\sigma} \ar[r]_{\sigma'} & {{\mathcal X}}' \ar[r]_\nu & {{\mathcal X}}}\end{gathered}$$ ]{} and set $A':=A\times_{{{\mathcal X}}} {{\mathcal X}}'$. Then $\sigma'$ induces an isomorphism $K({{\mathcal X}}')\xrightarrow{\cong} K({{\mathcal Y}})$. Furthermore, ${{\mathcal X}}'\xrightarrow{\nu} {{\mathcal X}}$ and and $A'\xrightarrow{\nu|_A} A$ are homeomorphisms. Thus $H^1({{\mathcal X}}^u)=H^1(\bar A)\xrightarrow{\nu^*} H^1(({{\mathcal X}}')^u) =H^1(\bar{A'})$ is an isomorphism. On the other hand, since $\sigma'_*{{\mathbb Q}}_\ell={{\mathbb Q}}_\ell$, the Leray spectral sequence for $\sigma'$ applied to $H^1({{\mathcal Y}}^u)$ yields an inclusion $H^1(({{\mathcal X}}')^u)=H^1(\bar{A'})\xrightarrow{{\rm inj}} H^1({{\mathcal Y}}^u)=H^1(\bar B)$. This finishes the proof.
[99]{}
Artin, M.: Algebraization of Formal Moduli: II. Existence of Modifications, Ann. of Math. (2) [**91**]{} (1970), 88–135. Blickle, M., Esnault, H.: Rational singularities and rational points, preprint 2006, 12 pages, appears in the volume dedicated to F. Bogomolov, Pure and Applied Math. Quarterly. de Jong, A. J.: Smoothness, semi-stability and alterations, Publ. Math. IHES [**83**]{} (1996), 51–93. de Jong, A. J.: Families of curves and alterations, Ann. Inst. Fourier [**47**]{} no2 (1997), 599–621. Deligne, P.: Théorème d’intégralité, Appendix to Katz, N.: Le niveau de la cohomologie des intersections complètes, Exposé XXI in SGA 7, Lect. Notes Math. vol. [**340**]{}, 363–400, Berlin Heidelberg New York Springer 1973. Deligne, P.: La conjecture de Weil, II. Publ. Math. IHES [**52**]{} (1981), 137-252. Esnault, H.: Deligne’s integrality theorem in unequal characteristic and rational points over finite fields, with an appendix with P. Deligne, Ann. of Math. [**164**]{} (2006), 715–730. Esnault, H.: Coniveau over $\frak{p}$-adic fields and points over finite fields, preprint 2007, 5 pages, to appear in C. R. Acad. Sci. Paris. Fujiwara, K.: A Proof of the Absolute Purity Conjecture (after Gabber), in Algebraic Geometry 2000, Azumino, Advanced Studies in Pure Mathematics [**36**]{} (2002), Mathematical Society of Japan, 153–183. Grothendieck, A.: Éléments de Géométrie Algébrique (EGA): [**III**]{} (2): Étude cohomologique des faisceaux cohérents, Publ. Math. IHES [**17**]{} (1963)\
[**IV**]{} (3): Études locales des schémas et des morphismes de schémas, Publ. Math. IHES [**28**]{} (1966). Keel, S.: Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) [**149**]{} (1999), no.1, 253–286. Milne, J.: Lectures on Étale Cohomology, v2 02, August 9 (1998), http://www.jmilne.org/math/.
[^1]: Partially supported by the DFG Leibniz Preis and the Amercian Institute for Mathematics
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epsf
Introduction
============
This paper is concerned with domain growth in far from equilibrium systems. This is a subject of increasing interest both for the wide range of behaviors and for the large number of applications, which range from phase separation in mixtures to island formation and coarsening during deposition of a thin film or submonolayer [@bartelt92; @bales], among other systems.
Our aim is to discuss a series of one-dimensional exclusion models with particle diffusion, reversible or irreversible attachment to clusters and deposition mechanisms that represent, for example, volume reduction effects after cluster coalescence. Diffusion processes tend to bring these systems to equilibrium steady states, but pressure and other external influences may drive the system to new steady states. Though not usually related to a specific real problem, these one-dimensional models may reveal interesting features that help to understand more complex and realistic surface and bulk systems [@sollich; @jackle], with the advantage of being more tractable both analytically and numerically. We will discuss a series of plausible physical situations in systems with diffusion and mechanisms that drive them out of equilibrium, in order to understand the details of domain growth and convergence to steady states, if it occurs.
In the first model, hereafter called model I, a fixed fraction $\rho$ of a one-dimensional lattice is randomly filled with hard core particles. The diffusion rates are $r=1$ when they are free, i. e. when they have two empty nearest-neighbor sites, and $r=\epsilon\sim e^{-E/T}$ (where $E$ is the related energy barrier) when they have one occupied nearest neighbor site (Fig. 1a). For $\epsilon\ll 1$ ($T\to 0$), the average aggregates’ length grows as $t^{1/3}$ in a long time range, and eventually approaches saturation at $\sim {\epsilon}^{-1/2}$ with a slow $t^{-1/2}$ decay (Sec. \[secdiffusiononly\]). In the limit $\epsilon\to 0$, this model is equivalent to the Ising model with Kawasaki dynamics previously studied by Cornell et al [@cornell], who focused on its zero temperature features. However, the dynamic rules are mainly motivated by the Clarke-Vvedensky model for thin films or submonolayer growth [@cv], excluding the deposition processes. In the simplest versions of that model, an isolated adatom has to overcome an energy barrier $E_s$ to diffuse, while when it is attached to $n$ nearest neighbors the energy barrier increases to $E_s+nE_b$, where $E_b$ is a bonding energy. This model and related ones were already intensively studied in two dimensions during the deposition process [@cv; @barkema; @ratch], but a few works have considered the post-deposition coarsening dynamics [@lam].
Subsequently we will generalize the previous hard core dynamical system by introducing deposition of particles (see e. g. Refs. [@evans] and [@robinbook]), but allowing deposition only at (empty) sites with at least one empty nearest neighbor (Fig. 1b). In this model (referred to as model II), the domain coalescence, which generates larger vacancies between aggregates, is followed by a density increase. The exclusion of deposition at single holes between clusters represent the geometrical frustration of real systems. In Sec. \[secdiffusiondep\], we will show that this model exhibits a $t^{1/2}$ domain growth. This is among other results we have obtained by simulation studies, which are presented for both models I and II in Secs. \[secdiffusiononly\] and \[secdiffusiondep\], respectively.
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Our models share some aspects with diffusion limited coalescence models [@abad; @benavraham; @zhong] and with fragmentation-aggregation models [@suphriaetal; @suphriarobin]. They are similarly described in terms of cluster or interval probabilities, and like the fragmentation models they are amenable to analytic investigation based on an independent cluster approximation (the independent interval approximation to the joint cluster length probability occurring in the master equation). We use this approach to explain properties of models I and II, including distributions of cluster size (Secs. \[secdiffusiononly\] and \[secdiffusiondep\]). Further, a simple scaling picture can be developed in order to describe the basic domain growth laws; we use this at the beginning of Sec. \[secdiffusiononly\].
Model I: diffusion, detachment and reattachment of particles {#secdiffusiononly}
============================================================
Processes {#Processdifonly}
---------
The model studied in this section has the particle hopping processes depicted in Fig. 1a. Isolated particles hop symmetrically on a chain at unit rate (“diffusion”), while a single particle with a left hand / right hand neighbor can hop to an empty right / left neighbor with rate $\epsilon$ (detachment). So clusters evolve by detachment and reattachment of particles. The model is of exclusion type: no site can accommodate more than one particle.
This model is clearly particle-conserving, so density $\rho$ is fixed. The case $\epsilon\ll 1$ is of particular interest since, as reported in the simulation studies below and explained in the following subsection, very large clusters emerge.
Simulations {#Simulationsdifonly}
-----------
We simulated model I in one-dimensional lattices of length $L=8000$. This length is sufficiently large to ensure that finite size effects are negligible, as shown by comparisons of some results with data from lattices with $L=16000$ (particularly for the smallest values of $\epsilon$ this comparison is essential).
Initially, the lattice is randomly filled with a density of particles $\rho$. We simulated three values of the density, $\rho = 0.1$, $\rho = 0.5$ and $\rho = 0.9$, which are representative of the range of intermediate densities, i. e. densities not too small ($\rho\approx 0$) nor too large ($\rho\approx 1$). For $\rho = 0.1$ and $\rho = 0.9$, we considered several values of the diffusion rate $\epsilon$ ranging from $\epsilon =
{10}^{-1}$ to $\epsilon = {10}^{-3}$, and for $\rho = 0.5$ we performed simulations until $\epsilon = {10}^{-5}$.
The sequence of characteristic behaviors of model I, as shown by simulation results, are: $(i)$ early fast attachment of isolated particles to each other to form clusters; $(ii)$ an intermediate regime in which detachment sets in, allowing further coarsening; $(iii)$ finally, there is a diffusive approach to a saturated state where the clusters have a large steady mean size that depends on $\epsilon$.
The three regimes are well separated at small $\epsilon$. This is illustrated in the plot of $\log_{10}{d}$ versus time $\log_{10}{t}$, shown in Fig. 2, where $d$ is the mean size of clusters of two or more particles; $d$ is given in terms of the probability $P_t(m)$ that an arbitrarily chosen cluster has size (or mass) $m$ at time $t$ by $$d = {{\sum_{m=2}^{\infty}{mP_t\left( m\right)}}\over
{\sum_{m=2}^{\infty}{P_t\left( m\right)}}} .
\label{defd}$$
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The early time dependence of $d$ in region $(i)$ (at small $\epsilon$) starts with a characteristic increase with rate proportional to a high power of $\epsilon$, and then crosses over to a form allowing data collapse in terms of the reduced time variable $\epsilon t$, as shown in Fig. 3.
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In region $(ii)$, $d$ increases as $$d \sim t^{\alpha} .
\label{defalpha}$$ The apparent exponent $\alpha_{eff}$, defined as the local slope of the $\log{d}\times\log{t}$ plot, was calculated numerically. $\alpha_{eff}$ is shown in Fig. 4 as a function of ${\left( \epsilon t\right)}^{-1}$ for three different values of $\epsilon$. It appears to approach the value $\alpha = 1/3$ in the limit of small $\epsilon$ and correspondingly large $t$, which is consistent with the prediction of a simple scaling description (next subsection).
Fig. 5 shows the diffusive approach of the mean cluster size to its saturation value $d_\infty$. This approach is well described by $d = d_\infty - {C\over{t^{1/2}}}$, for $t\to\infty$, with $C$ constant. The dependence on $\epsilon$ of the saturation value $d_\infty$ is illustrated in Fig. 6 for $\rho=0.5$. The least squares fit in Fig. 6 gives $$d_\infty \approx 0.72\epsilon^{-1/2} + 1.93,
\label{scalingdinf}$$ in which the dominant (proportional to $\epsilon^{-1/2}$) and the sub-dominant (additive constant) terms were estimated. Like Eq. (\[defalpha\]) with $\alpha = 1/3$, this result follows from the analytic work in Sec. \[Theorydifonly\].
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Theory {#Theorydifonly}
------
The characteristic results just described have been interpreted by simple heuristic scaling arguments and by detailed analytic studies starting from the master equation and employing an independent interval approximation. This second approach is capable of yielding cluster length distributions and their evolution.
To begin with, we focus on the asymptotic cluster size at small $\epsilon$. This asymptotics occurs in the regime where lone particles are rare, and those that are present are in the process of reattaching themselves to the clusters they came from or to a neighboring one. The second case provides the sharing which sets the mean cluster size $d$. At densities of order $\rho\sim
1/2$, the cluster size is roughly of the order of cluster separation (see Fig. 7a). Thus the equilibrium of detachment time ($1/\epsilon$) and time of diffusion to a neighboring cluster ($\sim d^2$) gives the observed saturation result $$d \sim \epsilon^{-1/2} .
\label{scalingdinf2}$$ This argument can be generalized rather obviously to explain the $t^{-1/2}$ approach to saturation.
A more interesting application is the explanation of the early cluster size growth law (Eq. \[defalpha\]). Here, unlike the saturation just described, the cluster separations are such that the detached particle is likely to return and reattach many times before it eventually diffuses to the next cluster (Figs. 7a and 7b). Its likelihood of returning to the origin means that the detachment rate $\epsilon$ needs to be replaced by an effective rate $\tilde{\epsilon} = \epsilon P_{mig}$, where $P_{mig}$ is the (migration) probability that a freely diffusing particle does not return to the detachment site before diffusing the distance $\sim d$ to the next cluster. In other words (see Fig. 7b), this is the probability that a free particle at position $y=1$ does not return to the origin ($y=0$) before a time of order $d^2$, which is the typical time for diffusion along a distance $d$. Considering that $$Q(y,t) = {y\over{{\left( 4\pi D\right)}^{1/2}}} t^{-3/2}
\exp{\left( - {{y^2}\over{4Dt}}\right)}
\label{probfirstpassage}$$ is the probability that the first passage of a random walker at point $y$ occurs at time $t$ [@montroll] ($D$ is the diffusion coefficient), $P_{mig}$ is given by $$P_{mig} \sim \int_{d^2}^{\infty}{Q\left( 1,t\right) dt} \sim d^{-1} .
\label{probmig}$$
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In terms of the effective rate $\tilde{\epsilon}$, the required time for a particle to transfer to the next cluster is of order $1/\tilde{\epsilon}$. Thus, the time required for doubling the size $d$ of a cluster by successive gain and loss of particles is $d^2/\tilde{\epsilon}\sim d^3/\epsilon$. So the cluster growth proceeds according to $${d\over{dt}}d \sim {d\over{\left( d^3/\epsilon\right)}} ,
\label{eqdif_difonly}$$ and hence $$d\sim {\left(\epsilon t\right)}^{1/3} .
\label{scalingii}$$ This explains the behavior seen in the simulations (Figs. 3 and 4). The situation is analogous to domain scaling in Ising chains where, with Kawasaki dynamics [@cornell], spins split off from domain edges and migrate across to increase the domain size by one lattice unit.
We turn next to the more powerful analysis starting from a version of the master equation, which can provide a full description of the process. This is more easily set up by reformulating the process using a column picture, in which a column of height $m$ represents a cluster of size $m$, and then the original detachment and diffusion processes correspond to those shown in Fig. 8. Since one cluster has two edges but corresponds to a single column, the one-particle detachment rate in the column picture is $$\gamma = 2\epsilon .
\label{rates}$$
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We denote by $P_t(m)$ the probability that a randomly chosen cluster (equivalently, column) has size $m$ at time $t$. Then the gain/loss from in/out processes provides the following master equation, in an independent interval approximation in which joint probabilities are factorized: $$\begin{aligned}
&&P_{t+1}(m) - P_t(m) = {\cal A}_m \equiv
\nonumber\\
&&P_t(m-1)\theta (m-1)
\left[ \gamma\sum_{m'\geq 2}{P_t(m')} + 1\times P_t(m'=1) \right] +
\nonumber\\
&&P_t(m+1) \left[ \gamma\theta (m+1-2) + 1\times \delta_{m,0} \right] -
\nonumber\\
&&P_t(m) [ \gamma\theta (m-2) + 1\times \delta_{m,1} +
\nonumber\\
&&\gamma\sum_{m'\geq 2}{P_t(m')} + 1\times P_t(m'=1) ] .
\label{masterdifonly}\end{aligned}$$ The corresponding equation for the generating function $$G_t(s) \equiv \sum_{m=0}{P_t(m) s^m}
\label{generating}$$ is $$\begin{aligned}
G_{t+1}(s) &=& G_t(s) \left[ 1 + s a(t) + {\gamma\over s} - \gamma - a(t)
\right]
\nonumber\\
&&+ s(\gamma -1)P_t(1) +
\nonumber\\
&&\left[ \gamma P_t(0) + (1-\gamma)P_t(1) \right] - {\gamma\over s}P_t(0) ,
\label{gendifonly}\end{aligned}$$ where $a(t) = \gamma \left[ 1-P_t(0) \right] + (1-\gamma) P_t(1)$. It is easy to check probability and mass conservation using $G_t(0)$ and $G'_t(0)$.
The steady state distribution $P(m)$ and generating function $G(s)$ resulting from Eq. (\[gendifonly\]) are given by $$G(s) = {\left( \gamma - sa\right)}^{-1}
\left[ \gamma P(0) - s(1-\gamma)P(1) \right] ,
\label{gsteadydifonly}$$ $$P(m) = {\left( {A\over\gamma}\right)}^{m-1}P(1) , m>1 ,
\label{psteadydifonly}$$ and $$P(1) = A P(0) ,
\label{p1difonly}$$ with $A = \gamma \left[ 1-P(0)\right]
{\left[ 1 - (1-\gamma )P(0)\right]}^{-1}$. So the steady state cluster size distribution is exponential. The mean size of multi-particle clusters (Eq. \[defd\]) and the mean mass $\langle m\rangle \equiv\sum_{m=0}^{\infty}{mP\left( m\right)}$ are then obtainable in terms of $P(0)$, as is the density $\rho$. So, in particular the mean cluster length $d\equiv \langle m\rangle$ can be found in terms of $\rho$. The result simplifies at small $\gamma$ (small $\epsilon$) to $$\begin{aligned}
d &=& \gamma^{-1/2}
{\left[ {\rho\over{\left( 1-\rho\right)}}\right]}^{1/2} +
{{\left( {\rho/{\left( 1-\rho\right)}}+3\right)}\over 2}
=
\nonumber\\
&&\epsilon^{-1/2}
{\left[ {\rho\over{2\left( 1-\rho\right)}}\right]}^{1/2} +
{{\left( {\rho/{\left( 1-\rho\right)}}+3\right)}\over 2} ,
\label{dsteadydifonly}\end{aligned}$$ where the dominant and the first sub-dominant terms are shown. This form is consistent with the scaling result (\[scalingdinf2\]) and is in very good agreement with the simulation result (\[scalingdinf\]), including the sub-dominant constant term: Eq. (\[dsteadydifonly\]) gives $d=0.7071\dots\epsilon^{-1/2}+2$ for $\rho=0.5$ (see also Fig. 6). In the same limit, this is also, apart from a numerical factor, the characteristic size in the exponential cluster mass distribution.
Model II: diffusion and deposition of particles {#secdiffusiondep}
===============================================
Processes {#Processdepdif}
---------
Model II is a generalization of model I, different only by having the deposition processes depicted in Fig. 1b, in addition to the diffusion and detachment processes of Fig. 1a. This makes the model non-particle-conserving, which leads to continued coarsening and other scaling properties and crossover.
Simulations {#Simulationsdepdif}
-----------
The characteristic behavior of model II, as exhibited by simulation results, is as follows. For initial densities not too near $\rho =1$, there is: (i) an early regime of rapid filling, due to deposition, and cluster evolution due to both processes; (ii) an intermediate regime where deposition slows because of the scarcity of deposition sites due to the increased density - the exclusion constraint of course applies. The slow detachment process allows redistribution of particles, opening up new deposition sites and allowing the continually slowing coarsening (with no saturation as $\rho <1$).
Fig. 9 shows simulation results for the evolution of the mean cluster size $d$. That plot shows that $d(t)$ is well fitted by the form $$d(t) = B t^{1/2} \left( 1 + Ct^{-1/2} + \dots \right) .
\label{ddiffusiondep}$$ In Fig. 10 we show the ratio between the estimates of the amplitude $B$ and $\epsilon^{1/2}$ for several values of $\epsilon$. Those results give $$B(\epsilon) \sim b \epsilon^{1/2} ,
\label{scalingB}$$ with negligible corrections to scaling, where $b=0.252\pm 0.002$. This result is in accord with theoretical analysis given in the next subsection, including the estimate of the amplitude $b$.
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The dependence of the evolving density on time $t$ and rate $\epsilon$ has also been studied. The simulation results shown in Fig. 11 imply that the density is a function of the scaling variable ${\left( \epsilon t\right)
}^{1/2}$ and, at very long times, it converges to $1$ as $$1-\rho \sim {\left( \epsilon t\right) }^{-1/2} .
\label{scalingrho}$$ This form is also in agreement with the theoretical analysis below.
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Theory {#Theorydepdif}
------
The characteristics presented in the foregoing subsection can be interpreted using an analytic investigation along the lines of the detailed discussion given in Sec. \[Theorydifonly\].
We have to include the effects of the extra deposition process, which leads to the decrease of the total number of clusters and of the number of holes between the clusters as time increases. On the other hand, the length $L$ of the line in which particles are deposited and diffuse is kept constant. Consequently, in order to adopt the column picture of Sec. II.3 (see Fig. 8), it is necessary to consider that the length $L_0$ of the corresponding column problem decreases in time (these lengths are related as $L_0=L-M$, where $M$ is the total mass or total number of particles, for periodic boundary conditions).
The evolution equation here is written for cluster numbers as $$N(m,t+1) - N(m,t) = L_0 \left( {\cal A}_m + {\cal B}_m \right) ,
\label{masterdiffusiondep}$$ where the diffusion contribution ${\cal A}_m$ is given in Eq. (\[masterdifonly\]) and the deposition contribution is $$\begin{aligned}
&&{\cal B}_m = P_t(0) [ 2 \theta\left( m-2\right) P_t\left( m-1\right) -
\nonumber\\
&&2 \theta\left( m-1\right) P_t\left( m\right) + \delta_{m,1}P_t\left( 0\right)
- 2\delta_{m,0} ] .
\label{contribdep}\end{aligned}$$ The length of the lattice in which the column problem is defined varies due to deposition as $${{L_0(t+1)-L_0(t)}\over{L_0(t)}} = -P_t(0) \left[ 2-P_t(0)\right] .
\label{lenght}$$ In these equations, the cluster probability is $$P_t(m) = {{N(m,t)}\over{L_0(t)}} .
\label{probdifdep}$$ They preserve conservation of probability, but mass is no longer conserved.
The resulting equation is similar to ones occurring in coalescence models [@abad; @benavraham; @zhong]. From this we expect that a large time and small $\epsilon$ limit discussed subsequently is equivalent to the model in Ref. [@abad]. Our approach, which exploits the generating function method, becomes equivalent, in the scaling limit, though in a conjugate space, to continuum approximations used in the coalescence studies of Refs. [@abad] and [@zhong].
Now the generating function (Eq. \[generating\]) satisfies $$\begin{aligned}
&&G_{t+1}(s) {\left[ 1-P_t(0)\right]}^2 - G_t(s) =
\nonumber\\
&&(s-1) \left[ G_t(s) \left( a(t)-{\gamma\over s}\right) + (\gamma -1)P_t(1) +
{\gamma\over s}P_t(0) \right] +
\nonumber\\
&&P_t(0) \left[ 2(s-1)G_t(s) - sP_t(0) + 2\left( P_t(0)-1\right) \right] .
\label{gendifdep}\end{aligned}$$ In the right hand site of Eq. (\[gendifdep\]), the first term corresponds to diffusion processes and the second one to deposition processes.
Because deposition slowly fills the system, we expect the configurations to coarsen and presumably to go into some scaling asymptotics where mass scales with some power of $t$, and $P_t(m)$ and $G_t(s)$ each become one-variable scaling functions. So we look for a long time scaling solution of the above equation.
At long times, the finite difference $G_{t+1}(s)-G_t(s)$ in Eq. (\[gendifdep\]) can be taken as a derivative. The scaling variable will be some combination of $t$ (large) and $u\equiv 1-s$ (small), the latter because large cluster sizes arise from structure in $G_t(s)$ at $s\approx 1$. The variable $u$ is actually conjugate to $m$ (see below). Coarsening will correspond to the scale of $m$ as $t^z$, with some power $z$, in which case the one-variable form will be $$G_t(s) = u^\alpha f\left( ut^z\right) ,
\label{scalinggendifdep}$$ with some function $f$. Normalization requires $\alpha = 0$ and $f(0) = 1$. In the scaling limit, the relationship of the generating function to the probability $P_t(m)$ requires the latter to be of the form $$P_t(m) = {1\over{t^z}} g\left( {m\over {t^z}}\right) ,
\label{scalingpdifdep}$$ with $$f(x) = \int_0^\infty{ g(y) e^{-xy} dy} .
\label{deff}$$ It turns out that the consistent scaling solution has $g(0)=0$, so the $1/t^z$ contribution to $P_t(0)$ vanishes, leaving a leading term of lower-than-scaling order, $$P_t(0) = {{c/2}\over{t^{2z}}} ,
\label{Pt0depdiff}$$ where $c$ is a constant. Eq. (\[gendifdep\]) leads to the dynamical exponent $$z=1/2
\label{zdepdiff}$$ and to the following equation for the one-variable scaling function: $$xf'(x) - 2cf(x) -2\gamma x^2 f(x) + 2c = 0 .
\label{eqdiff}$$
Even without solving Eq. (\[eqdiff\]) we can infer that $$P_t(0) = {{c/2}\over t} ,
\label{p0difdep}$$ $${\langle m\rangle}_t \sim \gamma^{1/2}t^{1/2} \sim \epsilon^{1/2}t^{1/2}
\label{mdifdep}$$ and $$1-\rho_t \sim \gamma^{-1/2}t^{-1/2} \sim \epsilon^{-1/2}t^{-1/2} .
\label{rhodifdep}$$ These hold in the long time scaling limit we have introduced and agree with the observed simulation results in Eqs. (\[ddiffusiondep\]) , (\[scalingB\]) and (\[scalingrho\]).
Eq. (\[eqdiff\]) can be formally solved for the scaling function $f(x)$ by using the variable $\zeta = x^2$ and considering the function $f(x) x^{-2c}$. The result is $$f(x) = {c\over\gamma}\int_0^\infty{ {\left( 1+{v\over\gamma}\right)}^{-c-1}
e^{-vx^2}dv } .
\label{f}$$ The large $x$ expansion of $f(x)$ and Eq. (\[deff\]) provides the small $y$ expansion of $g(y)$: $$g(y) = {1\over{\gamma^{1/2}}} {{\cal G}{\left(
{y\over{\gamma^{1/2}}}\right) }} ,
\label{g}$$ where $${\cal G}(u) = \sum_{m=0}^\infty{ {{c(c+1)\dots (c+m)}\over{(2m+1)!}}
{\left( -1\right)}^m u^{2m+1} } .
\label{calG}$$ This confirms that $g(0)=0$. The cluster distribution has the following form, in terms of the odd function ${\cal G}$: $$P_t(m) = {1\over{m^*(t)}} {{\cal G}{\left( {m\over{m^*(t)}}\right) }} ,
\label{scalingPtm}$$ where $m^* = {\left( \gamma t\right)}^{1/2}\sim {\left( \epsilon
t\right)}^{1/2}$. It explains the scaling variable ${\left( \epsilon
t\right)}^{1/2}$ used to collapse simulation data in Fig. 11.
The conditions that ${\cal G}$ must be non-negative and normalisable are satisfied with $c=1/2$ in Eq. (\[Pt0depdiff\]), which leads to $${\cal G}(u) = {u\over 2}e^{-{\left( u/2\right)}^2} .
\label{calGfinal}$$ The mean cluster mass (cluster length in the original problem) is easily obtained as $${\langle m\rangle}_t \approx \sqrt{\pi} {\left( \gamma t\right)}^{1/2} .
\label{mdifdepfinal}$$ Considering relation (\[rates\]), the amplitude of cluster length scaling is $B=\sqrt{2\pi}\epsilon^{1/2}\approx 2.507\epsilon^{1/2}$. It quantitatively agrees with the result obtained in simulations (Sec. \[Simulationsdepdif\]).
Conclusion {#secconclusion}
==========
We studied two one-dimensional exclusion models with particle diffusion, reversible or irreversible attachment to clusters and deposition mechanisms.
In model I, starting from a randomly filled lattice, only particle diffusion is allowed, with small detachment rates $\epsilon$ for particles at the edges of the clusters. Simulation results show an initial regime with formation of small clusters, a regime of cluster size growth as $d\sim t^{1/3}$ and a regime of cluster size saturation at $d\sim
\epsilon^{-1/2}$. These results can be explained using heuristic scaling arguments. The analytical treatment of the master equation with an independent cluster approximation for joint probabilities distributions predicts a saturation cluster size in quantitative agreement with numerical data.
Model II generalizes model I in having also particle deposition: this is allowed only at empty sites with one or two empty nearest neighbors. Simulation results show continuous coarsening with a $t^{1/2}$ increase of the average cluster size and an increase of the density with $t^{-1/2}$ corrections. These scaling forms are justified by analytical investigations again using an independent cluster approximation, which provides good quantitative agreement with the simulations.
We expect that the models presented above and the combination of different methods to explain their scaling behaviors can be used to understand further non-equilibrium systems. Of particular interest would be the extension of theoretical methods (e. g. scaling approaches) to two-dimensional systems such as adatom islands on surfaces or the extension of the one-dimensional models to include other mechanisms that drive the systems to new non-equilibrium steady states or which lead to anomalous coarsening.
FDAA Reis thanks the Department of Theoretical Physics at Oxford University, where part of this work was done, for the hospitality, and acknowledges support by CNPq and FINEP (brazilian agencies).
RB Stinchcombe acknowledges support from the EPSRC under the Oxford Condensed Matter Theory Grants, numbers GR/R83712/01 and GR/M04426.
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M. C. Bartelt and J. W. Evans, Phys. Rev. B [**46**]{}, 12675 (1992). G. S. Bales and A. Zangwill, Phys. Rev. B [**55**]{}, 1973 (1997). P. Sollich and M. R. Evans, Phys. Rev. Lett. [**83**]{}, 3238 (1999). J. Jäckle and S. Eisinger, Z. Phys. B [**84**]{}, 115 (1991). S. J. Cornell, K. Kaski, and R. B. Stinchcombe, Phys. Rev. B [**44**]{}, 12263 (1991). S. Clarke and D. D. Vvedensky, J. Appl. Phys. [bf 63]{}, 2272 (1988). O. Barkema, O. Biham, M. Breeman, D. O. Boerma, and G. Vidali, Surf. Sci. [**306**]{}, L569 (1994). C. Ratsch, A. Zangwill, P. Smilauer, and D. D. Vvedensky, Phys. Rev. Lett. [**72**]{}, 3194 (1994). P.-M. Lam, D. Bayayoko, and X.-Y. Hu, Surf. Sci. [**429**]{}, 161 (1999). I. Koponen, M. Rusanen, and J. Heinonen, Phys. Rev. E [**58**]{}, 4037 (1998). J. W. Evans, Rev. Mod. Phys. [**65**]{}, 1281 (1993). R. Stinchcombe, in [*Jamming and Rheology*]{}, eds. A. J. Liu and S. R. Nagel (Taylor and Francis, London and New York, 2001). S. N. Majumdar, S. Krishnamurthy, and M. Barma, Phys. Rev. Lett. [**81**]{}, 3691 (1998). S. Krishnamurthy and R. B. Stinchcombe, to be published (2004). E. Abad, T. Masser, and D. ben-Avraham, J. Phys. A [**35**]{}, 1483 (2002). D. ben-Avraham, M. A. Burschka, and C. R. Doering, J. Stat. Phys. [**60**]{}, 695 (1990) D. Zhong and D. ben-Avraham, J. Phys. A [**28**]{}, 33 (1995). E. W. Montroll and B. J. West, in [*Fluctuation phenomena*]{}, eds. E. W. Montroll and J. L. Lebowitz (North-Holland, Amsterdam, 1979).
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abstract: '**The emerging field of phase-coherent caloritronics (from the Latin word “calor”, i.e., heat) is based on the possibility to control heat currents using the phase difference of the superconducting order parameter. The goal is to design and implement thermal devices able to master energy transfer with a degree of accuracy approaching the one reached for charge transport by contemporary electronic components. This can be obtained by exploiting the macroscopic quantum coherence intrinsic to superconducting condensates, which manifests itself through the Josephson and the proximity effect. Here, we review recent experimental results obtained in the realization of heat interferometers and thermal rectifiers, and discuss a few proposals for exotic non-linear phase-coherent caloritronic devices, such as thermal transistors, solid-state memories, phase-coherent heat splitters, microwave refrigerators, thermal engines and heat valves. Besides being very attractive from the fundamental physics point of view, these systems are expected to have a vast impact on many cryogenic microcircuits requiring energy management, and possibly lay the first stone for the foundation of electronic thermal logic.**'
author:
- Antonio Fornieri
- Francesco Giazotto
title: 'Towards phase-coherent caloritronics in superconducting circuits'
---
In the last decades, the impressive evolution of modern electronics has reached a point where quantum effects and phase coherence are ordinarily exploited to study exotic phenomena at the nanoscale under controlled and adjustable conditions. Only very recently, instead, scientists have started to exploit the great potentialities offered by nanotechnology for the investigation and control of heat currents (the branch of science called “caloritronics”). Interesting advances in the understanding of fundamental properties of thermal transport have been obtained in experiments involving atomic or molecular junctions [@DubiRev]. A few works [@Schwab; @Meschke; @Jezouin] have shown that heat flow has a quantum limit - just as the electric current - and that this limit does not depend on the nature of heat carriers. Furthermore, a remarkable amount of theoretical and experimental studies has been focused on the interaction between heat and spin currents in thermoelectric devices [@BauerRev].
From the point of view of applications, the largest effort has been put into electronic and phononic thermometry or refrigeration [@GiazottoRev; @MuhonenRev; @MottonenArx], but the most intriguing and ambitious goal has always been the full control of heat currents, aiming to emulate the accuracy regularly obtained for charge transport in modern electronic devices. This attracting possibility was first envisioned by a conspicuous amount of theoretical works designing non-linear phononic devices [@LiRev]. However, the practical realization of these structures still appears challenging, hampering significant improvements of the first promising results [@Zettl; @Tian]. An appealing alternative is represented by the notable ingredient of phase coherence, whose role in thermal transport was almost unknown until ten years ago, with just the exceptions of Refs. . This gap was filled by the birth of *phase-coherent caloritronics* [@Meschke; @GiazottoNature; @MartinezRev], a young field of nanoscience that takes advantage of the long-range phase coherence of the superconducting condensate to manipulate electronic and photonic heat currents in solid-state mesoscopic circuits. As we shall argue, superconducting phase coherence represents a unique control knob for heat flows and allows to design several non-linear caloritronic devices to obtain an unprecedented thermal management at the nanoscale.
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The physical picture at the basis of phase-coherent caloritronics is represented schematically in Fig. \[Fig1\]a. The fundamental idea is to exploit a suitable physical effect that depends on the superconducting phase difference $\varphi$ to control the electronic heat flow ($J_{\rm left}$ and $J_{\rm right}$) between two electronic thermal reservoirs residing at temperatures $T_{\rm hot}>T_{\rm cold}$. Towards this end, three main different approaches can be followed. As shown in Fig. \[Fig1\]b, the first possibility consists in the phase control of heat transported by electrons in a temperature-biased Josephson tunnel junction (JJ), since the electronic thermal current $J_{\rm SIS}$ flowing through the junction depends on the macroscopic phase difference between the superconducting electrodes S [@MakiGriffin; @GiazottoAPL; @GiazottoNature]. Although this method has been the most prolific from the experimental point of view [@GiazottoNature; @MartinezNature; @MartinezNatRect; @FornieriNature; @FornieriArxiv], it allows to only partially modulate $J_{\rm SIS}$, as it will be discussed in the following.
The second approach, instead, is sketched in Fig. \[Fig1\]c and relies on the phase manipulation of the electronic thermal conductivity and the electron-phonon coupling in phase-engineered superconducting proximity systems. Here, two normal metal (N) reservoirs are tunnel-coupled to an other normal layer (S’) in which superconducting correlations are induced thanks to the proximity effect. As we shall argue, S’ can be inserted in a superconducting loop forming a superconducting quantum interference proximity transistor (SQUIPT) [@SQUIPT1; @SQUIPT2; @SQUIPT3], which can be used to tune the phase difference across the proximized layer and therefore its density of states (DOS) $\mathcal{N}(\varphi)$ [@Zhou; @leSueur]. The latter directly affects the thermal properties of S’, leading to the phase manipulation of the electronic heat currents exchanged with the reservoirs $J_{\rm S'IN}(\varphi)$ and with the lattice phonons $J_{\rm e-phon}(\varphi)$. Even though an experimental proof is still lacking, this method could in principle provide variations of the thermal conductance of several orders of magnitude [@StrambiniAPL].
Last approach is sketched in Fig. \[Fig1\]d and is based on the regulation of the energy exchanged between electrons and photons thanks to the phase tuning of the coupling with the electromagnetic environment. In contrast to the previous cases, in general the reservoirs are *not* in galvanic contact and the photonic heat current $J_{\rm e-phot}(\varphi)$ is controlled by an intermediate superconducting circuit \[for instance, a direct current superconducting quantum interference device (DC SQUID)\] inductively or capacitively coupled to the reservoirs [@Ojanen; @Pascal]. The SQUID is characterized by a magnetic-flux-dependent impedance $Z_{\rm SQUID}(\Phi)$ that can act as a *contactless* knob for $J_{\rm e-phot}(\varphi)$. This method has been already demonstrated experimentally [@Meschke], but its potential is far from being fully exploited.
All the options listed above can be implemented in the framework of quasiequilibrium regime [@GiazottoRev]: since the electron-phonon coupling in metals is strongly suppressed at temperatures below 1 K, one can inject a Joule power in the structure, thereby driving the electronic system into a Fermi distribution function characterized by a temperature $T_{\rm e}$ that can be significantly different from that of the lattice phonons. In our case, the latter are fully thermalized with the substrate phonons residing at the bath temperature $T_{\rm bath}$, thanks to the vanishing Kapitza resistance between thin metallic films and the substrate [@GiazottoNature; @Wellstood; @MartinezNature; @MartinezNatRect; @FornieriNature; @FornieriArxiv]. Typically, the superconducting parts are implemented by aluminum (Al) electrodes, which are well known to form high-quality tunnel junctions. At low temperatures, the heat current $J_{\rm e-phon}$ released by the electrons to the phonon bath is exponentially suppressed by the superconducting energy gap [@Timofeev1]. On the other hand, N electrodes are usually made of copper (Cu) or manganese-doped aluminum (Al$_{0.98}$Mn$_{0.02}$) [@Meschke; @GiazottoNature; @MartinezNature; @MartinezNatRect; @FornieriNature; @FornieriArxiv]. The former material is particularly suited to be coupled to Al leads in order to form superconducting proximity structures [@SQUIPT1; @SQUIPT2; @SQUIPT3] and is characterized by $J_{\rm e-phon}\sim (T_{\rm e}^5-T_{\rm bath}^5)$ [@GiazottoRev; @Wellstood]. In Al$_{0.98}$Mn$_{0.02}$, instead, $J_{\rm e-phon}\sim (T_{\rm e}^6-T_{\rm bath}^6)$ [@Maasilta], thus reducing phononic losses at low temperatures. Moreover, its good oxidation properties make Al$_{0.98}$Mn$_{0.02}$ very useful to realize NIN and NIS structures [@MartinezNature; @MartinezNatRect; @FornieriNature]. In the following sections, we shall analyze more deeply each different approach, reviewing the major experimental achievements and the existing proposals for novel caloritronic devices.
Josephson tunnel circuits {#josephson-tunnel-circuits .unnumbered}
=========================
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Just three years after the prediction of the Josephson effect [@Josephson], Maki and Griffin [@MakiGriffin] calculated the expression accounting for the electronic heat current flowing through a temperature-biased SIS JJ (where S stands for superconductor and I for insulator) [@Guttman; @GiazottoAPL; @Zhao]: $$J_{\rm SIS}(\varphi)=J_{\rm qp}-J_{\rm int}\rm\; cos \varphi.\label{Jtot}$$ Equation \[Jtot\] contains interesting information about the impact of the Josephson effect on thermal transport. First of all, $J_{\rm SIS}$ consists of two components: the former accounts for the heat carried by quasiparticles and represents an incoherent flow of energy from the hot to the cold reservoir [@GiazottoRev; @Tinkham]. On the other hand, $J_{\rm int}$ is the thermal counterpart of the “quasiparticle-pair interference” term that contributes also to the *electrical* current tunneling through a JJ [@Barone; @Pop]. It stems from energy-carrying tunneling processes involving concomitant creation and destruction of Cooper pairs on both sides of the junctions [@Barone; @Guttman] and is therefore regulated by the phase difference $\varphi$ between the two superconducting condensates. Depending on $\varphi$, this phase-coherent component can flow in the opposite direction with respect to that imposed by the temperature gradient, although the total $J_{\rm SIS}$ follows the second principle of thermodynamics, i.e., $J_{\rm qp}$ is always greater than $J_{\rm int}$. As mentioned in the introduction, this inequality represents the main limit of this approach based on Josephson tunnel junctions, which can only partially modulate electronic heat currents.
All these features have been investigated and confirmed experimentally in different interferometer-like structures, which are schematically depicted in Fig. \[Fig2\]. The first three devices have a similar structure: two normal metal electrodes N are used as electronic thermal reservoirs connected by a superconducting central part that forms the core of the interferometer. The reservoirs are also connected to superconducting wires that create SINIS junctions, which can be used as Joule heaters or thermometers (not shown) [@GiazottoRev]. By injecting a Joule power, one can heat the electrons in the source up to a temperature $T_{\rm hot}$, so as to elevate the quasiparticle temperature $T_1$ of the upper branch S$_1$ of the interferometers above $T_{\rm bath}$. On the contrary, the lower branch S$_2$ is thermally anchored to the bath temperature, thanks to its large volume. In this way, it is possible to obtain a substantial thermal gradient between S$_1$ and S$_2$, thereby generating finite heat currents $J_{\rm SIS}$ flowing through the JJs that form the central part of the structure. By applying an external magnetic flux $\Phi$ one can tune the phase polarization of the JJs [@Tinkham] and therefore manipulate $J_{\rm int}$, leading to phase-coherent oscillations of the drain temperature $T_{\rm drain}$.
{width="70.00000%"}
Figure \[Fig2\]a displays the schematic of the thermal counterpart of a symmetric DC SQUID, which offered the first experimental proof of Eq. \[Jtot\] (Ref. ). By imposing a thermal gradient across the device and varying $\Phi$, the authors observed modulations of $T_{\rm drain}$ as large as 21 mK at $T_{\rm bath}=235$ mK and $\langle T_{\rm drain}\rangle \simeq 310$ mK, as shown in Fig. \[Fig2\]b. These modulations stem from the interference between the phase-coherent components of the heat currents flowing across the JJs of the SQUID, and can be fitted with a thermal model accounting for the predominant energy-exchange mechanisms present in the system (see black line in Fig. \[Fig2\]b). The model is based on the conservation of energy and imposes that in stationary conditions the sum of the incoming and outgoing thermal currents for each electrode of the structure must be equal to zero [@GiazottoNature]. The excellent agreement between the experiment and the theory confirms the physical picture described above.
The complementary demonstration of the phase-coherent nature of $J_{\rm int}$ was obtained one year later, with the observation of a thermal diffraction pattern in an extended rectangular JJ [@MartinezNature], as depicted in Fig. \[Fig2\]c. The temperature-biased JJ between S$_1$ and S$_2$ is threaded by a magnetic flux controlled by an in-plane magnetic field $H$. This generates quantum diffraction for the heat current $J_{\rm SIS}$ and produces an archetypal Fraunhofer-like pattern \[$\propto |\mathrm{sin}(\pi \Phi/\Phi_0)/(\pi \Phi/\Phi_0)|$, where $\Phi_0$ is the superconducting flux quantum (Ref. )\] for the electronic drain temperature, as shown in Fig. \[Fig2\]d [@GiazottoDiff; @MartinezNature]. In analogy to what was done in the ’60s for the electrical Josephson current, this experiment confirmed unequivocally the thermal current-phase relationship expressed by Eq. \[Jtot\].
Although these interferometers embody the simplest structures to obtain the experimental demonstration of the prediction by Maki and Griffin, they are not ideal systems to achieve the full control of $J_{\rm int}$. Indeed, unwanted structural asymmetries (for instance, a difference in the normal-state resistance of the JJs) can reduce severely the visibility of the oscillations of a conventional single-loop SQUID [@MartinezDLoop]. To avoid this, one can replace one of the JJs with an additional SQUID, thereby realizing the Josephson thermal modulator sketched in Fig. \[Fig2\]e [@FornieriNature], in which in principle two magnetic fluxes $\Phi_1$ and $\Phi_2$ can be driven independently. This device enables the generation of exotic thermal interference patterns, which are characterized by large oscillation amplitudes, high sensitivities to magnetic flux variations, and a 99 % modulation of $J_{\rm int}$ despite the presence of a non-negligible junction asymmetry [@FornieriNature]. Figure \[Fig2\]f shows a detail around $\Phi=0$ of the temperature oscillations resulting from the double loop geometry. Foremost, this system showed a perfect correspondence in the phase engineering of charge and thermal transport [@FornieriNature], opening the way for the conception of more sophisticated phase-coherent caloritronic devices where thermal currents can be manipulated at will. In this perspective, the Josephson modulator could be the core of a thermal splitter [@BosisioPRB; @BenAbdallahAPL], able to control the amount of energy transferred among several terminals residing at different temperatures. Or even more interesting, the magnetic fluxes threading the loops could be driven independently with the help of superconducting on-chip coils. This would allow to control separately the phase-biasing of the loops, with the possibility to perform closed cycles in the flux parameter space. Combining this possibility with the thermal rectifying properties of a JJ formed by S leads with different energy gaps [@MartinezAPL] would open many opportunities to realize heat pumps [@Ren], microwave coolers [@Valenzuela; @SolinasPRB] or time-dependent thermal engines [@Campisi; @Niskanen; @Quan].
The most recent achievement in mastering $J_{\rm int}$ is represented by a $0-\pi$ phase-controllable thermal JJ. As depicted in Fig. \[Fig2\]g, the latter is embedded in a “pseudo” radio frequency (rf) SQUID containing three JJs, one of which supports a lower Josephson critical current with respect to the others [@FornieriPRB]. This configuration enables the phase-biasing of the S$_1$IS$_2$ JJ from 0 to $\pi$ (when the flux is varied from 0 to $\Phi_0/2$), thereby allowing to minimize or maximize $J_{\rm SIS}$ and to obtain unprecedented temperature modulation amplitudes ($\sim 100$ mK at $T_{\rm bath}=25$ mK and $\langle T_{\rm 1}\rangle \simeq 570$ mK, as shown in Fig. \[Fig2\]h) and sensitivities to the magnetic flux exceeding 1 K$/\Phi_0$ [@FornieriArxiv]. The fully superconducting nature of the device allows to efficiently suppress the influence of the electron-phonon coupling, leading to a remarkably high maximum operational temperature of 800 mK. Yet, as it will be clear from the next paragraphs, this structure realizes the fundamental requirement to obtain negative differential thermal conductance (NDTC) which is at the basis of non-linear thermal devices, such as tunnel heat diodes [@MartinezAPL; @FornieriRev], thermal switches and transistors [@FornieriPRB].
Figure \[Fig3\]a shows a possible design for a thermal transistor, in particular for a thermal modulator. The latter consists of a three-terminal device in which two N electrodes acting as source and gate are tunnel coupled to the S$_1$IS$_2$ JJ and reside at temperatures $T_{\rm hot}$ and $T_{\rm gate}$, respectively. The phase polarization of the JJ can be controlled from 0 to $\pi$ by the “pseudo” rf SQUID that we just discussed. We also define $J_{\rm source}$ as the thermal current flowing from the source to S$_1$, $J_{\rm gate}$ as the heat current flowing from the gate to S$_1$ and, lastly, $J_{\rm drain}\equiv J_{\rm SIS}$ represents the thermal flow across the JJ. Now, it is worth noting that besides phase-coherence the heat current $J_{\rm drain}$ exhibits another important feature. Figure \[Fig3\]b shows the calculated behavior of $J_{\rm drain}$ as a function of $T_1$ for a set drain temperature $T_{\rm drain}$ and $\delta=\Delta_2(0)/\Delta_1(0)=0.75$ \[being $\Delta_{1,2}(T_{1,2})$ the temperature-dependent energy gaps of S$_1$ and S$_2$, respectively [@Tinkham]\]. When $\Delta_1(T_1)=\Delta_2(T_2)$, the heat current presents a sharp peak for $\varphi \neq 0$, which is due to the matching of the singularities in the superconducting densities of states [@Barone]. At higher values of $T_1$ we reach the condition in which $\Delta_1<\Delta_2$ and the thermal transport across the JJ is reduced, giving rise to a region of NDTC. This effect is maximum for $\varphi=\pi$, while the peak is perfectly canceled by $J_{\rm int}$ for $\varphi=0$ [@FornieriPRB]. However, we can notice that the value of $J_{\rm drain}$ is always positive, confirming that $J_{\rm qp}$ is always greater than $J_{\rm int}$. A realistic system able to detect NDTC is the thermal counterpart of a tunnel diode (as the one envisioned in Ref. ), which could also serve as a solid-state thermal memory device. Here, we just show how NDTC in a thermal modulator can generate *heat amplification*: changes in $J_{\rm gate}$ can induce a larger change in $J_{\rm source}$ and $J_{\rm drain}$, leading to a heat amplification factor $\alpha\equiv|\partial J_{\rm source,drain}/\partial J_{\rm gate}|>1$. Figure \[Fig3\]c shows the predicted behavior of $J_{\rm source}$ and $J_{\rm drain}$ when $T_{\rm hot}>T_{\rm drain}$ and $T_{\rm gate}$ is varied as a control knob. It is clear how in region I the device can significantly reduce both $J_{\rm source}$ and $J_{\rm drain}$ while $J_{\rm gate}$ remains close to zero and almost constant. A more quantitative analysis tells us that in this region the heat amplification factor tends to infinity, but $\alpha$ turns out to be greater than 1 also in region II, which is several hundreds of mK wide [@FornieriPRB]. Alternatively, it was shown that a thermal amplifier (able to increase both $J_{\rm source}$ and $J_{\rm drain}$ as $T_{\rm gate}$ is raised above $T_{\rm drain}$) can also be designed by simply switching the position of the JJ and the N source electrode [@FornieriPRB]. Even though the variations of $J_{\rm source}$ and $J_{\rm drain}$ appear to be smaller than their absolute values, these devices represent the most accessible systems in order to obtain the first experimental demonstration of heat amplification. In the last section, we will show how a more sophisticated device based on thermoelectric effect can represent a further step towards the realization of an efficient phase-coherent thermal amplifier [@PaolucciAmpl]. The last structure we wish to present in this section is a hybrid thermal rectifier, a device that allows heat to flow preferentially in one direction. Even though the system is not phase-coherent, it proves the potential of superconducting hybrid circuits as one of the best platforms to manipulate heat currents. The structure (whose schematic is shown in Fig. \[Fig3\]d) consists of a N$_1$IN$_2$ISIN$_3$ chain that joins two theoretical proposals [@MartinezAPL; @FornieriAPL], and exhibits two different regimes of rectification. The first is based on the different temperature dependence of the DOSes of the N$_2$ and S electrodes. Indeed, the temperature dependence of the superconducting energy gap breaks the directional symmetry of thermal transport through the simple N$_2$IS junction [@MartinezAPL; @GiazottoRectAPL]. The second and most efficient mechanism relies on the asymmetric release of energy from the device to the thermal bath thanks to the N$_4$ probe acting as a cooling fin [@FornieriAPL]. In the experiment, the authors imposed a temperature gradient across the device by setting the electronic temperature of N$_1$ and N$_3$ to $T_{\rm hot}$ in the forward and reverse thermal bias configuration, respectively (in Fig. \[Fig3\]d the device is depicted in the forward configuration). Afterwards, they measured the output temperature of the structure $T_{\rm fw}$ ($T_{\rm rev}$) of the electrode N$_3$ (N$_1$) in the forward (reverse) configuration [@MartinezNatRect]. As shown in Fig. \[Fig3\]e, a maximum $\delta T=T_{\rm rev}-T_{\rm fw}$ exceeding 60 mK was observed at $T_{\rm bath}=50$ mK and $T_{\rm hot} \simeq 360$ mK (full circles, left axis). The analysis of the data provided a rectification ratio $\mathcal{R}=J_{\rm fw}/J_{\rm rev}$ (solid line, right axis), where $J_{\rm fw}$ and $J_{\rm rev}$ are the heat currents flowing from S to N$_3$ and from N$_2$ to N$_1$ in the two different temperature bias configurations. A significant maximum value of $\mathcal{R}\simeq 140$ was obtained, which outscored previous experimental results by more than two orders of magnitude [@Zettl; @Tian]. Finally it is worthwhile to emphasize that this device can be easily modified and combined with the interferometers discussed above to realize phase-coherent thermal rectifiers [@MartinezAPL; @FornieriRev].
Superconducting proximity structures {#superconducting-proximity-structures .unnumbered}
====================================
![**Superconducting proximity structures.** **a.** Scheme of a system that can be exploited to detect the phase-cherent modulation of the heat current $J_{\rm e-phon}$ due to the electron-phonon coupling in a proximity wire S’ (made of Cu) inserted in a superconducting loop S (made of Al with $T_{\rm c}=1.3$ K). The electrode S’ can be tunnel coupled to other superconducting electrodes that can be used as Joule heaters and thermometers (not shown) [@GiazottoRev]. **b.** Calculated electronic temperature $T_1$ of the S’ electrode as a function of the magnetic flux $\Phi$ threading the loop for different values of the normalized injected Joule power $J_{\rm in}^{\rm norm}$ (for details, see Ref. ) at $T_{\rm bath}=50$ mK. **c.** Schematic representation of a proximity heat valve. The structure is similar to the one shown in panel a, but with a normal metal probe N (made of Al$_{0.98}$Mn$_{0.02}$) connected to S’ by means of a tunnel junction. When S and S’ are heated up to the electronic temperature $T_{\rm hot}$, the heat current $J_{\rm S'IN}$ flows across the tunnel junction S’IN and its amplitude depends on the magnetic flux $\Phi$. **d.** Calculated electronic temperature $T_1$ of the N electrode vs. $\Phi$ for different values of $T_{\rm hot}$ at $T_{\rm bath}=20$ mK. In panels a, b, c and d, we assumed $T_{\rm hot}>T_1>T_{\rm bath}$. **e.** Normalized thermal conductance $\kappa/\kappa^{\rm N}$ of the S’IN junction vs. $T\equiv (T_{\rm hot} + T_1)/2$ calculated for $T_{\rm hot} - T_1 \ll T$ and for different values of $\Phi$. Here, $\kappa^{\rm N}$ is the normal state conductance. **f.** Normalized thermal conductance $\kappa/ \kappa^0$ as a function of $T$ for different values of $\Phi$, where $\kappa^0$ is the thermal conductance calculated for $\Phi=0$. In panels e and f, the Dynes parameter in S’ is assumed to be $\Gamma=10^{-4} E_{\rm g}$ [@Dynes; @StrambiniAPL].\[Fig4\]](Fig4.pdf){width="0.98\columnwidth"}
As mentioned in the introduction, the second approach to phase-coherent caloritronics is based on the superconducting proximity effect [@DeGennes]. When a normal metal (S’) is brought in contact with a superconductor, the superconducting order parameter leaks out to the normal side. The main consequences of this effect are the changes in the local DOS [@Usadel; @Zhou; @leSueur] and the induction of a finite pair amplitude in the normal metal. For instance, in a SS’S JJ the proximity effect generates a phase-tunable minigap $E_{\rm g}(\varphi)$ in the DOS of the weak-link that has a maximum for $\varphi=0$ and vanishes for $\varphi=\pi$ [@leSueur; @Virtanen]. This offers the opportunity to manipulate in a continuous fashion the thermal properties of the S’ electrode, including its electronic entropy and specific heat [@RabaniJAP; @RabaniPRB], which can be varied from those of a superconductor to those of a normal metal. Even more interesting, also the relaxation mechanisms existing in S’ can be controlled by tuning the value of $E_{\rm g}$ [@HeikkilaPRB]. Indeed, as noted in the introduction, the presence of the energy gap in the DOS exponentially suppresses the electron-phonon coupling in a superconductor [@Timofeev1; @Wellstood].
Figure \[Fig4\]a shows a possible configuration to detect the phase modulation of the electron-phonon coupling in a S’ weak-link assumed to be made of Cu. The latter interrupts an Al loop S (with $T_{\rm c}=1.3$ K) that is used to phase-polarize the JJ by means of an external magnetic flux $\Phi$ [@Tinkham]. When $|\Phi|=k \Phi_0$ (being $k$ an integer), $E_{\rm g}$ is maximum and by heating S’ the heat current $J_{\rm e-phon}$ released by electrons to the phonon bath is minimized. Thus, for these values of $\Phi$ the S’ electronic temperature $T_1$ reaches a maximum, whereas it shows a minimum for semi-integer values of the flux quantum (i.e., for $E_{\rm g}=0$), as shown in Fig. \[Fig4\]b. The periodic modulations of $T_1$ can exhibit a remarkable maximum amplitude of $\sim 30$ mK at $T_{\rm bath}=50$ mK and for $\langle T_1 \rangle = 90$ mK [@HeikkilaPRB], but it is possible to envision an even more efficient way to exploit the phase-dependence of $E_{\rm g}$.
Indeed, one can think of the superconducting energy gap as a barrier for energy-carrying quasiparticles. The possibility to tune phase-coherently the height of this barrier leads promptly to the design of a valve for electronic heat currents [@StrambiniAPL]. The proposed structure is shown on Fig. \[Fig4\]c, and consists of a SQUIPT [@SQUIPT1; @SQUIPT2; @SQUIPT3], i.e. an Al loop S interrupted by a normal wire S’ (made of Cu), which is tunnel coupled to a normal metal probe N (made of Al$_{0.98}$Mn$_{0.02}$). If the electronic temperature of S and S’ is raised up to $T_{\rm hot}$, we can obtain modulations of the N temperature $T_1$ with amplitudes exceeding 100 mK at $T_{\rm bath}= 20$ mK and $\langle T_1 \rangle = 100$ mK.
We stress that, although not yet experimentally proven, the approach described in this section has the potential to be very effective thanks to the ability to vary the thermal conductance of the S’IN junction by several orders of magnitude, as shown in Figs. \[Fig4\]e and \[Fig4\]f. This is in contrast to previous realizations of thermal Andreev interferometers [@Chandrasekhar1; @Chandrasekhar2; @Vinokur] and to the Josephson circuits analyzed in the previous section, which are not able to control the incoherent component of the heat current $J_{\rm qp}$.
Photonic heat transistors {#photonic-heat-transistors .unnumbered}
=========================
![**Photonic heat transistors.** **a.** Scheme of the first system used to demonstrate phase-coherent photonic heat conduction and its quantum limit. Two N reservoirs (made of palladium-gold) residing at $T_1>T_2>T_{\rm bath}$ are coupled by an intermediate circuit consisting of two DC SQUIDs made of Al, connected by superconducting lines *in clean contact* with the reservoirs. These lines act as ideal insulators for galvanic thermal transport between the N electrodes at low temperatures, thanks to the Andreev reflection mechanism [@Tinkham; @Zhao] and the presence of the energy gap. The magnetic flux $\Phi$ piercing the loops can vary the Josephson inductance of the SQUIDs, thus regulating the electromagnetic coupling between the N electrodes, and modulating the photonic heat current $J_{\rm e-phot}(\Phi)$. **b.** Electronic temperature $T_1$ modulations as a function of $\Phi$ for a parasitic injected power of $1$ fW and for different values of $T_{\rm bath}$. Data are taken from Ref. . **c.** Sketch of a non-galvanic photon thermal transistor. In this case, the intermediate circuit is a DC SQUID (made of Al) *inductively* or *capacitevily* coupled to the N reservoirs (made of Al$_{0.98}$Mn$_{0.02}$). The temperature of the right reservoir $T_2$ can be controlled by the change of the Josephson inductance when $\Phi$ is varied. **d.** Calculated magnetic interference pattern of $T_2$ for different values of $T_{\rm hot}$ at $T_{\rm bath}=10$ mK in the case of an inductive coupling. Here, we set a mutual inductance of $500$ pH, the capacitance of the SQUID JJs of $2$ fF and the maximum Josephson inductance of $250$ pH. In all the panels we assumed $T_{\rm hot}>T_1>T_2>T_{\rm bath}$. \[Fig5\]](Fig5.pdf){width="1\columnwidth"}
Last approach to obtain phase control of heat currents is through the transport of thermal photons, which becomes the dominant mechanism when the phononic and the electronic channels are frozen [@schmidt; @Meschke]. Figure \[Fig5\]a shows the first experimental configuration that was used to detect and phase control the photonic thermal transport between two N reservoirs with finite resistances $R_1$ and $R_2$ [@Meschke]. When the electronic temperature of the left reservoir is raised up to $T_1>T_{\rm bath}$, the electromagnetic noise power radiated from this resistor generates a net photonic power $J_{\rm e-phot}$ flowing towards the other reservoir residing at temperature $T_2$ (with $T_1>T_2>T_{\rm bath}$). Since the dimensions of the circuit are typically much smaller than the photon thermal wavelength ($\lambda_{\rm phot}>1$ cm at temperatures below 1 K) [@schmidt; @Meschke], it is possible to consider the structure as a lumped series of equivalent circuits [@schmidt; @Pascal; @Paolucci]. It has been shown that this circuital approach is equivalent to the analysis employing nonequilibrium Green’s functions [@Ojanen]. It can be demonstrated that $J_{\rm e-phot}$ is proportional to a frequency-dependent transfer function [@schmidt; @Pascal]: $$\mathcal{T}(\omega)=\frac{4 \Re [Z_1 (\omega)] \Re [Z_2 (\omega)]}{|Z_{\rm tot}(\omega)|^2},$$ where $Z_{1,2}(\omega)$ are the impedances of the N reservoirs, $Z_{\rm tot}(\omega)=Z_1(\omega)+Z_2(\omega)+Z_{\rm c}(\omega)$ is the total series impedance of the circuit and $Z_{\rm c}(\omega)$ is the impedance of the coupling circuit. In the case depicted in Fig. \[Fig5\]a, the coupling circuit consists of two DC SQUIDs connected to the N electrodes \[made of palladium-gold, with $J_{\rm e-ph}\sim (T_{\rm e}^5-T_{\rm bath}^5)$\] through superconducting Al lines, which act as ideal thermal insulators at low temperatures, thanks to the Andreev reflection mechanism [@Tinkham; @Zhao] and the presence of the energy gap. The SQUIDs are equivalent to two LC circuits with a variable Josephson inductance that depends on the magnetic flux $\Phi$ piercing the loops. Thus, also $\mathcal{T}$ and $J_{\rm e-phot}$ become flux-dependent, allowing a phase control of the photonic heat flux across the circuit. In this way, modulations of $T_1$ (up to $\sim 6$ mK of amplitude) as a function of $\Phi$ have been observed up to $T_{\rm bath}\simeq 150$ mK and for $\langle T_1 \rangle$ slightly higher than $T_{\rm bath}$, as shown in Fig. \[Fig5\]b. This experiment was one of the first demonstrations of phase-coherent caloritronics, and proved that photonic heat conduction can reach the quantum limit when $Z_1=Z_2$ and $Z_{\rm c}$ is minimized [@Meschke].
One step beyond this experiment is based on the concept of a fully contactless photonic heat transistor [@Ojanen; @Pascal; @Paolucci], schematically represented in Fig. \[Fig5\]c. In this case, the N reservoirs (assumed to be made of Al$_{0.98}$Mn$_{0.02}$) are *inductively* or *capacitively* coupled to the intermediate circuit, which is based on a DC SQUID. The latter can be treated as a purely reactive LC circuit with a variable Josephson inductance. By heating the left reservoir up to $T_{\rm hot}$ yields flux-dependent $T_2$ modulations in the other remote N electrode, as displayed by the calculated curves in Fig. \[Fig5\]d. These theoretical curves have been obtained by assuming an inductive coupling with a mutual inductance of $500$ pH for different values of $T_{\rm hot}$ at $T_{\rm bath}=10$ mK and show remarkably large modulations up to 200 mK for $T_{\rm hot} =700$ mK and $\langle T_2 \rangle =170$ mK. Similar results can be obtained via a capacitive coupling, which turns out to be even more efficient and accessible from the fabrication point of view [@Pascal; @Paolucci].
The photonic method opens the way to the investigation and exploitation of non-galvanic thermal transport, which would lead to wireless electronic cooling and to the remote control of noise and decoherence in mesoscopic quantum circuits. Although its effectiveness is maximized in the case of N leads, it has been predicted that the photonic coupling between S leads could still produce a significant thermal flow (two orders of magnitude higher than that produced by the electron-phonon coupling) [@BosisioPRB2]. This possibility could lead to non-galvanic refrigerators for superconducting quantum circuits, in which a precise qubit initialization is required. For instance, a SINIS cooler [@GiazottoRev; @MuhonenRev] could be capacitively coupled by means of a DC SQUID to a part of the superconducting qubit structure. For the same purpose, a notable alternative approach has been experimentally demonstrated by exploiting photon assisted tunneling in a NIS junction, which can be used as a quantum circuit refrigerator [@MottonenArx]. It is also worth mentioning recent experimental results that proved quantum-limited heat conduction over macroscopic distances (up to a meter) [@Partanen], leading to even more possibilities for remote cooling. Finally, the same kind of systems could also be the basis of photonic thermal rectifiers [@Ruokola].
Applications and future directions {#applications-and-future-directions .unnumbered}
==================================
![**Future directions.** **a.** Schematic representation of a microwave Josephson refrigerator, which consists of a DC SQUID formed by two superconductors S$_1$ and S$_2$ biased by a small current $I_{\rm bias}$. The superconducting loop is pierced by a time-dependent magnetic flux $\Phi (t)$ that can induce a finite voltage bias across the device, which generates a heat current $J_{\rm SIS,AC}$ flowing through the JJs. The latter is used to cool the electronic temperature $T_1$ of S$_1$ below $T_{\rm bath}$. In order to obtain an efficient cooling power, it is necessary to have $\delta\equiv \Delta_2(0)/\Delta_1(0)>1$ and a finite capacitance of the JJs forming the SQUID to rectify $J_{\rm SIS,AC}$ [@SolinasPRB]. **b.** Sketch of a phase-coherent thermal amplifier based on the thermoelectric effect in a normal metal-ferromagnetic insulator-superconductor junction (assumed to be made of Cu-europium sulfide-Al). When the input temperature $T_{\rm in}$ is increased above $T_{\rm bath}$, a finite thermoelectric current can flow through a closed circuit including a superconducting coil. By means of the mutual inductance $M$, the latter generates a finite flux $\Phi$ that can control the thermal current flowing across a temperature-biased heat valve (presented in Fig. \[Fig4\]c). The S and S’ parts of the valve are biased at $T_{\rm supply}$, while the N electrode resides at the output temperature of the amplifier $T_{\rm out}$ [@PaolucciAmpl]. **c.** Calculated $T_{\rm out}$ vs. $T_{\rm in}$ for different values of $M$ at $T_{\rm bath}=10$ mK and $T_{\rm supply}=250$ mK. **d.** Temperature differential gain $G=dT_{\rm out}/dT_{\rm in}$ as a function of $T_{\rm in}$ calculated for the curves shown in panel c. In panels c and d the arrows indicate increasing values of $M$. \[Fig6\]](Fig6.pdf){width="1\columnwidth"}
All the structures discussed above can be implemented by conventional nanofabrication techniques, i.e., electron beam lithography, shadow mask evaporation of metals and *in-situ* oxidation [@Meschke; @GiazottoNature; @MartinezNature; @MartinezNatRect; @FornieriNature; @FornieriArxiv]. They would join the well-known systems used for heating, cooling and thermometry in mesoscopic superconducting circuits [@GiazottoRev], which now embody a unique playground for investigating heat transport at the nanoscale. Moreover, this platform has already demonstrated that high levels of thermal isolation from the environment are achievable [@Wei; @Govenius], showing significant room for improvement in the performance of caloritronic devices.
Phase-coherent caloritronics would offer many new possibilities to this field, as for instance the microwave Josephson refrigerator [@SolinasPRB] shown in Fig. \[Fig6\]a. The latter consists of a DC SQUID made of two different superconductors S$_1$ and S$_2$ characterized by $\delta>1$ (indicating the thermal asymmetry in the system) and pierced by a time-dependent magnetic flux $\Phi(t)$. The latter induces phase oscillations across the JJs, which generate a finite voltage bias across the device and leads to the active cooling of S$_1$. The simplicity and the scalability of this system make it attractive for many superconducting quantum circuits that could be cooled at distance, thanks to high frequency modulations of $\Phi(t)$. Together with thermal rectifiers and non-galvanic photonic refrigerators, this system could be useful to evacuate unwanted hot quasiparticles from qubit architectures in preparation for quantum operations, thus improving their performance against decoherence. Beyond quantum information [@NielsenChuang], phase-coherent caloritronics would certainly benefit many fields of nanoscience requiring an accurate administration of heat, including solid-state cooling [@GiazottoRev], energy harvesting, thermal isolation and radiation detection [@GiazottoRev]. The advent of heat transistors and thermal memories could also pave the way to a new field called thermal logic [@LiRev], in which information is transferred, processed and stored under the form of energy. The latter could represent one of the most intriguing opportunities to exploit the (otherwise wasted) power dissipated by electronic circuits. In this context, another interesting proposal is an Al DC SQUID with a non-negligible inductance $L$. For significant values of $L$, the temperature interference pattern generated by the SQUID should exhibit a remarkable hysteresis, that could be used to obtain a thermal memory device [@Guarcello; @Guarcello2].
Furthermore, caloritronic superconducting circuits might be combined with hybrid platforms based on, e.g., semiconducting nanowires [@Mourik; @Mastomaki], graphene [@Yokohama; @Rainis; @Paolucci], ferromagnets or ferromagnetic insulators (FI) [@Kawabata; @GiazottoBergeret1; @GiazottoBergeret2; @GiazottoBergeret3], topological insulators [@Zhang; @Ren1; @Ren2; @Sothmann1; @Sothmann2] or low dimensional electronic devices to enhance their functionalities. For instance, Ref. theoretically analyzes a photonic heat transistor based on graphene reservoirs, whose carrier densities can be independently tuned thanks to two electrostatic gates. The latter can therefore control the photonic transfer function and the electron-phonon coupling in each reservoir, leading to increased temperature modulation amplitudes [@Paolucci]. Even more interesting, Fig. \[Fig6\]b shows how the combination of a N-FI-S junction (assumed to be made of Cu-europium sulfide-Al) and a proximity heat valve, as the one discussed previously, can be used to design a very efficient temperature amplifier [@PaolucciAmpl]. When the input temperature $T_{\rm in}$ of the N electrode is increased above $T_{\rm bath}$, a finite thermoelectric current $I_{\rm T}$ can flow through a closed circuit including a superconducting coil. By means of the mutual inductance $M$, the latter generates a finite flux $\Phi$ that can control the thermal current flowing across a temperature-biased heat valve (presented in Fig. \[Fig4\]c). The S and S’ parts of the valve are biased at $T_{\rm supply}>T_{\rm bath}$, while the N electrode resides at the output temperature of the amplifier $T_{\rm out}$. Fig. \[Fig6\]c displays the behavior of $T_{\rm out}$ as a function of $T_{\rm in}$ for different values of $M$ between the coil and the heat valve at $T_{\rm bath}=10$ mK and $T_{\rm supply}=250$ mK. While the minimum and the maximum values of $T_{\rm out}$ are determined by $T_{\rm bath}$ and $T_{\rm supply}$, the value of $T_{\rm in}$ corresponding to the maximum output temperature decreases as the inductive coupling is raised. The temperature differential gain $G=dT_{\rm out}/dT_{\rm in}$ is shown in Fig. \[Fig6\]d and exceeds 10 in a large range of parameters. Although appearing more challenging from a fabrication point of view, this device can provide output temperatures in the same range as its input temperatures, thus representing a crucial element for the realization of thermal logic gates [@LiRev].
Finally, phase-coherent caloritronics can shed light on several fundamental energy- and heat-related phenomena at the nanoscale, such as quantum thermodynamics [@PekolaNat], heat transport in topological states of matter [@Zhang; @Ren1; @Ren2; @Sothmann1; @Sothmann2] and give rise to phase-coherent thermoelectric effects in superconducting hybrid circuits [@HeikkilaGiazotto; @GiazottoMoodera; @GiazottoDubi].
References {#references .unnumbered}
==========
[99]{} Dubi, Y. & Di Ventra, M. Heat flow and thermoelectricity in atomic and molecular junctions. *Rev. Mod. Phys.* **83**, 131-155 (2011). Schwab, K., Henriksen, E. A., Worlock, J. M. & Roukes, M. L. Measurement of the quantum of thermal conductance. *Nature* **404**, 974-977 (2000). Meschke, M., Guichard, W. & Pekola, J. P. Single-mode heat conduction by photons. *Nature* **444**, 187-190 (2006). Jezouin, *et al.* Quantum limit of heat flow across a single electronic channel. *Science* **342**, 601-604 (2013). Bauer, G. E. W., Saitoh, E. & van Wees, B. J. Spin caloritronics. *Nat. Mater.* **11**, 391–399 (2012). Giazotto, F., Heikkilä, T. T., Luukanen, A., Savin, A. M. & Pekola, J. P. Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications. *Rev. Mod. Phys.* **78**, 217-274 (2006). Muhonen, J. T., Meschke, M. & Pekola, J. P. Micrometre-scale refrigerators. *Rep. Prog. Phys.* **75**, 046501 (2012). Tan, K. Y. *et al.* Quantum circuit refrigerator. *Nat. Commun.* **8**, 15189 (2017). Li, N. *et al*. Phononics: manipulating heat flow with electronic analogs and beyond. *Rev. Mod. Phys.* **84**, 1045-1066 (2012). Chang, C. W., Okawa, D., Majumdar, A. & Zettl, A. Solid-state thermal rectifier. *Science* **314**, 1121–1124 (2006). Tian, H. *et al.* A novel solid-state thermal rectifier based on reduced graphene oxide. *Sci. Rep.* **2**, 523 (2012). Eom, J., Chien, C.-J., & Chandrasekhar, V. Phase dependent thermopower in Andreev interferometers. *Phys. Rev. Lett.* **81**, 437-440 (1998). Dikin, D. A., Jung, S., & Chandrasekhar, V. Low-temperature thermal properties of mesoscopic normal-metal/superconductor heterostructures. *Phys. Rev. B* **65**, 012511 (2001). Bezuglyi, E. V. & Vinokur, V. Heat transport in proximity structures. *Phys. Rev. Lett.* **91**, 137002 (2003). Giazotto, F. & Martínez-Pérez, M. J. The Josephson heat interferometer. *Nature* **492**, 401-405 (2012). Martínez-Pérez, M. J., Solinas, P. & Giazotto, F. Coherent caloritronics in Josephson-based nanocircuits. *J. Low Temp. Phys.* **175**, 813-837 (2014). Maki, K. & Griffin, A. Entropy transport between two superconductors by electron tunneling. *Phys. Rev. Lett.* **15**, 921-923 (1965). Giazotto, F. & Martínez-Pérez, M. J. Phase-controlled superconducting heat-flux quantum modulator. *Appl. Phys. Lett.* **101**, 102601 (2012). Martínez-Pérez, M. J. & Giazotto, F. A quantum diffractor for thermal flux. *Nat. Commun.* **5**, 3579 (2014). Martínez-Pérez, M. J., Fornieri, A. & Giazotto, F. Rectification of electronic heat current by a hybrid thermal diode. *Nat. Nanotechnol.* **10**, 303-307 (2015). Fornieri, A., Blanc, C., Bosisio, R., D’Ambrosio, S. & Giazotto, F. Nanoscale phase engineering of thermal transport with a Josephson heat modulator. *Nat. Nanotechnol.* **11**, 258-262 (2016). Fornieri, A., Timossi, G., Virtanen, P., Solinas, P. & Giazotto, F. 0-$\pi$ phase-controllable *thermal* Josephson junction. *Nat. Nanotechnol.* **12**, 425-429 (2017). Giazotto, F., Peltonen, J. T., Meschke, M. & Pekola, J. P. Superconducting quantum interference proximity transistor. *Nat. Phys.* **6**, 254-259 (2010). Ronzani, A., Altimiras, C. & Giazotto, F. Highly sensitive superconducting quantum-interference proximity transistor. *Phys. Rev. Appl.* **2**, 024005 (2014). D’Ambrosio, S., Meissner, M., Blanc, C., Ronzani, A. & Giazotto, F. Normal metal tunnel junction-based superconducting quantum interference proximity transistor. *Appl. Phys. Lett.* **107**, 113110 (2015). Zhou, F., Charlat, P., Spivak, B. & Pannetier, B. Density of states in superconductor-normal metal-superconductor junctions. *J. Low Temp. Phys.* **110**, 841-850 (1998). le Sueur, H., Joyez, P., Pothier, H., Urbina, C. & Esteve, D. Phase controlled superconducting proximity effect probed by tunneling spectroscopy. *Phys. Rev. Lett.* **100**, 197002 (2008). Strambini, E., Bergeret, F. S. & Giazotto, F. Proximity nanovalve with large phase-tunable thermal conductance. *Appl. Phys. Lett.* **105**, 082601 (2014). Ojanen, T. & Jauho, A.-P. Mesoscopic photon heat transistor. *Phys. Rev. Lett.* **100**, 155902 (2008). Pascal, L. M. A., Courtois, H. & Hekking, F. W. J. Circuit approach to photonic heat transport. *Phys. Rev. B* **83**, 125113 (2011). Wellstood, F. C., Urbina, C. & Clarke, J. Hot-electron effects in metals. *Phys. Rev. B* **49**, 5942-5955 (1994). Timofeev, A. V. *et al*. Recombination-limited energy relaxation in a Bardeen-Cooper-Schrieffer superconductor. *Phys. Rev. Lett.* **102**, 017003 (2009). Taskinen, L. J. & Maasilta, I. J. Improving the performance of hot-electron bolometers and solid state coolers with disordered alloys. Appl. Phys. Lett. **89**, 143511 (2006). Josephson, B. D. Possible new effects in superconductive tunneling. *Phys. Lett.* **1**, 251-253 (1962). Guttman, G. D., Nathanson, B., Ben-Jacob, E. & Bergman, D. J. Phase-dependent thermal transport in Josephson junctions. *Phys. Rev. B* **55**, 3849-3855 (1997). Zhao, E., Löfwander, T. & Sauls, J. A. Heat transport through Josephson point contacts. *Phys. Rev. B* **69**, 134503 (2004). Tinkham, M. *Introduction to Superconductivity* (McGraw-Hill, 1996) and references therein. A. Barone and G. Paternò, *Physics and Applications of the Josephson Effect* (Wiley, New York, 1982). I. M. Pop *et al.* Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles. *Nature* **508**, 369-372 (2014). Giazotto, F., Martínez-Pérez, M. J. & Solinas, P. Coherent diffraction of thermal currents. *Phys. Rev. B* **88**, 094506 (2013). Martínez-Pérez, M. J. & Giazotto, F. Fully balanced heat interferometer. *Appl. Phys. Lett.* **102**, 092602 (2013). Bosisio, R. *et al*. A magnetic thermal switch for heat management at the nanoscale. *Phys. Rev. B* **91**, 205420 (2015). Ben-Abdallah, P., Belarouci, A., Frechette, L. & Biehs, S.-A. Heat flux splitter for near-field thermal radiation. *Appl. Phys. Lett.* **107**, 053109 (2015). Martínez-Pérez, M. J. & Giazotto, F. Efficient phase-tunable Josephson thermal rectifier. *Appl. Phys. Lett.* **102**, 182602 (2013). Ren, J., Hänggi, P. & Li, B. Berry-phase-induced heat pumping and its impact on the fluctuation theorem. *Phys. Rev. Lett.* **104**, 170601 (2010). Valenzuela, S. O. *et al.* Microwave-induced cooling of a superconducting qubit. *Science* **314**, 1589-1592 (2006). Solinas, P., Bosisio, R. & Giazotto, F. Microwave quantum refrigeration based on the Josephson effect. *Phys. Rev. B* **93**, 224521 (2016). Campisi, M., Pekola, J. & Fazio, R. Nonequilibrium fluctuations in quantum heat engines: theory, example, and possible solid state experiments. *New J. Phys.* **17**, 035012 (2015). Niskanen, A. O., Nakamura, Y. & Pekola J. Information entropic superconducting microcooler. *Phys. Rev. B* **76**, 174523 (2007). Quan, H. T., Wang, Y. D., Liu, Yu-xi, Sun, C. P. & Nori, F. Maxwell’s Demon Assisted Thermodynamic Cycle in Superconducting Quantum Circuits. *Phys. Rev. Lett.* **97**, 180402 (2006). Fornieri, A., Timossi, G., Bosisio, R., Solinas, P. & Giazotto, F. Negative differential thermal conductance and heat amplification in superconducting hybrid devices. *Phys. Rev. B* **93**, 134508 (2016). Fornieri, A., Martínez-Pérez, M. J. & Giazotto, F. Electronic heat current rectification in hybrid superconducting devices. *AIP Adv.* **5**, 053301 (2015). Paolucci, F., Marchegiani, G., Strambini, E. & Giazotto, F. Phase-coherent temperature amplifier. arXiv:1612.0017 (2016). Fornieri, A., Martínez-Pérez, M. J. & Giazotto, F. A normal metal tunnel-junction heat diode. *Appl. Phys. Lett.* **104**, 183108 (2014). Giazotto, F. & Bergeret, F. S. Thermal rectification of electrons in hybrid normal metal-superconductor nanojunctions. *Appl. Phys. Lett.* **103**, 242602 (2013). de Gennes, P. G. & Saint-James, D. Elementary excitations in the vicinity of a normal metal-superconducting metal contact. *Phys. Lett.* **4**, 151-152 (1963). Usadel, K. D. Generalized Diffusion Equation for Superconducting Alloys. *Phys. Rev. Lett.* **25**, 507-509 (1970). Virtanen, P., Ronzani, A. & Giazotto, F. Spectral characteristics of a fully-superconducting SQUIPT. *Phys. Rev. B* **95**, 144512 (2017). Rabani, H., Taddei, F., Bourgeois, O., Fazio, R. & Giazotto, F. Phase-dependent electronic specific heat of mesoscopic Josephson junctions. *Phys. Rev. B* **78**, 012503 (2008). Rabani, H., Taddei, F., Giazotto, F. & Fazio, R. Influence of interface transmissivity and inelastic scattering on the electronic entropy and specific heat of diffusive superconductor-normal metal-superconductor Josephson junctions. *J. Appl. Phys.* **105**, 093904 (2009). Heikkilä, T. T. & Giazotto, F. Phase sensitive electron-phonon coupling in a superconducting proximity structure. *Phys. Rev. B* **79**, 094514 (2009). R. C. Dynes, V. Narayanamurty & J. P. Garno. Direct measurement of quasiparticle-lifetime broadening in a strong-coupled superconductor. *Phys. Rev. Lett.* **41**, 1509-1512 (1978). Schmidt, D. R., Schoelkopf, R. J. & Cleland, A. N. Photon-mediated thermal relaxation of electrons in nanostructures. *Phys. Rev. Lett.* **93**, 045901 (2004). Paolucci, F., Timossi, G., Solinas, P. & Giazotto, F. Coherent manipulation of thermal transport by tunable electron-photon and electron-phonon interaction. *J. Appl. Phys.* **121**, 244305 (2017). Bosisio, R., Solinas, P., Braggio, A. & Giazotto, F. Photonic heat conduction in Josephson-coupled Bardeen-Cooper-Schrieffer superconductors. *Phys. Rev. B* **93**, 144512 (2016). Partanen, M. *et al.* Quantum-limited heat conduction over macroscopic distances. *Nat. Phys.* **12**, 460-464 (2016). Ruokola, T., Ojanen, T. & Jauho, A.-P. Thermal rectification in nonlinear quantum circuits. *Phys. Rev. B* **79**, 144306 (2009). Wei, J. *et al.* Ultrasensitive hot-electron nanobolometers for terahertz astrophysics. *Nat. Nanotechnol.* **3**, 496-500 (2008). Govenius, J., Lake, R. E., Tan, K. Y. & Möttönen, M. Detection of zeptojoule microwave pulses using electrothermal feedback in proximity-induced Josephson junctions.. *Phys. Rev. Lett.* **117**, 030802 (2016). Nielsen, M. A. & Chuang, I. L. *Quantum Computation and Quantum Information*, (Cambridge University Press, 2002). Guarcello, C., Solinas, P., Di Ventra, M. & Giazotto, F. Hysteretic superconducting heat-flux quantum modulator. *Phys. Rev. Applied* **7**, 044021 (2017). Guarcello, C., Solinas, P., Braggio, A., Di Ventra, M. & Giazotto, F. A Josephson thermal memory. arXiv:1706.05323 (2017). Mourik, V. *et al.* Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. *Science* **336**, 1003–1007 (2012). Mastomäki, J. *et al.* InAs nanowire superconducting tunnel junctions: spectroscopy, thermometry and nanorefrigeration. *Nano Res.*, doi:10.1007/s12274-017-1558-7 (2017). Yokoyama, T., Linder, J. & Sudbø, A. Heat transport by Dirac fermions in normal/superconducting graphene junctions. *Phys. Rev. B* **77**, 132503 (2008). Rainis, D., Taddei, F., Dolcini, F., Polini, M. & Fazio, R. Andreev reflection in graphene nanoribbons. *Phys. Rev. B* **79**, 115131 (2009). Kawabata, S., Ozaeta, A., Vasenko, A. S., Hekking, F. W. J. & Bergeret, F. S. Efficient electron refrigeration using superconductor/spin-filter devices. *Appl. Phys. Lett.* **103**, 032602 (2013). Bergeret, F. S. & Giazotto, F. Manifestation of a spin-splitting field in a thermally biased Josephson junction. *Phys. Rev. B* **89**, 054505 (2014). Bergeret, F. S. & Giazotto, F. Phase-dependent heat transport through magnetic Josephson tunnel junctions. *Phys. Rev. B* **88**, 014515 (2013). Giazotto, F. & Bergeret, F. S. Phase-tunable colossal magnetothermal resistance in ferromagnetic Josephson valves. *Appl. Phys. Lett.* **102**, 132603 (2013). Wang, Z., Qi, X.-L. & Zhang, S.-C. Topological field theory and thermal responses of interacting topological superconductors. *Phys. Rev. B* **84**, 014527 (2011). Ren, J. & Zhu, J.-X. Anomalous energy transport across topological insulator superconductor junctions. *Phys. Rev. B* **87**, 165121 (2013). Ren, J. & Zhu, J.-X. Asymmetric Andreev reflection induced electrical and thermal Hall-like effects in metal/anisotropic superconductor junctions. *Phys. Rev. B* **89**, 064512 (2014). Sothmann, B. & Hankiewicz, E. M. Fingerprint of topological Andreev bound states in phase-dependent heat transport. *Phys. Rev. B* **94**, 081407(R) (2016). Sothmann, B., Giazotto, F. & Hankiewicz, E. M. High-efficiency thermal switch based on topological Josephson junctions. *New J. Phys.* **19**, 023056 (2017). Pekola, J. Towards quantum thermodynamics in electronic circuits. *Nat. Phys.* **11**, 118-123 (2015). Giazotto, F., Heikkilä, T. T. & Bergeret, F. S. Very Large Thermophase in Ferromagnetic Josephson Junctions. *Phys. Rev. Lett.* **114**, 067001 (2015). Giazotto, F., Robinson, J. W. A., Moodera, J. S. & Bergeret, F. S. Proposal for a phase-coherent thermoelectric transistor. *Appl. Phys. Lett.* **105**, 062602 (2014). Kleeorin, Y., Meir, Y., Giazotto, F. & Dubi, Y. Large tunable thermophase in superconductor - quantum dot - superconductor Josephson junctions. *Sci. Rep.* **6**, 35116 (2016).
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank J. P. Pekola and M. Meschke for providing experimental data. We also thank F. Paolucci, G. Timossi, E. Strambini and L. Casparis for fruitful discussions. The MIUR-FIRB2013–Project Coca (grant no. RBFR1379UX) and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 615187 - COMANCHE are acknowledged for partial financial support.
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abstract: 'The hydrogen plasma is studied in the very high density (atomic and metallic) regime by extensive ab initio Molecular Dynamics simulations. Protons are treated classically, and electrons in the Born- Oppenheimer framework, within the local density approximation (LDA) to density functional theory. Densities and temperatures studied fall within the strong coupling regime of the protons. We address the question of the validity of linear screening, and we find it to yield a reasonably good description up to $r_s\approx 0.5$, but already too crude for $r_s=1$ (with $r_s=(3/4\pi\rho)^{1/3}$ the ion sphere radius). These values are typical of Jovian planets interiors. Finite-size and Brillouin zone sampling effects in metallic systems are studied and shown to be very delicate also in the fluid (liquid metal) phase. We analyse the low-temperature phase diagram and the melting transition. A remarkably fast decrease of the melting temperature with decreasing density is found, up to a point when it becomes comparable to the Fermi temperature of the protons. The possible vicinity of a triple point bcc-hcp(fcc)-liquid is discussed in the region of $r_s\approx 1.1$ and $T\approx 100~-~200~K$. The fluid phase is studied in detail for several temperatures. The structure of the fluid is found to be reminiscent of the underlying bcc (solid) phase. Proton-electron correlations show a weak temperature dependence, and proton-proton correlations exhibit a well-defined first coordination shell, thus characterizing fluid H in this regime as an atomic liquid. Diffusion coefficients are computed and compared to the values for the one-component plasma (OCP). Vibrational densities of states (VDOS) show a plasmon renormalization due to electron screening, and the presence of a plasmon-coupled single-particle mode up to very high temperatures. Collective modes are studied through dynamical structure factors. In close relationship with the VDOS, the simulations reveal the remarkable persistence of a weakly damped high-frequency ion acoustic mode, even under conditions of strong electron screening. The possibility of using this observation as a diagnostic for the plasma phase transition to the fluid molecular phase at lower densities is discussed.'
address:
- ' $^{\#}$ International Centre for Theoretical Physics, Strada Costiera 11, I-34014 Trieste, Italy'
- ' $^*$ Laboratoire de Physique (U.R.A. 1325 du CNRS), Ecole Normale Supérieure de Lyon, F-69364 Lyon Cedex 07, France'
author:
- 'Jorge Kohanoff$^{~\# *}$ and Jean-Pierre Hansen$^*$'
title: 'Statistical Properties of the Dense Hydrogen Plasma: an ab initio Molecular Dynamics Investigation'
---
Introduction and state of the art
=================================
The possibility of hydrogen metallization under high pressure was first discussed by Wigner and Huntington in the thirties [@wigner]. This particular subject was included in the more general conjecture that any system becomes metallic if sufficient pressure is applied. Electrons, which are bound in the isolated atomic or molecular species, hybridize in a condensed phase to form extended states which gather in energy bands. When pressure is applied, the bands widen due to the enhancement of the overlap between neighboring atoms’ orbitals, up to the point at which the energy gap between the valence and the conduction band vanishes, thus giving rise to metallic behavior. This enhancement of the overlap can also be viewed as the growing importance of the electronic kinetic energy relative to the ion-electron potential; the latter tends to bind electrons to the ions (also in the form of interatomic, intramolecular or intermolecular bonds). In the case of pure hydrogen, metallization was estimated to occur around 2.5 Mbars (1 Mbar = 100 GPa), i.e. at pressures that have become accessible to experiment in diamond anvil cells only very recently [@mao]. The absence of clear signs of metallic behaviour up to 2.9 Mbar [@cui; @ruoff] certainly adds to the challenge, and stimulates the continuous improvement of experimental setups in the search for the [*reluctant*]{} metallization.
An additional difficulty in the description of this particular metal-insulator phase transition arises from the fact that at low pressures the low-temperature phase is not a monoatomic but a molecular solid, i.e. an hexagonal close packed (hcp) arrangement of H$_2$ molecules [@mao]. In this sense, H behaves more like a halogen than like an alkali metal [@ashcroft]. Up to pressures of the order of 1 Mbar, experiment indicates that the hcp arrangement is preserved. At pressures of the order of 110 GPa, the low-pressure free-rotator phase of para-H$_2$ freezes into an orientationally ordered phase, whose nature is not yet fully understood. The observation of more than one vibron mode [@hanfland] precludes the hcp structure with all the molecules pointing along the c-axis. However, other more complex hcp structures with tilted molecules (herringbone-like) are compatible with experiment [@mao]. It is also possible that orientational order is not complete, thus giving rise to a sequence of weak phase transitions until the perfectly ordered phase is reached.
An intriguing phase transition at 150 GPa, signalled by a discontinuity in the molecular vibron [@hemley], has been ascribed either to a relative reorientation of the H$_2$ groups [@kaxiras], to metallization arising from an electronic band-overlap mechanism without molecular reorientation [@chacham], and to both occuring at the same time [@surh]. Density functional (DFT) calculations indicate that arrangements with complex molecular orientations (non parallel) are energetically preferred in the high pressure phase [@kaxiras]. Another study [@nagara] concentrated on other classes of structures including cubic $Pa3$ and rutile, and also two hcp structures, namely $Pca2_1$ and $P2_1/c$. The hexagonal $Pca2_1$ was found to be the favorable one above the 150 GPa transition. Very recently, Tse and Klug performed ab initio simulated annealing calculations with 96 H atoms in the supercell, and found as a ground state an orthorombic structure, instead of hcp, composed of groups of three strongly interacting $H_2$ molecules [@tse]. All these candidates were found to be insulating, and the reason for this was claimed to be the opening of a hybridization gap at the Fermi level [@mazin]. Diffusion Monte Carlo (DMC) calculations by Natoli et al. [@natoli1] have essentially confirmed results by Kaxiras et al. [@kaxiras], also at the quantitative level for the angles between different molecular units. They considered neither Nagara’s $Pca2_1$ nor Tse’s orthorombic structure. This DMC calculation also finds an insulating behaviour and, interestingly, zero-point-motion effects due to the protons appear to be quite structure independent. As a consequence, zero-point-motion would turn out to be irrelevant as regards the 150 GPa transition, contrarily to the claims of Surh et al. [@surh].
Well-converged local density functional (LDA) calculations supplemented with the zero-point-motion energy contribution taken from frozen-phonon calculations [@barbee], suggest that the molecular-hcp phase goes over directly to an atomic squeezed-hexagonal phase, by breaking the intramolecular bonds, at a pressure of 380 GPa. However, this calculation did not take into account non-hexagonal structures. Hence, also from the theoretical point of view, it is not clear up to now whether metallization happens already in the molecular phase, or whether it is a consequence of dissociation, thus occuring simultaneously with the molecular-atomic transition [@cui]. A rhombohedral structure was also postulated to supersede the squeezed-hexagonal phase at higher pressures (above 400 GPa) [@barcoh], while the transition to a body-centered-cubic (bcc) phase was located at around 1100 GPa [@barbee].
In general, static methods suffer from the drawback of having to carry out the investigation by [*guessing*]{} different structures. In this respect, the molecular dynamics results by Tse and Klug seem to be the most reliable. However, there is still some doubt because it is not obvious that their 96-atom supercell with $\Gamma$-point sampling and fixed cell volume and shape calculation is a sufficiently good description. A recently proposed method that combines state-of-the-art electronic structure calculations with variable cell shape Molecular Dynamics simulations [@CPPR], thus avoiding every undesired bias, is currently being used to investigate the low-temperature phase of H as a funcion of pressure [@ksct:unp].
It is important to remark that, in general, energy differences between different structures are quite small, so that different levels of approximation often lead to different ground-state structures. In this respect, fully quantum-mechanical DMC calculations are very indicative. Ceperley and Alder [@cepald87] located the molecular-to-atomic phase transiton at a pressure of about 300 GPa, by studying only cubic structures. Further refinements of these DMC calculations have shown how dramatic the effect of improving the description can be [@natoli2]. Eventually, it appears that the atomic ground state structure in the vicinity of the molecular-atomic transition is the diamond structure, but the energy difference with respect to others is very small and, in particular, with respect to $\beta$-Sn and hexagonal diamond, it is well within the error bars. The diamond structure was considered as a candidate for the ground state in one of the previous DFT calculations, but found to be less favorable than a distorted hexagonal phase [@barbee].
The monoatomic bcc structure is unequivocally identified as the ground state in the very high density limit. There, the electronic kinetic energy is largely dominant over the remaining contributions, such that the electrons behave essentially as free fermions; i.e. the electronic subsystem decouples from the protons and becomes a rigid, homogeneous electron gas, which acts only as a neutralizing background for the protons. This is well-known as the one-component plasma (OCP) model, whose classical version was shown to crystallize into the bcc structure. This follows from a simple calculation of the Madelung energy, although the energy differences relative to other structures turn out to be rather small. In the classical OCP, which was thoroughly studied during the past two decades [@baus], the bcc structure turns out to be the stable one also at finite temperature and up to the melting point. The quantum version of the OCP at finite temperature has not yet been studied in detail. A numerical study using the path integral Monte Carlo technique (PIMC) is currently under development [@cep:pc].
At higher temperatures the protons begin to behave as classical particles. The degeneracy temperature for the protons can be estimated to be of the order of $T_d^p\approx 180~K/r_s^2$, by comparing the mean interprotonic distance and the thermal de Broglie wavelength. The electron degeneracy temperature ($T_d^e$) lies well above (by a factor $m_p=1836$) such that, at temperatures lower than $T_d^e\approx 326000~K/r_s^2=1~{\rm Hartree}/r_s^2$, electrons can be considered to follow the protons adiabatically, always staying in the the ground state compatible with the current protonic configuration (Fermi temperatures are about twice these numbers, e.g. $T_F^p=326~K/r_s^2$). This latter approximation allowed Hohl et al. [@hohl] to examine the hot, dense [*molecular*]{} phase by means of ab initio Molecular Dynamics (AIMD) simulations, and also to make some progress regarding the low-temperature structure. The AIMD study of the [*atomic*]{} phase is the subject of this and of a previous publication [@kh:prl]. The effect of excited electronic states in the same regime has very recently been explored by a novel ab initio Molecular Dynamics scheme using Mermin’s density functional [@ali], and the fully-quantum PIMC method was applied to simulate the very high-temperature fluid phase [@carlo], where the number of excited electronic states involved becomes too large to be treated efficiently with Mermin’s functional. The low-temperature regime, where protons in turn become degenerate, requires different simulation techniques which are currently being developed [@kpc:unp].
High-temperature high-pressure measurements on Hydrogen have been very recently reported in shock wave experiments [@weir]. Pressures between 1 and 2 Mbar and temperatures of some thousand $K$ are now feasible and rather controllable, and further expansion of this range is in sight. In such experiments, Nellis et al. observed the metallization of fluid molecular Hydrogen at $P=1.4$ Mbar and $T=3000~K$, a temperature which is significantly lower than that predicted for the plasma phase transition by Chabrier et al. [@chabrier] on the basis of a multicomponent thermodynamic theory that employs an approximate equation of state. Although more accurate PIMC simulations have essentially confirmed the predictions of the theory, including the existence of a first order phase transition at high pressures and a somewhat lower value for the transition temeprature [@magro], the discrepancy with respect to shock wave experiments and AIMD results [@aliala] still holds. The latter would rather indicate a continuous transition in which metallization and dissociation of H$_2$ molecules are closely related phenomena. A possible scenario would be the existence of two phase transitions: a continuous one, at relatively low temperatures, where molecular Hydrogen already metallizes and begins to dissociate, and a discontinuous one at higher temperatures where a massive dissociation occurs with a concomitant jump in the electrical conductivity.
Hydrogen at finite temperatures and high densites consitutes, hence, a strongly coupled proton-electron plasma which is of great astrophysical interest since it is, in particular, the major constituent in Jovian planet interiors. Actually, the latter are basically H plasmas with a few percent admixture of He, and the statistical properties of the mixture (mixing/demixing transition) are of great importance to account for experimental observations and to establish models for the evolution of these planets [@chabrier1]. Nevertheless, the answer to many relevant questions in astrophysics requires a full understanding of the statistical properties of the pure main constituents. Hydrogen is present in both atomic and molecular fluid phases. These are separated by a boundary located at some distance from the center of the planet. The characteristics of this boundary, e.g. precise location, width, etc. depend on the density and temperature profile, i.e. on the equation of state. The matter of dissociation and metallization is particularly relevant because the large magnetic field measured in Jupiter would include a non neglgible component generated in the outer molecular fluid phase, provided that temperature is above the metallization threshold. To quantify this effect the important quantities are the dissociated fraction and the electrical conductivity, which can be obtained from simulations in the molecular phase.
The hydrogen plasma is perhaps the most fundamental, simple many-body system, with [*all*]{} the interactions (proton-proton, proton-electron and electron-electron) described exactly by the [*bare*]{} Coulomb potential. In spite of this, the phase diagram appears to be surprisingly rich. The purpose of this article is to present a detailed study of the very high pressure physical properties and phase diagram of this simple and fascinating system. We are going to deal always with atomic phases, the molecular phases appearing at lower densities than those studied here. In Section II we briefly describe some details of the simulations, and we address some important technical issues such as the interplay between pseudopotentials and basis set expansions. Section III is devoted to the analysis of the validity of linear response theory in its ability to describe proton-proton interactions in terms of an effective pair potential of the screened Coulomb form. In Section IV we address a crucial problem that arises in the ab initio simulation of metallic systems, in particular liquid metals, i.e. size effects and Fermi surface sampling. In Section V we enter directly into the properties of the Hydrogen plasma by describing the solid phases and the melting at very high pressure. Section VI is concerned with the fluid phase, which we characterize as atomic-like. Diffusion and vibrational properties are presented in Section VII, while Section VIII is devoted to a thorough study of the collective dynamics, as a approach to the metal-insulator transition coming from the metallic side. The behavior of the longitudinal collective mode is studied as a function of density and temperature, and proposed as a probe for the metal-insulator transition at finite temeprature. Finally, we conclude in Section IX. In the remaining part of this Section we define some useful parameters and ratios.
The plasma is made up of $n$ electrons and as many ions (protons) per unit volume, such that the usual dimensionless density parameter is $r_s=a/a_B=(3/4\pi n)^{1/3}/a_B$, where $a$ is the mean interionic distance (ion-sphere radius) and $a_B$ is the Bohr radius. Adopting atomic units throughout, the Fermi momentum is $k_F=(9/4\pi)^{1/3}/r_s$, and the Thomas-Fermi screening length is $k_{TF}=(12/\pi)^{1/3}/\sqrt{r_s}$. The dimensionless Coulomb coupling constant associated with the classical ions is $\Gamma=e^2/(ak_BT)$. Note that $k_BT=1/(\Gamma r_s)$, and that the electron degeneracy parameter is $\theta=T/T_F=2(4/9\pi)^{2/3}r_s/\Gamma$. Typical densities inside Jovian planets are between 1 and 10 $g/cm^3$ and temperatures are of the order of 100 to 10000 $K$, i.e. $r_s$ ranging from 0.5 to 1.5, and $\Gamma\approx 20-200$ [@baus]. Note that within this range $\theta\approx 0.015\ll~1$ and for the present calculations, whose aim is precisely to address some aspects of astrophysical plasmas, we can resort to the adiabatic approximation by assuming that the electrons are always in their instantaneous ground state for any given ionic configuration. Higher temperatures require the relaxation of this hypothesis.
Technical details of the simulations
====================================
Our simulations were carried out using the standard Car-Parrinello (CP) AIMD scheme [@carpar]. The ions were considered as classical point-like particles, whereas the electrons were described by density functional theory (DFT) within the local density approximation (LDA) [@ks]. The electronic density was constructed via Kohn-Sham single-electron orbitals, expanded in a plane wave basis set up to a specified energy cutoff $E_{\rm
cut}$ (see below). We have extensively studied a system consisting of 54 H atoms contained in a simple cubic simulation box under periodic boundary conditions (PBC), and we have also performed simulations for larger supercells, containing 128 and 162 H atoms, in order to analyse finite-size effects. The plane wave expansion was carried out around the $\Gamma$-point of the supercell’s Brillouin zone. Finite-size and Brillouin zone sampling effects will be analysed later, in a forthcoming section.
We used the exchange-correlation functional deduced from the results of quantum Monte Carlo calculations on the uniform electron gas [@cepald80], and the bare Coulomb potential for the ion-electron interaction. The singularity of the ion-electron Coulomb attraction implies that there is never absolute convergence of the electronic quantities as the energy cutoff for plane waves is increased. In particular, the cusp condition is never satisfied, the density reaching the origin with zero slope. This is because the Fourier expansion of $1/r$ is also a long-range function ($4\pi/g^2$), meaning that there always exists a spatial scale small enough to require a representation involving large-$g$ components. To satisfy the cusp condition, a different (localized) basis set is needed instead of plane waves, e.g. Slater-type orbitals [@harris]. A finite cutoff $g_{\rm cut}=\sqrt{2E_{\rm
cut}({\rm Ry})}$ translates into the following effective potential:
$$v_{\rm eff}(r)= \frac{1}{r}~\left(1-\frac{2}{\pi}\int_{g_{\rm cut}}^{\infty}
\frac{\sin(x)}{x}~dx\right)$$
This description is variational in the number of plane waves (i.e. in the energy cutoff) and, even if the energy never fully converges, other quantities, like the electronic distribution around the protons, are affected only in a tiny region around the nuclei, the remainder being essentially converged at a finite cutoff. This is shown in Fig. 1 for protons fixed at the ideal bcc lattice sites. The choice of the cutoff strongly depends on the average density: the lower $r_s$, the higher the cutoff. It is the number of plane waves which has to be kept approximately constant in order to achieve similar levels of convergence. In order to have the electronic density correctly described from less than one third of the nearest-neighbor ionic distance onward, we decided to vary the cutoff between 230 Ry for $r_s=0.5$ and 60 Ry for $r_s=1$ and 1.2. It can be seen in Fig. 1 ($r_s=0.5$) that differences in the proton-electron correlation function become noticeable at distances smaller than $0.5~a$, while the nearest neighbor distance in the bcc lattice is $d_{nn}=1.76~a$. It should be pointed out that a cutoff of 60 Ry is sufficient to achieve convergence in the properties of the H$_2$ molecule, like the proton-proton distance (1.5 a.u. = 0.79 Å) and the vibrational frequency (4160 cm$^{-1}$). The difference with respect to the experimental quantities (0.74 Å and 4400 cm$^{-1}$) may be ascribed exclusively to the LDA.
Isothermal ionic dynamics (particularly in the fluid phase) was achieved by using a Nosé-Hoover thermostat. The time step for integration of the CP equations of motion was chosen typically between 0.25 and 1.5 atomic units (depending on the temperature and density), i.e. roughly 10$^{-17}$ s, compared to the ionic plasma period $\tau_P\approx 155~r_s^{3/2}$ a.u. In some cases another Nosé thermostat was necessary to keep the fictitious kinetic energy of the electronic degrees of freedom at low values. Most runs extended over 0.4 to 0.6 psec, which amounts to more than 50 plasma oscillations. Before this, we allowed for thermal equilibration during an initial period of 0.2 psec in the solid phase, and 0.8 to 1.2 psec in the fluid phase. The densities $r_s=0.5$, $r_s=1$ and $r_s=1.2$ were explored in detail for temperatures in the range $100~K\leq T\leq
10000~K$, which correspond to the strong coupling regime ($\Gamma\geq 30$), and cover both fluid and solid phases of the ionic component. In particular, $r_s\approx 1$ and $T\approx 7000~K$ are typical conditions inside Jovian planets [@hubbard]. The simulations at $r_s=1.2$ were carried out only in the fluid phase because the stable solid has no longer the bcc symmetry, and thus it is not compatible with the choice of 54 H atoms in a simple cubic supercell.
Validity of Linear Response Theory
==================================
In the very high density limit ($r_s\to 0$) the electrons are barely polarized by the ionic charge distribution, due to their rapidly increasing Fermi energy, and the hydrogen plasma practically reduces to two decoupled components: a classical ionic “one-component-plasma” (OCP), and a degenerate, rigid electron “jellium”. Both systems have been extensively studied by classical [@baus] and quantum [@cepald80] computer simulations. For finite, but small $r_s$ ($r_s\ll 1$), the coupling between the two components may be treated perturbatively within linear screening theory [@galam]. However, due to the strength of the unscreened Coulomb interaction between protons and electrons, such a perturbative approach is expected to break down rapidly as $r_s$ increases. In this section we investigate the high density limit and the validity of the description of the hydrogen plasma in terms of a linear response picture (LRT) for the electronic component.
To this end we compare the proton-electron pair correlation function taking into account the full LDA response:
$$g_{pe}(r)=-\frac{V}{N^2}\int d\bfk~\langle\rho_i(\bfk)~\rho_e(-\bfk)\rangle
~j_0(kr)$$
and the one obtained by replacing the electronic charge density by its linear response expression in terms of the ionic charge density:
$$\rho_e(\bfk)=\chi_e(k)\left(-\frac{4\pi}{k^2}\right)\rho_i(\bfk)$$
where $\chi_e(k)$ is the static electron density response function. In eq. (2), $j_0(x)$ denotes the $l=0$ spherical Bessel function, $\rho_i(\bfk)$ is a Fourier component of the microscopic ionic density, and $\rho_e(-\bfk)=\rho_e^*(\bfk)$ is a Fourier component of the electronic density corresponding to the instantaneous ionic configuration $\{\bfrr_I\}$. The latter represents, within the adiabatic approximation, a ground-state expectation value.
For the density response we adopt the random phase approximation (RPA) corrected by the long wavelength limit of the local field function $G(k)$ [@baus].
$$\frac{4\pi}{k^2}\chi_e(k)=\frac{k_{TF}^2~l(k/k_F)}
{k^2+k_{TF}^2~l(k/k_F)[1-G(k/k_F)]}
\label{pot}$$
with $l(x)$ the Lindhard function and $k_{TF}$ the Thomas-Fermi wavenumber. For the local field correction we use the following expression, which is consistent with the LDA and Slater’s local exchange:
$$G(k/k_F)=\Big[\frac{1}{4}-\frac{\pi\lambda}{24}\left(r_s^3
\frac{d^2\epsilon_c(r_s)}{dr_s^2}-2r_s^2
\frac{d\epsilon_c(r_s)}{dr_s}\right)\Big]~\frac{k^2}{k_F^2}$$
where the correlation energy density $\epsilon_c(r_s)$ is that proposed by Perdew and Zunger [@pz], and $\lambda=(4/9\pi)^{1/3}$. At typical densities studied in this work the exchange contribution to $G(k/k_F)$ turns out to be largely dominant over correlation, and both of them represent a small correction relative to the RPA susceptibility. In fact, this is reasonable because the RPA (or Lindhard approximation) is known to be good in the high density limit.
The proton-electron distribution function reads accordingly:
$$\begin{aligned}
\lefteqn{g_{pe}^{LR}(r)= \frac{V}{N^2}\int d\bfk~
<\rho_i(\bfk)\rho_i(-\bfk)>~j_0(kr)}\nonumber \\ & \nonumber \\
&{\displaystyle\times\left(\frac{k_{TF}^2~l(k/k_F)}{k^2+k_{TF}^2~l(k/k_F)
[1-G(k/k_F)]}\right)}\end{aligned}$$
In Fig. 2 we compare the proton-electron radial distribution functions in the ideal bcc structure as obtained from the LDA and from LRT, for $r_s=0.5$ and $r_s=1$. As expected, $g_{pe}(r)$ is considerably more structured at the lower density. The LDA and LRT distribution functions are very close at $r_s=0.5$, but significant differences are clearly apparent at $r_s=1$, reaching a region well beyond the first ionic shell and signalling the break-down of the linear screening regime. When the ions are at finite temperature the differences are significantly enhanced, particularly at short distances. Also the location of the first minimum turns out to be shifted outwards by LRT in the fluid phase (unlike in the solid phase), thus indicating that the LRT description of the electronic charge distribution worsens for increasing temperature. This is shown in Fig. 3. The dashed curves have been obtained by averaging the adiabatic electronic charge distribution around the protons along the AIMD trajectory, assuming that the configurations generated in this way are also representative of the LRT. In fact, the magnitude of the differences observed in Fig. 3 suggests that the LRT trajectories will differ significantly from the LDA ones, implying that this averaged electronic distribution might be meaningless within LRT. In other words, the fully-consistent LRT is likely to be worse than the rasults presented here.
Following a novel procedure [@furio] we have fitted, for $r_s=1$, an effective proton-proton pair potential to our set of AIMD configurations generated at several temperatures spanning both solid and fluid phases. In Fig. 4 we compare this potential to the one obtained within LRT, i.e. by Fourier transforming $v_{\rm LRT}(k)=4\pi[1-\chi_e(k)]/k^2$, where $\chi_e(k)$ is given by expression (\[pot\]). The nearest neighbor distance in the bcc structure for $r_s=1$ is 1.76 a.u., and the position of the first peak in the proton-proton radial distribution function decreases down to values of the order of 1.5 a.u. in the fluid phase and upon heating (see below). This means that nearest pairs of protons will spend most of the time at distances of this order. Looking carefully at Fig. 4 it can be seen that, at those distances, the two potentials differ by more than 10%, the LDA one being steeper. On the other side, the LRT pair potential appears to be more long-ranged than the LDA one. This implies that the LDA pair potential has a characteristic screening length shorter than its LRT counterpart, i.e. screening is more efficient than linear at $r_s=1$.
The departure from the linear screening regime, also in the form of many-body effects beyond the pair potential approach (e.g. three-body terms or embedding functions), becomes more pronounced as $r_s$ is increased, leading to ground state-atomic phases other than bcc, fcc or hcp (e.g. hexagonal and diamond) and, eventually, to recombination and to the different H$_2$ molecular phases. The reason for this early departure from the LR regime has to be traced back to the unusual strength of the bare Coulomb interaction, arising from the absence of core electrons. In fact, other alkali metals can be reasonably well described by LRT at much lower electronic densities ($r_s> 3$) [@hafner]. Interestingly, the range of validity deduced here for the LRT is much wider than expected in previous theoretical work based on perturbative expansions [@galam], where $r_s=0.1$ was identified as the upper limit.
Size effects and Fermi surface sampling
=======================================
In the solid phase, the combination of $\Gamma$-point sampling and a finite system size is not expected to provide a very accurate description of the electronic component, because all the periodicities beyond the size of the supercell are not properly included. This is particularly important in metallic systems, where no point in the Brillouin zone can be taken as representative of the band structure of the bulk solid. The reason is that occupied states (contributing to the electronic density) and empty states (which do not contribute) coexist in the same band, corresponding to different k-points. A particular choice (e.g. the $\Gamma$-point and its refolded images) will sample the conduction band in some specific points, but the character of the Fermi surface could be misinterpreted if empty states are taken as if they were occupied and viceversa. For quasi-spherical Fermi surfaces this is unlikely to happen, but for transition metals or semimetals (like graphite) this effect is crucial, and a very fine sampling of the Brillouin zone is needed to obtain the right physics.
In the case of simple metals the effect of quantization of the electronic states in the simulation box is more important. The Brillouin zone of the unit cell is sampled with a finite number of points, which arise from the refolding of the $\Gamma$-point of the supercell. These points reflect the symmetries of the system in the sense that, if the $\Gamma$-point refolds onto some point $\bfk_0$, then all the points in the star of $\bfk_0$ must arise from some other refolding of $\Gamma$. Otherwise, the symmetry is broken and spurious forces appear that drive the system away from the symmetric ground state. Depending on the case at hand, the distortion can be rather large, especially if the symmetry of the supercell is very different from that of the unit cell. If the correct symmetry is used for the supercell then, since all the points in the star are equivalent, the eigenvalues associated with them are degenerate, thus giving rise to the formation of electronic shells. If a better sampling of the Brillouin zone is performed, or the size of the system is increased, new shells appear that eventually give rise to a continuous energy band. But for finite systems and restriction to the $\Gamma$-point, the electronic density of states consists of a set of discrete peaks (the shells). A problem arises with the highest occupied shell because in finite-size metallic systems it is partially occupied, unless a very fortuitous situation occurs. The immediate consequence is that it is not possible to fullfil the symmetry requirements by occupying all the points in the star with an [*integer*]{} number of electrons. Then, unless [*fractional*]{} occupation is introduced, the symmetry is broken even when the symmetry of the unit cell is mantained for the supercell. This is analogous to the Jahn-Teller effect in molecular systems, where a lower-energy state can be obtained by reducing the symmetry and breaking the electronic degeneracy. However, in the present case this effect is unphysical.
In a fluid phase these effects are normally expected to become less important because of the breakdown of the discrete translational invariance, as reflected in the very existence of a finite Brillouin zone via Bloch’s theorem. The unit cell becomes of infinite size and the Brillouin zone reduces to a single point, i.e. the $\Gamma$-point. However, for practical reasons, computer simulations are bound to represent the infinite (fluid) system with a limited number of particles (typically of the order of some hundreds in ab initio calculations), while repeating periodically the supercell (via PBC). In fact, this introduces a spurious Brillouin zone, which is associated with periodicities that are absent in the infinite fluid. The properties of the fluid are assumed to be recovered from the finite sample in terms of PBC combined with statistical averages, which in the case of our AIMD simulations are computed as time averages.
This is a reasonable justification for a purely classical system and also for liquid semiconductors but, for metallic systems, as soon as electronic states are introduced, the quantization of these states in the (unphysical) simulation box acquires a crucial role. The problem of partial occupation of the star persists in the fluid, in the sense that spurious forces appear that modify in a non-trivial way the structural and dynamical properties of the system.
A qualitative picture of the consequences of these observations can be obtained by comparing the proton-proton radial distribution function for systems of different sizes. A system of 54 H atoms (of valence charge equal to 1) is really a fortuitous case of compatibility of a closed electronic shell (the whole star is fully occupied) with a bcc atomic arrangement in a simple cubic supercell. In the case of 128 H atoms only 57 states are doubly occupied, and the next shell of 24 degenerate states has to accomodate 14 electrons (i.e. only 30% of the shell). On the contrary, 162 H atoms is not compatible with a bcc arrangement in a simple cubic supercell, but it closes the former electronic shell thus amounting to 81 doubly occupied states. Therefore, in the fluid phase it should behave essentially as the 54-atom supercell but better converged in system size (or k-points). In Fig. 5(a) we show the curves corresponding to these 3 different system sizes at $r_s=1$ and $T=1000~K$. In fact, up to the boundary of the 54-atom cell ($r=3~a.u.$) there is hardly any relevant difference between the $g_{\rm pp}(r)$ corresponding to 54 and 162-atom supercells. The same kind of picture holds also at higher temperatures, meaning that, as regards static thermodynamic properties, 54 H atoms already give a very reasonable picture.
Very different is the situation with open shell systems like the 128 H atoms one (short-dashed line in Fig. 5(a)). It is clear that this $g_{\rm pp}(r)$ has little to do with those of 54 and 162 atoms. The first peak is significantly lower and broader, and also its position is shifted downwards. The first valley is much shallower and, in practice, the pair distribution becomes structureless beyond it, while with 162 atoms the atomic shell structure is still visible at least up to the second valley. The 128-atom distribution resembles, in fact, the one that would have been obtained with closed shells at a higher temperature value.
The correct way to account for open shell structures at finite temperature is to doubly occupy the lowest-lying electronic states at every time step of the AIMD simulation, while keeping empty the rest of that shell. The curves in Fig. 5(a) have been obtained by applying the standard procedure of considering explicitly, in the description of the electronic component, a number of electronic states equal to one half of the number of electrons, i.e. including those strictly necessary. The justification for this approach is that typical temperatures are much lower than the Fermi temperature. Atomic motion leads to a degeneracy lifting of the order of a few percent of an eV, implying that the distance between electronic states is usually much larger than the width of the Fermi-Dirac distribution; this latter is of the order of $k_BT$, i.e. about 0.1 eV at $T=1000~K$. This leads to a symmetry breaking in the sampling of the Fermi surface, which is expected to be recovered in terms of statistical averaging. However, following the approach above, in practical simulation times we have not noticed a convergence to the closed-shell picture. In Fig. 6 we show the Fermi-Dirac distribution function for three different temperatures, namely 1000 $K
$, 5000 $K$, and 10000 $K$, compared to the electronic density of states averaged along an AIMD run at $T=1000~K$, and to a snapshot of the instantaneous Kohn-Sham eigenvalues. It is clear that the occupation numbers fall from 2 to 0 in an energy scale narrower than the thermal splitting of the eigenvalues.
The drawback of this kind of approach is that for open shells the ordering of the states within the highest (partially) occupied shell changes continuously during the MD evolution and, in particular, occupied states become empty and viceversa. The standard CP approach is not able to take into account this phenomenon, and this is one reason for the well-known failure in metallic systems, reflected in the energy transfer between electronic and ionic degrees of freedom (cooling the ions and heating the electronic orbitals). The double Nosé thermostat proposed by Blöchl and Parrinello [@bloechl] helps in fixing the temperature of each of the components, but still does not take into account level crossing effects. To our knowledge, there are three methods capable of solving this problem: one is to abandon the CP Lagrangian strategy in favor of a self-consistent minimization at each time step [@payne], a second one is to abandon the description of the electronic component in terms of single-particle orbitals to work directly with the electronic density which is, by definition, constructed with the lowest occupied states [@ali], and finally a third one consisting of a rigorous Lagrangian formulation which incorporates the occupation numbers (actually the conjugate variables, i.e. the Kohn-Sham eigenvalues) as dynamical variables [@hohlcar]. However, this latter procedure exhibits the odd feature that the fictitious kinetic energy of the electronic orbitals still increases at the expense of the ionic component, due to the appearance of low-energy excitations introduced by the dynamics of the eigenvalues.
An approximate solution can still be found within the Lagrangian approach by evolving explicitly the whole open shell, but initially occupying – with an integer number of electrons – only the lowest states of the this shell. This will approximately take into account level crossing in terms of mixing of states within the shell, during the time evolution. The very same mechanism that leads to energy transfer between ions and electronic orbitals, i.e. the vanishing energy gap [@pastore], is responsible for mixing occupied and empty orbitals, thus allowing for initially empty states (if explicitly included) to become occupied and viceversa. Statistical averaging completes the task by generating a more uniform sampling of the Fermi surface. Fig. 5(b) shows how the results obtained with 128 H atoms can reproduce approximately those obtained with 54 and 162 H atoms. It has to be pointed out that this procedure, although not rigorously justified, has the nice feature that the fictitious kinetic energy of the electronic orbitals is practically constant, behaving exactly as in systems with a gap, i.e. there is no energy exchange between ionic and electronic degrees of freedom.
Low-temperature phase: structure and melting
============================================
In the OCP ($r_s\to 0$) limit the ionic bcc lattice is known to be the stable crystalline structure up to melting (which occurs at $\Gamma\approx 180$ [@baus]). We have studied the stability of the bcc structure at low temperatures and finite $r_s$ by performing canonical MD simulations; initial conditions were constructed by giving the ions a small random displacement (about 3 % of the nearest neighbor distance $d_{nn}$) from their alleged equilibrium positions (the bcc lattice sites). The bcc structure was found to be dynamically stable against such displacements at least up to $r_s=0.5$. At $r_s=1$ the bcc structure was found to be unstable for $T<100~K$, where a close-packed structure appears to be favored. This is consistent with a description in terms of LRT. In fact, at high densities the effective LRT potential behaves essentially like a Yukawa potential
$$v_{SC}(r)=\frac{1}{r}\exp(-r~k_{TF})
\label{yukawa}$$
while Friedel oscillations are practically negligible. The phase diagram of a classical system of particles interacting via the above Yukawa potential (\[yukawa\]) has been extensively studied by computer simulations and lattice dynamics [@robbins]. These calculations point to a bcc-fcc phase transition when the density-dependent screening wavenumber increases, i.e. when the effective interaction becomes of shorter range. At $T=0$ the bcc structure is found to be stable up to $r_s\approx 0.6$, beyond which the fcc phase becomes the stable structure. The $r_s$ at coexistence shifts to higher values at finite temperatures, such that the system goes through a structural fcc-bcc phase transition as the temperature increases along an isochore.
The situation here is reminiscent of the behavior of alkali metals. Na exhibits an hcp ground state, while Li goes from bcc to fcc and eventually to hcp at very low T. The heavier alkalis K, Rb and Cs undergo a structural transition to fcc upon cooling below $T\approx 5~K$. In all these cases the entropic contribution of the bcc structure, arising from the valley in the phonon dispersion along the (110) direction, wins over at finite temperature and stabilizes this phase. This is exactly what we observe in our simulations for H, where at $r_s=1$ the bcc structure appears to be stable for $T>100~K$.
Still, finite energy cutoff, finite system size, and coarse Brillouin zone sampling may have a large influence on the stability of the ground state structure. The study of this part of the phase diagram deserves special attention because also zero-point motion effects on the protons have been shown to influence the stability of different structures at $T=0$ [@natoli2]. Disregarding the problem of zero-point energy (ZPE), and only as a check of the present calculations (which do not include ZPE), we have performed total energy full-potential Linear Muffin Tin Orbitals (FP-LMTO) calculations for solid, monoatomic H in the bcc, fcc and hcp structures, at $r_s=0.5$ and $r_s=1$. The energy differences turned out to be very small, but at $r_s=1$ the fcc and hcp structures are significantly lower in energy than the bcc. Moreover, the energy differences are enhanced if a coarse sampling of the Brillouin zone is performed. These calculations also identify the hcp structure as the lowest energy one, but the difference with respect to the fcc is within the accuracy of the calculations. The reason for this can be found in Fig. 4. The differences between fcc and hcp structures begin only at the level of third nearest neighbors, a region where the effective potential shows negligibly small Friedel oscillations.
Between $r_s=1$ and 1.2 a phase transition occurs that takes the system from hcp to a simple-hexagonal phase with a compressed $c/a$ ratio (squeezed-hexagonal). This is compatible with static total energy calculations by Barbee et al. [@barbee], and it is an additional confirmation of the breakdown of LRT, because pair potentials (like the LRT one) are not able to stabilize anisotropic structures like the simple-hexagonal. A more detailed study of the low-temperature atomic phases of Hydrogen is currently under way [@ksct:unp].
Next, the melting of the ionic crystal was investigated by gradually increasing the temperature and monitoring the time-dependent mean square displacement of the ions $<\mid\Delta\bfr(t)\mid^2>=
<\mid\bfr(t)-\bfr(0)\mid^2>$. In the crystalline phase, $<\mid\Delta\bfr(t)\mid^2>$ goes over to $2<\mid\delta\bfr\mid^2>$ for sufficiently long times, where $<\mid\delta\bfr\mid^2>=<\mid\bfr-\bfrr\mid^2>$ denotes the static mean square displacement ($\bfrr$ are the equilibrium positions of the ions). In the fluid phase diffusion sets in, so that $<\mid\Delta\bfr(t)\mid^2>=6Dt$ at long times, with $D$ the ionic diffusion constant. At $r_s=0.5$ diffusion was found to set in at $\Gamma\approx 230$, which may be identified with the limit of mechanical stability of the (overheated) metastable crystal. The thermodynamic transition occurs at lower temperature (higher $\Gamma$); its location may be estimated by assuming that the Lindemann ratio $L=(<\mid\delta\bfr\mid^2>)^{1/2}/d$ at melting is the same as for the OCP, i.e. $L\approx 0.15$ [@hong]. This leads to $\Gamma_m(r_s=0.5)\approx 290$ compared to $\Gamma_m(r_s=0)\approx 180$, indicating a strong influence of electron screening on the melting transition. This confirms recent predictions based on free energy comparisons, obtained by means of an approximate density functional theory [@hong].
We have also studied the melting transition at $r_s=1$ using the same procedure; the melting temperature drops sharply from $T_m(r_s=0.5)\approx
2200~K$ to $T_m(r_s=1)\approx 350~K$ ($\Gamma_m(r_s=1)\approx 930$). This indicates the possible proximity of a triple point bcc-hcp(fcc)-liquid, analogous to that found for Yukawa potentials. However, the nonlinearity of the screening at these values of $r_s$ is likely to bring the triple point from $r_s=3$ [@robbins] to $r_s\approx 1.1$. Moreover, the hcp structure goes over to a simple hexagonal one at $r_s$ somewhere between 1 and 1.2, and this has to be a consequence of the appearance of anisotropic forces, beyond the level of pair-wise additivity. The very low value of the melting temperature might also be related to the appearance of these forces. Interestingly, at $r_s=1$ the Fermi temperature of the ionic component is $T_F^p=326~K$, a value close to the melting point ($350~K$). Therefore, the influence of quantum effects for the protons on the melting transition cannot be ignored, and will probably also play an important role in the above structural phase transition. In particular, they might destabilize the hexagonal phase in favor of some more isotropic configuration which has a higher energy within a framework of classical protons.
The fluid phase: an atomic-like plasma
======================================
Turning to the fluid phase, a quantitative measure of ion-electron correlations is provided by the sphericalized average of the ion-electron pair correlation function $g_{pe}(r)$, as computed from eq. (2). The effect of temperature on $g_{pe}(r)$ is illustrated in Fig 7, for $r_s=0.5$ and $r_s=1$. The distribution function is seen to be remarkably insensitive to $T$ over the whole range of temperatures, covering the solid and fluid phases, as already noticed at lower $\Gamma$ (higher $T$) by Dharma-Wardana and Perrot in the framework of an approximate static DFT-HNC calculation [@dharma]. The observed weak temperature dependence implies that the main effect of ionic thermal motion (electrons are always at $T=0$), is to enhance the localization of the electronic charge close to the protons.
This behavior is to be contrasted with the predictions of higher-level theories that go beyond the Born-Oppenheimer approximation by including excited electronic states. Both, fully-quantum PIMC [@pierleon] and Mermin functional [@alavi] simulations imply that ion-electron correlations become weaker as temperature increases. This is to be intuitively expected, but for temperatures much higher than the ones studied here. Excited electronic states are much more insensitive to ionic polarization effects, because they correspond to larger kinetic energies. In this way, for temperatures larger than a threshold value that can be estimated around $\theta=0.1$, i.e. $T\approx 60000~K$ for $r_s=1$, the above localization effect due to ionic disorder starts to be compensated by the effect of electronic excitations, so that eventually the opposite trend will take over.
The ion-electron pair correlation functions for [*all*]{} temperatures are seen to intersect at a well-defined (reduced) distance from the proton site, irrespective of thermal ionic disorder and almost independently of density. We locate this value at $r^*\approx 1.3~a$, and notice that the ratio $r^*/d_{nn}\approx 0.73$ is related to the ratio of the location of the nodes corresponding to the first two spherical Bessel functions ($j_0(x)$ and $j_1(x)$). In fact, the electronic problem can be modelled, in a very crude approximation, as that of a particle in a spherical well; the corresponding radial solutions are precisely the spherical Bessel functions. Temperature effects can be mimicked by increasing the relative population of excited states with respect to $j_0(x)$. However, the location of the node of $j_1(x)$ (the leading excitation) relative to $j_0(x)$ (the ground state) does not depend on temperature. Since the location of the nodes is defined in units of the radius of the well, and this is identified with $d_{nn}$, which is proportional to $r_s$, the crossing should not depend significantly on density. The first maximum of $g_{pe}(r)$ is clearly associated with the location of the first coordination shell (the first maximum in $g_{pp}(r)$ – see below), which is quite natural since the electronic density peaks at the proton sites. It is interesting to notice that differences in the electronic screening properties between $r_s=1$ and $r_s=1.2$ are significative only in the vicinity of the protons, up to $r\approx 0.5~a$.
The proton-proton pair correlation function for $r_s=1$ is illustrated in Fig. 8(a), as a function of temperature. It can be observed that the first peak remains clamped at the nearest neighbor distance ($d_{nn}=1.76$) for temperatures below the mechanical stability limit – $T_s(r_s=1)\approx 500
K$ –, while in the fluid phase it shifts continuously to shorter distances. The same plot shows that the location of the first minimum is quite insensitive to temperature. This, together with the fact that equivalent results are found at $r_s=0.5$, defines quite univocally a first coordination shell of radius $r_{fcs}\approx 2.4~a$ (with $a$ the ion-sphere radius). The integrated number of particles is shown in Fig. 8(b) as a function of temperature. The main result is that the first coordination shell contains 14 atoms on average, implying that the short-range structure of the liquid is quite reminiscent of that of the solid, since the first coordination shell of the bcc structure (containing first and second nearest neighbors) also contains 14 atoms. Simulations performed with 162 H atoms show that this is a genuine feature and not an artifact of the small system size. A second coordination shell is also well-defined in the fluid phase provided that the temperature is low enough, i.e. $T<1500~K$ ($\Gamma>200$), as can be observed in Fig. 9. However, the fluid becomes structureless beyond the first coordination shell at temperatures of the order of 5000 K (at $r_s=1$). Summarizing, the fluid phase of the H plasma at moderately high temperatures and very high densities (typical of the inner H shell of Jovian planets) behaves like an atomic liquid with a well-defined first coordination shell.
The influence of electron screening is also apparent when comparing the ion-ion and charge-charge static structure factors $S_{ii}(k)$ and $S_{ZZ}(k)$. While at $r_s=0.5$, the two are nearly indistinguishable, the amplitudes of their main peaks differ significantly for $r_s=1$ (by roughly 9 %), and $r_s=1.2$ (12%). Due to the discrete sampling of the electronic density in reciprocal space, the curves are noisy and will not be reproduced here. The strong polarization of the electronic component leads to a damping of the local charge fluctuations, and hence to a reduction of $S_{ZZ}(k)$. Again, the presence of a well-defined first peak and valley in $S(k)$ is an indication that the fluid is well structured in this region of the parameter space.
Diffusion coefficients and Vibrational properties
=================================================
Our AIMD simulations give direct access to the ionic dynamics by analysing the time evolution of atomic coordinates and velocities. Diffusion coefficients have been calculated using the asymptotic relation $<\mid\Delta\bfr(t)\mid^2>=6Dt$; i.e. by measuring the slope of the mean square displacement of the atoms as a function of time. Our results are displayed in table I for $r_s=0.5$,$r_s=1$ and $r_s=1.2$, in reduced plasma units $D^*=D/a^2\omega_{\rm pl}$, where $\omega_{\rm pl}(r_s)=(3/M_I)^{1/2}~
r_s^{-3/2}$ is the bare ionic plasma frequency. The results are shown in a log-log plot in Fig. 10. The relationship between $D^*$ and $\Gamma=1/(r_sT)$ follows quite accurately a power-law of the type $D^*=D_o\Gamma^\alpha$. We have fitted our data to such an expression, obtaining for $r_s=1$ the following values: $D_o=(10.4\pm 1.4)$ and $\alpha=(-1.38\pm 0.07)$, and for $r_s=0.5$ the values: $D_o=(4.0\pm 1.5)$ and $\alpha=(-1.37\pm 0.09)$.
It is interesting to note that the diffusion coefficient follows the same relationship as in the OCP model. The OCP values for the parameters, fitted to classical MD simulations [@pollock] are $D_o=2.95$ and $\alpha=-1.33$. The value of the exponent seems to be unaffected by electronic screening, at least within the accuracy of our calculations. The prefactor, however, is clearly enhanced from its OCP value, and this can be readily understood in terms of the response of the electronic component to the motion of the protons. A rigid uniform electronic background (as in the OCP) does not have any influence on the dynamics of the protons. A polarizable background weakens the proton-proton interaction thus increasing the mobility of the ions. In fact, the results obtained at $r_s=0.5$ are quite close to the OCP values, represented by the dashed line in Fig. 10. The difference becomes much larger at $r_s=1$, where $D^*$ differs from its OCP values by a factor of 3; this contrasts with the Thomas-Fermi MD results of Zérah et al. [@zerah], who observed a much milder effect (a factor 1.4 at $\Gamma=50$ and $r_s=1$). Therefore, the diffusion coefficient appears to be very sensitive to the treatment of electron screening. The case of $r_s=1.2$ is slightly different, because the exponent appears to be larger than the OCP value. However, it can be seen that the error bars are also compatible with an OCP-like power-law, represented by a line parallel to that of the other densities.
The diffusion coefficient is also related to the ion velocity autocorrelation function (VACF), which, in the OCP limit, exhibits a striking oscillatory behavior due to a strong coupling of the single-particle motion to the collective ionic plasma oscillations [@pollock]. Such oscillations were recently shown to persist at finite $r_s$, by MD simulations using the approximate Thomas-Fermi kinetic energy functional instead of the Kohn-Sham version [@zerah]. The present ab initio calculations qualitatively confirm this behavior. In Fig. 11 we present the VACF for a typical simulation in the solid phase ($T=300~K$) and then for three different temperatures in the fluid phase. The latter were computed in the supercell containing 162 atoms. Finite size effects are not very significant as regards the general features of the VACF. However, a small frequency shift is observed, and decorrelation happens faster in the larger sample. It is interesting to note that the fastest oscillation, i.e. the one associated with the ion plasma oscillations, is essentially temperature-independent.
The power spectra of the ionic VACF are plotted in Fig. 12 for $r_s=1$, at several temperatures. The spectra exhibit a high-frequency peak (or shoulder at the highest temperature) at a frequency which amounts to 55 % of the bare ion plasma frequency ($\omega_{\rm pl}$). The power spectra for $r_s=0.5$ are similar, with the difference that the high-frequency peak occurs now at a value which is 70 % of the bare plasma frequency [@kh:prl]. As expected, electron screening shifts the vibrational spectra to lower frequencies. Temperature, however, does not affect the position of the high-frequency peak, which remains the only well-defined feature at high temperatures, while the rest of the spectrum merges into a structureless continuum. These spectra were obtained from 54 H atom simulations. The 162 H atom sample yields a frequency 5 % lower than the 54 H atom one.
Collective modes: a signature of the metal-insulator transition
===============================================================
The oscillations in the VACF point to a long-lived longitudinal collective mode, related to the ionic plasmon mode of the OCP [@pollock]. We have computed the charge autocorrelation function:
$$F_{ZZ}(\bfk,t)=\frac{1}{N}<\rho_Z(\bfk,t)~\rho_Z(-\bfk,0)>$$
where the Fourier components of the microscopic charge density are:
$$\rho_Z(\bfk,t)=\rho_i(\bfk,t)-\rho_e(\bfk,t)$$
with $\rho_i$ and $\rho_e$ the AIMD-generated time-dependent densities. In keeping with the Born-Oppenheimer approximation, $\rho_e(\bfk,t)$ is a Fourier component of the expectation value of the electronic density for the instantaneous ion configuration. In practice, we computed the average of $F_{ZZ}(\bfk,t)$ over the shell of equal-modulus $\bfk$-vectors
$$F_{ZZ}(k,t)=\sum_{\vert\bfk\vert=k}~F_{ZZ}(\bfk,t)$$
From this, we computed the dynamical structure factor $S_{ZZ}(k,\omega)$ by Fourier transforming $F_{ZZ}(k,t)$. In the $r_s=0$ (OCP) limit, where the electrons form a uniform (non polarizable) background, $S_{ZZ}(k,\omega)$, which there reduces to the dynamical structure factor of the bare ions, is simply the k-dependent spectrum of the ionic plasma oscillations [@pollock]. In the long wavelength limit the mode is undamped, and its characteristic frequency is the ion plasma frequency $\omega_{\rm pl}$. Adiabatic electron polarization transforms this ionic plasmon (or optic) mode into an acoustic mode for any finite value of $r_s$ [@postogna; @barrat]. This mode is to be identified with the familiar low-frequency ion-acoustic mode. Only if the system were treated as a fully dynamical ion-electron plasma, would the high-frequency plasma oscillation mode appear, which is related to the fast electronic motions. This mode is obviously not accessible by adiabatic MD simulations. The conjectured scenario is confirmed by the results of our AIMD simulations. The dynamical structure factor $S_{ZZ}(k,\omega)$ was computed for the smallest wavenumber k compatible with the PBC ($ka=1.031$ for the 54 atom system) and for selected larger wavenumbers ($ka<3$). The resulting $S_{ZZ}(k,\omega)$ for $r_s=0.5$, $r_s=1$ and $r_s=1.2$ are shown in Fig. 13, for the smallest available wavenumber, and at a temperature just above melting. The sharp peaks are characteristic of the long-lived (weakly damped) mode anticipated above. The peaks shift to lower frequencies as $r_s$ increases due to enhanced electron screening, and in accord with the behavior of the plasmon-like peak in the spectrum of the VACF (the single-particle excitation coupled to the collective plasmon mode). The k-dependence of the spectrum $S_{ZZ}(k,\omega)$ is illustrated in Fig. 14, for $r_s=1$; the resulting dispersion curve is shown in fig. 15. A striking feature is the nearly constant width of the resonance peaks for $ka<1.5$, pointing to a nearly k-independent damping mechanism. The damping increases dramatically at larger wavenumbers, while the dispersion curve bends over; the behaviour is reminiscent of that observed for a classical fluid of atoms interacting via an effective Yukawa potential [@barrat]. The bending over may be regarded as a remnant of the negative dispersion of the plasmon mode observed in the strongly-coupled OCP [@pollock]. From the initial slope of the dispersion relation, we estimate a sound velocity of $c_s\approx 70$ km/s at $r_s=1$, which is is consistent with the extrapolation of very recent results by Alavi et al. to the ultra-high density regime [@aliala]. Sound velocities are relevant to the determination of global free oscillations of Jovian planets, which have been recently measured for Jupiter [@mosser].
The remarkable feature is that this collective mode is much sharper than the usual sound mode observed in metals at comparable wavenumbers. Moreover, the peaks do not shift significantly with temperature, although they broaden. However, particularly for low values of $k$, the signature of the collective mode can be detectable up to quite high temperatures (well above 3000 $K$). This is seen in Fig. 16, where we plot the dynamical structure factor for $ka=1.238$ (i.e. just before the bending-down point) in the 162-atom sample at $r_s=1$ and for three different temperatures. It is interesting to note that this is precisely the temperature range where a transition is expected to occur to the molecular (H$_2$) fluid phase at lower densities [@weir]. The observed collective behavior may be regarded as characteristic of the metallic phase of hydrogen, and is expected to change dramatically at the transition towards the molecular phase, which begins to show up at low temperatures at $r_s\approx 1.3$. Thus, an analysis of $S_{ZZ}(k,\omega)$ may provide an efficient diagnostic to locate the plasma phase transition at finite temperature.
Conclusions
===========
The main conclusions to be drawn from the present AIMD simulations of the hydrogen plasma in the high-density ($r_s\leq 1.2$) regime may be summarized as follows:
a\) Due to the significant spacing between the quantized electronic states in the vicinity of the Fermi surface, the N-dependence of the statistical averages must be treated with great care, in order to extract meaningful results.
b\) A linear-response treatment of the ion-electron correlations yields reasonable results at $r_s=0.5$, but becomes rapidly unreliable at lower densities.
c\) The bcc structure, which is the stable low temperature solid phase at least up to $r_s=0.5$, becomes unstable at lower densities, where hcp and simple-hexagonal phases appear. More work is needed to determine the full low-temperature phase diagram, also including zero-point-motion effects.
d\) The melting temperature drops sharply with decreasing density, due to the enhanced efficiency of electron screening of the effective interaction between ions. New interesting physics is likely to arise in the region of $r_s\approx 1.1$ and $T\approx 100~-~200~K$, where the existence of a bcc-hcp(fcc)-liquid triple point is argued, in a region where quantum effects in the protons are non-negligible.
e\) The fluid metallic phase behaves very much like a simple atomic liquid from a structural point of view, but the longitudinal collective dynamics of the ions retain a strong plasma-like character at intermediate wavenumbers. This reflects itself in unusually sharp peaks in the charge-fluctuation spectrum, which are gradually shifted to lower frequencies with decreasing density, as a result of electron screening. The damping, however, appears to be surprisingly insensitive to density, but is significantly enhanced by temperature. The acoustic character of the longitudinal mode is recovered at sufficiently small wavenumbers ($ka<1$), in qualitative agreement with a simple linear screening picture. A strong damping of the mode at intermediate wavenumbers should be a clear-cut signature of the plasma-to-molecular phase transition, which is expected to start at $T=0$ around $r_s=1.3$, and to move to finite temperatures of the order of a few thousand $K$ at lower densities ($r_s>1.3$) [@weir].
f\) The single-particle motion of the ions couples to the longitudinal collective mode, and reflects itself in a striking oscillatory behaviour of the velocity autocorrelation function, which is reminiscent of the behavior of the OCP, despite the action of strong electron screening. The resulting ionic self-diffusion constant is strongly enhanced at lower densities, for identical values of the plasma coupling constant $\Gamma$, but follows a power law similar to that observed in the OCP.
The present AIMD simulations will be extended to lower densities, in order to characterize the plasma-to-molecular phase transition, starting from the high density, metallic side.
One of us (JK) would like to thank Furio Ercolessi for facilitating his code to fit the two-body potential, and Ruben Weht for helping with the FP-LMTO calculations. We acknowledge helpful discussions with Ali Alavi, Detlef Hohl, Hong Xu, Gilles Zerah, Stephane Bernard, Carlo Pierleoni, Pietro Ballone, Erio Tosatti, Giorgio Pastore and Sandro Scandolo.
E. Wigner and H. B. Huntington, J. Chem. Phys. [**3**]{}, 764 (1935); H. K. Mao and R. J. Hemley, Science [**244**]{}, 1462 (1989). See H. K. Mao and R. J. Hemley, Rev. Mod. Phys. [**66**]{}, 671 (1994) and references therein. See also I. F. Silvera in [*Simple Molecular Systems at very High Pressure*]{}, eds. A. Polian, P. Loubeyre and N. Bocarra (Plenum, NY, 1989). L. Cui, N. H. Chen and I. F. Silvera, Phys. Rev. Lett. [**74**]{}, 4011 (1995). A. Ruoff, private communication. N. W. Ashcroft, Physics World, July 1995, p. 43. M. Hanfland, R. J. Hemley and H. K. Mao, Phys. Rev. Lett. [**70**]{},3760 (1993). R. J. Hemley and H. K. Mao, Phys. Rev. Lett. [**61**]{}, 857 (1988). E. Kaxiras, J. Broughton and R. J. Hemley, Phys. Rev. Lett. [**67**]{}, 1138 (1991); E. Kaxiras and J. Broughton, Europhys. Lett. [**17**]{}, 151 (1992). H. Chacham and S. G. Louie, Phys. Rev. Lett. [**66**]{}, 64 (1991). M. P. Surh, T. W. Barbee III and C. Mailhiot, Phys. Rev. Lett. [**70**]{}, 4090 (1993). H. Nagara and T. Nakamura, Phys. Rev. Lett. [**68**]{}, 2468 (1992). J. S. Tse and D. D. Klug, Nature (in press). I. I. Mazin and R. E. Cohen, Phys. Rev. B [**52**]{}, R8597 (1995). V. Natoli, R. M. Martin and D. M. Ceperley, Phys. Rev. Lett. [**70**]{}, 1952 (1995). T. W. Barbee III, A. Garcia, M. L. Cohen and J. L. Martins, Phys. Rev. Lett. [**62**]{}, 1150 (1989). T. W. Barbee III and M. L. Cohen, Phys. Rev. B [**44**]{}, 11563 (1991). P. Focher, G. L. Chiarotti, M. Bernasconi, E. Tosatti and M. Parrinello, Europhys. Lett. [**36**]{}, 345 (1994). J. Kohanoff, S. Scandolo, G. L. Chiarotti and E. Tosatti, unpublished. D. M. Ceperley and B. J. Alder, Phys. Rev. B [**36**]{}, 2092 (1987). V. Natoli, R. M. Martin and D. M. Ceperley, Phys. Rev. Lett. [**70**]{}, 1952 (1993). M. Baus and J.-P. Hansen, Phys. Reports [**59**]{}, 1 (1980); S. Ichimaru, H. Iyetomi and S. Tanaka, [*ibid*]{} [**149**]{}, 91 (1987). D. M. Ceperley, private communication. D. Hohl, V. Natoli, D. M. Ceperley and R. M. Martin, Phys. Rev. Lett. [**71**]{}, 541 (1993). J. Kohanoff and J.-P. Hansen, Phys. Rev. Lett. [**74**]{}, 626 (1995). A. Alavi, J. Kohanoff, M. Parrinello and D. Frenkel, Phys. Rev. Lett. [**73**]{}, 2599 (1994). C. Pierleoni, D. M. Ceperley, B. Bernu and W. R. Magro, Phys. Rev. Lett. [**73**]{}, 2145 (1994). J. Kohanoff, C. Pierleoni and D. M. Ceperley (unpublished). W. J. Nellis, M. Ross and N. C. Holmes, Science [**269**]{}, 1249 (1995), S. T. Weir, A. C. Mitchell, and W. J. Nellis, preprint. G. Chabrier, D. Saumon, W. B. Hubbard, and J. I. Lunine, Astrophys. J. [**391**]{}, 817 (1992), and references therein. W. R. Magro, D. M. Ceperley, C. Pierleoni, and B. Bernu, Phys. Rev. Lett [**76**]{}, 1240 (1996). A. Alavi, M. Parrinello and D. Frenkel, Science [**269**]{}, 1252 (1995). D. Saumon, W. B. Hubbard, G. Chabrier, and H. M. van Horn, Astrophys. J. [**391**]{}, 827 (1992). R. Car and M. Parrinello, Phy. Rev. Lett. [**55**]{}, 2471 (1985); for a recent review see G. Galli and M. Parrinello in [*Computer Simulations in Materials Science*]{}, p. 282, M. Meyer and V. Pontikis (eds.) (Kluwer, Dodrecht, 1991). For a comprehensive treatise see, e.g. R. G. Parr and W. Yang, [*Density Functional Theory of Atoms and Molecules*]{}, O.U.P. (1989). H. Xu, J.-P. Hansen and D. Chandler, Europhys. Lett. [**26**]{}, 419 (1994). D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. [**45**]{}, 566 (1980); D. M. Ceperley, Phys. Rev. B [**18**]{}, 3126 (1978). Z. Lin and J. Harris, J. Phys.: Condens. Matter [**4**]{}, 1055 (1992). W. B. Hubbard and M. S. Marley in [*Strongly Coupled Plasma Physics*]{}, NATO ASI Series B [**154**]{}, eds. D. Rogers and H. E. Dewitt. S. Galam and J.-P. Hansen, Phys. Rev. A [**14**]{}, 816 (1976). J. P. Perdew and A. Zunger, Phys. Rev. B [**23**]{}, 5048 (1981). F. Ercolessi and G. B. Adams, Europhys. Lett. [**26**]{}, 583 (1994). J. Hafner, [*From Hamiltonians to Phase Diagrams*]{}, Springer Series in Solid-State Sciences [**70**]{} (Springer, Berlin, 1987). P. Blöchl and M. Parrinello, Phys. Rev. B [**45**]{}, 9413 (1992). T. A. Arias, M. C. Payne and J. D. Joannopoulos, Phys. Rev. Lett. [**69**]{}, 1077 (1992). M. P. Grumbach, D. Hohl, R. M. Martin, and R. Car, J. Phys.: Condens. Matter [**6**]{}, 1999 (1994). G. Pastore, E. Smargiassi and F. Buda, Phys. Rev. A [**44**]{}, 6334 (1991). M. O. Robbins, K. Kremer and G. S. Grest, J. Chem. Phys. [**88**]{}, 3286 (1988). M. W. C. Dharma-wardana and F. Perrot, Phys. Rev. A [**26**]{}, 2096 (1982). C. Pierleoni, private communication. A. Alavi, private communication. J. P. Hansen, I. R. McDonald and E. L. Pollock, Phys. Rev. A [**11**]{}, 1025 (1975). G. Zérah, J. Clérouin and E. L. Pollock, Phys. Rev. Lett. [**69**]{}, 446 (1992). F. Postogna and M. P. Tosi, Nuovo Cimento [**55 B**]{}, 399 (1980). J. L. Barrat, J. P. Hansen and H. Totsuji, J. Phys. C [**21**]{}, 4511 (1988). B. Mosser et al., Astron. Astrophys. [**267**]{}, 604 (1993).
$r_s$ $\Gamma$ $D^*$ T (K)
------- ---------- ------- -------
0.5 65 .0122 10000
0.5 90 .0090 7500
0.5 130 .0052 5000
0.5 210 .0025 3100
1.0 108 .0163 3000
1.0 130 .0119 2500
1.0 163 .0096 2000
1.0 217 .0066 1500
1.0 326 .0034 1000
1.2 136 .0173 2000
1.2 181 .0114 1500
1.2 217 .0072 1000
1.2 272 .0056 800
: Diffusion coefficient as a function of the plasma coupling parameter $\Gamma$. The error in the determination of the diffusion coefficients is $\Delta D^*=0.001$.
|
---
abstract: 'We propose a self-consistent generalized quantum master equation (GQME) to describe electron transport through molecular junctions. In a previous study \[M.Esposito and M.Galperin. Phys. Rev. B **79**, 205303 (2009)\], we derived a time-nonlocal GQME to cure the lack of broadening effects in Redfield theory. To do so, the free evolution used in the Born-Markov approximation to close the Redfield equation was replaced by a standard Redfield evolution. In the present paper, we propose a backward Redfield evolution leading to a time-local GQME which allows for a self-consistent procedure of the GQME generator. This approach is approximate but properly reproduces the nonequilibrium steady state density matrix and the currents of an exactly solvable model. The approach is less accurate for higher moments such as the noise.'
author:
- Massimiliano Esposito
- Michael Galperin
bibliography:
- 'akb.bib'
title: 'A self-consistent quantum master equation approach to molecular transport'
---
\[intro\]Introduction
=====================
Recent advances in the experimental capabilities for constructing molecular junctions and measuring their response to external perturbations create new challenges for an adequate theoretical description of open quantum systems far from equilibrium.[@NitzanRatner; @LindsayRatner; @our_jpcm; @our_review] In molecular junctions, most of the interesting applications are concerned with either strongly correlated or resonant tunneling regimes (or both). In these situations, conventional perturbation theory and/or effective mean-field theories (e.g. Hartree-Fock, GW, DFT) are inapplicable and may lead to unphysical predictions.[@Millis]
In molecular junctions, molecules are sensible to processes such as oxidation, reduction, and excitations. This makes the description of transport in the language of the many-body states of the isolated molecules a convenient tool for molecules weakly coupled to the contacts. Furthermore, sophisticated quantum chemistry methods use many-body states to describe molecular electronic structures.[@Jensen] The dressed state representation,[@Nitzanbook] often employed in quantum chemistry, is another example of many-body states formulation. The standard nonequilibrium Green function (NEGF) approach,[@Danielewicz; @RammerSmith; @HaugJauhobook] is formulated in the language of elementary excitations and is therefore not well suited for a many-body states description. A suitable alternative, the Hubbard NEGF approach, [@Sandalov; @Fransson; @Hubbard] still displays inconsistencies for low level of approximation that need to be resolved (see Ref. for a discussion).
The simplest approach to transport at the molecular state level is the Redfield quantum master equation (QME) originally developed in the context of NMR[@Redfield57] and later in many other fields[@Nitzanbook; @Breuer02; @GaspNaga99] including transport through quantum junctions.[@Yan05; @Esposito06; @vonOppen09] It is derived using the Born-Markov approximation in order to get a closed evolution equation for the reduced density matrix. The rotating wave (or secular) approximation (RWA)[@Breuer02; @Esposito06] is often invoked when the molecular levels are well separated so that the molecule Bohr frequencies evolve fast compared to the relaxation time scale induced by the coupling to the contact. This leads to a QME which has a simple physical interpretation in the molecular eigenbasis. Indeed, populations satisfy a rate equation with rates given by Fermi golden rule (Pauli equation) and the coherences are each independently exponentially damped. As a result, in nonequilibrium steady-states, all coherence effects (in the molecular eigenbasis) are lost. In its general form, the Markovian Redfield equation predicts steady-state coherences that can be inaccurate and even sometimes lead to unphysical results.[@Tannor97a; @Tannor97b; @Esposito06; @Fleming09] Furthermore, broadening effects are totally absent from the Redfield description.[@Wacker05; @Neuhauser04; @Wegewijs08; @EspoGalpPRB]. In order to overcome some of these difficulties, we proposed in our previous work[@EspoGalpPRB] a generalized QME capable of predicting the broadening of the molecular levels in an approximate way. The key idea was to replace the assumption of free molecular evolution inside the kernel (which is second order in the molecule-contact interaction) used in the Born-Markov approximation by the Redfield evolution. The resulting equation was however non-local in time. Here we extend our consideration by proposing a kind of time-reversed Redfield evolution that leads to a time local generalized QME. This procedure is rewarding because it allows to formulate a practical self-consistent scheme to calculate the generator of the QME.
The structure of the paper is the following. Section \[model\] introduces a model of molecular junction. In section \[qme\], we briefly review our generalized QME derived earlier and introduce the self-consistent scheme. Section \[num\] presents numerical examples and compares the results obtained within the scheme to a numerically exact approach. Conclusions are drawn in section \[conclude\].
\[model\]Model
==============
We consider a molecule ($M$) coupled to two metal contacts ($L$ and $R$) each at its own equilibrium. The Hamiltonian of the system is $$\label{H}
\hat H=\hat H_M + \sum_{K=L,R}\left(\hat H_K + \hat V_{KM}\right)$$ The contacts are assumed to be reservoirs of free charge carriers $$\label{HK}
\hat H_K = \sum_{k\in K}\varepsilon_k\hat c_k^\dagger\hat c_k$$ where $K=L,R$ and $\hat c_k^\dagger$ ($\hat c_k$) are creation (annihilation) operators for an electron in state $k$.
The molecular Hamiltonian is represented in terms of the many-body states $\{|M\rangle\}$ of the isolated molecule $$\label{HM}
\hat H_M = \sum_{M_1,M_2} |M_1\rangle H^{(M)}_{M_1M_2} \langle M_2|
\equiv \sum_{M_1,M_2} H^{(M)}_{M_1M_2} \hat X_{M_1,M_2}$$ where $\hat X_{M_1,M_2}$ are projection (or Hubbard) operators. In particular, in the eigenbasis representation, the molecular Hamiltonian reads $$\label{HMeig}
\hat H_M = \sum_M E_M\hat X_{M,M}$$ where $E_M$ are the eigenenergies. Note that the molecular many-body states $|M\rangle$ are characterized by all the relevant quantum numbers describing the state of an isolated molecule or a dressed state.
The coupling between the molecule and the contacts is introduced in the usual way with hopping terms for the electrons moving from the contact to the molecule and vice versa $$\hat V_{KM} = \sum_{k\in K}\sum_\mathcal{M}\left(
V_{\mathcal{M}k}\hat X_\mathcal{M}^\dagger\hat c_k +
V_{k\mathcal{M}}\hat c_k^\dagger\hat X_\mathcal{M} \right)$$ Here $$\label{M}
\mathcal M \equiv(N_f\, s_f,N_i\, s_i)$$ is a transition from the molecular state $M_i\equiv N_i\, s_i$ to the molecular state $M_f\equiv N_f\, s_f$. The number of electrons on the molecule is $N_f=N_i-1$ and $s_i$ ($s_f$) is the set of all the quantum numbers characterizing the molecular state in the charging block $N_i$ ($N_f$), so that $\hat X_\mathcal{M}\equiv\hat X_{M_f,M_i}$.
\[qme\]Generalized quantum master equation
==========================================
In this section we start by reviewing the derivation of the generalized QME introduced in our previous work. We then introduce our new way of closing the master equation by using a time-reversed effective evolution, and demonstrate how the resulting equation can lead to a self-consistent scheme to calculate the propagator of the molecular density matrix.
\[derivation\]Exact equation of motion {#derivationexact-equation-of-motion .unnumbered}
--------------------------------------
We start by deriving exact equation of motion (EOM) for the quantity $$\label{X21}
\langle\hat X_{21}(t)\rangle \equiv
\mbox{Tr}\left[e^{+i\hat H(t-t_0)}\hat X_{21}e^{-i\hat H(t-t_0)}
\hat\rho_0\right] \equiv
\sigma_{12}(t)$$ where $|1\rangle$ and $|2\rangle$ are many-body molecular states, $t_0$ is a starting point of the evolution usually taken at the infinite past $t_0\to-\infty$, and $\hat\rho_0$ is the initial density operator for the whole system (molecule and contacts). We note that $$\label{DM}
\sigma_{12}(t) \equiv \langle 1|\,\mbox{Tr}_K\left[\hat\rho(t)\right]\,
|2\rangle$$ is a matrix element of the reduced density operator of the molecule $\hat\sigma$ at time $t$ which is obtained by tracing out the contact degrees of freedom from the full density matrix.
The Heisenberg EOM for the Hubbard operator in Eq.(\[X21\]) yields an [*exact*]{} EOM for the reduced density matrix (see Ref. for details of derivation) which in molecular eigenbasis reads $$\begin{aligned}
\label{EOM}
&\frac{d}{dt}\sigma_{12}(t) = -i\left(E_1-E_2\right)\sigma_{12}(t)
-\sum_\mathcal{M}\sum_s\int_{-\infty}^t dt'
\nonumber \\
&i\left\{(-1)^{N_1-N_2}\left[\Sigma^{<}_{(N_1-1\, s,1)\mathcal{M}}(t-t')
\langle\hat X_\mathcal{M}(t')\,\hat X^\dagger_{N_1-1\, s,2}(t)\rangle
\nonumber \right.\right. \\ & \qquad\qquad \left.
+\langle\hat X_{N_2-1\, s,1}(t)\,\hat X^\dagger_\mathcal{M}(t')\rangle
\Sigma^{<}_{\mathcal{M}(N_2-1\, s,2)}(t'-t)\right]
\nonumber \\
&\quad -\Sigma^{<}_{(2,N_2+1\, s)\mathcal{M}}(t-t')
\langle\hat X_\mathcal{M}(t')\,\hat X^\dagger_{1,N_2+1\, s}(t)\rangle
\nonumber \\ &\quad
+\langle\hat X^\dagger_\mathcal{M}(t')\,\hat X_{N_2-1\, s,1}(t)\rangle
\Sigma^{>}_{\mathcal{M}(N_2-1\, s,2)}(t'-t)
\nonumber \\
&\quad +\Sigma^{>}_{(N_2-1\, s,1)\mathcal{M}}(t-t')
\langle\hat X^\dagger_{N_1-1\, s,2}(t)\,\hat X_\mathcal{M}(t')\rangle
\nonumber \\ &\quad
-\langle\hat X_{2,N_1+1\, s}(t)\,\hat X^\dagger_\mathcal{M}(t')\rangle
\Sigma^{<}_{\mathcal{M}(1,N_1+1\, s)}(t'-t)
\nonumber \\ &
-(-1)^{N_1-N_2}\left[\Sigma^{>}_{(2,N_2+1\, s)\mathcal{M}}(t-t')
\langle\hat X^\dagger_{1,N_2+1\, s}(t)\,\hat X_\mathcal{M}(t')\rangle
\right. \nonumber \\ &\left.\left. \qquad\qquad
+\langle\hat X^\dagger_\mathcal{M}(t')\,\hat X_{2,N_1+1\, s}(t)\rangle
\Sigma^{>}_{\mathcal{M}(1,N_1+1\, s)}(t'-t) \right]\right\}\end{aligned}$$ Here, $\mathcal{M}$ is defined in Eq.(\[M\]), $\Sigma^{>\,(<)}_{\mathcal{M}_1\mathcal{M}_2}$ are the greater (lesser) projections of a self-energy due to the coupling to the contacts $$\label{SE}
\Sigma_{\mathcal{M}_1\mathcal{M}_2}(\tau_1,\tau_2) \equiv
\sum_{K=L,R} \sum_{k\in K} V_{\mathcal{M}_1k}\, g_k(\tau_1,\tau_2)\,
V_{k\mathcal{M}_2}$$ where $g_k(\tau_1,\tau_2)\equiv-i\langle T_c\hat c_k(\tau_1)\,
\hat c_k^\dagger(\tau_2)\rangle$ is the free electron Green function ($T_c$ is the contour ordering operator). Dissipation matrix $\Gamma$ is introduced in a usual way $$\Gamma_{\mathcal{M}_1\mathcal{M}_2}(E) =
i\left[\Sigma^r_{\mathcal{M}_1\mathcal{M}_2}(E)
-\Sigma^a_{\mathcal{M}_1\mathcal{M}_2}(E)\right]$$ and within wide band approximation assumed below is energy independent.
Eq.(\[EOM\]) is [*exact*]{} but does not have a closed form in terms of the reduced density matrix $\sigma_{12}$. Indeed, its right hand side contains two-time correlation functions of Hubbard operators. Attempt to write EOM for the latter would lead to expressions having in their right hand side correlation functions of higher order involving both, molecular Hubbard operators and creation (annihilation) operators for the electrons in the contacts, at different times. This would generate an infinite hierarchy of EOMs.
\[GQME\]Generalized time-nonlocal QME {#gqmegeneralized-time-nonlocal-qme .unnumbered}
-------------------------------------
The usual strategy involves closing the system of equations by [*approximately*]{} representing high order correlation function in terms of correlation function(s) of lower order. In particular, closing Eq.(\[EOM\]) for the case when $M$ are effective single particle orbitals at the first step (i.e. reducing the two-time correlation function in the right hand side to a single time average) is known as the generalized Kadanoff-Baym ansatz (GKBA).[@HaugJauhobook] In our previous work [@EspoGalpPRB], we introduced the analogue of the GKBA in Liouville space where $M$ are molecular many-body states: $$\begin{aligned}
\label{GKBA}
&\langle\hat X_{12}(t)\hat X^\dagger_{34}(t')\rangle
\approx
\nonumber \\ &
i\sum_{M_1,M_2}\left[
\mathcal{G}^r_{12,M_1M_2}(t-t')\langle\hat X_{M_1M_2}(t')\,
\hat X^\dagger_{34}(t')\rangle
\right. \\ & \left. \qquad\quad
-\langle \hat X_{12}(t)\,
\hat X^\dagger_{M_1M_2}(t)\rangle\mathcal{G}^a_{M_1M_2,34}(t-t')
\right]
\nonumber\end{aligned}$$ A similar expression holds for $\langle\hat X^\dagger_{34}(t')\, \hat X_{12}(t)\rangle$. Here $\mathcal{G}^{r\, (a)}$ are retarded (advanced) Green functions in Liouville space $$\begin{aligned}
\mathcal{G}^r_{12,34}(t) &\equiv -i\theta(t) \ll \hat X_{21}\hat I_B \rvert
e^{-i\mathcal{L}t}\rvert\hat X_{43}\hat\rho_B^{eq}\gg
\nonumber \\
&\equiv -i\theta(t)\ll\hat X_{21}\rvert e^{-i\mathcal{L}_{eff}t}
\rvert\hat X_{43}\gg
\\
\mathcal{G}^a_{12,34}(t) &\equiv -i\theta(-t) \ll \hat X_{34}\hat I_B \rvert
e^{i\mathcal{L}t}\rvert\hat X_{12}\hat\rho_B^{eq}\gg
\nonumber \\
&\equiv i\theta(-t) \ll\hat X_{34}\rvert e^{i\mathcal{L}_{eff}t}
\lvert \hat X_{12}\gg\end{aligned}$$ where $\hat\rho_B^{eq}\equiv\hat\rho_L^{eq}\hat\rho_R^{eq}$ is the equilibrium density operator for the contacts and $\hat I_B$ is unity operator in the contacts subspace. Using (\[GKBA\]) in (\[EOM\]), leads to a time-nonlocal generalized QME given by Eq.(35) in Ref. .
\[antiGQME\]Generalized time-local QME {#antigqmegeneralized-time-local-qme .unnumbered}
--------------------------------------
We now propose an alternative way to close Eq.(\[EOM\]) by reducing the two-time correlation function in the right hand side of (\[EOM\]) to the later rather than earlier time. Indeed, since the whole system evolution is time-reversible (one has to be careful with spin and magnetic field, though), and since the reduction of a two-time correlation function to a single-time average is an approximation, the reduction of the correlation function to a later time does not a priori seem to be worse than to the earlier one.
The two-time correlation function can be [*exactly*]{} expressed in Liouville space as $$\begin{aligned}
\label{corr}
&\langle\hat X_{12}(t)\,\hat X^\dagger_{34}(t')\rangle =
\nonumber \\ & \qquad
\theta(t-t')\ll\hat X_{34}\hat I_B\rvert e^{-i\mathcal{L}(t'-t)}
\lvert \hat\rho(t)\hat X_{12}\gg
\\ & \quad
+ \theta(t'-t)\ll \hat X_{12}^\dagger\hat I_B\rvert e^{-i\mathcal{L}(t-t')}
\lvert \hat X_{34}^\dagger\hat\rho(t')\gg
\nonumber\end{aligned}$$ Then using the projection superoperator introduced in Ref. $$\mathcal{P} = \sum_{M_1,M_2} \lvert\hat X_{M_1,M_2}\hat\rho_B^{eq}\gg\,
\ll\hat X_{M_1,M_2}\hat I_B\rvert$$ one gets the alternative ansatz $$\begin{aligned}
\label{anti_GKBA}
&\langle\hat X_{12}(t)\hat X^\dagger_{34}(t')\rangle \approx
\nonumber \\ &
i\sum_{M_1,M_2}\left[
\tilde{\mathcal{G}}{}^r_{12,M_1M_2}(t-t')
\langle\hat X_{M_1M_2}(t')\,\hat X^\dagger_{34}(t')\rangle
\right. \\ & \left. \qquad\quad
- \langle \hat X_{12}(t)\,\hat X^\dagger_{M_1M_2}(t)\rangle
\tilde{\mathcal{G}}{}^a_{M_1M_2,34}(t-t')
\right]
\nonumber\end{aligned}$$ and a similar expression for $\langle\hat X^\dagger_{34}(t')\, \hat X_{12}(t) \rangle$. Here we introduced the retarded and advanced Green functions on the Keldysh anti-contour[@Banyai] $$\begin{aligned}
\label{Gr_akb}
\tilde{\mathcal{G}}{}^r_{12,34}(t) &\equiv -i\theta(-t)
\ll \hat X_{21}\hat I_B \rvert e^{-i\mathcal{L}t}\rvert\hat X_{43}
\hat\rho_B^{eq}\gg
\nonumber \\
&\equiv -i\theta(-t)\ll\hat X_{21}\rvert e^{-i\ {}^\theta\mathcal{L}_{eff}t}
\rvert\hat X_{43}\gg
\\
\label{Ga_akb}
\tilde{\mathcal{G}}{}^a_{12,34}(t) &\equiv i\theta(t)
\ll\hat X_{34}\hat I_B \rvert e^{i\mathcal{L}t}\rvert\hat X_{12}
\hat\rho_B^{eq}\gg
\nonumber \\
&\equiv i\theta(t) \ll\hat X_{34}\rvert e^{i\ {}^\theta\mathcal{L}_{eff}t}
\lvert \hat X_{12}\gg\end{aligned}$$ where ${}^\theta\mathcal{L}_{eff}$ is the effective Liouvillian generating the time-reversed evolution. Its connection to the effective Liouvillian $\mathcal{L}_{eff}$ generating forward-time evolution is[@SJ1; @SJ2] $$\label{Lconnect}
{}^\theta\mathcal{L}=\mathcal{L}^{*}$$ Note difference in sign with Refs. which is due to our definition of evolution operator as $e^{-i\ {}^\theta\mathcal{L}_{eff}t}$ instead of $e^{{}^\theta\mathcal{L}_{eff}t}$ in Refs. .
Applying (\[anti\_GKBA\]) to (\[EOM\]) leads to a time-local version of the generalized QME $$\label{sGQME}
\frac{d}{dt}\sigma_{12}(t) = \sum_{3,4}\mathcal{L}_{12,34}\, \sigma_{34}(t)$$ where in the molecular eigenbasis $$\begin{aligned}
\label{LGQME}
&\mathcal{L}_{12,34} = -i(E_1-E_2)\,\delta_{1,3}\,\delta_{2,4}
-\sum_\mathcal{M}\int_{-\infty}^{+\infty}dt
\nonumber \\ & \left\{
\ (-1)^{N_1-N_2}\,\delta_{N_1-1,N_3}\,\delta_{N_2-1,N_4}\times
\right. \nonumber \\ &
\left[\tilde{\mathcal{G}}{}^a_{(3,1)\mathcal{M}}(t)\,
\Sigma^{<}_{\mathcal{M}(4,2)}(-t) - \Sigma^{<}_{(3,1)\mathcal{M}}(t)\,
\tilde{\mathcal{G}}{}^r_{\mathcal{M}(4,2)}(-t)\right]
\nonumber \\ &
-(-1)^{N_1-N_2}\,\delta_{N_1+1,N_3}\,\delta_{N_2+1,N_4}\times
\nonumber \\ &
\left[\tilde{\mathcal{G}}{}^a_{(2,4)\mathcal{M}}(t)\,
\Sigma^{>}_{\mathcal{M}(1,3)}(-t) - \Sigma^{>}_{(2,4)\mathcal{M}}(t)\,
\tilde{\mathcal{G}}{}^r_{\mathcal{M}(1,3)}(-t)\right]
\nonumber \\ &
+\delta_{1,3}\,\delta_{N_2,N_4}\sum_s\left[
\tilde{\mathcal{G}}{}^a_{(N_2-1\, s,4)\mathcal{M}}(t)\,
\Sigma^{>}_{\mathcal{M}(N_2-1\, s,2)}(-t)
\right. \nonumber \\ &\left.\qquad\qquad +
\Sigma^{<}_{(2,N_2+1\, s)\mathcal{M}}(t)\,
\tilde{\mathcal{G}}{}^r_{\mathcal{M}(4,N_2+1\, s)}(-t)\right]
\nonumber \\ &
-\delta_{2,4}\,\delta_{N_1,N_3}\sum_s \left[
\tilde{\mathcal{G}}{}^a_{(3,N_1+1\, s)\mathcal{M}}(t)\,
\Sigma^{<}_{\mathcal{M}(1,N_1+1\, s)}(-t)
\right. \nonumber \\ &\left.\left. \qquad\qquad +
\Sigma^{>}_{(N_1-1\, s,1)\mathcal{M}}(t)\,
\tilde{\mathcal{G}}{}^r_{\mathcal{M}(N_1-1\, s,3)}(-t)
\right] \right\}\end{aligned}$$ Note that this expression for the effective Liouvillian in an arbitrary basis will differ from Eq.(\[LGQME\]) only in the free evolution term.
If the QME, Eqs. (\[sGQME\]) and (\[LGQME\]), is used with the free molecular evolution instead of the effective dynamics in Eqs. (\[Gr\_akb\]) and (\[Ga\_akb\]), one gets the standard Markovian Redfield QME. The RWA is justified when $\Gamma_{mn}\ll|\varepsilon_a-\varepsilon_b|$ ($m,n\in\{a,b\}$) in the molecular eigenbasis and consist in neglecting the nondiagonal of $\Gamma_{mn}$ (here the standard effective single orbital formulation is used)[@Esposito06]. As a result, coherences become decoupled from population in the molecular eigenbasis. The former die off at steady state and later obey a Pauli rate equation. When the RWA is not justified, coherence effects in the molecular eigenbasis cannot be neglected.
It should be clear from our derivation that time-local form Eq.(\[sGQME\]) of the generalized QME should have a similar degree of accuracy as the time-nonlocal version proposed earlier.[@EspoGalpPRB] However, the present form is more suitable for realistic calculations, but most important it naturally suggests a self-consistent procedure to evaluate the effective Liouvillian. Indeed, the effective Liouvillian, Eq.(\[LGQME\]), depends on the Liouville space Green functions, Eqs. (\[Gr\_akb\]) and (\[Ga\_akb\]), which in turn depend on the Liouvillian through the connection (\[Lconnect\]).
Expression for current {#expression-for-current .unnumbered}
----------------------
The general expression for the time-dependent current at the contact-molecule interface $K$ can be calculated within the non-equilibrium Hubbard Green function approach as[@JauhoMeirWingreen; @Hubbard] $$\begin{aligned}
I_K(t) &= 2\,\mbox{Im}\int_{-\infty}^t dt'\,\sum_{\mathcal{M},\mathcal{M}'}
\nonumber \\ &
\left[ \
\langle\hat X_\mathcal{M}(t)\,\hat X^\dagger_{\mathcal{M}'}(t')\rangle
\Sigma^{<\, K}_{\mathcal{M}'\mathcal{M}}(t'-t)
\right. \\ & \left. +
\langle\hat X^\dagger_{\mathcal{M}'}(t')\,\hat X_\mathcal{M}(t)\rangle
\Sigma^{>\, K}_{\mathcal{M}'\mathcal{M}}(t'-t)
\right]
\nonumber\end{aligned}$$ Using our ansatz, Eq.(\[anti\_GKBA\]), leads to the expression $$\begin{aligned}
\label{I_anti_GKBA}
& I_K(t) = -2\,\mbox{Re}\int_{-\infty}^{+\infty}dt'
\sum_{\mathcal{M},\mathcal{M}'}\sum_s
\\ & \left[
\sigma_{N\, s,N\, s_f}(t)\,
\tilde{\mathcal{G}}{}^a_{(N\, s,N+1\, s_i)\mathcal{M}'}(t-t')\,
\Sigma^{<\, K}_{\mathcal{M}'\mathcal{M}}(t'-t)
\right. \nonumber \\ & \left.
+
\sigma_{N+1\, s_i,N+1\, s}(t)\,
\tilde{\mathcal{G}}{}^a_{(N\, s_f,N+1\, s)\mathcal{M}'}(t-t')\,
\Sigma^{>\, K}_{\mathcal{M}'\mathcal{M}}(t'-t)
\right]
\nonumber\end{aligned}$$ where $\mathcal{M}\equiv (N\, s_f,N+1\, s_i)$.
Full counting statistics {#full-counting-statistics .unnumbered}
------------------------
The theory of full counting statistics (FCS) was initially proposed by Levitov and Lesovik,[@LevitovLesovik1; @LevitovLesovik2] and became popular in the molecular electronics community when shot noise in molecular junctions became experimentally measurable.[@Ruitenbeek] The theoretical formalism for FCS within the QME was developed in Ref. . Within the FCS the evolution operator is dressed by counting field(s) $\lambda$ which track the exchange of electrons between the molecule and the contact(s) $$\hat V^{\lambda}_{KM} = \sum_{k\in K}\sum_\mathcal{M}\left(
V_{\mathcal{M}k}e^{i\lambda/2}\hat X_\mathcal{M}^\dagger\hat c_k +
V_{k\mathcal{M}}e^{-i\lambda/2}\hat c_k^\dagger\hat X_\mathcal{M} \right)$$ Below we assume that counting starts at time $t_0$.
The dressed Liouville equation for the total density matrix takes the form $$|\hat\rho_\lambda(t)\gg=\exp[-i\mathcal{L}_\lambda(t-t_0)]\,|\hat\rho(t_0)\gg$$ As usual initial condition is assumed to be a direct product of the molecular and bath density matrices $$|\hat\rho(t_0)\gg = |\hat\sigma(t_0)\,\hat\rho_B^{eq}\gg$$ so that dressed evolution of the system DM becomes $$|\hat\sigma_\lambda(t)\gg = \exp[-i\mathcal{L}_{eff,\lambda}(t-t_0)]\,
|\hat\sigma(t_0)\gg$$ The FCS is given by the generating function $$G(t,\lambda) \equiv \ll\hat I|\hat\rho_\lambda(t)\gg$$ The long time limit of the logarithm of the generating function, $$S(\lambda) \equiv \lim_{t\to\infty} \frac{1}{t}\ln G(t,\lambda),$$ provides information on the steady-state cumulants of the FCS $$C_n\equiv\frac{d^n}{d(i\lambda)^n}S(\lambda)$$ In particular, the steady-state current is given by the first cumulant and the second cumulant yields to the zero-frequency shot noise. Using the spectral decomposition of the effective Liouvillian $$\mathcal{L}_{eff,\lambda} = \sum_\gamma |R_\gamma(\lambda)\gg\,
\nu_\gamma(\lambda)\,\ll L_\gamma(\lambda)|$$ and since in the long time limit only one eigenmode $\nu_0(\lambda)$ survives, we get $$\begin{aligned}
\label{I_FCS}
I &= -\frac{d}{d\lambda}\nu_0(\lambda)
\\
\label{S_FCS}
S &= i\frac{d^2}{d\lambda^2}\nu_0(\lambda)\end{aligned}$$ We note that both (\[I\_anti\_GKBA\]) and (\[I\_FCS\]) provide the same steady-state current.
\[num\]Numerical examples
=========================
We now compare the results predicted by our new self-consistent approach with the standard Markovian Redfield equation and the nonequilibrium Green function results. The simplest model which can be treated by these three methods, while providing information on both populations and coherences, is a non-interacting two-level bridge (TLB) between metallic contacts.
The bridge part of the model has 4 many-body states: $\lvert 0\rangle\equiv\lvert 0,0\rangle$, $\lvert a\rangle\equiv\lvert 1,0\rangle$, $\lvert b\rangle\equiv\lvert 0,1\rangle$, and $\lvert 2\rangle\equiv\lvert 1,1\rangle$, where $\lvert n_a,n_b\rangle$ indicates number of electrons $n_{a,b}=\{0,1\}$ on the level $a$ and $b$, respectively. The relevant single electron transitions $\mathcal{M}$ are: $(0,a)$, $(0,b)$, $(b,2)$, and $(a,2)$. They are connected to the second quantized (single-particle) excitation operators used in standard GF approaches by $$\begin{aligned}
\label{da}
\hat d_a &= \hat X_{0a} + \hat X_{b2} \\
\label{db}
\hat d_b &= \hat X_{0b} - \hat X_{a2}\end{aligned}$$ where $\hat d^\dagger_{a,b}$ ($\hat d_{a,b}$) are the creation (annihilation) operators for an electron on level $a$ and $b$, respectively.
The molecular Hamiltonian, Eq.(\[HM\]), in this case takes the form $$\label{HM_TLB}
\hat H_M = \varepsilon_a\hat X_{aa} + \varepsilon_b\hat X_{bb}
+ t\left(\hat X_{ab}+\hat X_{ba}\right)$$ The model was treated extensively within the QME approach.[@Novotny02; @Esposito06; @Esposito07; @Esposito08]
The TLB model is easily exactly solved using NEGF. The connection, Eqs. (\[da\]) and (\[db\]), provides partial information about the many-body state populations but full information about coherences, which makes the comparison between the different approaches meaningful. In particular, $$\begin{aligned}
\label{Gaa}
-iG^{<}_{aa}(t,t) &= \sigma_{00}(t) + \sigma_{bb}(t) \\
\label{Gbb}
-iG^{<}_{bb}(t,t) &= \sigma_{00}(t) + \sigma_{aa}(t) \\
\label{Gab}
-iG^{<}_{ab}(t,t) &= \sigma_{ab}(t)\end{aligned}$$
Inter-level coherence in the TLB is either due to hoping matrix element $t$ or due to the coupling of the two levels via a common bath. The latter interference enters through the non-diagonal elements of the bridge-contact coupling matrix $$\begin{aligned}
\Gamma^K_{ab} &\equiv \sum_{k\in K} V_{ak}V_{kb}\delta(E-\varepsilon_k)
\nonumber \\
&\equiv \ \sum_{k\in K} V_{(oa)k}V_{k(0b)}\delta(E-\varepsilon_k)
\\
&\equiv - \sum_{k\in K} V_{(b2)k}V_{k(a2)}\delta(E-\varepsilon_k) \end{aligned}$$ where $K=L,R$. The matrix is energy independent in the wide-band limit assumed below. We intentionally perform simulations at low temperature $T=10$ K to avoid broadening of the bridge levels due to artificially high values of the temperature. We focus on the relevant temperature regime for molecular junctions: $\Gamma \gg k_BT$.
![\[f1\] (Color online) Two-level bridge by Redfield QME (dash-dotted line, green), NEGF (dashed line, red), and self-consistent GQME (solid line, red) approaches. Shown are (a) probabilities, (b) coherence, and (c) current vs. bias. Far off-resonant local basis treatment. See text for parameters. ](fig1){width="\linewidth"}
We start our consideration from a simple far off-resonant case, where $\Gamma_{mn}\ll|\varepsilon_a-\varepsilon_b|$ ($m,n\in\{a,b\}$). Here our consideration is done in a local molecular basis ($t\neq 0$). The parameters of the calculation are the following: level positions $\varepsilon_a=0.2$ eV and $\varepsilon_b=0.5$ eV, inter-level hopping $t=0.1$, escape rates for both levels into both contacts $\Gamma^K_{aa}=\Gamma^K_{bb}=0.1$ eV ($K=L,R$), and level mixing due to coupling to the contacts $\Gamma^K_{ab}=\Gamma^K_{ba}=0.05$ eV. The Fermi energy in the absence of bias is taken as zero, $E_F=0$, and the voltage division factor is $1$, i.e. $\mu_L=E_F+|e|V$ and $\mu_R=E_F$. The NEGF calculation is performed on an energy grid spanning the range from $-10$ to $10$ eV with step $10^{-4}$ eV. Both the Redfield QME and our scheme are expected to work properly in this region of parameters. Figure \[f1\] compares the results of the Redfield QME and our new scheme to the exact NEGF results for the model. The main graph and the inset in Fig.\[f1\]a display populations, Eqs. (\[Gaa\]) and (\[Gbb\]). The real part of the coherence, Eq.(\[Gab\]), is shown in Fig.\[f1\]b. The imaginary part of the coherence is zero in this case (not shown). As discussed earlier,[@EspoGalpPRB] the Redfield QME approach misses the information about the broadening of the molecular levels. Our approach accurately recovers this information. Both populations and coherence are well reproduced by our GQME as well as the current-voltage characteristics (see Fig.\[f1\]c). Qualitatively, the predictions of the Redfield QME approach are also correct here.
![\[f2\] (Color online) Two-level bridge by Redfield QME (dash-dotted line, green), NEGF (dashed line, red), and self-consistent GQME (solid line, red) approaches. Shown are (a) probabilities, (b) coherence, and (c) current vs. bias. Case of strong mixing due to coupling to the contact(s). See text for parameters. ](fig2){width="\linewidth"}
We next consider the case of strong mixing due to coupling to the contact(s). Parameters of the calculation are $t=0$, $\Gamma^L_{aa}=0.3$ eV, $\Gamma^R_{aa}=0.048$ eV, $\Gamma^L_{bb}=0.2$ eV, $\Gamma^R_{bb}=0.1125$ eV, $\Gamma^L_{ab}=\Gamma^L_{ba}=0.12$ eV, and $\Gamma^R_{ab}=\Gamma^R_{ba}=0.15$ eV. The other parameters are the same as in Fig.\[f1\]. Figure \[f2\] compares the Redfield QME and our self-consistent GQME scheme with the exact NEGF results. Our scheme still reproduces populations (Fig.\[f2\]a), coherences (Fig.\[f2\]b), and current (Fig.\[f2\]c) accurately. The Redfield QME however fails to reproduce, even qualitatively, the populations (see Fig.\[f2\]a) and the real part of the coherence (see Fig.\[f2\]b). This is due to coherence effects through the bath which are not properly captured by the Redfield QME.
![\[f3\] (Color online) Two-level bridge by Redfield QME (dash-dotted line, green), NEGF (dashed line, red), and self-consistent GQME (solid line, red) approaches. Shown are (a) current and (b) conductance vs. bias. Resonant, $\varepsilon_a=\varepsilon_b$, consideration in a local basis with only level $a$ coupled to contacts, $\Gamma^K_{aa}\neq 0$ and $\Gamma^{K}_{bb}=\Gamma^K_{ab}=\Gamma^K_{ba}=0$ ($K=L,R)$. See text for parameters. ](fig3){width="\linewidth"}
Interference effects in molecular systems were observed experimentally for electron transfer[@Joachim] and molecular junction currents[@MayorWeber] involving derivatives of benzene connected in meta or para position. They were also extensively discussed in theoretical literature.[@Mazumdar07; @RatnerJACS07; @RatnerJACS08com; @RatnerJACS08; @RatnerJCP08; @RatnerCPC09] Figure \[f3\] shows current (a) and conductance (b) vs. bias, for a model system in which destructive interferences can be experimentally observed and which has been discussed earlier in the literature.[@RatnerCPC09] This model is a two-level system with only one of the levels, $a$, attached to both $L$ and $R$ contacts. The other level, $b$, is coupled to the level $a$ through a hopping element $t$. Level $b$ is not directly attached to any contacts. As a result, the tunneling electron has two possible pathways to be transferred from contact $L$ to contact $R$ via the system: one directly through level $a$ and the other by exploring level $b$ on its way. Interference between the two paths leads to destructive interference in the transport characteristics, which reveals itself as a dip in the conductance. Parameters of the calculation are $\varepsilon_a=\varepsilon_b=0.25$ eV, $t=0.05$ eV, $\Gamma^L_{aa}=\Gamma^R_{aa}=0.2$ eV and $\Gamma^K_{ab}=\Gamma^K_{ba}=\Gamma^K_{bb}=0$. Fig.\[f3\]b shows that destructive interference in conductance is accurately captured by our self-consistent GQME approach but not by the Redfield QME.
![\[f4\] (Color online) Two-level bridge by Redfield QME (dash-dotted line, green), NEGF (dashed line, red), and self-consistent GQME (solid line, red) approaches. Shown are (a) shot noise and (b) differential shot noise vs. bias. See text for parameters. ](fig4){width="\linewidth"}
Finally, we present the results of calculation for the zero-frequency shot noise. The shot noise was calculated using the FCS approach, which for the Redfield QME and our self-consistent scheme yields to expression (\[S\_FCS\]). The FCS within NEGF was discussed in our previous publication.[@spin_pump] Note that the FCS for a two-level bridge within QME was also discussed in Ref. We intentionally consider two independent levels (no coupling either within the system or through the bath) to demonstrate the general problem that usual QME schemes have in representing higher moments. We assume that the level $\varepsilon_a=0.5$ eV is coupled symmetrically to both contacts $\Gamma^L_{aa}=\Gamma^R_{aa}=0.1$ eV. The other level $\varepsilon_b=1.5$ eV is coupled asymmetrically $\Gamma^L_{bb}=10\Gamma^R_{bb}=0.1$ eV. Other parameters are $t=0$, $\Gamma^K_{ab}=\Gamma^K_{ba}=0$, and $T=10$ K.
Figure \[f4\] compares the results of the Redfield QME and the self-consistent GQME with the exact results provided by NEGF. It is known[@our_noise] that the differential shot noise yields a two-peak structure for a symmetrically coupled molecule, with only a single peak observed for highly asymmetric coupling. One sees that the latter peak (centered around $1.5$ eV in Fig.\[f4\]b) is reproduced quite well by our self-consistent approach. The double well structure is however missed by both the Redfield QME and our approach (see peak centered aroundi $0.5$ eV). Cumulants beyond first order can be very sensible to approximations and our level of theory, which neglects molecule-contacts correlations beyond second order, has to be improved to properly reproduce them.
Summarizing, within our simple two-level model of molecular junction, the comparison of the proposed self-consistent GQME and the Markovian Redfield QME to the exact results provided by the NEGF shows that the self-consistent GQME is highly accurate even when the Redfield QME fails qualitatively. We find that the self-consistent GQME will only fail in the region of parameters where the inter-level coherence due to the coupling to contacts in the molecular eigenbasis is bigger than the distance between the molecular eigenstates and is of the order of the molecule-contact coupling strength (i.e. exactly at resonance). As a result, in most relevant practical calculations of transport in molecular junctions, we expect our approach to be accurate. The ability of the proposed scheme to treat molecular transport in the language of many-body states of the isolated molecule and its ability to properly account for interference effects makes it a valuable practical tool for ab initio calculations of transport in molecular junctions.
\[conclude\]Conclusion
======================
We present a practical scheme for molecular transport calculation. The scheme is based on a time-local generalized quantum master equation obtained by closing the exact EOM for Hubbard operators by employing a time-reversed evolution ansatz on the Keldysh anti-contour, similar to generalized Kadanoff-Baym ansatz introduced in our previous publication. We note that the approximations involved in derivation of the time-local equation are essentially the same as for the earlier (more traditional) time-nonlocal GQME. The time-locality of the GQME allows us to formulate a feasible self-consistent scheme to calculate the time dependent molecular density matrix and the current in term of molecular many-body states. We find that the convergence of the self consistent method for a simple two-level bridge model is achieved within two iterative steps. The results of the calculation for TLB within self-consistent GQME are compared to Redfield QME approach, and to exact NEGF results. We demonstrate that our scheme (contrary to the usual QME result) properly captures populations and coherences. In particular, destructive interference effects in the molecular devices previously discussed in the literature can now be properly described in the many-body states language, which makes the scheme a valuable tool for practical ab initio calculations. Current-voltage characteristics are reproduced with high accuracy. We are able to calculate the molecular device characteristics in the experimentally relevant regime of low temperatures. We find that the scheme breaks down in the region of the parameters where coherences in the system eigenbasis (i.e. coherences introduced through non-diagonal elements of molecule-contact coupling matrix $\Gamma$) are both bigger than the inter-level separation and are of the order of the escape rates (diagonal elements of the molecule-contact coupling matrix $\Gamma$). Thus, we expect the scheme to be a valuable tool for most practical transport calculations in molecular junctions.
The formulation of a scheme capable of reproducing higher moments of the full counting statistics in the many-body state language that goes beyond the Redfield QME approach[@EspositoReview] is the goal of future research.
M.E. is supported by the Belgian Federal Government (IAP project “NOSY"). M.G. gratefully acknowledges support of the UCSD (Startup Fund) and US-Israel Binational Science Foundation.
|
---
abstract: |
We calculate the independent helicity amplitudes in the decays $B \to K^*
\ell^+ \ell^-$ and $B \to \rho \ell \nu_\ell$ in the so-called Large-Energy-Effective-Theory (LEET). Taking into account the dominant $O(\alpha_s)$ and $SU(3)$ symmetry-breaking effects, we calculate various single (and total) distributions in these decays making use of the presently available data and decay form factors calculated in the QCD sum rule approach. Differential decay rates in the dilepton invariant mass and the Forward-Backward asymmetry in $B \to K^* \ell^+ \ell^-$ are worked out. Measurements of the ratios $R_i(s) \equiv d
\Gamma_{H_i}(s)(B \to K^* \ell^+ \ell^-)/ d \Gamma_{H_i}(s)(B \to \rho
\ell \nu_\ell)$, involving the helicity amplitudes $H_i(s)$, $i=0,+1,
-1$, as precision tests of the standard model in semileptonic rare $B$-decays are emphasized. We argue that $R_0(s)$ and $R_{-}(s)$ can be used to determine the CKM ratio $\vert V_{ub}\vert/\vert V_{ts} \vert$ and search for new physics, where the later is illustrated by supersymmetry.
author:
- |
A. Salim Safir\
Deutsches Elektronen-Synchrotron, DESY, D-22603 Hamburg, Germany\
E-mail : safir@mail.desy.de
title: |
DESY 02-143\
hep-ph/0209191\
September 2002
**$B \to K^* \ell^+ \ell^-\big (\rho~ \ell \nu_{\ell}\big)$ helicity analysis in the LEET[^1]**
---
Introduction
============
Rare $B$ decays involving flavour-changing-neutral-current (FCNC) transitions, such as $b \to s \gamma$ and $b \to s \ell^+ \ell^-$, have received a lot of theoretical interest [@Greub:1999sv], especially after the first measurements reported by the CLEO collaboration [@Alam:1995aw] of the $B \to X_s \gamma$ decay. The current world average based on the improved measurements by the CLEO [@Chen:2001fj], ALEPH [@alephbsg] and BELLE [@bellebsg] collaborations, ${\cal B}(B \to X_s \gamma)=(3.22 \pm 0.40) \times 10^{-4}$, is in good agreement with the estimates of the standard model (SM) [@Chetyrkin:1997vx; @Kagan:1999ym; @Gambino:2001ew], which we shall take as ${\cal B}(B \to X_s \gamma)=(3.50 \pm 0.50) \times 10^{-4}$, reflecting the parametric uncertainties dominated by the scheme-dependence of the quark masses. The decay $B \to X_s \gamma$ also provides useful constraints on the parameters of the supersymmetric theories, which in the context of the minimal supersymmetric standard model (MSSM) have been recently updated in [@Ali:2001jg].
Exclusive decays involving the $b \to s \gamma$ transition are best exemplified by the decay $B \to K^* \gamma$, which have been measured with a typical accuracy of $\pm 10\%$, the current branching ratios being [@Chen:2001fj; @TajimaH:2001; @Aubert:2001] ${\cal B}(B^\pm \to K^{*\pm} \gamma)=(3.82 \pm 0.47) \times 10^{-5}$ and ${\cal B}(B^0 \to K^{* 0} \gamma)=(4.44 \pm 0.35) \times 10^{-5}$. These decays have been analyzed recently [@Ali:2001ez; @Beneke:2001at; @Bosch:2001gv], by taking into account $O(\alpha_s)$ corrections, henceforth referred to as the next-to-leading-order (NLO) estimates, in the large-energy-effective-theory (LEET) limit [@Dugan:1990de; @Charles:1998dr]. As this framework does not predict the decay form factors, which have to be supplied from outside, consistency of NLO-LEET estimates with current data constrains the magnetic moment form factor in $B \to K^* \gamma$ in the range $T_1^{K^*}(0)=0.27 \pm 0.04$. These values are somewhat lower than the corresponding estimates in the lattice-QCD framework, yielding [@DelDebbio:1997kr] $T_1^{K^*}(0)=0.32^{+0.04}_{-0.02}$, and in the light cone QCD sum rule approach, which give typically $T_1^{K^*}(0)=0.38 \pm 0.05$ [@Ball:1998kk; @Ali:1999mm]. (Earlier lattice-QCD results on $B \to K^* \gamma$ form factors are reviewed in [@Soni:1995qq].) It is imperative to check the consistency of the NLO-LEET estimates, as this would provide a crucial test of the ideas on QCD-factorization, formulated in the context of non-leptonic exclusive $B$-decays [@Beneke:1999br], but which have also been invoked in the study of exclusive rare $B$-decays [@Ali:2001ez; @Beneke:2001at; @Bosch:2001gv].
The exclusive decays $B \to K^* \ell^+ \ell^-$, $\ell^\pm =e^\pm,\mu^\pm$ have also been studied in the NLO-LEET approach in [@Beneke:2001wa; @Beneke:2001at]. In this case, the LEET symmetry brings an enormous simplicity, reducing the number of independent form factors from seven to only two, corresponding to the transverse and longitudinal polarization of the virtual photon in the underlying process $B \to K^* \gamma^*$, called hereafter $\xi_\perp^{(K^*)}(q^2)$ and $\xi_{||}^{(K^*)}(q^2)$. The same symmetry reduces the number of independent form factors in the decays $B \to \rho \ell \nu_\ell$ from four to two. Moreover, in the $q^2$-range where the large energy limit holds, the two set of form factors are equal to each other, up to $SU(3)$-breaking corrections, which are already calculated in specific theoretical frameworks. Thus, knowing $V_{ub}$ precisely, one can make theoretically robust predictions for the rare $B$-decay $B \to K^* \ell^+ \ell^-$ from the measured $B \to \rho
\ell \nu_\ell$ decay in the SM. The LEET symmetries are broken by QCD interactions and the leading $O(\alpha_s)$ corrections in perturbation theory are known [@Beneke:2001wa; @Beneke:2001at].
In this talk we present the results of [@Ali:2002qc], where a systematic analysis of the various independent helicity amplitudes in the decays $B \to K^* \ell^+ \ell^-$ and $B \to \rho \ell \nu_\ell$ were performed in the NLO accuracy in the large energy limit. We recall that a decomposition of the final state $B \to K^* (\to K \pi) \ell^+ \ell^-$ in terms of the helicity amplitudes $H_\pm^{L,R}(q^2)$ and $H_0^{L,R}(q^2)$, without the explicit $O(\alpha_s)$ corrections, was undertaken in a number of papers [@Melikhov:1998cd; @Aliev:1999gp; @Kim:2000dq; @Kim:2001xu; @Nguyen:2001zu; @Chen:2002bq]. Combining the analysis of the decay modes $B \to K^* \ell^+ \ell^-$ and $B \to \rho \ell \nu_\ell$, we show that the ratios of differential decay rates involving definite helicity states, $R_{-}(s)$ and $R_{0}(s)$, can be used for testing the SM precisely.
General framework
=================
At the quark level, the rare semileptonic decay $b \to s~ \ell^+ \ell^-$ can be described in terms of the effective Hamiltonian $${\cal H}_{eff} = - \frac{G_F}{\sqrt{2}} \lambda_t
\sum_{i=1}^{10} C_{i}(\mu) {\cal O}_i(\mu) \; ,
\label{eq:he}$$ where $\lambda_t= V_{t s}^\ast V_{tb}$ are the CKM matrix elements [@ckm] and $G_F$ is the Fermi coupling constant. Following the notation and the convention used in ref. [@Ali:2002qc], the above Hamiltonian leads to the following free quark decay amplitude[^2]: $$\begin{aligned}
{\cal M} &=& \frac{G_F \alpha_{em}}{\sqrt{2} \pi} \lambda_t \,
\Big\{ C_{9} [ \bar{s} \gamma_{\mu} L b ] [ \bar{\ell} \gamma^{\mu} \l]
\label{eq:m}\\
&&
\hspace{-1.5cm}
+ C_{10} [ \bar{s} \gamma_{\mu} L b ] [ \bar{\ell}
\gamma^\mu \gamma_5 \l ]
- 2 \hat{m_{b}} C_{7}^{\bf eff} [ \bar{s} i \sigma_{\mu \nu} {\hat{q^{\nu}}\over\hat{s}} R b ]
[ \bar{\l} \gamma^{\mu} \l ]\Big\} .\nn\end{aligned}$$ Here, $L/R \equiv {(1 \mp \gamma_5)}/2$, $s=q^2$, $\sigma_{\mu \nu}={i\over
2}[\gamma_{\mu},\gamma_{\nu}]$ and $q_{\mu}=(p_{+} +p_{-})_{\mu}$, where $p_{\pm}$ are the four-momenta of the leptons. Since we are including the next-to-leading corrections into our analysis, we will take the NLO $\overline{MS}$ definition of the $b$-quark mass $m_b \equiv m_b(\mu)$ and the Wilson coefficients in next-to-leading-logarithmic order given in Table 1 in [@Ali:2002qc]. Exclusive $B\to V$ transitions[^3] are described by the matrix elements of the quark operators in Eq. (\[eq:m\]) over meson states, which can be parameterized in terms of the full QCD form factors (called in the literature $A_0(q^2), A_1(q^2), A_2(q^2), V(q^2), T_1(q^2), T_2(q^2), T_3(q^2)$).
However, the factorization Ansatz enables one to relate in the restricted kinematic region[^4], the form factors in full QCD and the two corresponding LEET form factors, namely $\xi^{(V)}_\perp(q^2)$ and $\xi^{(V)}_{||}(q^2)$ [@Beneke:2001wa; @Beneke:2001at]; $$\begin{aligned}
f_k(q^2)&=& C_{\perp}\xi^{(V)}_\perp (q^2) + C_{||}\xi^{(V)}_{||}(q^2) \nn\\
&&+ \Phi_B \otimes T_k \otimes \Phi_{(V)}~,
\label{eq:fact}\end{aligned}$$ where the quantities $C_i$ $(i=\perp, \parallel)$ encode the perturbative improvements of the factorized part $$C_i=C_i^{(0)} + \frac{\alpha_s}{\pi} C_i^{(1)} + ... ,
\nonumber$$ and $T_k$ is the hard spectator kernel (regulated so as to be free of the end-point singularities), representing the non-factorizable perturbative corrections, with the direct product understood as a convolution of $T_k$ with the light-cone distribution amplitudes of the $B$ meson ($\Phi_B$) and the vector meson ($\Phi_V$). With this Ansatz, it is a straightforward exercise to implement the $O(\alpha_s)$-improvements in the various helicity amplitudes. For further details we refer to [@Ali:2002qc].
Lacking a complete solution of non-perturbative QCD, one has to rely on certain approximate methods to calculate the above form factors. We take the ones given in [@Ali:1999mm], obtained in the framework of Light-cone QCD sum rules. The corresponding LEET form factors $\xi^{(K^*)}_{\perp}(s)$ and $\xi^{(K^*)}_{||}(s)$ are illustrated in ref .[@Ali:2002qc]. The range $\xi^{(K^*)}_{\perp}(s)=0.28\pm 0.04$ is determined by the $B \rightarrow \ K^{*}\gamma$ decay rate, calculated in the LEET approach in NLO order [@Ali:2001ez; @Bosch:2001gv; @Beneke:2001at] and current data. This gives somewhat smaller values for $T_1(0)$ and $T_2(0)$ than the ones estimated with the QCD sum rules.
$B \rightarrow \ K^{*} \ell^{+} \ell^{-}$ helicity analysis
===========================================================
Using the $B \rightarrow \ K^{*}(\to K \pi) \ell^{+} \ell^{-}$ helicity amplitudes [@Kim:2000dq], namely $H^{(K^*)}_{i}(s)~~~
(i=0,\pm 1)$ , the dilepton invariant mass spectrum reads $$\begin{aligned}
{d{\cal B} \over ds} &=& \tau_B {\alpha_{em}^2 G_{F}^2\over 384 \pi^5}
\sqrt{\lambda} {m_{b}^2\over m_{B}^3 } \lambda_t^2
\sum_{i=0,\pm1}|H^{(K^*)}_{i}(s)|^2,\nn\\
&=& {d{\cal B}_{|H_{+}|^2} \over ds}+{d{\cal B}_{|H_{-}|^2} \over ds}+{d{\cal B}_{|H_{0}|^2} \over ds}.\end{aligned}$$ where $ \lambda=\left[{1\over 4}(m_{B}^2-m_{V}^2-s)^2 - m_{V}^2\
s\right]$. Using the input parameters presented in [@Ali:2002qc], we have plotted in Fig. (\[figdBrK\*\]), the dilepton invariant mass spectrum $d{\cal B}_{|H_{\{0,\pm\}}|^2}/ ds$ and the total dilepton invariant mass, showing in each case the leading order and the next-to-leading order results.
[-0.8cm ${\cal H}_{-}^{(K^*)}$]{} [ $ 10^{-3}\times{\cal H}_{+}^{(K^*)}$]{}\
[-0.8cm ${\cal H}_{0}^{(K^*)}$]{} [ ${\cal H}^{(K^*)}$]{}
As can be seen from Fig. (\[figdBrK\*\]) the total decay rate is dominated by the contribution from the helicity $|H_{-}|$ component, whereas the contribution proportional to the helicity amplitude $H_{+}(s)$ is negligible. The next-to-leading order correction to the lepton invariant mass spectrum in $ B \rightarrow K^* \ell^+ \ell^-$ is significant in the low dilepton mass region ($s \leq 2$ GeV$^2$), but small beyond that shown for the anticipated validity of the LEET theory ($s \leq 8$ GeV$^2$). Theoretical uncertainty in our prediction is mainly due to the form factors, and to a lesser extent due to the parameters $\lambda_{B,+}^{-1}$ and the $B$-decay constant, $f_B$. Besides the differential branching ratio, $B \to K^* \ell^+ \ell^-$ decay offers other distributions (with different combinations of Wilson coefficients) to be measured. An interesting quantity is the Forward-Backward (FB) asymmetry defined in [@Ali:1991is] $$\frac{\d \a_{\rm FB}}{\d \sh} \equiv
-\int_0^{\uh(\sh)} \d\uh \frac{\d^2\gl}{\d\uh~ \d\sh}
+ \int_{-\uh(\sh)}^0 \d\uh \frac{\d^2\gl}{\d\uh~ \d\sh} \; .
\label{eq:dfba}$$ Where the kinematic variable $\hat{u} \equiv (\hat{p}_{B} -
\hat{p}_-)^2 - (\hat{p}_{B} -\hat{p}_+)^2$ [^5]. Our results for FBA are shown in Fig. \[FigFBA\] in the LO and NLO accuracy.
![ *FB-asymmetry at NLO order (solid center line) and LO (dashed). The band reflects the theoretical uncertainties from the input parameters.*[]{data-label="FigFBA"}](SUSY02FB.eps){width="8cm" height="4cm"}
At the LO the location of the FB-asymmetry zero is $s_0\simeq3.4\, \mbox{GeV}^2$, which is substantially shifted at the NLO by $\sim 1\, \mbox{GeV}^2$ leading to $s_0\simeq 4.7\, \mbox{GeV}^2$. We essentially confirm the results obtained in the NLO-LEET context by [*Beneke et al.*]{} [@Beneke:2001at].
$B \rightarrow \rho \ell \nu_{\ell}$ helicity analysis
======================================================
The helicity amplitudes for $B\rightarrow \rho (\rightarrow
\pi^{+} \pi^{-}) \ell \nu_{\ell}$, namely $H^{(\rho)}_{i}(s)~~~(i=0,\pm 1)$, can in turn be related to the two axial-vector form factors, $A_{1}(s)$ and $A_{2}(s)$, and the vector form factor, $V(s)$, which appear in the hadronic current [@Richman:wm]. The $B\rightarrow \rho (\rightarrow \pi^{+} \pi^{-}) \ell
\nu_{\ell}$ total branching decay rate[^6] can be expressed in terms of the corresponding helicity amplitudes as [@Korner:1989qb; @Richman:wm] $$\begin{aligned}
{d{\cal B} \over ds} &=& \tau_B {G_F^2\ s\ \sqrt{\lambda} \over 96
m_{B}^3 \pi^4} |V_{ub}|^2
\sum_{i=0,\pm1}|H^{(\rho)}_{i}(s)|^2,\nn\\
{} &=& {d{\cal B}_{|H_{0}|^2} \over ds}+{d{\cal B}_{|H_{+}|^2} \over ds}+{d{\cal B}_{|H_{-}|^2}
\over ds} ~.\label{dgamads2}\end{aligned}$$ The contributions from the $\vert H^{(\rho)}_{-}(s)\vert^2$, $\vert
H^{(\rho)}_{+}(s)\vert^2$, $\vert H^{(\rho)}_{0}(s)\vert^2$ and the total are shown in Fig. (\[dBHrho\]). Contrary to the $B \rightarrow K^* \ell^+ \ell^- $ decay rate, the $B \rightarrow \rho \ell \nu_\ell$ decay is dominated by the helicity-0 component. The impact of the NLO correction on the various branching ratios in $B \rightarrow \rho \ell
\nu_\ell$ is less significant than in the $B \rightarrow K^* \ell^+
\ell^- $ decay, reflecting the absence of the penguin-based amplitudes in the former decay.
![*Various individual helicity contributions $\Big({\cal
H}^{(\rho)}_{\{\pm,0\}}\equiv{d{\cal B}_{|H_{\{\pm,0\}}|^2} \over ds\
|V_{ub}|^2}\Big)$ and the sum $\Big({\cal H}^{(\rho)}\equiv{d{\cal B }
\over ds|V_{ub}|^2}\Big)$ to dilepton invariant mass distributions for $B \rightarrow \rho \ell \nu_{\ell}$ at NLO order (solid center line) and LO (dashed).The band reflects the theoretical uncertainties from input parameters.*[]{data-label="dBHrho"}](figr8.eps "fig:"){width="3.8cm" height="3cm"} [0.8cm $10^{-4}\times {\cal H}_{+}^{(\rho)}$]{} ![*Various individual helicity contributions $\Big({\cal
H}^{(\rho)}_{\{\pm,0\}}\equiv{d{\cal B}_{|H_{\{\pm,0\}}|^2} \over ds\
|V_{ub}|^2}\Big)$ and the sum $\Big({\cal H}^{(\rho)}\equiv{d{\cal B }
\over ds|V_{ub}|^2}\Big)$ to dilepton invariant mass distributions for $B \rightarrow \rho \ell \nu_{\ell}$ at NLO order (solid center line) and LO (dashed).The band reflects the theoretical uncertainties from input parameters.*[]{data-label="dBHrho"}](figr7.eps "fig:"){width="3.8cm" height="3cm"}\
[-0.5cm ${\cal H}^{(\rho)}$]{} ![*Various individual helicity contributions $\Big({\cal
H}^{(\rho)}_{\{\pm,0\}}\equiv{d{\cal B}_{|H_{\{\pm,0\}}|^2} \over ds\
|V_{ub}|^2}\Big)$ and the sum $\Big({\cal H}^{(\rho)}\equiv{d{\cal B }
\over ds|V_{ub}|^2}\Big)$ to dilepton invariant mass distributions for $B \rightarrow \rho \ell \nu_{\ell}$ at NLO order (solid center line) and LO (dashed).The band reflects the theoretical uncertainties from input parameters.*[]{data-label="dBHrho"}](figr9.eps "fig:"){width="3.8cm" height="3cm"} [0.8cm ${\cal H}_{-}^{(\rho)}$]{} ![*Various individual helicity contributions $\Big({\cal
H}^{(\rho)}_{\{\pm,0\}}\equiv{d{\cal B}_{|H_{\{\pm,0\}}|^2} \over ds\
|V_{ub}|^2}\Big)$ and the sum $\Big({\cal H}^{(\rho)}\equiv{d{\cal B }
\over ds|V_{ub}|^2}\Big)$ to dilepton invariant mass distributions for $B \rightarrow \rho \ell \nu_{\ell}$ at NLO order (solid center line) and LO (dashed).The band reflects the theoretical uncertainties from input parameters.*[]{data-label="dBHrho"}](figr6.eps "fig:"){width="3.8cm" height="3cm"}
Concerning the $B \rightarrow \rho \ell \nu_\ell$ form factors, one has to consider the SU(3)-breaking effects in relating them to the corresponding form factors in $B \to K^* \ell^+ \ell^-$. For the form factors in full QCD, they have been evaluated within the QCD sum-rules [@Ali:vd]. These can be taken to hold also for the ratio of the LEET form factors. Thus, we take $$\xi^{(\rho)}_{\perp,||}(s)={\xi^{(K^*)}_{\perp,||}(s) \over
\zeta_{SU(3)}}~.
\label{eq:paraball}$$ While admitting that this is a somewhat simplified picture, as the effect of $SU(3)$-breaking is also present in the $s$-dependent functions, but checking numerically the functions resulting from Eq. (\[eq:paraball\]) with the ones worked out for the full QCD form factors in the QCD sum-rule approach in [@Ball:1998kk], we find that the two descriptions are rather close numerically in the region of interest of $s$.
Determination of $|V_{ub}|/|V_{ts}|$
=====================================
The measurement of exclusive $B \rightarrow \rho \ell \nu_{\ell}$ decays is one of the major goals of B physics. It provides a good tool for the extraction of $|V_{ub}|$, provided the form factors can be either measured precisely or calculated from first principles, such as the lattice-QCD framework. To reduce the non-perturbative uncertainty in the extraction of $V_{ub}$, we propose to study the ratios of the differential decay rates in $B \to \rho \ell \nu_\ell$ and $B \to K^* \ell^+
\ell^-$ involving definite helicity states. These $s$-dependent ratios $R_{i}(s)$, $(i=0,-1,+1)$ are defined as follows: $$\begin{aligned}
R_{i}(s) ={{d\Gamma_{H_i}^{B \rightarrow K^{*} \ \ell^{+} \ell^{-}}/ds}
\over{d\Gamma_{H_i}^{B \rightarrow \rho \ \ell \nu_{\ell}}/ds}}
\label{Ratio} \end{aligned}$$ The ratio $R_{-}(s)$ suggests itself as the most interesting one, as the form factor dependence essentially cancels. From this, one can measure the ratio $\vert V_{ts} \vert/\vert V_{ub}\vert$. In Fig. \[RVub\], we plot $R_{-}(s)$ and $R_{0}(s)$ for three representative values of the CKM ratio $R_b = \vert V_{ub}\vert/\vert V_{tb} V_{ts}^* \vert =
\vert V_{ub}\vert/\vert V_{cb}\vert =0.08$, $0.094$, and $0.11$.
![*The Ratios $R_-(s)$ (left-hand plot) and $R_0(s)$ (right-hand plot) with three indicated values of the CKM ratio $R_{b} \equiv |V_{ub}|/|V_{tb}V_{ts}^*|$. The bands reflect the theoretical uncertainty from $\zeta_{SU(3)}=1.3 \pm
0.06$ and $\xi^{(K^*)}_{{\perp}}(0)=0.28\pm 0.04$.*[]{data-label="RVub"}](SUSY02Rminus.eps "fig:"){width="3.8cm" height="3cm"} ![*The Ratios $R_-(s)$ (left-hand plot) and $R_0(s)$ (right-hand plot) with three indicated values of the CKM ratio $R_{b} \equiv |V_{ub}|/|V_{tb}V_{ts}^*|$. The bands reflect the theoretical uncertainty from $\zeta_{SU(3)}=1.3 \pm
0.06$ and $\xi^{(K^*)}_{{\perp}}(0)=0.28\pm 0.04$.*[]{data-label="RVub"}](SUSY02R0.eps "fig:"){width="3.8cm" height="3cm"}
However, as we remarked earlier, the ratio $R_{-}(s)$ may be statistically limited due to the dominance of the decay $B \to \rho \ell \nu_\ell$ by the Helicity-$0$ component. Hence, we also show the ratio $R_{0}(s)$, where the form factor dependence does not cancel. For the LEET form factors used here, the compounded theoretical uncertainty is shown by the shaded regions. This figure suggests that high statistics experiments may be able to determine the CKM-ratio from measuring $R_{0}(s)$ at a competitive level compared to the other methods [*en vogue*]{} in experimental studies.
SUSY effect in $B \rightarrow K^{*} \ \ell^{+} \ell^{-}$
===========================================================
In order to look for new physics in $B \rightarrow K^{*} \ell^{+}
\ell^{-}$, we propose to study the ratio $R_{\{0,-\}}(s)$, introduced in Eq. (\[Ratio\]). To illustrate generic SUSY effects in ${B \rightarrow K^{*} \ell^{+} \ell^{-}}$, we note that the Wilson coefficients $C^{\bf eff}_7$, $C^{\bf eff}_8$, $C_9$ and $C_{10}$ receive additional contributions from the supersymmetric particles. We incorporate these effects by assuming that the ratios of the Wilson coefficients in these theories and the SM deviate from 1. These ratios are defined as follows: $$\begin{aligned}
r_{k}(\mu) = {C_{k}^{SUSY}\over C_{k}^{SM}},~~~ (k= 7, \cdots, 10).
\label{rk} \end{aligned}$$ They depend on the renormalization scale (except for $C_{10}$), for which we take $\mu =m_{b,pole}$. For the sake of illustration, we use representative values for the large(small)-$\tan \beta$ SUGRA model, in which the ratios $r_7$ and $r_8$ actually change (keep) their signs. The supersymmetric effects on the other two Wilson coefficients $C_9$ and $C_{10}$ are generally small in the SUGRA models, leaving $r_9$ and $r_{10}$ practically unchanged from their SM value. To be specific, we take [^7] $r_{7} = -1.2,\ \ r_{8} = -1,\ \ \ r_{9} = 1.03,\ \ r_{10} = 1.0~$ ($r_{7} = 1.1,\ \ r_{8} = 1.4,\ \ \ r_{9} = 1.03,\ \ r_{10} = 1.0$).
![ *The Ratio $R_{\{-,0\}}(s)$ in the SM with $R_b=0.094$, $\zeta_{SU(3)}=1.3$, $\xi^{(K^*)}_{{\perp}}(0)=0.28$ and in SUGRA, with $(r_7,\ r_8)=(1.1,\
1.4)$ (left-hand plots) and $(r_7,\ r_8)=( -1.2,\ -1)$ (right-hand plots). The SM and the SUGRA contributions are represented respectively by the shaded area and the solid curve. The shaded area depicts the theoretical uncertainty on $\zeta_{SU(3)}=1.3 \pm 0.06$ and on $\xi^{(K^*)}_{{\perp}}(0)=0.28\pm 0.04$.*[]{data-label="RSMetSUGRA"}](SUSY02RminusSUGRApos.eps "fig:"){width="3.8cm" height="3cm"} ![ *The Ratio $R_{\{-,0\}}(s)$ in the SM with $R_b=0.094$, $\zeta_{SU(3)}=1.3$, $\xi^{(K^*)}_{{\perp}}(0)=0.28$ and in SUGRA, with $(r_7,\ r_8)=(1.1,\
1.4)$ (left-hand plots) and $(r_7,\ r_8)=( -1.2,\ -1)$ (right-hand plots). The SM and the SUGRA contributions are represented respectively by the shaded area and the solid curve. The shaded area depicts the theoretical uncertainty on $\zeta_{SU(3)}=1.3 \pm 0.06$ and on $\xi^{(K^*)}_{{\perp}}(0)=0.28\pm 0.04$.*[]{data-label="RSMetSUGRA"}](SUSY02RminusSUGRAneg.eps "fig:"){width="3.8cm" height="3cm"}\
![ *The Ratio $R_{\{-,0\}}(s)$ in the SM with $R_b=0.094$, $\zeta_{SU(3)}=1.3$, $\xi^{(K^*)}_{{\perp}}(0)=0.28$ and in SUGRA, with $(r_7,\ r_8)=(1.1,\
1.4)$ (left-hand plots) and $(r_7,\ r_8)=( -1.2,\ -1)$ (right-hand plots). The SM and the SUGRA contributions are represented respectively by the shaded area and the solid curve. The shaded area depicts the theoretical uncertainty on $\zeta_{SU(3)}=1.3 \pm 0.06$ and on $\xi^{(K^*)}_{{\perp}}(0)=0.28\pm 0.04$.*[]{data-label="RSMetSUGRA"}](SUSY02RzeroSUGRApos.eps "fig:"){width="3.8cm" height="3cm"} ![ *The Ratio $R_{\{-,0\}}(s)$ in the SM with $R_b=0.094$, $\zeta_{SU(3)}=1.3$, $\xi^{(K^*)}_{{\perp}}(0)=0.28$ and in SUGRA, with $(r_7,\ r_8)=(1.1,\
1.4)$ (left-hand plots) and $(r_7,\ r_8)=( -1.2,\ -1)$ (right-hand plots). The SM and the SUGRA contributions are represented respectively by the shaded area and the solid curve. The shaded area depicts the theoretical uncertainty on $\zeta_{SU(3)}=1.3 \pm 0.06$ and on $\xi^{(K^*)}_{{\perp}}(0)=0.28\pm 0.04$.*[]{data-label="RSMetSUGRA"}](SUSY02RzeroSUGRAneg.eps "fig:"){width="3.8cm" height="3cm"}
In Fig. \[RSMetSUGRA\], we present a comparative study of the SM and SUGRA partial distribution for $H_{-}$ and $H_{0}$. In doing this, we also show the attendant theoretical uncertainties for the SM, worked out in the LEET approach. For these distributions, we have used the form factors from [@Ali:1999mm] with the SU(3)-symmetry breaking parameter taken in the range $\zeta_{SU(3)}=1.3\pm 0.06$.
From Fig. \[RSMetSUGRA\]-[*left-hand plots*]{}, where $r_{k}>0$, it is difficult to work out a signal of new physics from the SM picture. There is no surprise to be expected, due to the fact that in these scenario the corresponding ratio $r_k$ is approximatively one, which makes the SUGRA picture almost the same as in the SM one. However, Fig. \[RSMetSUGRA\]-[*right-hand plots*]{} with $(r_7,\ r_8)<0$ illustrates clearly that despite non-perturbative uncertainties, it is possible, in principle, in the low $s$ region to distinguish between the SM and a SUGRA-type models, provided the ratios $r_k$ differ sufficiently from 1.
Summary
=======
In this talk, we have reported an $O(\alpha_s)$-improved analysis of the various helicity components in the decays $B \to K^* \ell^+ \ell^-$ and $B \to \rho \ell \nu_\ell$, carried out in the context of the Large-Energy-Effective-Theory. The results presented here make use of the form factors calculated in the QCD sum rule approach. The LEET form factor $\xi_\perp^{(K^*)}(0)$ is constrained by current data on $B
\to K^* \gamma$. As the theoretical analysis is restricted to the lower part of the dilepton invariant mass region in $B \to K^* \ell^+ \ell^-$, typically $s \leq 8$ GeV$^2$, errors in this form factor are not expected to severely limit theoretical precision. This implies that distributions involving the $H_{-}(s)$ helicity component can be calculated reliably. Precise measurements of the two LEET form factors $\xi_{\perp}^{(\rho)}(s)$ and $\xi_{\parallel}^{(\rho)}(s)$ in the decays $B \to \rho \ell \nu_\ell$ can be used to largely reduce the residual model dependence. With the assumed form factors, we have worked out a number of single (and total) distributions in $B \to \rho
\ell \nu_\ell$, which need to be confronted with data. We also show the $O(\alpha_s)$ effects on the FB-asymmetry, confirming essentially the result found in [@Beneke:2001at]. Combining the analysis of the decay modes $B \to K^* \ell^+ \ell^-$ and $B \to \rho \ell \nu_\ell$, we show that the ratios of differential decay rates involving definite helicity states, $R_{-}(s)$ and $R_{0}(s)$, can be used for testing the SM precisely. We work out the dependence of these ratios on the CKM matrix elements $\vert V_{ub}\vert/\vert V_{ts}\vert$.
We have also analyzed possible effects on these ratios from New Physics contributions, exemplified by representative values for the effective Wilson coefficients in SUGRA models. Finally, we remark that the current experimental limits on $B \to
K^* \ell^+ \ell^-$ decay (and the observed $B \to X_s \ell^+ \ell^-$ and $B \to K \ell^+ \ell^-$ decays) [@bellebsll; @babarbsll; @Affolder:1999eb; @Anderson:2001nt] are already probing the SM-sensitivity. With the integrated luminosities over the next couple of years at the $B$ factories, the helicity analysis in $B \to \rho \ell \nu_\ell$ and $B \to K^* \ell^+ \ell^-$ decays presented here can be carried out experimentally.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a great pleasure to thank Ahmed Ali for the fruitful collaboration and useful remarks on the manuscript. For the work presented here, I also gratefully acknowledge DESY for financial support.
[99]{} A. Ali and A. S. Safir, \[hep-ph/0205254\]. See, e.g., C. Greub, talk given at the 8th International Symposium on Heavy Flavour Physics, Southampton, England, 25-29 Jul 1999. \[hep-ph/9911348\]. M. S. Alam [*et al.*]{} \[CLEO Coll.\], Phys. Rev. Lett. [**74**]{} (1995) 2885. S. Chen [*et al.*]{} \[CLEO Coll.\], Phys. Rev. Lett. [**87**]{} (2001) 251807 \[hep-ex/0108032\]. R. Barate [*et al.*]{} \[ALEPH Coll.\], Phys. Lett. [**B429**]{}, (1998) 169. K. Abe [*et al.*]{} \[BELLE Coll.\], Phys. Lett. [**B511**]{} (2001) 151 \[hep-ex/0103042\]. K. Chetyrkin [*et al.*]{}, Phys. Lett. [**B400**]{} (1997) 206; E: [**B425**]{} (1998) 414 \[hep-ph/9612313\]. A. L. Kagan and M. Neubert, Eur. Phys. J. [**C7**]{}, 5 (1999) \[hep-ph/9805303\]. P. Gambino and M. Misiak, Nucl. Phys. [**B611**]{} (2001) 338 \[hep-ph/0104034\]. A. Ali [*et al.*]{}, Phys. Rev. D [**66**]{} (2002) 034002 \[hep-ph/0112300\]. H. Tajima \[BELLE Coll.\], Plenary Talk, XX International Symposium on Lepton and Photon Interactions at High Energies, Rome 23 - 28 Jul 2001. B. Aubert [*et al.*]{} \[BABAR Coll.\] Phys. Rev. Lett. [**88**]{} (2002) 101805 \[hep-ex/0110065\]. A. Ali and A. Y. Parkhomenko, Eur. Phys. J. [**C23**]{} (2002) 89 \[hep-ph/0105302\]. M. Beneke, T. Feldmann and D. Seidel Nucl. Phys. [**B612**]{} (2001) 25 \[hep-ph/0106067\]. S. W. Bosch and G. Buchalla, Nucl. Phys. [**B621**]{} (2002) 459 \[hep-ph/0106081\]. M. J. Dugan and B. Grinstein, Phys. Lett. [**B255**]{}, 583 (1991). J. Charles [*et al.*]{}, Phys. Rev. [**D60**]{} (1999) 014001 \[hep-ph/9812358\]. L. Del Debbio [*et al.*]{}, \[UKQCD Coll.\], Phys. Lett. [**B416**]{} (1998) 392 \[hep-lat/9708008\]. A. Ali [*et al.*]{}, Phys. Rev. [**D 61**]{} (2000) 074024 \[hep-ph/9910221\]. P. Ball and V. M. Braun, Phys. Rev. [**D58**]{} (1998) 094016 \[hep-ph/9805422\]. A. Soni, Nucl. Phys. Proc. Suppl. [**47**]{} (1996), 43 \[hep-lat/9510036\]. M. Beneke [*et al.*]{}, Phys. Rev. Lett. [**83**]{} (1999) 1914 \[hep-ph/9905312\]. M. Beneke and T. Feldmann, Nucl. Phys. [**B592**]{} (2001) 3 \[hep-ph/0008255\]. D. Melikhov, N. Nikitin and S. Simula, Phys. Lett. B [**442**]{} (1998) 381 \[hep-ph/9807464\]. T. M. Aliev, C. S. Kim and Y. G. Kim, Phys. Rev. D [**62**]{} (2000) 014026 \[hep-ph/9910501\]. C. S. Kim [*et al.*]{}, Phys. Rev. [**D62**]{} (2000) 034013 \[hep-ph/0001151\]. C. S. Kim [*et al.*]{} Phys. Rev. [**D64**]{} (2001) 094014 \[hep-ph/0102168\]. X. S. Nguyen and X. Y. Pham, \[hep-ph/0110284\]. C. H. Chen and C. Q. Geng, \[hep-ph/0203003\]. N. Cabibbo, Phys. Rev. Lett. [**10**]{}, 531 (1963),\
M. Kobayashi and T. Maskawa, Prog. Theor. Phys. [**49**]{}, 652 (1973). A. Ali, T. Mannel and T. Morozumi, Phys. Lett. [**B273**]{} (1991) 505. J. D. Richman and P. R. Burchat, Rev. Mod. Phys. [**67**]{} (1995) 893 \[hep-ph/9508250\]. J. G. Körner and G. A. Schuler, Z. Phys. C [**46**]{} (1990) 93; Phys. Lett. B [**226**]{} (1989) 185. A. Ali, V. M. Braun and H. Simma, Z. Phys. [**C63**]{} (1994) 437 \[hep-ph/9401277\]. K. Abe [*et al.*]{} \[Belle Coll.\], BELLE-CONF-0110 \[hep-ex/0107072\];\
K. Abe [*et al.*]{} \[BELLE Coll.\], Phys. Rev. Lett. [**88**]{} (2002) 021801 \[hep-ph/0109026\]. B. Aubert [*et al.*]{} \[BABAR Coll.\], BABAR-CONF-01/24 \[hep-ex/0107026\]. T. Affolder [*et al.*]{} \[CDF Coll.\], Phys. Rev. Lett. [**83**]{} (1999) 3378 \[hep-ex/9905004\]. S. Anderson [*et al.*]{} \[CLEO Coll.\], Phys. Rev. Lett. [**87**]{} (2001) 181803 \[hep-ex/0106060\].
[^1]: Talk presented at the $10^{th}$ International Conference on Supersymmetry and Unification of Fundamental Interactions (SUSY02), Hamburg, Germany, 10-23 June 2002. Based on work together with Ahmed Ali [@Ali:2002qc].
[^2]: We put $m_s/m_b = 0$ and the hat denotes normalization in terms of the $B$-meson mass, $m_B$, e.g. $\hat{s}=s/m_B^2$, $\hat{m}_b =m_b/m_B$.
[^3]: $V$ stands for the corresponding vector meson.
[^4]: For the $B\to K^* \ell^+ \ell^-$ decay, this region is identified as $s \leq \mbox{GeV}^2$.
[^5]: which are bounded as $ (2 \hat{m}_{\ell})^2 \leq \hat{s} \leq (1 - \hat{m}_{K^*})^2$, $ -\hat{u}(\hat{s}) \leq \hat{u} \leq \hat{u}(\hat{s})$, with $\hat{m}_{\ell}=m_{\ell}/m_{B}$ and $\hat{u}(\hat{s})={2\over m_{B}^2}
\sqrt{\lambda(1-4{\hat{m}_{l}^2\over\hat{s}})}$.
[^6]: we put ${\cal B}(\rho \rightarrow \pi^+ \pi^-)=100\%.$
[^7]: We thank Enrico Lunghi for providing us with these numbers.
|
---
abstract: 'If the Dark Matter (DM) in the Universe has interactions with the standard-model particle, the pair annihilation may give the imprints in the cosmic ray. In this paper we study the pair annihilation processes of the DM, which is neutral, however has the electroweak (EW) gauge non-singlet. In this estimation the non-relativistic (NR) effective theory in the EW sector is a suitable technique. We find that if the DM mass is larger than about 1 TeV, the attractive Yukawa potentials induced by the EW gauge bosons have significant effects on the DM annihilation processes, and the cross sections may be enhanced by several orders of magnitude, due to the zero energy resonance under the potentials. Especially, the annihilation to two $\gamma$’s might have a comparable cross section to other tree-level processes, while the cross section under the conventional calculation is suppressed by a loop factor. We also discuss future sensitivities to the $\gamma$ ray from the galactic center by the GLAST satellite detector and the Air Cerenkov Telescope (ACT) arrays.'
author:
- 'Junji Hisano,$^1$ Shigeki Matsumoto,'
- 'and Mihoko M. Nojiri'
title: Explosive Dark Matter Annihilation
---
Nature of the Dark Matter (DM) in the Universe is an important problem in both particle physics and cosmology. The Weakly-Interacting Massive Particle (WIMP), $\chi^0$, is a good candidate for the DM. It works as the cold dark matter in the structure formation in the Universe. High resolution $N$-body simulations show that the cold dark matter hypothesis explains well the structure larger than about 1 Mpc [@nbody]. Also, the WMAP measured the cosmological abundance precisely as $\Omega_{DM}=0.27\pm0.04$ [@wmap]. Now we know the gravitational property of the DM in the structure formation and the abundance and distribution in the cosmological scale. The next questions are the constituent of the DM and the distribution in the galactic scale.
If the DM is SU(2)$_L$ non-singlet, a pair of the DM could annihilate into the standard-model (SM) particles with significant cross sections [@DMCR]. We call such DM’s as electroweak-interacting massive particle (EWIMP) DM in this paper. The detection of exotic cosmic ray fluxes, such as positron, anti-proton and $\gamma$ ray, may be a feasible technique to search for the DM’s. Since some DM candidates in the supersymmetric (SUSY) models have interactions with the SM particles, these annihilation processes are extensively studied. Especially, excess of monochromatic $\gamma$ ray due to the pair annihilation is a robust signal if observed, because the diffused $\gamma$-ray background must have a continuous energy spectrum [@Bergstrom:1997fj]. Searches for the exotic $\gamma$ ray from the galactic center, the galactic halo, or even from extra galaxies are ones of the projects in the GLAST satellite detector and the big Air Cerenkov Telescope (ACT) arrays such as CANGAROO III, HESS, MAGIC and VERITAS.
In the previous estimates, the cross sections for the EWIMP are evaluated at the leading order in the perturbation. However, the DM is non-relativistic (NR) in the current Universe. In this case, if the EWIMP mass $m$ is much heavier than EW scale, the EWIMP wave function may be deformed under the Yukawa potentials induced by the EW gauge boson exchanges and it may give a non-negligible effect in the annihilation processes. Furthermore, the neutral EWIMP should has a charged SU(2)$_L$ partner, $\chi^{\pm}$. When the EWIMP is heavier than EW scale, their masses are almost degenerate, and the unsuppressed transition between the two-body states of $2\chi^0$ and $\chi^-\chi^+$ may play an important role in the $2\chi^0$ pair annihilation.
In this letter we reevaluate the pair annihilation cross sections of the EWIMP’s, for the two cases that the DM is a component of two SU(2)$_L$-doublet fermions or of an SU(2)$_L$-triplet fermion. These correspond to the Higgsino-like and Wino-like DM’s in the SUSY models, respectively. Most interesting fact we find is that the annihilation cross sections to the gauge boson pairs for the SU(2)$_L$-doublet (triplet) DM suffer from a zero energy resonance around $m\simeq 6(2)$ TeV, whose binding energy is zero [@landau] under the potential. Therefore, the cross sections would be enhanced significantly compared with ones in the perturbative estimations for $m{ \mathop{}_{\textstyle \sim}^{\textstyle >} }1 (0.5)$ TeV. Furthermore, it is found that the cross section for $2\chi^0\rightarrow 2\gamma$, which is usually suppressed by a one-loop factor, becomes comparable to the other tree-level processes, such as $2\chi^0\rightarrow W^+W^-$, around the resonance. This means that the mixing between the two-body states of $\chi^-\chi^+$ and $2
\chi^0$ is maximal under the potential. Due to the explosive enhancement of the cross sections, the SU(2)$_L$-triplet DM is already partially constrained by the EGRET observation of the $\gamma$ ray from the galactic center, and the future $\gamma$ ray searches may have sensitivity to the heavier EWIMP DM.
First, we summarize properties of the EWIMP DM’s. If the DM has a vector coupling to the $Z$ boson, the current bound from the direct DM searches through the spin-independent interaction [@ds] is stringent. This means that the EWIMP DM should be a Majorana fermion or a real scalar if it is relatively light. Here we consider a former case for simplicity.
A simple example for the EWIMP DM’s is a neutral component of an SU(2)$_L$-triplet fermion ($T$) whose hypercharge is zero. This corresponds to the Wino-like LSP in the SUSY models. It is accompanied with the a charged fermion, $\chi^{\pm}$. While $\chi^0$ and $\chi^{\pm}$ are almost degenerate in mass in the SU(2)$_L$ symmetric limit, the EW symmetry breaking by the Higgs field, $h$, generates the mass splitting, $\delta m$. If $\delta m$ comes from the radiative correction, $\delta m\simeq 1/2\alpha_2(m_W-c^2_W m_Z)\sim 0.18$ MeV for $m \gg m_W$ and $m_Z$. Here, $m_W$ and $m_Z$ are the $W$ and $Z$ boson masses, respectively, and $c_W(\equiv \cos\theta_W)$ is for the Weinberg angle. Effective higher-dimensional operators, such as $h^4
T^2/\Lambda^3$, also generate $\delta m$, however they are suppressed by the new particle mass scale $\Lambda$. The thermal relic density of the DM with mass around 1.7 TeV is consistent to the WMAP data.
Another example for the EWIMP DM’s is a neutral component of a pair of SU(2)$_L$-doublet fermions ($D$ and $D'$) with the hypercharges $\pm 1/2$. This corresponds to the Higgsino-like LSP in the SUSY models. The $\chi^0$ is accompanied with a neutral Majorana fermion, $\chi^{\prime 0}$, in addition to a charged Dirac fermion, $\chi^{\pm}$. They are again degenerated in mass in the $SU(2)_L$ symmetric limit. The mass difference is generated by the effective operators, such as $ h^2 D^2/\Lambda$, via the EW symmetry breaking. The thermal relic density of the DM explains the WMAP data when the mass is around 0.6 TeV.
In the current Universe the DM is expected to be highly non-relativistic as mentioned before. In this case, the perturbative pair annihilation cross sections of the EWIMP DM may have bad behaviors if the DM mass is heavier than the weak scale. One of the example is the annihilation cross section to $2\gamma$ at the leading order. The process is induced at one-loop level, and the cross section is $4(1/4)\pi
\alpha^2\alpha_2^2/m_W^2$ for the SU(2)$_L$-triplet (doublet) DM in the SU(2)$_L$ symmetric limit. The cross section is not suppressed by $1/m^2$, and the perturbative unitarity is violated when $m$ is heavy enough.
The NR effective theory is useful to evaluate the cross sections in the NR limit. In Ref. [@Hisano:2002fk] we studied the NR effective theory for the EWIMP in a perturbative way and found that the trouble in the cross section to $2\gamma$ is related to the threshold singularity. In order to evaluate the cross section quantitatively, we have to calculate the cross section non-perturbatively using the NR effective theory [^1].
For evaluation of the annihilation cross sections for heavy EWIMP, we need to solve the EWIMP wave function under the EW potential. In this paper, we show the formulae for evaluating the cross sections in the SU(2)$_L$-triplet DM case. Those for the SU(2)$_L$-doublet case will be shown in the further publications [@HMN].
The NR effective Lagrangian for two-body states, $\phi_N({\bf
r})(\simeq 1/2 \chi^0 \chi^0)$ and $\phi_C({\bf r})(\simeq
\chi^-\chi^+)$, is given as $${\cal L}
=
\frac12 {\bf \Phi}^{T}({\bf r})
\left(\left(E+\frac{\nabla^2}{m}\right) {\bf 1}
-{\bf V}(r)+2 i {\bf \Gamma} \delta^3({\bf r})\right)
{\bf \Phi}({\bf r})~,
\label{Leff}$$ where ${\bf \Phi}({\bf r}) = (\phi_C({\bf r}),~\phi_N({\bf r}))$, ${\bf r}$ is the relative coordinate ($r=|{\bf r}|$), and $E$ is the internal energy of the two-body state. The EW potential ${\bf V}(r)$ is $${\bf V}(r)
=
\left(
\begin{array}{cc}
2 \delta m
- {\displaystyle\frac{\alpha}{r}}
- {\alpha_2 c_W^2} {\displaystyle\frac{e^{-m_Zr}}{r}}
&
- {\sqrt{2}\alpha_2}{\displaystyle\frac{e^{-m_Wr}}{r}}
\\
- {\sqrt{2}\alpha_2}{\displaystyle\frac{e^{-m_Wr}}{r}}
&
0
\end{array}
\right).$$ In this equation we keep only $2\delta m$ in (1,1) components in order to calculate the DM annihilation rate up to $O(\sqrt{\delta m/m})$ [@Hisano:2002fk]. ${\bf \Gamma}$ is the absorptive part of the two-point functions. Note that a factor of $1/2$ ($1/\sqrt{2}$) is multiplied for ${\bf
V}_{22}$ and ${\bf \Gamma}_{22}$ (${\bf V}_{12}$ and ${\bf
\Gamma}_{12}$) since $\phi_N$ is a two-body state of identical particles. Thus, ${\bf \Gamma}_{22}$ (${\bf \Gamma}_{11}$) is the tree-level annihilation cross section multiplied by the relative velocity $v$ and $1/2(1)$. Since the SU(2)$_L$-triplet DM is assumed to be a Majorana fermion, the $^1S$-wave gauge contribution to ${\bf \Gamma}$ is relevant to the NR annihilation, and then, $${\bf \Gamma} =
\frac{\pi\alpha_2^2}{m^2}
\left(
\begin{array}{cc}
\frac{3}2 &\frac1{2\sqrt{2}} \\
\frac1{2\sqrt{2}} & 1
\end{array}
\right)~.$$
The annihilation cross section of $\chi^-\chi^+$ or $2\chi^0$ to the EW gauge boson pair can be expressed using the two-by-two Green function, ${\bf G}({\bf r},{\bf r^\prime})$, which is given by $$\begin{aligned}
&\left(\left(E+{\displaystyle\frac{\nabla^2}{m}}\right) {\bf 1}-{\bf V}(r)+2 i {\bf \Gamma}
\delta^3({\bf r})\right){\bf G}({\bf r},{\bf r^\prime})&
\nonumber\\
&= \delta^3({\bf r}-{\bf r^\prime})
{\bf 1}~.&
\label{eqgr}\end{aligned}$$ Due to the optical theorem, the long-distance (wave function) and the short-distance (annihilation) effects can be factorized [@Bodwin:1994jh]. The annihilation cross sections to $VV^\prime$ ($V,V^\prime= W,Z,\gamma$) are written as $$\begin{aligned}
&(\sigma v)_{VV^\prime} = c_i \sum_{ab} {\bf \Gamma}_{ab}|_{VV^\prime}& \times
A_a A_b^\star~,\end{aligned}$$ where $i$ represents the initial state ($i=0$ and $\pm$ for $2\chi^0$ and $\chi^-\chi^+$ pair annihilation, respectively) and $A_a =\int d^3
r~ {\rm e}^{-i {\bf k} {\bf r}} (E +{\nabla^2}/{M}) {\bf G}_{ai}({\bf
r},0)$ with $k=\sqrt{m E}=mv/2$. Here $c_0=2$ and $c_{\pm}=1$, where $c_0$ is a factor needed to compensate the symmetric factor for ${\bf
\Gamma}$ and ${\bf V}$. ${\bf
\Gamma}_{ab}|_{VV^\prime}$ is the contribution to ${\bf \Gamma}_{ab}$ from the final states $VV^\prime$. It is clear that if the long-distance effect is negligible, $(\sigma v)_{VV^\prime}= c_i {\bf
\Gamma}_{ii}|_{VV^\prime}$.
The $S$-wave annihilation is dominant in the NR annihilation. Thus, the Green function is reduced to ${\bf G}({\bf r},{\bf r^\prime}) ={\bf
g}(r,r^\prime)/rr^\prime$. Similar to the case in one-flavor system, we find that ${\bf g}(r,r^\prime)/rr^\prime$ is expressed by the independent solutions of the homogeneous part of the Eq. (\[eqgr\]), ${\bf g}_>(r)/r$ and ${\bf
g}_<(r)/r$, as $$\begin{aligned}
{\bf g}(r,r^\prime)
&=&
\frac{m}{4\pi}
{\bf g}_>(r) {\bf g}_<^T(r^\prime)\theta(r-r^\prime)
\nonumber\\
&&+
\frac{m}{4\pi}
{\bf g}_<(r) {\bf g}_>^T(r^\prime)\theta(r^\prime-r)~.\end{aligned}$$ The solutions ${\bf g}_>(r)$ and ${\bf g}_<(r)$ are also two-by-two matrices since $\Phi({\bf r})$ has two degrees of freedom. The boundary conditions at $r=0$ are ${\bf g}_<(r)|_{r\rightarrow 0} = {\bf 0}$, ${\bf g}_<^\prime(r)|_{r\rightarrow 0} = \bf{1}$, and $ {\bf g}_>(r)
|_{r\rightarrow 0} = {\bf 1}$. In the following, we assume $E<2\delta
m$ so that a pair annihilation of $\chi^0$ does not produce on-shell $\chi^-\chi^+$. As the result, $${\bf g}_>(r) |_{r\rightarrow \infty} =
\left(\begin{array}{cc}
0&0\\
d_{1} {\rm e}^{ik r}& d_{2} {\rm e}^{ik r}
\end{array}
\right)~.$$ In this case, the $\chi^0$-pair annihilation cross sections are $(\sigma v)_{VV^\prime}= c_i \sum_{ab} {\bf \Gamma}_{ab}|_{VV^\prime}
d_{a} d_{b}^{\star}$, as expected. It is enough to calculate $d$ in order to evaluate the cross sections.
In Fig. (\[fig1\]) we show the annihilation cross sections of the SU(2)$_L$-triplet DM pair to $2\gamma$ and $W^+W^-$ as functions of $m$. We evaluated the cross section numerically. Here, we take $v/c=10^{-3}$, which is the typical averaged velocity of the DM in our galaxy, and $\delta m=0.1$ GeV and 1 GeV. The perturbative cross sections are also plotted. Large $\delta m$ leads to unreliable numerical calculation for large $m$, and then some curves are terminated at some points. However, $\delta m$ should be suppressed around the regions.
When $m$ is around 100 GeV, the cross sections to $2 \gamma$ and $W^+W^-$ are almost the same as the perturbative ones. The cross section to $2\gamma$ is suppressed by a loop factor there. However, when $m{ \mathop{}_{\textstyle \sim}^{\textstyle >} }0.5$ TeV, the cross sections are significantly enhanced and have the resonance structure. Especially, the cross section to $2\gamma$ becomes comparable to that to $W^+W^-$ around the resonance. This suggests that the $2\chi^0$ state is strongly mixed with $\chi^+\chi^-$.
The qualitative behavior of the cross sections around the first resonance may be understood by approximating the EW potential by a well potential. Taking $c_W=1$ for simplicity, the EW potential is approximated as $${\bf V}(r)
=
\left(
\begin{array}{cc}
2 \delta m
- b_1{\alpha_2 } m_W
&
- b_1{\sqrt{2}\alpha_2} m_W
\\
- b_1{\sqrt{2}\alpha_2} m_W
&
0
\end{array}
\right)~,$$ for $r<R(\equiv ({b_2} m_W)^{-1})$. Here, $b_1$ and $b_2$ are numerical constants. By comparing the annihilation cross sections to $2
\gamma$ in this potential and in the perturbative calculation for small $m$, we find $b_1=8/9$ and $b_2=2/3$. Under this potential, two-body states $2\chi^0$ and $\chi^-\chi^+$ have the attractive and repulsive states, whose potential energies are $\lambda_{\pm}=1/2
({\bf V}_{11}\pm\sqrt{{\bf V}^2_{11}+4{\bf V}^2_{12}})$ with ${\bf
V}_{ij}(i,j=1,2)$ elements in ${\bf V}$. The attractive state is $-\sin\theta
\phi_C+\cos\theta \phi_N$ with $\tan^2\theta=\lambda_-/\lambda_+$.
When $\delta m\ll b_1\alpha_2 m_W/2(\sim 1$ GeV), $\theta$ is not suppressed by $\delta m$ and $\chi^-\chi^+$ and $2\chi^0$ are mixed under the potential. In this case, the cross section to $2\gamma$ is given as $$\begin{aligned}
(\sigma v)_{2\gamma}
&=&
\frac{4\pi\alpha^2}{9 m^2}\left(
\frac1{\cos(k_-R)}-\frac1{\cosh(k_+R)}\right)^2~,
\label{appro}\end{aligned}$$ where $k_\pm^2 = |\lambda_\pm| m$. Here, we neglect the ${\bf \Gamma}$ term contribution to the wave function for simplicity and take $E\simeq 0$. The cross section (\[appro\]) is reduced to $4\pi\alpha^2\alpha_2^2/m_W^2$ for ${\alpha_2 m}{ \mathop{}_{\textstyle \sim}^{\textstyle <} }m_W$. On the other hand, it is not suppressed by a one-loop factor for ${\alpha_2 m}{ \mathop{}_{\textstyle \sim}^{\textstyle >} }m_W$ and has a correct behavior as $\sim 1/m^2$ in a heavy $m$ limit. When $k_-R=(2n-1)\pi/2$ ($n=1,2,\cdots$), the zero energy resonance, whose binding energy is zero, appears and the cross section is enhanced significantly. In Fig. (\[fig1\]), the $n$-th zero energy resonance appears at $m=m^{(n)}\sim n^2\times m^{(1)}$, while the well potential predicts $m^{(n)}\sim (2n-1)^2
\times m^{(1)}$. We guess that the Yukawa potential might be approximated better by the Coulomb potential for the higher zero energy resonances.
When the zero energy resonance exits, the cross sections $\sigma v$ are proportional to $v^{-2}$ for $v\ll 1$. However, this is not a signature for breakdown of the unitarity. We find from study in the one-flavor system under the well potential $V$ that when $v \ll m V \Gamma$, $\sigma v$ is saturated by the finite width $\Gamma$ and the unitarity is not broken.
We also show the annihilation cross sections for the SU(2)$_L$-doublet DM in Fig. (\[fig1\]). The SU(2)$_L$-doublet DM has the smaller gauge charges compared with the SU(2)$_L$-triplet DM. As the result, the cross section is smaller, and the first zero energy resonance appears at 5 TeV.
The enhancement for the DM annihilation rates gives significant impacts on the indirect searches for the DM in the cosmic ray. In the following we discuss the $\gamma$ ray search from the DM annihilation in the galactic center and the future prospects.
The line $\gamma$ from the pair annihilation to $2\gamma$ or $Z\gamma$ at the galactic center is a robust signal for the DM. Also, $W^-W^+$ and $2 Z^0$ final states produce the continuum $\gamma$ spectrum through $\pi^0\rightarrow 2\gamma$, and the observation may constrain the EWIMP DM. The $\gamma$ flux, $\Psi_\gamma(E)$, is given as $$\begin{aligned}
&{\displaystyle\frac{d\Psi_\gamma(E)}{d E}} =
9.3 \times 10^{-12} {\rm cm}^{-2} {\rm
sec}^{-1}{\rm GeV^{-1}} \times \bar{J} \Delta \Omega&
\nonumber\\
&\times\left(
{\displaystyle\frac{100{\rm GeV}}{m}}
\right)^2
{\displaystyle\sum_{VV^\prime}}
{\displaystyle\frac{dN^{{VV^\prime}}}{dE}}
\left(
{\displaystyle
\frac{\left<\sigma v\right>_{VV^\prime}}
{10^{-27} {\rm cm^3 sec^{-1}}}
}\right)~.&\end{aligned}$$ where $N^{{VV^\prime}}$ is the number of photons from the final state $VV^\prime$ and $\left<\sigma v\right>$ is the averaged cross section by the velocity distribution function. Note that the angular acceptance of the detector, $\Delta\Omega$, is $10^{-3}$ typically for the ACT detectors. The flux depends on the halo DM density profile $\rho$ through $$\bar{J} \Delta \Omega = \frac{1}{8.5{\rm kpc}}\int_{\begin{array}{l}
\mbox{l.o.s}\\
\Delta \Omega
\end{array}
} d\Omega dl~
\left(\frac{\rho}{0.3~{\rm GeV cm^{-3}}}\right)^2~,$$ where the integral is along the line of sight. $\bar{J}$ is studied for various halo models, and $3{ \mathop{}_{\textstyle \sim}^{\textstyle <} }\bar{J} { \mathop{}_{\textstyle \sim}^{\textstyle <} }10^5$ [@Bergstrom:1997fj]. The cuspy structures in the halo density profiles, which are suggested by the $N$-body simulations, tend to give larger numbers to $\bar{J}$. In the following we take a moderate value as $\bar{J}=500$, which is typical for the NFW profile [@nfw].
In Fig. (\[fig2\]a) we show the line $\gamma$ flux from the galactic center in cases of the SU(2)$_L$-triplet and doublet DM’s. Here, we take $\delta m=0.1,1,10$ GeV. We also show the flux obtained by the leading-order calculation for comparison. The ACT detectors have high sensitivity for the TeV-scale $\gamma$ ray. MAGIC and VERITAS in the north hemisphere might reach to $10^{-14}$cm$^{-2}$s$^{-1}$ at the TeV scale while CANGAROO III and HESS in the south hemisphere to $10^{-13}$cm$^{-2}$s$^{-1}$ [@Bergstrom:1997fj]. These ACT detectors are expected to cover the broad region.
In Fig. (\[fig2\]b), the contour plot of the continuum $\gamma$ flux from the galactic center is presented. For ${dN^{{VV^\prime}}}/{dE}$ we use the fitting functions given in [@Bergstrom:1997fj]. Shaded regions correspond to $S/B>1$. In order to evaluate the background $B$, we assume a power low fall-off in the energy for the diffused $\gamma$ ray flux $\Psi_{BG}(E)$ as ${d\Psi_{BG}(E)}/{dE} =9.1\times
10^{-5} {\rm cm^{-2} sec^{-1} GeV^{-1}} \times ({{E}/{\rm
1~GeV}})^{-2.7} \Delta \Omega$ [@Bergstrom:1997fj]. The EGRET experiment has observed the diffused $\gamma$ ray emission from the galactic center up to about 10 GeV [@egret]. Even the small regions around the resonances in addition to the triplet DM with $m_{\chi}=100$ GeV are already constrained by the EGRET observation. The GLAST satellite detector, which will detect $\gamma$ ray with 1 GeV$<E<$300 GeV, have more sensitivity to the around the region.
\
This work is supported in part by the Grant-in-Aid for Science Research, Ministry of Education, Science and Culture, Japan (No.15540255, No.13135207 and No.14046225 for JH and No.14540260 and 14046210 for MMN).
[99]{}
For review, J. R. Primack, arXiv:astro-ph/0205391.
D. N. Spergel [*et al.*]{}, arXiv:astro-ph/0302209; C. L. Bennett [*et al.*]{}, arXiv:astro-ph/0302207.
For reviews, G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. [**267**]{}, 195 (1996); L. Bergstrom, Rept. Prog. Phys. [**63**]{}, 793 (2000)
L. Bergstrom, P. Ullio and J. H. Buckley, Astropart. Phys. [**9**]{}, 137 (1998).
L.D. Landau, [*Quantum Mechanics: Non-Relativistic Theory, Volume 3, Third Edition*]{} (Elsevier Science Ltd).
D. S. Akerib [*et al.*]{} \[CDMS Collaboration\], arXiv:hep-ex/0306001; A. Benoit [*et al.*]{} \[the EDELWEISS Collaboration\], Phys. Lett. B [**545**]{}, 43 (2002).
J. Hisano, S. Matsumoto and M. M. Nojiri, Phys. Rev. D [**67**]{}, 075014 (2003).
J. Hisano, S. Matsumoto, M. M. Nojiri, and O. Saito, in preparation.
G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D [**51**]{}, 1125 (1995).
J.F. Navarro, C.S. Frenk, and S.D.M. Whilte, Astrophys. J. [**490**]{}, 493 (1997).
J.H. Hunter [*et al*]{}, Astrophys. J. [**481**]{}, 205 (1997).
[^1]: The long-distance effect is negligible for the DM annihilation in the early universe since the the velocity at the decoupling temperature is larger than $\alpha_2$.
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